{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r07-finite-flow-network-0001","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-flow-network"],"title_zh":"有限流網路","stem_zh":"finite loop-free directed network、逐edge capacity與逐internal-vertex conservation。操作=flow-network-qualification；資料={\"vertices\":[\"s0\",\"a0\",\"t0\",\"u0\"],\"edges\":[[\"s0\",\"a0\"],[\"a0\",\"s0\"],[\"a0\",\"t0\"],[\"t0\",\"a0\"]],\"source\":\"s0\",\"sink\":\"t0\",\"capacities\":[5,5,5,5],\"flows\":[4,2,3,1]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([a0, s0, t0, u0], s0, t0, [[[a0, s0], 5, 2], [[a0, t0], 5, 3], [[s0, a0], 5, 4], [[t0, a0], 5, 1]], 2, 是, finite_loop_free_simple_digraph_with_explicit_antiparallel_convention_zero_capacities_and_disconnected_vertices_allowed)","explanation_zh":"Wave6第七層核對=([a0, s0, t0, u0], s0, t0, [[[a0, s0], 5, 2], [[a0, t0], 5, 3], [[s0, a0], 5, 4], [[t0, a0], 5, 1]], 2, 是, finite_loop_free_simple_digraph_with_explicit_antiparallel_convention_zero_capacities_and_disconnected_vertices_allowed)；依完整明示資料獨立重算；antiparallel edges明示，zero capacity與disconnected vertices允許；flow value由source與sink交叉計算；feasible不等於maximum，不提前宣稱min-cut equality或augmenting-path optimality。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限流網路"]},"variant":{"family_id":"qf-mod44-flow-network-qualification","seed":202611137001,"parameters":{"vertices":["s0","a0","t0","u0"],"edges":[["s0","a0"],["a0","s0"],["a0","t0"],["t0","a0"]],"source":"s0","sink":"t0","capacities":[5,5,5,5],"flows":[4,2,3,1],"domain":"Exact_finite_flow_network","operation":"flow-network-qualification","semantic_template":"qf-mod44-flow-network-qualification/{\"capacities\":[[5,1],[5,1],[5,1],[5,1]],\"edges\":[[\"a0\",\"s0\"],[\"a0\",\"t0\"],[\"s0\",\"a0\"],[\"t0\",\"a0\"]],\"flows\":[[2,1],[3,1],[4,1],[1,1]],\"sink\":\"t0\",\"source\":\"s0\",\"vertices\":[\"a0\",\"s0\",\"t0\",\"u0\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、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-20261113-r07-finite-flow-network-0002","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-flow-network"],"title_zh":"有限流網路","stem_zh":"finite loop-free directed network、逐edge capacity與逐internal-vertex conservation。操作=flow-network-qualification；資料={\"vertices\":[\"s1\",\"a1\",\"t1\",\"u1\"],\"edges\":[[\"s1\",\"a1\"],[\"a1\",\"s1\"],[\"a1\",\"t1\"],[\"t1\",\"a1\"]],\"source\":\"s1\",\"sink\":\"t1\",\"capacities\":[10,10,10,10],\"flows\":[8,4,6,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([a1, s1, t1, u1], s1, t1, [[[a1, s1], 10, 4], [[a1, t1], 10, 6], [[s1, a1], 10, 8], [[t1, a1], 10, 2]], 2, 是, finite_loop_free_simple_digraph_with_explicit_antiparallel_convention_zero_capacities_and_disconnected_vertices_allowed)","explanation_zh":"Wave6第七層核對=([a1, s1, t1, u1], s1, t1, [[[a1, s1], 10, 4], [[a1, t1], 10, 6], [[s1, a1], 10, 8], [[t1, a1], 10, 2]], 2, 是, finite_loop_free_simple_digraph_with_explicit_antiparallel_convention_zero_capacities_and_disconnected_vertices_allowed)；依完整明示資料獨立重算；antiparallel edges明示，zero capacity與disconnected vertices允許；flow value由source與sink交叉計算；feasible不等於maximum，不提前宣稱min-cut equality或augmenting-path optimality。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限流網路"]},"variant":{"family_id":"qf-mod44-flow-network-qualification","seed":202611137002,"parameters":{"vertices":["s1","a1","t1","u1"],"edges":[["s1","a1"],["a1","s1"],["a1","t1"],["t1","a1"]],"source":"s1","sink":"t1","capacities":[10,10,10,10],"flows":[8,4,6,2],"domain":"Exact_finite_flow_network","operation":"flow-network-qualification","semantic_template":"qf-mod44-flow-network-qualification/{\"capacities\":[[10,1],[10,1],[10,1],[10,1]],\"edges\":[[\"a1\",\"s1\"],[\"a1\",\"t1\"],[\"s1\",\"a1\"],[\"t1\",\"a1\"]],\"flows\":[[4,1],[6,1],[8,1],[2,1]],\"sink\":\"t1\",\"source\":\"s1\",\"vertices\":[\"a1\",\"s1\",\"t1\",\"u1\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、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-20261113-r07-finite-flow-network-0003","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-flow-network"],"title_zh":"有限流網路","stem_zh":"finite loop-free directed network、逐edge capacity與逐internal-vertex conservation。操作=flow-network-qualification；資料={\"vertices\":[\"s2\",\"a2\",\"t2\",\"u2\"],\"edges\":[[\"s2\",\"a2\"],[\"a2\",\"s2\"],[\"a2\",\"t2\"],[\"t2\",\"a2\"]],\"source\":\"s2\",\"sink\":\"t2\",\"capacities\":[15,15,15,15],\"flows\":[12,6,9,3]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([a2, s2, t2, u2], s2, t2, [[[a2, s2], 15, 6], [[a2, t2], 15, 9], [[s2, a2], 15, 12], [[t2, a2], 15, 3]], 2, 是, finite_loop_free_simple_digraph_with_explicit_antiparallel_convention_zero_capacities_and_disconnected_vertices_allowed)","explanation_zh":"Wave6第七層核對=([a2, s2, t2, u2], s2, t2, [[[a2, s2], 15, 6], [[a2, t2], 15, 9], [[s2, a2], 15, 12], [[t2, a2], 15, 3]], 2, 是, finite_loop_free_simple_digraph_with_explicit_antiparallel_convention_zero_capacities_and_disconnected_vertices_allowed)；依完整明示資料獨立重算；antiparallel edges明示，zero capacity與disconnected vertices允許；flow value由source與sink交叉計算；feasible不等於maximum，不提前宣稱min-cut equality或augmenting-path optimality。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限流網路"]},"variant":{"family_id":"qf-mod44-flow-network-qualification","seed":202611137003,"parameters":{"vertices":["s2","a2","t2","u2"],"edges":[["s2","a2"],["a2","s2"],["a2","t2"],["t2","a2"]],"source":"s2","sink":"t2","capacities":[15,15,15,15],"flows":[12,6,9,3],"domain":"Exact_finite_flow_network","operation":"flow-network-qualification","semantic_template":"qf-mod44-flow-network-qualification/{\"capacities\":[[15,1],[15,1],[15,1],[15,1]],\"edges\":[[\"a2\",\"s2\"],[\"a2\",\"t2\"],[\"s2\",\"a2\"],[\"t2\",\"a2\"]],\"flows\":[[6,1],[9,1],[12,1],[3,1]],\"sink\":\"t2\",\"source\":\"s2\",\"vertices\":[\"a2\",\"s2\",\"t2\",\"u2\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、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-20261113-r07-finite-flow-network-0004","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-flow-network"],"title_zh":"有限流網路","stem_zh":"finite loop-free directed network、逐edge capacity與逐internal-vertex conservation。操作=flow-network-qualification；資料={\"vertices\":[\"s3\",\"a3\",\"t3\",\"u3\"],\"edges\":[[\"s3\",\"a3\"],[\"a3\",\"s3\"],[\"a3\",\"t3\"],[\"t3\",\"a3\"]],\"source\":\"s3\",\"sink\":\"t3\",\"capacities\":[20,20,20,20],\"flows\":[16,8,12,4]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([a3, s3, t3, u3], s3, t3, [[[a3, s3], 20, 8], [[a3, t3], 20, 12], [[s3, a3], 20, 16], [[t3, a3], 20, 4]], 2, 是, finite_loop_free_simple_digraph_with_explicit_antiparallel_convention_zero_capacities_and_disconnected_vertices_allowed)","explanation_zh":"Wave6第七層核對=([a3, s3, t3, u3], s3, t3, [[[a3, s3], 20, 8], [[a3, t3], 20, 12], [[s3, a3], 20, 16], [[t3, a3], 20, 4]], 2, 是, finite_loop_free_simple_digraph_with_explicit_antiparallel_convention_zero_capacities_and_disconnected_vertices_allowed)；依完整明示資料獨立重算；antiparallel edges明示，zero capacity與disconnected vertices允許；flow value由source與sink交叉計算；feasible不等於maximum，不提前宣稱min-cut equality或augmenting-path optimality。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限流網路"]},"variant":{"family_id":"qf-mod44-flow-network-qualification","seed":202611137004,"parameters":{"vertices":["s3","a3","t3","u3"],"edges":[["s3","a3"],["a3","s3"],["a3","t3"],["t3","a3"]],"source":"s3","sink":"t3","capacities":[20,20,20,20],"flows":[16,8,12,4],"domain":"Exact_finite_flow_network","operation":"flow-network-qualification","semantic_template":"qf-mod44-flow-network-qualification/{\"capacities\":[[20,1],[20,1],[20,1],[20,1]],\"edges\":[[\"a3\",\"s3\"],[\"a3\",\"t3\"],[\"s3\",\"a3\"],[\"t3\",\"a3\"]],\"flows\":[[8,1],[12,1],[16,1],[4,1]],\"sink\":\"t3\",\"source\":\"s3\",\"vertices\":[\"a3\",\"s3\",\"t3\",\"u3\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、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-20261113-r07-finite-flow-network-0005","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-flow-network"],"title_zh":"有限流網路","stem_zh":"finite loop-free directed network、逐edge capacity與逐internal-vertex conservation。操作=flow-network-qualification；資料={\"vertices\":[\"s4\",\"a4\",\"t4\",\"u4\"],\"edges\":[[\"s4\",\"a4\"],[\"a4\",\"s4\"],[\"a4\",\"t4\"],[\"t4\",\"a4\"]],\"source\":\"s4\",\"sink\":\"t4\",\"capacities\":[25,25,25,25],\"flows\":[20,10,15,5]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([a4, s4, t4, u4], s4, t4, [[[a4, s4], 25, 10], [[a4, t4], 25, 15], [[s4, a4], 25, 20], [[t4, a4], 25, 5]], 2, 是, finite_loop_free_simple_digraph_with_explicit_antiparallel_convention_zero_capacities_and_disconnected_vertices_allowed)","explanation_zh":"Wave6第七層核對=([a4, s4, t4, u4], s4, t4, [[[a4, s4], 25, 10], [[a4, t4], 25, 15], [[s4, a4], 25, 20], [[t4, a4], 25, 5]], 2, 是, finite_loop_free_simple_digraph_with_explicit_antiparallel_convention_zero_capacities_and_disconnected_vertices_allowed)；依完整明示資料獨立重算；antiparallel edges明示，zero capacity與disconnected vertices允許；flow value由source與sink交叉計算；feasible不等於maximum，不提前宣稱min-cut equality或augmenting-path optimality。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限流網路"]},"variant":{"family_id":"qf-mod44-flow-network-qualification","seed":202611137005,"parameters":{"vertices":["s4","a4","t4","u4"],"edges":[["s4","a4"],["a4","s4"],["a4","t4"],["t4","a4"]],"source":"s4","sink":"t4","capacities":[25,25,25,25],"flows":[20,10,15,5],"domain":"Exact_finite_flow_network","operation":"flow-network-qualification","semantic_template":"qf-mod44-flow-network-qualification/{\"capacities\":[[25,1],[25,1],[25,1],[25,1]],\"edges\":[[\"a4\",\"s4\"],[\"a4\",\"t4\"],[\"s4\",\"a4\"],[\"t4\",\"a4\"]],\"flows\":[[10,1],[15,1],[20,1],[5,1]],\"sink\":\"t4\",\"source\":\"s4\",\"vertices\":[\"a4\",\"s4\",\"t4\",\"u4\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、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-20261113-r07-finite-flow-network-0006","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-flow-network"],"title_zh":"有限流網路","stem_zh":"finite loop-free directed network、逐edge capacity與逐internal-vertex conservation。操作=flow-network-qualification；資料={\"vertices\":[\"s5\",\"a5\",\"t5\",\"u5\"],\"edges\":[[\"s5\",\"a5\"],[\"a5\",\"s5\"],[\"a5\",\"t5\"],[\"t5\",\"a5\"]],\"source\":\"s5\",\"sink\":\"t5\",\"capacities\":[30,30,30,30],\"flows\":[24,12,18,6]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([a5, s5, t5, u5], s5, t5, [[[a5, s5], 30, 12], [[a5, t5], 30, 18], [[s5, a5], 30, 24], [[t5, a5], 30, 6]], 2, 是, finite_loop_free_simple_digraph_with_explicit_antiparallel_convention_zero_capacities_and_disconnected_vertices_allowed)","explanation_zh":"Wave6第七層核對=([a5, s5, t5, u5], s5, t5, [[[a5, s5], 30, 12], [[a5, t5], 30, 18], [[s5, a5], 30, 24], [[t5, a5], 30, 6]], 2, 是, finite_loop_free_simple_digraph_with_explicit_antiparallel_convention_zero_capacities_and_disconnected_vertices_allowed)；依完整明示資料獨立重算；antiparallel edges明示，zero capacity與disconnected vertices允許；flow value由source與sink交叉計算；feasible不等於maximum，不提前宣稱min-cut equality或augmenting-path optimality。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限流網路"]},"variant":{"family_id":"qf-mod44-flow-network-qualification","seed":202611137006,"parameters":{"vertices":["s5","a5","t5","u5"],"edges":[["s5","a5"],["a5","s5"],["a5","t5"],["t5","a5"]],"source":"s5","sink":"t5","capacities":[30,30,30,30],"flows":[24,12,18,6],"domain":"Exact_finite_flow_network","operation":"flow-network-qualification","semantic_template":"qf-mod44-flow-network-qualification/{\"capacities\":[[30,1],[30,1],[30,1],[30,1]],\"edges\":[[\"a5\",\"s5\"],[\"a5\",\"t5\"],[\"s5\",\"a5\"],[\"t5\",\"a5\"]],\"flows\":[[12,1],[18,1],[24,1],[6,1]],\"sink\":\"t5\",\"source\":\"s5\",\"vertices\":[\"a5\",\"s5\",\"t5\",\"u5\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、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-20261113-r07-finite-flow-network-0007","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-flow-network"],"title_zh":"有限流網路","stem_zh":"finite loop-free directed network、逐edge capacity與逐internal-vertex conservation。操作=flow-network-qualification；資料={\"vertices\":[\"s6\",\"a6\",\"t6\",\"u6\"],\"edges\":[[\"s6\",\"a6\"],[\"a6\",\"s6\"],[\"a6\",\"t6\"],[\"t6\",\"a6\"]],\"source\":\"s6\",\"sink\":\"t6\",\"capacities\":[35,35,35,35],\"flows\":[28,14,21,7]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([a6, s6, t6, u6], s6, t6, [[[a6, s6], 35, 14], [[a6, t6], 35, 21], [[s6, a6], 35, 28], [[t6, a6], 35, 7]], 2, 是, finite_loop_free_simple_digraph_with_explicit_antiparallel_convention_zero_capacities_and_disconnected_vertices_allowed)","explanation_zh":"Wave6第七層核對=([a6, s6, t6, u6], s6, t6, [[[a6, s6], 35, 14], [[a6, t6], 35, 21], [[s6, a6], 35, 28], [[t6, a6], 35, 7]], 2, 是, finite_loop_free_simple_digraph_with_explicit_antiparallel_convention_zero_capacities_and_disconnected_vertices_allowed)；依完整明示資料獨立重算；antiparallel edges明示，zero capacity與disconnected vertices允許；flow value由source與sink交叉計算；feasible不等於maximum，不提前宣稱min-cut equality或augmenting-path optimality。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限流網路"]},"variant":{"family_id":"qf-mod44-flow-network-qualification","seed":202611137007,"parameters":{"vertices":["s6","a6","t6","u6"],"edges":[["s6","a6"],["a6","s6"],["a6","t6"],["t6","a6"]],"source":"s6","sink":"t6","capacities":[35,35,35,35],"flows":[28,14,21,7],"domain":"Exact_finite_flow_network","operation":"flow-network-qualification","semantic_template":"qf-mod44-flow-network-qualification/{\"capacities\":[[35,1],[35,1],[35,1],[35,1]],\"edges\":[[\"a6\",\"s6\"],[\"a6\",\"t6\"],[\"s6\",\"a6\"],[\"t6\",\"a6\"]],\"flows\":[[14,1],[21,1],[28,1],[7,1]],\"sink\":\"t6\",\"source\":\"s6\",\"vertices\":[\"a6\",\"s6\",\"t6\",\"u6\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、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-20261113-r07-finite-flow-network-0008","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-flow-network"],"title_zh":"有限流網路","stem_zh":"finite loop-free directed network、逐edge capacity與逐internal-vertex conservation。操作=flow-network-qualification；資料={\"vertices\":[\"s7\",\"a7\",\"t7\",\"u7\"],\"edges\":[[\"s7\",\"a7\"],[\"a7\",\"s7\"],[\"a7\",\"t7\"],[\"t7\",\"a7\"]],\"source\":\"s7\",\"sink\":\"t7\",\"capacities\":[40,40,40,40],\"flows\":[32,16,24,8]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([a7, s7, t7, u7], s7, t7, [[[a7, s7], 40, 16], [[a7, t7], 40, 24], [[s7, a7], 40, 32], [[t7, a7], 40, 8]], 2, 是, finite_loop_free_simple_digraph_with_explicit_antiparallel_convention_zero_capacities_and_disconnected_vertices_allowed)","explanation_zh":"Wave6第七層核對=([a7, s7, t7, u7], s7, t7, [[[a7, s7], 40, 16], [[a7, t7], 40, 24], [[s7, a7], 40, 32], [[t7, a7], 40, 8]], 2, 是, finite_loop_free_simple_digraph_with_explicit_antiparallel_convention_zero_capacities_and_disconnected_vertices_allowed)；依完整明示資料獨立重算；antiparallel edges明示，zero capacity與disconnected vertices允許；flow value由source與sink交叉計算；feasible不等於maximum，不提前宣稱min-cut equality或augmenting-path optimality。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限流網路"]},"variant":{"family_id":"qf-mod44-flow-network-qualification","seed":202611137008,"parameters":{"vertices":["s7","a7","t7","u7"],"edges":[["s7","a7"],["a7","s7"],["a7","t7"],["t7","a7"]],"source":"s7","sink":"t7","capacities":[40,40,40,40],"flows":[32,16,24,8],"domain":"Exact_finite_flow_network","operation":"flow-network-qualification","semantic_template":"qf-mod44-flow-network-qualification/{\"capacities\":[[40,1],[40,1],[40,1],[40,1]],\"edges\":[[\"a7\",\"s7\"],[\"a7\",\"t7\"],[\"s7\",\"a7\"],[\"t7\",\"a7\"]],\"flows\":[[16,1],[24,1],[32,1],[8,1]],\"sink\":\"t7\",\"source\":\"s7\",\"vertices\":[\"a7\",\"s7\",\"t7\",\"u7\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、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-20261113-r07-finite-flow-network-0009","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-flow-network"],"title_zh":"有限流網路","stem_zh":"finite loop-free directed network、逐edge capacity與逐internal-vertex conservation。操作=flow-network-qualification；資料={\"vertices\":[\"s8\",\"a8\",\"t8\",\"u8\"],\"edges\":[[\"s8\",\"a8\"],[\"a8\",\"s8\"],[\"a8\",\"t8\"],[\"t8\",\"a8\"]],\"source\":\"s8\",\"sink\":\"t8\",\"capacities\":[45,45,45,45],\"flows\":[36,18,27,9]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([a8, s8, t8, u8], s8, t8, [[[a8, s8], 45, 18], [[a8, t8], 45, 27], [[s8, a8], 45, 36], [[t8, a8], 45, 9]], 2, 是, finite_loop_free_simple_digraph_with_explicit_antiparallel_convention_zero_capacities_and_disconnected_vertices_allowed)","explanation_zh":"Wave6第七層核對=([a8, s8, t8, u8], s8, t8, [[[a8, s8], 45, 18], [[a8, t8], 45, 27], [[s8, a8], 45, 36], [[t8, a8], 45, 9]], 2, 是, finite_loop_free_simple_digraph_with_explicit_antiparallel_convention_zero_capacities_and_disconnected_vertices_allowed)；依完整明示資料獨立重算；antiparallel edges明示，zero capacity與disconnected vertices允許；flow value由source與sink交叉計算；feasible不等於maximum，不提前宣稱min-cut equality或augmenting-path optimality。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限流網路"]},"variant":{"family_id":"qf-mod44-flow-network-qualification","seed":202611137009,"parameters":{"vertices":["s8","a8","t8","u8"],"edges":[["s8","a8"],["a8","s8"],["a8","t8"],["t8","a8"]],"source":"s8","sink":"t8","capacities":[45,45,45,45],"flows":[36,18,27,9],"domain":"Exact_finite_flow_network","operation":"flow-network-qualification","semantic_template":"qf-mod44-flow-network-qualification/{\"capacities\":[[45,1],[45,1],[45,1],[45,1]],\"edges\":[[\"a8\",\"s8\"],[\"a8\",\"t8\"],[\"s8\",\"a8\"],[\"t8\",\"a8\"]],\"flows\":[[18,1],[27,1],[36,1],[9,1]],\"sink\":\"t8\",\"source\":\"s8\",\"vertices\":[\"a8\",\"s8\",\"t8\",\"u8\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、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-20261113-r07-finite-flow-network-0010","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-flow-network"],"title_zh":"有限流網路","stem_zh":"finite loop-free directed network、逐edge capacity與逐internal-vertex conservation。操作=flow-network-qualification；資料={\"vertices\":[\"s9\",\"a9\",\"t9\",\"u9\"],\"edges\":[[\"s9\",\"a9\"],[\"a9\",\"s9\"],[\"a9\",\"t9\"],[\"t9\",\"a9\"]],\"source\":\"s9\",\"sink\":\"t9\",\"capacities\":[50,50,50,50],\"flows\":[40,20,30,10]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([a9, s9, t9, u9], s9, t9, [[[a9, s9], 50, 20], [[a9, t9], 50, 30], [[s9, a9], 50, 40], [[t9, a9], 50, 10]], 2, 是, finite_loop_free_simple_digraph_with_explicit_antiparallel_convention_zero_capacities_and_disconnected_vertices_allowed)","explanation_zh":"Wave6第七層核對=([a9, s9, t9, u9], s9, t9, [[[a9, s9], 50, 20], [[a9, t9], 50, 30], [[s9, a9], 50, 40], [[t9, a9], 50, 10]], 2, 是, finite_loop_free_simple_digraph_with_explicit_antiparallel_convention_zero_capacities_and_disconnected_vertices_allowed)；依完整明示資料獨立重算；antiparallel edges明示，zero capacity與disconnected vertices允許；flow value由source與sink交叉計算；feasible不等於maximum，不提前宣稱min-cut equality或augmenting-path optimality。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限流網路"]},"variant":{"family_id":"qf-mod44-flow-network-qualification","seed":202611137010,"parameters":{"vertices":["s9","a9","t9","u9"],"edges":[["s9","a9"],["a9","s9"],["a9","t9"],["t9","a9"]],"source":"s9","sink":"t9","capacities":[50,50,50,50],"flows":[40,20,30,10],"domain":"Exact_finite_flow_network","operation":"flow-network-qualification","semantic_template":"qf-mod44-flow-network-qualification/{\"capacities\":[[50,1],[50,1],[50,1],[50,1]],\"edges\":[[\"a9\",\"s9\"],[\"a9\",\"t9\"],[\"s9\",\"a9\"],[\"t9\",\"a9\"]],\"flows\":[[20,1],[30,1],[40,1],[10,1]],\"sink\":\"t9\",\"source\":\"s9\",\"vertices\":[\"a9\",\"s9\",\"t9\",\"u9\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、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-20261113-r07-finite-flow-network-0011","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-flow-network"],"title_zh":"有限流網路","stem_zh":"finite loop-free directed network、逐edge capacity與逐internal-vertex conservation。操作=flow-network-qualification；資料={\"vertices\":[\"s10\",\"a10\",\"t10\",\"u10\"],\"edges\":[[\"s10\",\"a10\"],[\"a10\",\"s10\"],[\"a10\",\"t10\"],[\"t10\",\"a10\"]],\"source\":\"s10\",\"sink\":\"t10\",\"capacities\":[55,55,55,55],\"flows\":[44,22,33,11]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([a10, s10, t10, u10], s10, t10, [[[a10, s10], 55, 22], [[a10, t10], 55, 33], [[s10, a10], 55, 44], [[t10, a10], 55, 11]], 2, 是, finite_loop_free_simple_digraph_with_explicit_antiparallel_convention_zero_capacities_and_disconnected_vertices_allowed)","explanation_zh":"Wave6第七層核對=([a10, s10, t10, u10], s10, t10, [[[a10, s10], 55, 22], [[a10, t10], 55, 33], [[s10, a10], 55, 44], [[t10, a10], 55, 11]], 2, 是, finite_loop_free_simple_digraph_with_explicit_antiparallel_convention_zero_capacities_and_disconnected_vertices_allowed)；依完整明示資料獨立重算；antiparallel edges明示，zero capacity與disconnected vertices允許；flow value由source與sink交叉計算；feasible不等於maximum，不提前宣稱min-cut equality或augmenting-path optimality。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限流網路"]},"variant":{"family_id":"qf-mod44-flow-network-qualification","seed":202611137011,"parameters":{"vertices":["s10","a10","t10","u10"],"edges":[["s10","a10"],["a10","s10"],["a10","t10"],["t10","a10"]],"source":"s10","sink":"t10","capacities":[55,55,55,55],"flows":[44,22,33,11],"domain":"Exact_finite_flow_network","operation":"flow-network-qualification","semantic_template":"qf-mod44-flow-network-qualification/{\"capacities\":[[55,1],[55,1],[55,1],[55,1]],\"edges\":[[\"a10\",\"s10\"],[\"a10\",\"t10\"],[\"s10\",\"a10\"],[\"t10\",\"a10\"]],\"flows\":[[22,1],[33,1],[44,1],[11,1]],\"sink\":\"t10\",\"source\":\"s10\",\"vertices\":[\"a10\",\"s10\",\"t10\",\"u10\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、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-20261113-r07-finite-flow-network-0012","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-flow-network"],"title_zh":"有限流網路","stem_zh":"finite loop-free directed network、逐edge capacity與逐internal-vertex conservation。操作=flow-network-qualification；資料={\"vertices\":[\"s11\",\"a11\",\"t11\",\"u11\"],\"edges\":[[\"s11\",\"a11\"],[\"a11\",\"s11\"],[\"a11\",\"t11\"],[\"t11\",\"a11\"]],\"source\":\"s11\",\"sink\":\"t11\",\"capacities\":[60,60,60,60],\"flows\":[48,24,36,12]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([a11, s11, t11, u11], s11, t11, [[[a11, s11], 60, 24], [[a11, t11], 60, 36], [[s11, a11], 60, 48], [[t11, a11], 60, 12]], 2, 是, finite_loop_free_simple_digraph_with_explicit_antiparallel_convention_zero_capacities_and_disconnected_vertices_allowed)","explanation_zh":"Wave6第七層核對=([a11, s11, t11, u11], s11, t11, [[[a11, s11], 60, 24], [[a11, t11], 60, 36], [[s11, a11], 60, 48], [[t11, a11], 60, 12]], 2, 是, finite_loop_free_simple_digraph_with_explicit_antiparallel_convention_zero_capacities_and_disconnected_vertices_allowed)；依完整明示資料獨立重算；antiparallel edges明示，zero capacity與disconnected vertices允許；flow value由source與sink交叉計算；feasible不等於maximum，不提前宣稱min-cut equality或augmenting-path optimality。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限流網路"]},"variant":{"family_id":"qf-mod44-flow-network-qualification","seed":202611137012,"parameters":{"vertices":["s11","a11","t11","u11"],"edges":[["s11","a11"],["a11","s11"],["a11","t11"],["t11","a11"]],"source":"s11","sink":"t11","capacities":[60,60,60,60],"flows":[48,24,36,12],"domain":"Exact_finite_flow_network","operation":"flow-network-qualification","semantic_template":"qf-mod44-flow-network-qualification/{\"capacities\":[[60,1],[60,1],[60,1],[60,1]],\"edges\":[[\"a11\",\"s11\"],[\"a11\",\"t11\"],[\"s11\",\"a11\"],[\"t11\",\"a11\"]],\"flows\":[[24,1],[36,1],[48,1],[12,1]],\"sink\":\"t11\",\"source\":\"s11\",\"vertices\":[\"a11\",\"s11\",\"t11\",\"u11\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、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-20261113-r07-finite-flow-network-0013","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-flow-network"],"title_zh":"有限流網路","stem_zh":"finite loop-free directed network、逐edge capacity與逐internal-vertex conservation。操作=flow-network-qualification；資料={\"vertices\":[\"s12\",\"a12\",\"t12\",\"u12\"],\"edges\":[[\"s12\",\"a12\"],[\"a12\",\"s12\"],[\"a12\",\"t12\"],[\"t12\",\"a12\"]],\"source\":\"s12\",\"sink\":\"t12\",\"capacities\":[65,65,65,65],\"flows\":[52,26,39,13]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([a12, s12, t12, u12], s12, t12, [[[a12, s12], 65, 26], [[a12, t12], 65, 39], [[s12, a12], 65, 52], [[t12, a12], 65, 13]], 2, 是, finite_loop_free_simple_digraph_with_explicit_antiparallel_convention_zero_capacities_and_disconnected_vertices_allowed)","explanation_zh":"Wave6第七層核對=([a12, s12, t12, u12], s12, t12, [[[a12, s12], 65, 26], [[a12, t12], 65, 39], [[s12, a12], 65, 52], [[t12, a12], 65, 13]], 2, 是, finite_loop_free_simple_digraph_with_explicit_antiparallel_convention_zero_capacities_and_disconnected_vertices_allowed)；依完整明示資料獨立重算；antiparallel edges明示，zero capacity與disconnected vertices允許；flow value由source與sink交叉計算；feasible不等於maximum，不提前宣稱min-cut equality或augmenting-path optimality。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限流網路"]},"variant":{"family_id":"qf-mod44-flow-network-qualification","seed":202611137013,"parameters":{"vertices":["s12","a12","t12","u12"],"edges":[["s12","a12"],["a12","s12"],["a12","t12"],["t12","a12"]],"source":"s12","sink":"t12","capacities":[65,65,65,65],"flows":[52,26,39,13],"domain":"Exact_finite_flow_network","operation":"flow-network-qualification","semantic_template":"qf-mod44-flow-network-qualification/{\"capacities\":[[65,1],[65,1],[65,1],[65,1]],\"edges\":[[\"a12\",\"s12\"],[\"a12\",\"t12\"],[\"s12\",\"a12\"],[\"t12\",\"a12\"]],\"flows\":[[26,1],[39,1],[52,1],[13,1]],\"sink\":\"t12\",\"source\":\"s12\",\"vertices\":[\"a12\",\"s12\",\"t12\",\"u12\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、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-20261113-r07-finite-flow-network-0014","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-flow-network"],"title_zh":"有限流網路","stem_zh":"finite loop-free directed network、逐edge capacity與逐internal-vertex conservation。操作=flow-network-qualification；資料={\"vertices\":[\"s13\",\"a13\",\"t13\",\"u13\"],\"edges\":[[\"s13\",\"a13\"],[\"a13\",\"s13\"],[\"a13\",\"t13\"],[\"t13\",\"a13\"]],\"source\":\"s13\",\"sink\":\"t13\",\"capacities\":[70,70,70,70],\"flows\":[56,28,42,14]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([a13, s13, t13, u13], s13, t13, [[[a13, s13], 70, 28], [[a13, t13], 70, 42], [[s13, a13], 70, 56], [[t13, a13], 70, 14]], 2, 是, finite_loop_free_simple_digraph_with_explicit_antiparallel_convention_zero_capacities_and_disconnected_vertices_allowed)","explanation_zh":"Wave6第七層核對=([a13, s13, t13, u13], s13, t13, [[[a13, s13], 70, 28], [[a13, t13], 70, 42], [[s13, a13], 70, 56], [[t13, a13], 70, 14]], 2, 是, finite_loop_free_simple_digraph_with_explicit_antiparallel_convention_zero_capacities_and_disconnected_vertices_allowed)；依完整明示資料獨立重算；antiparallel edges明示，zero capacity與disconnected vertices允許；flow value由source與sink交叉計算；feasible不等於maximum，不提前宣稱min-cut equality或augmenting-path optimality。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限流網路"]},"variant":{"family_id":"qf-mod44-flow-network-qualification","seed":202611137014,"parameters":{"vertices":["s13","a13","t13","u13"],"edges":[["s13","a13"],["a13","s13"],["a13","t13"],["t13","a13"]],"source":"s13","sink":"t13","capacities":[70,70,70,70],"flows":[56,28,42,14],"domain":"Exact_finite_flow_network","operation":"flow-network-qualification","semantic_template":"qf-mod44-flow-network-qualification/{\"capacities\":[[70,1],[70,1],[70,1],[70,1]],\"edges\":[[\"a13\",\"s13\"],[\"a13\",\"t13\"],[\"s13\",\"a13\"],[\"t13\",\"a13\"]],\"flows\":[[28,1],[42,1],[56,1],[14,1]],\"sink\":\"t13\",\"source\":\"s13\",\"vertices\":[\"a13\",\"s13\",\"t13\",\"u13\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、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-20261113-r07-finite-flow-network-0015","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-flow-network"],"title_zh":"有限流網路","stem_zh":"finite loop-free directed network、逐edge capacity與逐internal-vertex conservation。操作=flow-network-qualification；資料={\"vertices\":[\"s14\",\"a14\",\"t14\",\"u14\"],\"edges\":[[\"s14\",\"a14\"],[\"a14\",\"s14\"],[\"a14\",\"t14\"],[\"t14\",\"a14\"]],\"source\":\"s14\",\"sink\":\"t14\",\"capacities\":[75,75,75,75],\"flows\":[60,30,45,15]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([a14, s14, t14, u14], s14, t14, [[[a14, s14], 75, 30], [[a14, t14], 75, 45], [[s14, a14], 75, 60], [[t14, a14], 75, 15]], 2, 是, finite_loop_free_simple_digraph_with_explicit_antiparallel_convention_zero_capacities_and_disconnected_vertices_allowed)","explanation_zh":"Wave6第七層核對=([a14, s14, t14, u14], s14, t14, [[[a14, s14], 75, 30], [[a14, t14], 75, 45], [[s14, a14], 75, 60], [[t14, a14], 75, 15]], 2, 是, finite_loop_free_simple_digraph_with_explicit_antiparallel_convention_zero_capacities_and_disconnected_vertices_allowed)；依完整明示資料獨立重算；antiparallel edges明示，zero capacity與disconnected vertices允許；flow value由source與sink交叉計算；feasible不等於maximum，不提前宣稱min-cut equality或augmenting-path optimality。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限流網路"]},"variant":{"family_id":"qf-mod44-flow-network-qualification","seed":202611137015,"parameters":{"vertices":["s14","a14","t14","u14"],"edges":[["s14","a14"],["a14","s14"],["a14","t14"],["t14","a14"]],"source":"s14","sink":"t14","capacities":[75,75,75,75],"flows":[60,30,45,15],"domain":"Exact_finite_flow_network","operation":"flow-network-qualification","semantic_template":"qf-mod44-flow-network-qualification/{\"capacities\":[[75,1],[75,1],[75,1],[75,1]],\"edges\":[[\"a14\",\"s14\"],[\"a14\",\"t14\"],[\"s14\",\"a14\"],[\"t14\",\"a14\"]],\"flows\":[[30,1],[45,1],[60,1],[15,1]],\"sink\":\"t14\",\"source\":\"s14\",\"vertices\":[\"a14\",\"s14\",\"t14\",\"u14\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、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-20261113-r07-finite-flow-network-0016","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-flow-network"],"title_zh":"有限流網路","stem_zh":"finite loop-free directed network、逐edge capacity與逐internal-vertex conservation。操作=flow-network-qualification；資料={\"vertices\":[\"s15\",\"a15\",\"t15\",\"u15\"],\"edges\":[[\"s15\",\"a15\"],[\"a15\",\"s15\"],[\"a15\",\"t15\"],[\"t15\",\"a15\"]],\"source\":\"s15\",\"sink\":\"t15\",\"capacities\":[80,80,80,80],\"flows\":[64,32,48,16]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([a15, s15, t15, u15], s15, t15, [[[a15, s15], 80, 32], [[a15, t15], 80, 48], [[s15, a15], 80, 64], [[t15, a15], 80, 16]], 2, 是, finite_loop_free_simple_digraph_with_explicit_antiparallel_convention_zero_capacities_and_disconnected_vertices_allowed)","explanation_zh":"Wave6第七層核對=([a15, s15, t15, u15], s15, t15, [[[a15, s15], 80, 32], [[a15, t15], 80, 48], [[s15, a15], 80, 64], [[t15, a15], 80, 16]], 2, 是, finite_loop_free_simple_digraph_with_explicit_antiparallel_convention_zero_capacities_and_disconnected_vertices_allowed)；依完整明示資料獨立重算；antiparallel edges明示，zero capacity與disconnected vertices允許；flow value由source與sink交叉計算；feasible不等於maximum，不提前宣稱min-cut equality或augmenting-path optimality。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限流網路"]},"variant":{"family_id":"qf-mod44-flow-network-qualification","seed":202611137016,"parameters":{"vertices":["s15","a15","t15","u15"],"edges":[["s15","a15"],["a15","s15"],["a15","t15"],["t15","a15"]],"source":"s15","sink":"t15","capacities":[80,80,80,80],"flows":[64,32,48,16],"domain":"Exact_finite_flow_network","operation":"flow-network-qualification","semantic_template":"qf-mod44-flow-network-qualification/{\"capacities\":[[80,1],[80,1],[80,1],[80,1]],\"edges\":[[\"a15\",\"s15\"],[\"a15\",\"t15\"],[\"s15\",\"a15\"],[\"t15\",\"a15\"]],\"flows\":[[32,1],[48,1],[64,1],[16,1]],\"sink\":\"t15\",\"source\":\"s15\",\"vertices\":[\"a15\",\"s15\",\"t15\",\"u15\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、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-20261113-r07-finite-flow-network-0017","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-flow-network"],"title_zh":"有限流網路","stem_zh":"finite loop-free directed network、逐edge capacity與逐internal-vertex conservation。操作=flow-network-qualification；資料={\"vertices\":[\"s16\",\"a16\",\"t16\",\"u16\"],\"edges\":[[\"s16\",\"a16\"],[\"a16\",\"s16\"],[\"a16\",\"t16\"],[\"t16\",\"a16\"]],\"source\":\"s16\",\"sink\":\"t16\",\"capacities\":[85,85,85,85],\"flows\":[68,34,51,17]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([a16, s16, t16, u16], s16, t16, [[[a16, s16], 85, 34], [[a16, t16], 85, 51], [[s16, a16], 85, 68], [[t16, a16], 85, 17]], 2, 是, finite_loop_free_simple_digraph_with_explicit_antiparallel_convention_zero_capacities_and_disconnected_vertices_allowed)","explanation_zh":"Wave6第七層核對=([a16, s16, t16, u16], s16, t16, [[[a16, s16], 85, 34], [[a16, t16], 85, 51], [[s16, a16], 85, 68], [[t16, a16], 85, 17]], 2, 是, finite_loop_free_simple_digraph_with_explicit_antiparallel_convention_zero_capacities_and_disconnected_vertices_allowed)；依完整明示資料獨立重算；antiparallel edges明示，zero capacity與disconnected vertices允許；flow value由source與sink交叉計算；feasible不等於maximum，不提前宣稱min-cut equality或augmenting-path optimality。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限流網路"]},"variant":{"family_id":"qf-mod44-flow-network-qualification","seed":202611137017,"parameters":{"vertices":["s16","a16","t16","u16"],"edges":[["s16","a16"],["a16","s16"],["a16","t16"],["t16","a16"]],"source":"s16","sink":"t16","capacities":[85,85,85,85],"flows":[68,34,51,17],"domain":"Exact_finite_flow_network","operation":"flow-network-qualification","semantic_template":"qf-mod44-flow-network-qualification/{\"capacities\":[[85,1],[85,1],[85,1],[85,1]],\"edges\":[[\"a16\",\"s16\"],[\"a16\",\"t16\"],[\"s16\",\"a16\"],[\"t16\",\"a16\"]],\"flows\":[[34,1],[51,1],[68,1],[17,1]],\"sink\":\"t16\",\"source\":\"s16\",\"vertices\":[\"a16\",\"s16\",\"t16\",\"u16\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、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-20261113-r07-finite-flow-network-0018","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-flow-network"],"title_zh":"有限流網路","stem_zh":"finite loop-free directed network、逐edge capacity與逐internal-vertex conservation。操作=flow-network-qualification；資料={\"vertices\":[\"s17\",\"a17\",\"t17\",\"u17\"],\"edges\":[[\"s17\",\"a17\"],[\"a17\",\"s17\"],[\"a17\",\"t17\"],[\"t17\",\"a17\"]],\"source\":\"s17\",\"sink\":\"t17\",\"capacities\":[90,90,90,90],\"flows\":[72,36,54,18]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([a17, s17, t17, u17], s17, t17, [[[a17, s17], 90, 36], [[a17, t17], 90, 54], [[s17, a17], 90, 72], [[t17, a17], 90, 18]], 2, 是, finite_loop_free_simple_digraph_with_explicit_antiparallel_convention_zero_capacities_and_disconnected_vertices_allowed)","explanation_zh":"Wave6第七層核對=([a17, s17, t17, u17], s17, t17, [[[a17, s17], 90, 36], [[a17, t17], 90, 54], [[s17, a17], 90, 72], [[t17, a17], 90, 18]], 2, 是, finite_loop_free_simple_digraph_with_explicit_antiparallel_convention_zero_capacities_and_disconnected_vertices_allowed)；依完整明示資料獨立重算；antiparallel edges明示，zero capacity與disconnected vertices允許；flow value由source與sink交叉計算；feasible不等於maximum，不提前宣稱min-cut equality或augmenting-path optimality。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限流網路"]},"variant":{"family_id":"qf-mod44-flow-network-qualification","seed":202611137018,"parameters":{"vertices":["s17","a17","t17","u17"],"edges":[["s17","a17"],["a17","s17"],["a17","t17"],["t17","a17"]],"source":"s17","sink":"t17","capacities":[90,90,90,90],"flows":[72,36,54,18],"domain":"Exact_finite_flow_network","operation":"flow-network-qualification","semantic_template":"qf-mod44-flow-network-qualification/{\"capacities\":[[90,1],[90,1],[90,1],[90,1]],\"edges\":[[\"a17\",\"s17\"],[\"a17\",\"t17\"],[\"s17\",\"a17\"],[\"t17\",\"a17\"]],\"flows\":[[36,1],[54,1],[72,1],[18,1]],\"sink\":\"t17\",\"source\":\"s17\",\"vertices\":[\"a17\",\"s17\",\"t17\",\"u17\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、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-20261113-r07-finite-flow-network-0019","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-flow-network"],"title_zh":"有限流網路","stem_zh":"finite loop-free directed network、逐edge capacity與逐internal-vertex conservation。操作=flow-network-qualification；資料={\"vertices\":[\"s18\",\"a18\",\"t18\",\"u18\"],\"edges\":[[\"s18\",\"a18\"],[\"a18\",\"s18\"],[\"a18\",\"t18\"],[\"t18\",\"a18\"]],\"source\":\"s18\",\"sink\":\"t18\",\"capacities\":[95,95,95,95],\"flows\":[76,38,57,19]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([a18, s18, t18, u18], s18, t18, [[[a18, s18], 95, 38], [[a18, t18], 95, 57], [[s18, a18], 95, 76], [[t18, a18], 95, 19]], 2, 是, finite_loop_free_simple_digraph_with_explicit_antiparallel_convention_zero_capacities_and_disconnected_vertices_allowed)","explanation_zh":"Wave6第七層核對=([a18, s18, t18, u18], s18, t18, [[[a18, s18], 95, 38], [[a18, t18], 95, 57], [[s18, a18], 95, 76], [[t18, a18], 95, 19]], 2, 是, finite_loop_free_simple_digraph_with_explicit_antiparallel_convention_zero_capacities_and_disconnected_vertices_allowed)；依完整明示資料獨立重算；antiparallel edges明示，zero capacity與disconnected vertices允許；flow value由source與sink交叉計算；feasible不等於maximum，不提前宣稱min-cut equality或augmenting-path optimality。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限流網路"]},"variant":{"family_id":"qf-mod44-flow-network-qualification","seed":202611137019,"parameters":{"vertices":["s18","a18","t18","u18"],"edges":[["s18","a18"],["a18","s18"],["a18","t18"],["t18","a18"]],"source":"s18","sink":"t18","capacities":[95,95,95,95],"flows":[76,38,57,19],"domain":"Exact_finite_flow_network","operation":"flow-network-qualification","semantic_template":"qf-mod44-flow-network-qualification/{\"capacities\":[[95,1],[95,1],[95,1],[95,1]],\"edges\":[[\"a18\",\"s18\"],[\"a18\",\"t18\"],[\"s18\",\"a18\"],[\"t18\",\"a18\"]],\"flows\":[[38,1],[57,1],[76,1],[19,1]],\"sink\":\"t18\",\"source\":\"s18\",\"vertices\":[\"a18\",\"s18\",\"t18\",\"u18\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、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-20261113-r07-finite-flow-network-0020","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-flow-network"],"title_zh":"有限流網路","stem_zh":"finite loop-free directed network、逐edge capacity與逐internal-vertex conservation。操作=flow-network-qualification；資料={\"vertices\":[\"s19\",\"a19\",\"t19\",\"u19\"],\"edges\":[[\"s19\",\"a19\"],[\"a19\",\"s19\"],[\"a19\",\"t19\"],[\"t19\",\"a19\"]],\"source\":\"s19\",\"sink\":\"t19\",\"capacities\":[100,100,100,100],\"flows\":[80,40,60,20]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([a19, s19, t19, u19], s19, t19, [[[a19, s19], 100, 40], [[a19, t19], 100, 60], [[s19, a19], 100, 80], [[t19, a19], 100, 20]], 2, 是, finite_loop_free_simple_digraph_with_explicit_antiparallel_convention_zero_capacities_and_disconnected_vertices_allowed)","explanation_zh":"Wave6第七層核對=([a19, s19, t19, u19], s19, t19, [[[a19, s19], 100, 40], [[a19, t19], 100, 60], [[s19, a19], 100, 80], [[t19, a19], 100, 20]], 2, 是, finite_loop_free_simple_digraph_with_explicit_antiparallel_convention_zero_capacities_and_disconnected_vertices_allowed)；依完整明示資料獨立重算；antiparallel edges明示，zero capacity與disconnected vertices允許；flow value由source與sink交叉計算；feasible不等於maximum，不提前宣稱min-cut equality或augmenting-path optimality。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限流網路"]},"variant":{"family_id":"qf-mod44-flow-network-qualification","seed":202611137020,"parameters":{"vertices":["s19","a19","t19","u19"],"edges":[["s19","a19"],["a19","s19"],["a19","t19"],["t19","a19"]],"source":"s19","sink":"t19","capacities":[100,100,100,100],"flows":[80,40,60,20],"domain":"Exact_finite_flow_network","operation":"flow-network-qualification","semantic_template":"qf-mod44-flow-network-qualification/{\"capacities\":[[100,1],[100,1],[100,1],[100,1]],\"edges\":[[\"a19\",\"s19\"],[\"a19\",\"t19\"],[\"s19\",\"a19\"],[\"t19\",\"a19\"]],\"flows\":[[40,1],[60,1],[80,1],[20,1]],\"sink\":\"t19\",\"source\":\"s19\",\"vertices\":[\"a19\",\"s19\",\"t19\",\"u19\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、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-20261113-r07-finite-flow-network-0021","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-flow-network"],"title_zh":"有限流網路","stem_zh":"finite loop-free directed network、逐edge capacity與逐internal-vertex conservation。操作=flow-network-qualification；資料={\"vertices\":[\"s20\",\"a20\",\"t20\",\"u20\"],\"edges\":[[\"s20\",\"a20\"],[\"a20\",\"s20\"],[\"a20\",\"t20\"],[\"t20\",\"a20\"]],\"source\":\"s20\",\"sink\":\"t20\",\"capacities\":[105,105,105,105],\"flows\":[84,42,63,21]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([a20, s20, t20, u20], s20, t20, [[[a20, s20], 105, 42], [[a20, t20], 105, 63], [[s20, a20], 105, 84], [[t20, a20], 105, 21]], 2, 是, finite_loop_free_simple_digraph_with_explicit_antiparallel_convention_zero_capacities_and_disconnected_vertices_allowed)","explanation_zh":"Wave6第七層核對=([a20, s20, t20, u20], s20, t20, [[[a20, s20], 105, 42], [[a20, t20], 105, 63], [[s20, a20], 105, 84], [[t20, a20], 105, 21]], 2, 是, finite_loop_free_simple_digraph_with_explicit_antiparallel_convention_zero_capacities_and_disconnected_vertices_allowed)；依完整明示資料獨立重算；antiparallel edges明示，zero capacity與disconnected vertices允許；flow value由source與sink交叉計算；feasible不等於maximum，不提前宣稱min-cut equality或augmenting-path optimality。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限流網路"]},"variant":{"family_id":"qf-mod44-flow-network-qualification","seed":202611137021,"parameters":{"vertices":["s20","a20","t20","u20"],"edges":[["s20","a20"],["a20","s20"],["a20","t20"],["t20","a20"]],"source":"s20","sink":"t20","capacities":[105,105,105,105],"flows":[84,42,63,21],"domain":"Exact_finite_flow_network","operation":"flow-network-qualification","semantic_template":"qf-mod44-flow-network-qualification/{\"capacities\":[[105,1],[105,1],[105,1],[105,1]],\"edges\":[[\"a20\",\"s20\"],[\"a20\",\"t20\"],[\"s20\",\"a20\"],[\"t20\",\"a20\"]],\"flows\":[[42,1],[63,1],[84,1],[21,1]],\"sink\":\"t20\",\"source\":\"s20\",\"vertices\":[\"a20\",\"s20\",\"t20\",\"u20\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、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-20261113-r07-finite-flow-network-0022","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-flow-network"],"title_zh":"有限流網路","stem_zh":"finite loop-free directed network、逐edge capacity與逐internal-vertex conservation。操作=flow-network-qualification；資料={\"vertices\":[\"s21\",\"a21\",\"t21\",\"u21\"],\"edges\":[[\"s21\",\"a21\"],[\"a21\",\"s21\"],[\"a21\",\"t21\"],[\"t21\",\"a21\"]],\"source\":\"s21\",\"sink\":\"t21\",\"capacities\":[110,110,110,110],\"flows\":[88,44,66,22]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([a21, s21, t21, u21], s21, t21, [[[a21, s21], 110, 44], [[a21, t21], 110, 66], [[s21, a21], 110, 88], [[t21, a21], 110, 22]], 2, 是, finite_loop_free_simple_digraph_with_explicit_antiparallel_convention_zero_capacities_and_disconnected_vertices_allowed)","explanation_zh":"Wave6第七層核對=([a21, s21, t21, u21], s21, t21, [[[a21, s21], 110, 44], [[a21, t21], 110, 66], [[s21, a21], 110, 88], [[t21, a21], 110, 22]], 2, 是, finite_loop_free_simple_digraph_with_explicit_antiparallel_convention_zero_capacities_and_disconnected_vertices_allowed)；依完整明示資料獨立重算；antiparallel edges明示，zero capacity與disconnected vertices允許；flow value由source與sink交叉計算；feasible不等於maximum，不提前宣稱min-cut equality或augmenting-path optimality。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限流網路"]},"variant":{"family_id":"qf-mod44-flow-network-qualification","seed":202611137022,"parameters":{"vertices":["s21","a21","t21","u21"],"edges":[["s21","a21"],["a21","s21"],["a21","t21"],["t21","a21"]],"source":"s21","sink":"t21","capacities":[110,110,110,110],"flows":[88,44,66,22],"domain":"Exact_finite_flow_network","operation":"flow-network-qualification","semantic_template":"qf-mod44-flow-network-qualification/{\"capacities\":[[110,1],[110,1],[110,1],[110,1]],\"edges\":[[\"a21\",\"s21\"],[\"a21\",\"t21\"],[\"s21\",\"a21\"],[\"t21\",\"a21\"]],\"flows\":[[44,1],[66,1],[88,1],[22,1]],\"sink\":\"t21\",\"source\":\"s21\",\"vertices\":[\"a21\",\"s21\",\"t21\",\"u21\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、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-20261113-r07-finite-flow-network-0023","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-flow-network"],"title_zh":"有限流網路","stem_zh":"finite loop-free directed network、逐edge capacity與逐internal-vertex conservation。操作=flow-network-qualification；資料={\"vertices\":[\"s22\",\"a22\",\"t22\",\"u22\"],\"edges\":[[\"s22\",\"a22\"],[\"a22\",\"s22\"],[\"a22\",\"t22\"],[\"t22\",\"a22\"]],\"source\":\"s22\",\"sink\":\"t22\",\"capacities\":[115,115,115,115],\"flows\":[92,46,69,23]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([a22, s22, t22, u22], s22, t22, [[[a22, s22], 115, 46], [[a22, t22], 115, 69], [[s22, a22], 115, 92], [[t22, a22], 115, 23]], 2, 是, finite_loop_free_simple_digraph_with_explicit_antiparallel_convention_zero_capacities_and_disconnected_vertices_allowed)","explanation_zh":"Wave6第七層核對=([a22, s22, t22, u22], s22, t22, [[[a22, s22], 115, 46], [[a22, t22], 115, 69], [[s22, a22], 115, 92], [[t22, a22], 115, 23]], 2, 是, finite_loop_free_simple_digraph_with_explicit_antiparallel_convention_zero_capacities_and_disconnected_vertices_allowed)；依完整明示資料獨立重算；antiparallel edges明示，zero capacity與disconnected vertices允許；flow value由source與sink交叉計算；feasible不等於maximum，不提前宣稱min-cut equality或augmenting-path optimality。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限流網路"]},"variant":{"family_id":"qf-mod44-flow-network-qualification","seed":202611137023,"parameters":{"vertices":["s22","a22","t22","u22"],"edges":[["s22","a22"],["a22","s22"],["a22","t22"],["t22","a22"]],"source":"s22","sink":"t22","capacities":[115,115,115,115],"flows":[92,46,69,23],"domain":"Exact_finite_flow_network","operation":"flow-network-qualification","semantic_template":"qf-mod44-flow-network-qualification/{\"capacities\":[[115,1],[115,1],[115,1],[115,1]],\"edges\":[[\"a22\",\"s22\"],[\"a22\",\"t22\"],[\"s22\",\"a22\"],[\"t22\",\"a22\"]],\"flows\":[[46,1],[69,1],[92,1],[23,1]],\"sink\":\"t22\",\"source\":\"s22\",\"vertices\":[\"a22\",\"s22\",\"t22\",\"u22\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、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-20261113-r07-finite-flow-network-0024","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-flow-network"],"title_zh":"有限流網路","stem_zh":"finite loop-free directed network、逐edge capacity與逐internal-vertex conservation。操作=flow-network-qualification；資料={\"vertices\":[\"s23\",\"a23\",\"t23\",\"u23\"],\"edges\":[[\"s23\",\"a23\"],[\"a23\",\"s23\"],[\"a23\",\"t23\"],[\"t23\",\"a23\"]],\"source\":\"s23\",\"sink\":\"t23\",\"capacities\":[120,120,120,120],\"flows\":[96,48,72,24]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([a23, s23, t23, u23], s23, t23, [[[a23, s23], 120, 48], [[a23, t23], 120, 72], [[s23, a23], 120, 96], [[t23, a23], 120, 24]], 2, 是, finite_loop_free_simple_digraph_with_explicit_antiparallel_convention_zero_capacities_and_disconnected_vertices_allowed)","explanation_zh":"Wave6第七層核對=([a23, s23, t23, u23], s23, t23, [[[a23, s23], 120, 48], [[a23, t23], 120, 72], [[s23, a23], 120, 96], [[t23, a23], 120, 24]], 2, 是, finite_loop_free_simple_digraph_with_explicit_antiparallel_convention_zero_capacities_and_disconnected_vertices_allowed)；依完整明示資料獨立重算；antiparallel edges明示，zero capacity與disconnected vertices允許；flow value由source與sink交叉計算；feasible不等於maximum，不提前宣稱min-cut equality或augmenting-path optimality。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限流網路"]},"variant":{"family_id":"qf-mod44-flow-network-qualification","seed":202611137024,"parameters":{"vertices":["s23","a23","t23","u23"],"edges":[["s23","a23"],["a23","s23"],["a23","t23"],["t23","a23"]],"source":"s23","sink":"t23","capacities":[120,120,120,120],"flows":[96,48,72,24],"domain":"Exact_finite_flow_network","operation":"flow-network-qualification","semantic_template":"qf-mod44-flow-network-qualification/{\"capacities\":[[120,1],[120,1],[120,1],[120,1]],\"edges\":[[\"a23\",\"s23\"],[\"a23\",\"t23\"],[\"s23\",\"a23\"],[\"t23\",\"a23\"]],\"flows\":[[48,1],[72,1],[96,1],[24,1]],\"sink\":\"t23\",\"source\":\"s23\",\"vertices\":[\"a23\",\"s23\",\"t23\",\"u23\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、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-20261113-r07-finite-flow-network-0025","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-flow-network"],"title_zh":"有限流網路","stem_zh":"finite loop-free directed network、逐edge capacity與逐internal-vertex conservation。操作=flow-network-qualification；資料={\"vertices\":[\"s24\",\"a24\",\"t24\",\"u24\"],\"edges\":[[\"s24\",\"a24\"],[\"a24\",\"s24\"],[\"a24\",\"t24\"],[\"t24\",\"a24\"]],\"source\":\"s24\",\"sink\":\"t24\",\"capacities\":[125,125,125,125],\"flows\":[100,50,75,25]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([a24, s24, t24, u24], s24, t24, [[[a24, s24], 125, 50], [[a24, t24], 125, 75], [[s24, a24], 125, 100], [[t24, a24], 125, 25]], 2, 是, finite_loop_free_simple_digraph_with_explicit_antiparallel_convention_zero_capacities_and_disconnected_vertices_allowed)","explanation_zh":"Wave6第七層核對=([a24, s24, t24, u24], s24, t24, [[[a24, s24], 125, 50], [[a24, t24], 125, 75], [[s24, a24], 125, 100], [[t24, a24], 125, 25]], 2, 是, finite_loop_free_simple_digraph_with_explicit_antiparallel_convention_zero_capacities_and_disconnected_vertices_allowed)；依完整明示資料獨立重算；antiparallel edges明示，zero capacity與disconnected vertices允許；flow value由source與sink交叉計算；feasible不等於maximum，不提前宣稱min-cut equality或augmenting-path optimality。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限流網路"]},"variant":{"family_id":"qf-mod44-flow-network-qualification","seed":202611137025,"parameters":{"vertices":["s24","a24","t24","u24"],"edges":[["s24","a24"],["a24","s24"],["a24","t24"],["t24","a24"]],"source":"s24","sink":"t24","capacities":[125,125,125,125],"flows":[100,50,75,25],"domain":"Exact_finite_flow_network","operation":"flow-network-qualification","semantic_template":"qf-mod44-flow-network-qualification/{\"capacities\":[[125,1],[125,1],[125,1],[125,1]],\"edges\":[[\"a24\",\"s24\"],[\"a24\",\"t24\"],[\"s24\",\"a24\"],[\"t24\",\"a24\"]],\"flows\":[[50,1],[75,1],[100,1],[25,1]],\"sink\":\"t24\",\"source\":\"s24\",\"vertices\":[\"a24\",\"s24\",\"t24\",\"u24\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、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-20261113-r07-finite-flow-network-0026","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-flow-network"],"title_zh":"有限流網路","stem_zh":"finite loop-free directed network、逐edge capacity與逐internal-vertex conservation。操作=flow-capacity-audit；資料={\"vertices\":[\"s0\",\"a0\",\"t0\",\"u0\"],\"edges\":[[\"s0\",\"a0\"],[\"a0\",\"s0\"],[\"a0\",\"t0\"],[\"t0\",\"a0\"]],\"source\":\"s0\",\"sink\":\"t0\",\"capacities\":[5,5,5,5],\"flows\":[-1,2,3,1]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(s0, t0, [[[a0, s0], 5, 2], [[a0, t0], 5, 3], [[s0, a0], 5, -1], [[t0, a0], 5, 1]], [[[a0, s0], 2, 5, 是], [[a0, t0], 3, 5, 是], [[s0, a0], -1, 5, 否], [[t0, a0], 1, 5, 是]], 否, [[s0, a0], -1, 5, 否], every_edge_requires_zero_less_equal_flow_less_equal_capacity)","explanation_zh":"Wave6第七層核對=(s0, t0, [[[a0, s0], 5, 2], [[a0, t0], 5, 3], [[s0, a0], 5, -1], [[t0, a0], 5, 1]], [[[a0, s0], 2, 5, 是], [[a0, t0], 3, 5, 是], [[s0, a0], -1, 5, 否], [[t0, a0], 1, 5, 是]], 否, [[s0, a0], -1, 5, 否], every_edge_requires_zero_less_equal_flow_less_equal_capacity)；依完整明示資料獨立重算；antiparallel edges明示，zero capacity與disconnected vertices允許；flow value由source與sink交叉計算；feasible不等於maximum，不提前宣稱min-cut equality或augmenting-path optimality。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限流網路"]},"variant":{"family_id":"qf-mod44-flow-capacity-audit","seed":202611137026,"parameters":{"vertices":["s0","a0","t0","u0"],"edges":[["s0","a0"],["a0","s0"],["a0","t0"],["t0","a0"]],"source":"s0","sink":"t0","capacities":[5,5,5,5],"flows":[-1,2,3,1],"domain":"Exact_finite_flow_network","operation":"flow-capacity-audit","semantic_template":"qf-mod44-flow-capacity-audit/{\"capacities\":[[5,1],[5,1],[5,1],[5,1]],\"edges\":[[\"a0\",\"s0\"],[\"a0\",\"t0\"],[\"s0\",\"a0\"],[\"t0\",\"a0\"]],\"flows\":[[2,1],[3,1],[-1,1],[1,1]],\"sink\":\"t0\",\"source\":\"s0\",\"vertices\":[\"a0\",\"s0\",\"t0\",\"u0\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、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-20261113-r07-finite-flow-network-0027","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-flow-network"],"title_zh":"有限流網路","stem_zh":"finite loop-free directed network、逐edge capacity與逐internal-vertex conservation。操作=flow-capacity-audit；資料={\"vertices\":[\"s1\",\"a1\",\"t1\",\"u1\"],\"edges\":[[\"s1\",\"a1\"],[\"a1\",\"s1\"],[\"a1\",\"t1\"],[\"t1\",\"a1\"]],\"source\":\"s1\",\"sink\":\"t1\",\"capacities\":[10,10,10,10],\"flows\":[12,4,6,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(s1, t1, [[[a1, s1], 10, 4], [[a1, t1], 10, 6], [[s1, a1], 10, 12], [[t1, a1], 10, 2]], [[[a1, s1], 4, 10, 是], [[a1, t1], 6, 10, 是], [[s1, a1], 12, 10, 否], [[t1, a1], 2, 10, 是]], 否, [[s1, a1], 12, 10, 否], every_edge_requires_zero_less_equal_flow_less_equal_capacity)","explanation_zh":"Wave6第七層核對=(s1, t1, [[[a1, s1], 10, 4], [[a1, t1], 10, 6], [[s1, a1], 10, 12], [[t1, a1], 10, 2]], [[[a1, s1], 4, 10, 是], [[a1, t1], 6, 10, 是], [[s1, a1], 12, 10, 否], [[t1, a1], 2, 10, 是]], 否, [[s1, a1], 12, 10, 否], every_edge_requires_zero_less_equal_flow_less_equal_capacity)；依完整明示資料獨立重算；antiparallel edges明示，zero capacity與disconnected vertices允許；flow value由source與sink交叉計算；feasible不等於maximum，不提前宣稱min-cut equality或augmenting-path optimality。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限流網路"]},"variant":{"family_id":"qf-mod44-flow-capacity-audit","seed":202611137027,"parameters":{"vertices":["s1","a1","t1","u1"],"edges":[["s1","a1"],["a1","s1"],["a1","t1"],["t1","a1"]],"source":"s1","sink":"t1","capacities":[10,10,10,10],"flows":[12,4,6,2],"domain":"Exact_finite_flow_network","operation":"flow-capacity-audit","semantic_template":"qf-mod44-flow-capacity-audit/{\"capacities\":[[10,1],[10,1],[10,1],[10,1]],\"edges\":[[\"a1\",\"s1\"],[\"a1\",\"t1\"],[\"s1\",\"a1\"],[\"t1\",\"a1\"]],\"flows\":[[4,1],[6,1],[12,1],[2,1]],\"sink\":\"t1\",\"source\":\"s1\",\"vertices\":[\"a1\",\"s1\",\"t1\",\"u1\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、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-20261113-r07-finite-flow-network-0028","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-flow-network"],"title_zh":"有限流網路","stem_zh":"finite loop-free directed network、逐edge capacity與逐internal-vertex conservation。操作=flow-capacity-audit；資料={\"vertices\":[\"s2\",\"a2\",\"t2\",\"u2\"],\"edges\":[[\"s2\",\"a2\"],[\"a2\",\"s2\"],[\"a2\",\"t2\"],[\"t2\",\"a2\"]],\"source\":\"s2\",\"sink\":\"t2\",\"capacities\":[15,15,15,15],\"flows\":[12,6,9,3]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(s2, t2, [[[a2, s2], 15, 6], [[a2, t2], 15, 9], [[s2, a2], 15, 12], [[t2, a2], 15, 3]], [[[a2, s2], 6, 15, 是], [[a2, t2], 9, 15, 是], [[s2, a2], 12, 15, 是], [[t2, a2], 3, 15, 是]], 是, null, every_edge_requires_zero_less_equal_flow_less_equal_capacity)","explanation_zh":"Wave6第七層核對=(s2, t2, [[[a2, s2], 15, 6], [[a2, t2], 15, 9], [[s2, a2], 15, 12], [[t2, a2], 15, 3]], [[[a2, s2], 6, 15, 是], [[a2, t2], 9, 15, 是], [[s2, a2], 12, 15, 是], [[t2, a2], 3, 15, 是]], 是, null, every_edge_requires_zero_less_equal_flow_less_equal_capacity)；依完整明示資料獨立重算；antiparallel edges明示，zero capacity與disconnected vertices允許；flow value由source與sink交叉計算；feasible不等於maximum，不提前宣稱min-cut equality或augmenting-path optimality。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限流網路"]},"variant":{"family_id":"qf-mod44-flow-capacity-audit","seed":202611137028,"parameters":{"vertices":["s2","a2","t2","u2"],"edges":[["s2","a2"],["a2","s2"],["a2","t2"],["t2","a2"]],"source":"s2","sink":"t2","capacities":[15,15,15,15],"flows":[12,6,9,3],"domain":"Exact_finite_flow_network","operation":"flow-capacity-audit","semantic_template":"qf-mod44-flow-capacity-audit/{\"capacities\":[[15,1],[15,1],[15,1],[15,1]],\"edges\":[[\"a2\",\"s2\"],[\"a2\",\"t2\"],[\"s2\",\"a2\"],[\"t2\",\"a2\"]],\"flows\":[[6,1],[9,1],[12,1],[3,1]],\"sink\":\"t2\",\"source\":\"s2\",\"vertices\":[\"a2\",\"s2\",\"t2\",\"u2\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、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-20261113-r07-finite-flow-network-0029","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-flow-network"],"title_zh":"有限流網路","stem_zh":"finite loop-free directed network、逐edge capacity與逐internal-vertex conservation。操作=flow-capacity-audit；資料={\"vertices\":[\"s3\",\"a3\",\"t3\",\"u3\"],\"edges\":[[\"s3\",\"a3\"],[\"a3\",\"s3\"],[\"a3\",\"t3\"],[\"t3\",\"a3\"]],\"source\":\"s3\",\"sink\":\"t3\",\"capacities\":[20,20,20,20],\"flows\":[-4,8,12,4]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(s3, t3, [[[a3, s3], 20, 8], [[a3, t3], 20, 12], [[s3, a3], 20, -4], [[t3, a3], 20, 4]], [[[a3, s3], 8, 20, 是], [[a3, t3], 12, 20, 是], [[s3, a3], -4, 20, 否], [[t3, a3], 4, 20, 是]], 否, [[s3, a3], -4, 20, 否], every_edge_requires_zero_less_equal_flow_less_equal_capacity)","explanation_zh":"Wave6第七層核對=(s3, t3, [[[a3, s3], 20, 8], [[a3, t3], 20, 12], [[s3, a3], 20, -4], [[t3, a3], 20, 4]], [[[a3, s3], 8, 20, 是], [[a3, t3], 12, 20, 是], [[s3, a3], -4, 20, 否], [[t3, a3], 4, 20, 是]], 否, [[s3, a3], -4, 20, 否], every_edge_requires_zero_less_equal_flow_less_equal_capacity)；依完整明示資料獨立重算；antiparallel edges明示，zero capacity與disconnected vertices允許；flow value由source與sink交叉計算；feasible不等於maximum，不提前宣稱min-cut equality或augmenting-path optimality。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限流網路"]},"variant":{"family_id":"qf-mod44-flow-capacity-audit","seed":202611137029,"parameters":{"vertices":["s3","a3","t3","u3"],"edges":[["s3","a3"],["a3","s3"],["a3","t3"],["t3","a3"]],"source":"s3","sink":"t3","capacities":[20,20,20,20],"flows":[-4,8,12,4],"domain":"Exact_finite_flow_network","operation":"flow-capacity-audit","semantic_template":"qf-mod44-flow-capacity-audit/{\"capacities\":[[20,1],[20,1],[20,1],[20,1]],\"edges\":[[\"a3\",\"s3\"],[\"a3\",\"t3\"],[\"s3\",\"a3\"],[\"t3\",\"a3\"]],\"flows\":[[8,1],[12,1],[-4,1],[4,1]],\"sink\":\"t3\",\"source\":\"s3\",\"vertices\":[\"a3\",\"s3\",\"t3\",\"u3\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、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-20261113-r07-finite-flow-network-0030","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-flow-network"],"title_zh":"有限流網路","stem_zh":"finite loop-free directed network、逐edge capacity與逐internal-vertex conservation。操作=flow-capacity-audit；資料={\"vertices\":[\"s4\",\"a4\",\"t4\",\"u4\"],\"edges\":[[\"s4\",\"a4\"],[\"a4\",\"s4\"],[\"a4\",\"t4\"],[\"t4\",\"a4\"]],\"source\":\"s4\",\"sink\":\"t4\",\"capacities\":[25,25,25,25],\"flows\":[30,10,15,5]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(s4, t4, [[[a4, s4], 25, 10], [[a4, t4], 25, 15], [[s4, a4], 25, 30], [[t4, a4], 25, 5]], [[[a4, s4], 10, 25, 是], [[a4, t4], 15, 25, 是], [[s4, a4], 30, 25, 否], [[t4, a4], 5, 25, 是]], 否, [[s4, a4], 30, 25, 否], every_edge_requires_zero_less_equal_flow_less_equal_capacity)","explanation_zh":"Wave6第七層核對=(s4, t4, [[[a4, s4], 25, 10], [[a4, t4], 25, 15], [[s4, a4], 25, 30], [[t4, a4], 25, 5]], [[[a4, s4], 10, 25, 是], [[a4, t4], 15, 25, 是], [[s4, a4], 30, 25, 否], [[t4, a4], 5, 25, 是]], 否, [[s4, a4], 30, 25, 否], every_edge_requires_zero_less_equal_flow_less_equal_capacity)；依完整明示資料獨立重算；antiparallel edges明示，zero capacity與disconnected vertices允許；flow value由source與sink交叉計算；feasible不等於maximum，不提前宣稱min-cut equality或augmenting-path optimality。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限流網路"]},"variant":{"family_id":"qf-mod44-flow-capacity-audit","seed":202611137030,"parameters":{"vertices":["s4","a4","t4","u4"],"edges":[["s4","a4"],["a4","s4"],["a4","t4"],["t4","a4"]],"source":"s4","sink":"t4","capacities":[25,25,25,25],"flows":[30,10,15,5],"domain":"Exact_finite_flow_network","operation":"flow-capacity-audit","semantic_template":"qf-mod44-flow-capacity-audit/{\"capacities\":[[25,1],[25,1],[25,1],[25,1]],\"edges\":[[\"a4\",\"s4\"],[\"a4\",\"t4\"],[\"s4\",\"a4\"],[\"t4\",\"a4\"]],\"flows\":[[10,1],[15,1],[30,1],[5,1]],\"sink\":\"t4\",\"source\":\"s4\",\"vertices\":[\"a4\",\"s4\",\"t4\",\"u4\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、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-20261113-r07-finite-flow-network-0031","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-flow-network"],"title_zh":"有限流網路","stem_zh":"finite loop-free directed network、逐edge capacity與逐internal-vertex conservation。操作=flow-capacity-audit；資料={\"vertices\":[\"s5\",\"a5\",\"t5\",\"u5\"],\"edges\":[[\"s5\",\"a5\"],[\"a5\",\"s5\"],[\"a5\",\"t5\"],[\"t5\",\"a5\"]],\"source\":\"s5\",\"sink\":\"t5\",\"capacities\":[30,30,30,30],\"flows\":[24,12,18,6]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(s5, t5, [[[a5, s5], 30, 12], [[a5, t5], 30, 18], [[s5, a5], 30, 24], [[t5, a5], 30, 6]], [[[a5, s5], 12, 30, 是], [[a5, t5], 18, 30, 是], [[s5, a5], 24, 30, 是], [[t5, a5], 6, 30, 是]], 是, null, every_edge_requires_zero_less_equal_flow_less_equal_capacity)","explanation_zh":"Wave6第七層核對=(s5, t5, [[[a5, s5], 30, 12], [[a5, t5], 30, 18], [[s5, a5], 30, 24], [[t5, a5], 30, 6]], [[[a5, s5], 12, 30, 是], [[a5, t5], 18, 30, 是], [[s5, a5], 24, 30, 是], [[t5, a5], 6, 30, 是]], 是, null, every_edge_requires_zero_less_equal_flow_less_equal_capacity)；依完整明示資料獨立重算；antiparallel edges明示，zero capacity與disconnected vertices允許；flow value由source與sink交叉計算；feasible不等於maximum，不提前宣稱min-cut equality或augmenting-path optimality。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限流網路"]},"variant":{"family_id":"qf-mod44-flow-capacity-audit","seed":202611137031,"parameters":{"vertices":["s5","a5","t5","u5"],"edges":[["s5","a5"],["a5","s5"],["a5","t5"],["t5","a5"]],"source":"s5","sink":"t5","capacities":[30,30,30,30],"flows":[24,12,18,6],"domain":"Exact_finite_flow_network","operation":"flow-capacity-audit","semantic_template":"qf-mod44-flow-capacity-audit/{\"capacities\":[[30,1],[30,1],[30,1],[30,1]],\"edges\":[[\"a5\",\"s5\"],[\"a5\",\"t5\"],[\"s5\",\"a5\"],[\"t5\",\"a5\"]],\"flows\":[[12,1],[18,1],[24,1],[6,1]],\"sink\":\"t5\",\"source\":\"s5\",\"vertices\":[\"a5\",\"s5\",\"t5\",\"u5\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、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-20261113-r07-finite-flow-network-0032","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-flow-network"],"title_zh":"有限流網路","stem_zh":"finite loop-free directed network、逐edge capacity與逐internal-vertex conservation。操作=flow-capacity-audit；資料={\"vertices\":[\"s6\",\"a6\",\"t6\",\"u6\"],\"edges\":[[\"s6\",\"a6\"],[\"a6\",\"s6\"],[\"a6\",\"t6\"],[\"t6\",\"a6\"]],\"source\":\"s6\",\"sink\":\"t6\",\"capacities\":[35,35,35,35],\"flows\":[-7,14,21,7]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(s6, t6, [[[a6, s6], 35, 14], [[a6, t6], 35, 21], [[s6, a6], 35, -7], [[t6, a6], 35, 7]], [[[a6, s6], 14, 35, 是], [[a6, t6], 21, 35, 是], [[s6, a6], -7, 35, 否], [[t6, a6], 7, 35, 是]], 否, [[s6, a6], -7, 35, 否], every_edge_requires_zero_less_equal_flow_less_equal_capacity)","explanation_zh":"Wave6第七層核對=(s6, t6, [[[a6, s6], 35, 14], [[a6, t6], 35, 21], [[s6, a6], 35, -7], [[t6, a6], 35, 7]], [[[a6, s6], 14, 35, 是], [[a6, t6], 21, 35, 是], [[s6, a6], -7, 35, 否], [[t6, a6], 7, 35, 是]], 否, [[s6, a6], -7, 35, 否], every_edge_requires_zero_less_equal_flow_less_equal_capacity)；依完整明示資料獨立重算；antiparallel edges明示，zero capacity與disconnected vertices允許；flow value由source與sink交叉計算；feasible不等於maximum，不提前宣稱min-cut equality或augmenting-path optimality。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限流網路"]},"variant":{"family_id":"qf-mod44-flow-capacity-audit","seed":202611137032,"parameters":{"vertices":["s6","a6","t6","u6"],"edges":[["s6","a6"],["a6","s6"],["a6","t6"],["t6","a6"]],"source":"s6","sink":"t6","capacities":[35,35,35,35],"flows":[-7,14,21,7],"domain":"Exact_finite_flow_network","operation":"flow-capacity-audit","semantic_template":"qf-mod44-flow-capacity-audit/{\"capacities\":[[35,1],[35,1],[35,1],[35,1]],\"edges\":[[\"a6\",\"s6\"],[\"a6\",\"t6\"],[\"s6\",\"a6\"],[\"t6\",\"a6\"]],\"flows\":[[14,1],[21,1],[-7,1],[7,1]],\"sink\":\"t6\",\"source\":\"s6\",\"vertices\":[\"a6\",\"s6\",\"t6\",\"u6\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、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-20261113-r07-finite-flow-network-0033","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-flow-network"],"title_zh":"有限流網路","stem_zh":"finite loop-free directed network、逐edge capacity與逐internal-vertex conservation。操作=flow-capacity-audit；資料={\"vertices\":[\"s7\",\"a7\",\"t7\",\"u7\"],\"edges\":[[\"s7\",\"a7\"],[\"a7\",\"s7\"],[\"a7\",\"t7\"],[\"t7\",\"a7\"]],\"source\":\"s7\",\"sink\":\"t7\",\"capacities\":[40,40,40,40],\"flows\":[48,16,24,8]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(s7, t7, [[[a7, s7], 40, 16], [[a7, t7], 40, 24], [[s7, a7], 40, 48], [[t7, a7], 40, 8]], [[[a7, s7], 16, 40, 是], [[a7, t7], 24, 40, 是], [[s7, a7], 48, 40, 否], [[t7, a7], 8, 40, 是]], 否, [[s7, a7], 48, 40, 否], every_edge_requires_zero_less_equal_flow_less_equal_capacity)","explanation_zh":"Wave6第七層核對=(s7, t7, [[[a7, s7], 40, 16], [[a7, t7], 40, 24], [[s7, a7], 40, 48], [[t7, a7], 40, 8]], [[[a7, s7], 16, 40, 是], [[a7, t7], 24, 40, 是], [[s7, a7], 48, 40, 否], [[t7, a7], 8, 40, 是]], 否, [[s7, a7], 48, 40, 否], every_edge_requires_zero_less_equal_flow_less_equal_capacity)；依完整明示資料獨立重算；antiparallel edges明示，zero capacity與disconnected vertices允許；flow value由source與sink交叉計算；feasible不等於maximum，不提前宣稱min-cut equality或augmenting-path optimality。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限流網路"]},"variant":{"family_id":"qf-mod44-flow-capacity-audit","seed":202611137033,"parameters":{"vertices":["s7","a7","t7","u7"],"edges":[["s7","a7"],["a7","s7"],["a7","t7"],["t7","a7"]],"source":"s7","sink":"t7","capacities":[40,40,40,40],"flows":[48,16,24,8],"domain":"Exact_finite_flow_network","operation":"flow-capacity-audit","semantic_template":"qf-mod44-flow-capacity-audit/{\"capacities\":[[40,1],[40,1],[40,1],[40,1]],\"edges\":[[\"a7\",\"s7\"],[\"a7\",\"t7\"],[\"s7\",\"a7\"],[\"t7\",\"a7\"]],\"flows\":[[16,1],[24,1],[48,1],[8,1]],\"sink\":\"t7\",\"source\":\"s7\",\"vertices\":[\"a7\",\"s7\",\"t7\",\"u7\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、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-20261113-r07-finite-flow-network-0034","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-flow-network"],"title_zh":"有限流網路","stem_zh":"finite loop-free directed network、逐edge capacity與逐internal-vertex conservation。操作=flow-capacity-audit；資料={\"vertices\":[\"s8\",\"a8\",\"t8\",\"u8\"],\"edges\":[[\"s8\",\"a8\"],[\"a8\",\"s8\"],[\"a8\",\"t8\"],[\"t8\",\"a8\"]],\"source\":\"s8\",\"sink\":\"t8\",\"capacities\":[45,45,45,45],\"flows\":[36,18,27,9]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(s8, t8, [[[a8, s8], 45, 18], [[a8, t8], 45, 27], [[s8, a8], 45, 36], [[t8, a8], 45, 9]], [[[a8, s8], 18, 45, 是], [[a8, t8], 27, 45, 是], [[s8, a8], 36, 45, 是], [[t8, a8], 9, 45, 是]], 是, null, every_edge_requires_zero_less_equal_flow_less_equal_capacity)","explanation_zh":"Wave6第七層核對=(s8, t8, [[[a8, s8], 45, 18], [[a8, t8], 45, 27], [[s8, a8], 45, 36], [[t8, a8], 45, 9]], [[[a8, s8], 18, 45, 是], [[a8, t8], 27, 45, 是], [[s8, a8], 36, 45, 是], [[t8, a8], 9, 45, 是]], 是, null, every_edge_requires_zero_less_equal_flow_less_equal_capacity)；依完整明示資料獨立重算；antiparallel edges明示，zero capacity與disconnected vertices允許；flow value由source與sink交叉計算；feasible不等於maximum，不提前宣稱min-cut equality或augmenting-path optimality。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限流網路"]},"variant":{"family_id":"qf-mod44-flow-capacity-audit","seed":202611137034,"parameters":{"vertices":["s8","a8","t8","u8"],"edges":[["s8","a8"],["a8","s8"],["a8","t8"],["t8","a8"]],"source":"s8","sink":"t8","capacities":[45,45,45,45],"flows":[36,18,27,9],"domain":"Exact_finite_flow_network","operation":"flow-capacity-audit","semantic_template":"qf-mod44-flow-capacity-audit/{\"capacities\":[[45,1],[45,1],[45,1],[45,1]],\"edges\":[[\"a8\",\"s8\"],[\"a8\",\"t8\"],[\"s8\",\"a8\"],[\"t8\",\"a8\"]],\"flows\":[[18,1],[27,1],[36,1],[9,1]],\"sink\":\"t8\",\"source\":\"s8\",\"vertices\":[\"a8\",\"s8\",\"t8\",\"u8\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、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-20261113-r07-finite-flow-network-0035","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-flow-network"],"title_zh":"有限流網路","stem_zh":"finite loop-free directed network、逐edge capacity與逐internal-vertex conservation。操作=flow-capacity-audit；資料={\"vertices\":[\"s9\",\"a9\",\"t9\",\"u9\"],\"edges\":[[\"s9\",\"a9\"],[\"a9\",\"s9\"],[\"a9\",\"t9\"],[\"t9\",\"a9\"]],\"source\":\"s9\",\"sink\":\"t9\",\"capacities\":[50,50,50,50],\"flows\":[-10,20,30,10]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(s9, t9, [[[a9, s9], 50, 20], [[a9, t9], 50, 30], [[s9, a9], 50, -10], [[t9, a9], 50, 10]], [[[a9, s9], 20, 50, 是], [[a9, t9], 30, 50, 是], [[s9, a9], -10, 50, 否], [[t9, a9], 10, 50, 是]], 否, [[s9, a9], -10, 50, 否], every_edge_requires_zero_less_equal_flow_less_equal_capacity)","explanation_zh":"Wave6第七層核對=(s9, t9, [[[a9, s9], 50, 20], [[a9, t9], 50, 30], [[s9, a9], 50, -10], [[t9, a9], 50, 10]], [[[a9, s9], 20, 50, 是], [[a9, t9], 30, 50, 是], [[s9, a9], -10, 50, 否], [[t9, a9], 10, 50, 是]], 否, [[s9, a9], -10, 50, 否], every_edge_requires_zero_less_equal_flow_less_equal_capacity)；依完整明示資料獨立重算；antiparallel edges明示，zero capacity與disconnected vertices允許；flow value由source與sink交叉計算；feasible不等於maximum，不提前宣稱min-cut equality或augmenting-path optimality。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限流網路"]},"variant":{"family_id":"qf-mod44-flow-capacity-audit","seed":202611137035,"parameters":{"vertices":["s9","a9","t9","u9"],"edges":[["s9","a9"],["a9","s9"],["a9","t9"],["t9","a9"]],"source":"s9","sink":"t9","capacities":[50,50,50,50],"flows":[-10,20,30,10],"domain":"Exact_finite_flow_network","operation":"flow-capacity-audit","semantic_template":"qf-mod44-flow-capacity-audit/{\"capacities\":[[50,1],[50,1],[50,1],[50,1]],\"edges\":[[\"a9\",\"s9\"],[\"a9\",\"t9\"],[\"s9\",\"a9\"],[\"t9\",\"a9\"]],\"flows\":[[20,1],[30,1],[-10,1],[10,1]],\"sink\":\"t9\",\"source\":\"s9\",\"vertices\":[\"a9\",\"s9\",\"t9\",\"u9\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、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-20261113-r07-finite-flow-network-0036","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-flow-network"],"title_zh":"有限流網路","stem_zh":"finite loop-free directed network、逐edge capacity與逐internal-vertex conservation。操作=flow-capacity-audit；資料={\"vertices\":[\"s10\",\"a10\",\"t10\",\"u10\"],\"edges\":[[\"s10\",\"a10\"],[\"a10\",\"s10\"],[\"a10\",\"t10\"],[\"t10\",\"a10\"]],\"source\":\"s10\",\"sink\":\"t10\",\"capacities\":[55,55,55,55],\"flows\":[66,22,33,11]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(s10, t10, [[[a10, s10], 55, 22], [[a10, t10], 55, 33], [[s10, a10], 55, 66], [[t10, a10], 55, 11]], [[[a10, s10], 22, 55, 是], [[a10, t10], 33, 55, 是], [[s10, a10], 66, 55, 否], [[t10, a10], 11, 55, 是]], 否, [[s10, a10], 66, 55, 否], every_edge_requires_zero_less_equal_flow_less_equal_capacity)","explanation_zh":"Wave6第七層核對=(s10, t10, [[[a10, s10], 55, 22], [[a10, t10], 55, 33], [[s10, a10], 55, 66], [[t10, a10], 55, 11]], [[[a10, s10], 22, 55, 是], [[a10, t10], 33, 55, 是], [[s10, a10], 66, 55, 否], [[t10, a10], 11, 55, 是]], 否, [[s10, a10], 66, 55, 否], every_edge_requires_zero_less_equal_flow_less_equal_capacity)；依完整明示資料獨立重算；antiparallel edges明示，zero capacity與disconnected vertices允許；flow value由source與sink交叉計算；feasible不等於maximum，不提前宣稱min-cut equality或augmenting-path optimality。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限流網路"]},"variant":{"family_id":"qf-mod44-flow-capacity-audit","seed":202611137036,"parameters":{"vertices":["s10","a10","t10","u10"],"edges":[["s10","a10"],["a10","s10"],["a10","t10"],["t10","a10"]],"source":"s10","sink":"t10","capacities":[55,55,55,55],"flows":[66,22,33,11],"domain":"Exact_finite_flow_network","operation":"flow-capacity-audit","semantic_template":"qf-mod44-flow-capacity-audit/{\"capacities\":[[55,1],[55,1],[55,1],[55,1]],\"edges\":[[\"a10\",\"s10\"],[\"a10\",\"t10\"],[\"s10\",\"a10\"],[\"t10\",\"a10\"]],\"flows\":[[22,1],[33,1],[66,1],[11,1]],\"sink\":\"t10\",\"source\":\"s10\",\"vertices\":[\"a10\",\"s10\",\"t10\",\"u10\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、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-20261113-r07-finite-flow-network-0037","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-flow-network"],"title_zh":"有限流網路","stem_zh":"finite loop-free directed network、逐edge capacity與逐internal-vertex conservation。操作=flow-capacity-audit；資料={\"vertices\":[\"s11\",\"a11\",\"t11\",\"u11\"],\"edges\":[[\"s11\",\"a11\"],[\"a11\",\"s11\"],[\"a11\",\"t11\"],[\"t11\",\"a11\"]],\"source\":\"s11\",\"sink\":\"t11\",\"capacities\":[60,60,60,60],\"flows\":[48,24,36,12]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(s11, t11, [[[a11, s11], 60, 24], [[a11, t11], 60, 36], [[s11, a11], 60, 48], [[t11, a11], 60, 12]], [[[a11, s11], 24, 60, 是], [[a11, t11], 36, 60, 是], [[s11, a11], 48, 60, 是], [[t11, a11], 12, 60, 是]], 是, null, every_edge_requires_zero_less_equal_flow_less_equal_capacity)","explanation_zh":"Wave6第七層核對=(s11, t11, [[[a11, s11], 60, 24], [[a11, t11], 60, 36], [[s11, a11], 60, 48], [[t11, a11], 60, 12]], [[[a11, s11], 24, 60, 是], [[a11, t11], 36, 60, 是], [[s11, a11], 48, 60, 是], [[t11, a11], 12, 60, 是]], 是, null, every_edge_requires_zero_less_equal_flow_less_equal_capacity)；依完整明示資料獨立重算；antiparallel edges明示，zero capacity與disconnected vertices允許；flow value由source與sink交叉計算；feasible不等於maximum，不提前宣稱min-cut equality或augmenting-path optimality。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限流網路"]},"variant":{"family_id":"qf-mod44-flow-capacity-audit","seed":202611137037,"parameters":{"vertices":["s11","a11","t11","u11"],"edges":[["s11","a11"],["a11","s11"],["a11","t11"],["t11","a11"]],"source":"s11","sink":"t11","capacities":[60,60,60,60],"flows":[48,24,36,12],"domain":"Exact_finite_flow_network","operation":"flow-capacity-audit","semantic_template":"qf-mod44-flow-capacity-audit/{\"capacities\":[[60,1],[60,1],[60,1],[60,1]],\"edges\":[[\"a11\",\"s11\"],[\"a11\",\"t11\"],[\"s11\",\"a11\"],[\"t11\",\"a11\"]],\"flows\":[[24,1],[36,1],[48,1],[12,1]],\"sink\":\"t11\",\"source\":\"s11\",\"vertices\":[\"a11\",\"s11\",\"t11\",\"u11\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、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-20261113-r07-finite-flow-network-0038","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-flow-network"],"title_zh":"有限流網路","stem_zh":"finite loop-free directed network、逐edge capacity與逐internal-vertex conservation。操作=flow-capacity-audit；資料={\"vertices\":[\"s12\",\"a12\",\"t12\",\"u12\"],\"edges\":[[\"s12\",\"a12\"],[\"a12\",\"s12\"],[\"a12\",\"t12\"],[\"t12\",\"a12\"]],\"source\":\"s12\",\"sink\":\"t12\",\"capacities\":[65,65,65,65],\"flows\":[-13,26,39,13]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(s12, t12, [[[a12, s12], 65, 26], [[a12, t12], 65, 39], [[s12, a12], 65, -13], [[t12, a12], 65, 13]], [[[a12, s12], 26, 65, 是], [[a12, t12], 39, 65, 是], [[s12, a12], -13, 65, 否], [[t12, a12], 13, 65, 是]], 否, [[s12, a12], -13, 65, 否], every_edge_requires_zero_less_equal_flow_less_equal_capacity)","explanation_zh":"Wave6第七層核對=(s12, t12, [[[a12, s12], 65, 26], [[a12, t12], 65, 39], [[s12, a12], 65, -13], [[t12, a12], 65, 13]], [[[a12, s12], 26, 65, 是], [[a12, t12], 39, 65, 是], [[s12, a12], -13, 65, 否], [[t12, a12], 13, 65, 是]], 否, [[s12, a12], -13, 65, 否], every_edge_requires_zero_less_equal_flow_less_equal_capacity)；依完整明示資料獨立重算；antiparallel edges明示，zero capacity與disconnected vertices允許；flow value由source與sink交叉計算；feasible不等於maximum，不提前宣稱min-cut equality或augmenting-path optimality。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限流網路"]},"variant":{"family_id":"qf-mod44-flow-capacity-audit","seed":202611137038,"parameters":{"vertices":["s12","a12","t12","u12"],"edges":[["s12","a12"],["a12","s12"],["a12","t12"],["t12","a12"]],"source":"s12","sink":"t12","capacities":[65,65,65,65],"flows":[-13,26,39,13],"domain":"Exact_finite_flow_network","operation":"flow-capacity-audit","semantic_template":"qf-mod44-flow-capacity-audit/{\"capacities\":[[65,1],[65,1],[65,1],[65,1]],\"edges\":[[\"a12\",\"s12\"],[\"a12\",\"t12\"],[\"s12\",\"a12\"],[\"t12\",\"a12\"]],\"flows\":[[26,1],[39,1],[-13,1],[13,1]],\"sink\":\"t12\",\"source\":\"s12\",\"vertices\":[\"a12\",\"s12\",\"t12\",\"u12\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、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-20261113-r07-finite-flow-network-0039","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-flow-network"],"title_zh":"有限流網路","stem_zh":"finite loop-free directed network、逐edge capacity與逐internal-vertex conservation。操作=flow-capacity-audit；資料={\"vertices\":[\"s13\",\"a13\",\"t13\",\"u13\"],\"edges\":[[\"s13\",\"a13\"],[\"a13\",\"s13\"],[\"a13\",\"t13\"],[\"t13\",\"a13\"]],\"source\":\"s13\",\"sink\":\"t13\",\"capacities\":[70,70,70,70],\"flows\":[84,28,42,14]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(s13, t13, [[[a13, s13], 70, 28], [[a13, t13], 70, 42], [[s13, a13], 70, 84], [[t13, a13], 70, 14]], [[[a13, s13], 28, 70, 是], [[a13, t13], 42, 70, 是], [[s13, a13], 84, 70, 否], [[t13, a13], 14, 70, 是]], 否, [[s13, a13], 84, 70, 否], every_edge_requires_zero_less_equal_flow_less_equal_capacity)","explanation_zh":"Wave6第七層核對=(s13, t13, [[[a13, s13], 70, 28], [[a13, t13], 70, 42], [[s13, a13], 70, 84], [[t13, a13], 70, 14]], [[[a13, s13], 28, 70, 是], [[a13, t13], 42, 70, 是], [[s13, a13], 84, 70, 否], [[t13, a13], 14, 70, 是]], 否, [[s13, a13], 84, 70, 否], every_edge_requires_zero_less_equal_flow_less_equal_capacity)；依完整明示資料獨立重算；antiparallel edges明示，zero capacity與disconnected vertices允許；flow value由source與sink交叉計算；feasible不等於maximum，不提前宣稱min-cut equality或augmenting-path optimality。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限流網路"]},"variant":{"family_id":"qf-mod44-flow-capacity-audit","seed":202611137039,"parameters":{"vertices":["s13","a13","t13","u13"],"edges":[["s13","a13"],["a13","s13"],["a13","t13"],["t13","a13"]],"source":"s13","sink":"t13","capacities":[70,70,70,70],"flows":[84,28,42,14],"domain":"Exact_finite_flow_network","operation":"flow-capacity-audit","semantic_template":"qf-mod44-flow-capacity-audit/{\"capacities\":[[70,1],[70,1],[70,1],[70,1]],\"edges\":[[\"a13\",\"s13\"],[\"a13\",\"t13\"],[\"s13\",\"a13\"],[\"t13\",\"a13\"]],\"flows\":[[28,1],[42,1],[84,1],[14,1]],\"sink\":\"t13\",\"source\":\"s13\",\"vertices\":[\"a13\",\"s13\",\"t13\",\"u13\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、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-20261113-r07-finite-flow-network-0040","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-flow-network"],"title_zh":"有限流網路","stem_zh":"finite loop-free directed network、逐edge capacity與逐internal-vertex conservation。操作=flow-capacity-audit；資料={\"vertices\":[\"s14\",\"a14\",\"t14\",\"u14\"],\"edges\":[[\"s14\",\"a14\"],[\"a14\",\"s14\"],[\"a14\",\"t14\"],[\"t14\",\"a14\"]],\"source\":\"s14\",\"sink\":\"t14\",\"capacities\":[75,75,75,75],\"flows\":[60,30,45,15]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(s14, t14, [[[a14, s14], 75, 30], [[a14, t14], 75, 45], [[s14, a14], 75, 60], [[t14, a14], 75, 15]], [[[a14, s14], 30, 75, 是], [[a14, t14], 45, 75, 是], [[s14, a14], 60, 75, 是], [[t14, a14], 15, 75, 是]], 是, null, every_edge_requires_zero_less_equal_flow_less_equal_capacity)","explanation_zh":"Wave6第七層核對=(s14, t14, [[[a14, s14], 75, 30], [[a14, t14], 75, 45], [[s14, a14], 75, 60], [[t14, a14], 75, 15]], [[[a14, s14], 30, 75, 是], [[a14, t14], 45, 75, 是], [[s14, a14], 60, 75, 是], [[t14, a14], 15, 75, 是]], 是, null, every_edge_requires_zero_less_equal_flow_less_equal_capacity)；依完整明示資料獨立重算；antiparallel edges明示，zero capacity與disconnected vertices允許；flow value由source與sink交叉計算；feasible不等於maximum，不提前宣稱min-cut equality或augmenting-path optimality。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限流網路"]},"variant":{"family_id":"qf-mod44-flow-capacity-audit","seed":202611137040,"parameters":{"vertices":["s14","a14","t14","u14"],"edges":[["s14","a14"],["a14","s14"],["a14","t14"],["t14","a14"]],"source":"s14","sink":"t14","capacities":[75,75,75,75],"flows":[60,30,45,15],"domain":"Exact_finite_flow_network","operation":"flow-capacity-audit","semantic_template":"qf-mod44-flow-capacity-audit/{\"capacities\":[[75,1],[75,1],[75,1],[75,1]],\"edges\":[[\"a14\",\"s14\"],[\"a14\",\"t14\"],[\"s14\",\"a14\"],[\"t14\",\"a14\"]],\"flows\":[[30,1],[45,1],[60,1],[15,1]],\"sink\":\"t14\",\"source\":\"s14\",\"vertices\":[\"a14\",\"s14\",\"t14\",\"u14\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、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-20261113-r07-finite-flow-network-0041","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-flow-network"],"title_zh":"有限流網路","stem_zh":"finite loop-free directed network、逐edge capacity與逐internal-vertex conservation。操作=flow-capacity-audit；資料={\"vertices\":[\"s15\",\"a15\",\"t15\",\"u15\"],\"edges\":[[\"s15\",\"a15\"],[\"a15\",\"s15\"],[\"a15\",\"t15\"],[\"t15\",\"a15\"]],\"source\":\"s15\",\"sink\":\"t15\",\"capacities\":[80,80,80,80],\"flows\":[-16,32,48,16]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(s15, t15, [[[a15, s15], 80, 32], [[a15, t15], 80, 48], [[s15, a15], 80, -16], [[t15, a15], 80, 16]], [[[a15, s15], 32, 80, 是], [[a15, t15], 48, 80, 是], [[s15, a15], -16, 80, 否], [[t15, a15], 16, 80, 是]], 否, [[s15, a15], -16, 80, 否], every_edge_requires_zero_less_equal_flow_less_equal_capacity)","explanation_zh":"Wave6第七層核對=(s15, t15, [[[a15, s15], 80, 32], [[a15, t15], 80, 48], [[s15, a15], 80, -16], [[t15, a15], 80, 16]], [[[a15, s15], 32, 80, 是], [[a15, t15], 48, 80, 是], [[s15, a15], -16, 80, 否], [[t15, a15], 16, 80, 是]], 否, [[s15, a15], -16, 80, 否], every_edge_requires_zero_less_equal_flow_less_equal_capacity)；依完整明示資料獨立重算；antiparallel edges明示，zero capacity與disconnected vertices允許；flow value由source與sink交叉計算；feasible不等於maximum，不提前宣稱min-cut equality或augmenting-path optimality。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限流網路"]},"variant":{"family_id":"qf-mod44-flow-capacity-audit","seed":202611137041,"parameters":{"vertices":["s15","a15","t15","u15"],"edges":[["s15","a15"],["a15","s15"],["a15","t15"],["t15","a15"]],"source":"s15","sink":"t15","capacities":[80,80,80,80],"flows":[-16,32,48,16],"domain":"Exact_finite_flow_network","operation":"flow-capacity-audit","semantic_template":"qf-mod44-flow-capacity-audit/{\"capacities\":[[80,1],[80,1],[80,1],[80,1]],\"edges\":[[\"a15\",\"s15\"],[\"a15\",\"t15\"],[\"s15\",\"a15\"],[\"t15\",\"a15\"]],\"flows\":[[32,1],[48,1],[-16,1],[16,1]],\"sink\":\"t15\",\"source\":\"s15\",\"vertices\":[\"a15\",\"s15\",\"t15\",\"u15\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、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-20261113-r07-finite-flow-network-0042","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-flow-network"],"title_zh":"有限流網路","stem_zh":"finite loop-free directed network、逐edge capacity與逐internal-vertex conservation。操作=flow-capacity-audit；資料={\"vertices\":[\"s16\",\"a16\",\"t16\",\"u16\"],\"edges\":[[\"s16\",\"a16\"],[\"a16\",\"s16\"],[\"a16\",\"t16\"],[\"t16\",\"a16\"]],\"source\":\"s16\",\"sink\":\"t16\",\"capacities\":[85,85,85,85],\"flows\":[102,34,51,17]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(s16, t16, [[[a16, s16], 85, 34], [[a16, t16], 85, 51], [[s16, a16], 85, 102], [[t16, a16], 85, 17]], [[[a16, s16], 34, 85, 是], [[a16, t16], 51, 85, 是], [[s16, a16], 102, 85, 否], [[t16, a16], 17, 85, 是]], 否, [[s16, a16], 102, 85, 否], every_edge_requires_zero_less_equal_flow_less_equal_capacity)","explanation_zh":"Wave6第七層核對=(s16, t16, [[[a16, s16], 85, 34], [[a16, t16], 85, 51], [[s16, a16], 85, 102], [[t16, a16], 85, 17]], [[[a16, s16], 34, 85, 是], [[a16, t16], 51, 85, 是], [[s16, a16], 102, 85, 否], [[t16, a16], 17, 85, 是]], 否, [[s16, a16], 102, 85, 否], every_edge_requires_zero_less_equal_flow_less_equal_capacity)；依完整明示資料獨立重算；antiparallel edges明示，zero capacity與disconnected vertices允許；flow value由source與sink交叉計算；feasible不等於maximum，不提前宣稱min-cut equality或augmenting-path optimality。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限流網路"]},"variant":{"family_id":"qf-mod44-flow-capacity-audit","seed":202611137042,"parameters":{"vertices":["s16","a16","t16","u16"],"edges":[["s16","a16"],["a16","s16"],["a16","t16"],["t16","a16"]],"source":"s16","sink":"t16","capacities":[85,85,85,85],"flows":[102,34,51,17],"domain":"Exact_finite_flow_network","operation":"flow-capacity-audit","semantic_template":"qf-mod44-flow-capacity-audit/{\"capacities\":[[85,1],[85,1],[85,1],[85,1]],\"edges\":[[\"a16\",\"s16\"],[\"a16\",\"t16\"],[\"s16\",\"a16\"],[\"t16\",\"a16\"]],\"flows\":[[34,1],[51,1],[102,1],[17,1]],\"sink\":\"t16\",\"source\":\"s16\",\"vertices\":[\"a16\",\"s16\",\"t16\",\"u16\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、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-20261113-r07-finite-flow-network-0043","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-flow-network"],"title_zh":"有限流網路","stem_zh":"finite loop-free directed network、逐edge capacity與逐internal-vertex conservation。操作=flow-capacity-audit；資料={\"vertices\":[\"s17\",\"a17\",\"t17\",\"u17\"],\"edges\":[[\"s17\",\"a17\"],[\"a17\",\"s17\"],[\"a17\",\"t17\"],[\"t17\",\"a17\"]],\"source\":\"s17\",\"sink\":\"t17\",\"capacities\":[90,90,90,90],\"flows\":[72,36,54,18]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(s17, t17, [[[a17, s17], 90, 36], [[a17, t17], 90, 54], [[s17, a17], 90, 72], [[t17, a17], 90, 18]], [[[a17, s17], 36, 90, 是], [[a17, t17], 54, 90, 是], [[s17, a17], 72, 90, 是], [[t17, a17], 18, 90, 是]], 是, null, every_edge_requires_zero_less_equal_flow_less_equal_capacity)","explanation_zh":"Wave6第七層核對=(s17, t17, [[[a17, s17], 90, 36], [[a17, t17], 90, 54], [[s17, a17], 90, 72], [[t17, a17], 90, 18]], [[[a17, s17], 36, 90, 是], [[a17, t17], 54, 90, 是], [[s17, a17], 72, 90, 是], [[t17, a17], 18, 90, 是]], 是, null, every_edge_requires_zero_less_equal_flow_less_equal_capacity)；依完整明示資料獨立重算；antiparallel edges明示，zero capacity與disconnected vertices允許；flow value由source與sink交叉計算；feasible不等於maximum，不提前宣稱min-cut equality或augmenting-path optimality。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限流網路"]},"variant":{"family_id":"qf-mod44-flow-capacity-audit","seed":202611137043,"parameters":{"vertices":["s17","a17","t17","u17"],"edges":[["s17","a17"],["a17","s17"],["a17","t17"],["t17","a17"]],"source":"s17","sink":"t17","capacities":[90,90,90,90],"flows":[72,36,54,18],"domain":"Exact_finite_flow_network","operation":"flow-capacity-audit","semantic_template":"qf-mod44-flow-capacity-audit/{\"capacities\":[[90,1],[90,1],[90,1],[90,1]],\"edges\":[[\"a17\",\"s17\"],[\"a17\",\"t17\"],[\"s17\",\"a17\"],[\"t17\",\"a17\"]],\"flows\":[[36,1],[54,1],[72,1],[18,1]],\"sink\":\"t17\",\"source\":\"s17\",\"vertices\":[\"a17\",\"s17\",\"t17\",\"u17\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、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-20261113-r07-finite-flow-network-0044","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-flow-network"],"title_zh":"有限流網路","stem_zh":"finite loop-free directed network、逐edge capacity與逐internal-vertex conservation。操作=flow-capacity-audit；資料={\"vertices\":[\"s18\",\"a18\",\"t18\",\"u18\"],\"edges\":[[\"s18\",\"a18\"],[\"a18\",\"s18\"],[\"a18\",\"t18\"],[\"t18\",\"a18\"]],\"source\":\"s18\",\"sink\":\"t18\",\"capacities\":[95,95,95,95],\"flows\":[-19,38,57,19]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(s18, t18, [[[a18, s18], 95, 38], [[a18, t18], 95, 57], [[s18, a18], 95, -19], [[t18, a18], 95, 19]], [[[a18, s18], 38, 95, 是], [[a18, t18], 57, 95, 是], [[s18, a18], -19, 95, 否], [[t18, a18], 19, 95, 是]], 否, [[s18, a18], -19, 95, 否], every_edge_requires_zero_less_equal_flow_less_equal_capacity)","explanation_zh":"Wave6第七層核對=(s18, t18, [[[a18, s18], 95, 38], [[a18, t18], 95, 57], [[s18, a18], 95, -19], [[t18, a18], 95, 19]], [[[a18, s18], 38, 95, 是], [[a18, t18], 57, 95, 是], [[s18, a18], -19, 95, 否], [[t18, a18], 19, 95, 是]], 否, [[s18, a18], -19, 95, 否], every_edge_requires_zero_less_equal_flow_less_equal_capacity)；依完整明示資料獨立重算；antiparallel edges明示，zero capacity與disconnected vertices允許；flow value由source與sink交叉計算；feasible不等於maximum，不提前宣稱min-cut equality或augmenting-path optimality。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限流網路"]},"variant":{"family_id":"qf-mod44-flow-capacity-audit","seed":202611137044,"parameters":{"vertices":["s18","a18","t18","u18"],"edges":[["s18","a18"],["a18","s18"],["a18","t18"],["t18","a18"]],"source":"s18","sink":"t18","capacities":[95,95,95,95],"flows":[-19,38,57,19],"domain":"Exact_finite_flow_network","operation":"flow-capacity-audit","semantic_template":"qf-mod44-flow-capacity-audit/{\"capacities\":[[95,1],[95,1],[95,1],[95,1]],\"edges\":[[\"a18\",\"s18\"],[\"a18\",\"t18\"],[\"s18\",\"a18\"],[\"t18\",\"a18\"]],\"flows\":[[38,1],[57,1],[-19,1],[19,1]],\"sink\":\"t18\",\"source\":\"s18\",\"vertices\":[\"a18\",\"s18\",\"t18\",\"u18\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、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-20261113-r07-finite-flow-network-0045","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-flow-network"],"title_zh":"有限流網路","stem_zh":"finite loop-free directed network、逐edge capacity與逐internal-vertex conservation。操作=flow-capacity-audit；資料={\"vertices\":[\"s19\",\"a19\",\"t19\",\"u19\"],\"edges\":[[\"s19\",\"a19\"],[\"a19\",\"s19\"],[\"a19\",\"t19\"],[\"t19\",\"a19\"]],\"source\":\"s19\",\"sink\":\"t19\",\"capacities\":[100,100,100,100],\"flows\":[120,40,60,20]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(s19, t19, [[[a19, s19], 100, 40], [[a19, t19], 100, 60], [[s19, a19], 100, 120], [[t19, a19], 100, 20]], [[[a19, s19], 40, 100, 是], [[a19, t19], 60, 100, 是], [[s19, a19], 120, 100, 否], [[t19, a19], 20, 100, 是]], 否, [[s19, a19], 120, 100, 否], every_edge_requires_zero_less_equal_flow_less_equal_capacity)","explanation_zh":"Wave6第七層核對=(s19, t19, [[[a19, s19], 100, 40], [[a19, t19], 100, 60], [[s19, a19], 100, 120], [[t19, a19], 100, 20]], [[[a19, s19], 40, 100, 是], [[a19, t19], 60, 100, 是], [[s19, a19], 120, 100, 否], [[t19, a19], 20, 100, 是]], 否, [[s19, a19], 120, 100, 否], every_edge_requires_zero_less_equal_flow_less_equal_capacity)；依完整明示資料獨立重算；antiparallel edges明示，zero capacity與disconnected vertices允許；flow value由source與sink交叉計算；feasible不等於maximum，不提前宣稱min-cut equality或augmenting-path optimality。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限流網路"]},"variant":{"family_id":"qf-mod44-flow-capacity-audit","seed":202611137045,"parameters":{"vertices":["s19","a19","t19","u19"],"edges":[["s19","a19"],["a19","s19"],["a19","t19"],["t19","a19"]],"source":"s19","sink":"t19","capacities":[100,100,100,100],"flows":[120,40,60,20],"domain":"Exact_finite_flow_network","operation":"flow-capacity-audit","semantic_template":"qf-mod44-flow-capacity-audit/{\"capacities\":[[100,1],[100,1],[100,1],[100,1]],\"edges\":[[\"a19\",\"s19\"],[\"a19\",\"t19\"],[\"s19\",\"a19\"],[\"t19\",\"a19\"]],\"flows\":[[40,1],[60,1],[120,1],[20,1]],\"sink\":\"t19\",\"source\":\"s19\",\"vertices\":[\"a19\",\"s19\",\"t19\",\"u19\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、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-20261113-r07-finite-flow-network-0046","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-flow-network"],"title_zh":"有限流網路","stem_zh":"finite loop-free directed network、逐edge capacity與逐internal-vertex conservation。操作=flow-capacity-audit；資料={\"vertices\":[\"s20\",\"a20\",\"t20\",\"u20\"],\"edges\":[[\"s20\",\"a20\"],[\"a20\",\"s20\"],[\"a20\",\"t20\"],[\"t20\",\"a20\"]],\"source\":\"s20\",\"sink\":\"t20\",\"capacities\":[105,105,105,105],\"flows\":[84,42,63,21]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(s20, t20, [[[a20, s20], 105, 42], [[a20, t20], 105, 63], [[s20, a20], 105, 84], [[t20, a20], 105, 21]], [[[a20, s20], 42, 105, 是], [[a20, t20], 63, 105, 是], [[s20, a20], 84, 105, 是], [[t20, a20], 21, 105, 是]], 是, null, every_edge_requires_zero_less_equal_flow_less_equal_capacity)","explanation_zh":"Wave6第七層核對=(s20, t20, [[[a20, s20], 105, 42], [[a20, t20], 105, 63], [[s20, a20], 105, 84], [[t20, a20], 105, 21]], [[[a20, s20], 42, 105, 是], [[a20, t20], 63, 105, 是], [[s20, a20], 84, 105, 是], [[t20, a20], 21, 105, 是]], 是, null, every_edge_requires_zero_less_equal_flow_less_equal_capacity)；依完整明示資料獨立重算；antiparallel edges明示，zero capacity與disconnected vertices允許；flow value由source與sink交叉計算；feasible不等於maximum，不提前宣稱min-cut equality或augmenting-path optimality。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限流網路"]},"variant":{"family_id":"qf-mod44-flow-capacity-audit","seed":202611137046,"parameters":{"vertices":["s20","a20","t20","u20"],"edges":[["s20","a20"],["a20","s20"],["a20","t20"],["t20","a20"]],"source":"s20","sink":"t20","capacities":[105,105,105,105],"flows":[84,42,63,21],"domain":"Exact_finite_flow_network","operation":"flow-capacity-audit","semantic_template":"qf-mod44-flow-capacity-audit/{\"capacities\":[[105,1],[105,1],[105,1],[105,1]],\"edges\":[[\"a20\",\"s20\"],[\"a20\",\"t20\"],[\"s20\",\"a20\"],[\"t20\",\"a20\"]],\"flows\":[[42,1],[63,1],[84,1],[21,1]],\"sink\":\"t20\",\"source\":\"s20\",\"vertices\":[\"a20\",\"s20\",\"t20\",\"u20\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、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-20261113-r07-finite-flow-network-0047","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-flow-network"],"title_zh":"有限流網路","stem_zh":"finite loop-free directed network、逐edge capacity與逐internal-vertex conservation。操作=flow-capacity-audit；資料={\"vertices\":[\"s21\",\"a21\",\"t21\",\"u21\"],\"edges\":[[\"s21\",\"a21\"],[\"a21\",\"s21\"],[\"a21\",\"t21\"],[\"t21\",\"a21\"]],\"source\":\"s21\",\"sink\":\"t21\",\"capacities\":[110,110,110,110],\"flows\":[-22,44,66,22]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(s21, t21, [[[a21, s21], 110, 44], [[a21, t21], 110, 66], [[s21, a21], 110, -22], [[t21, a21], 110, 22]], [[[a21, s21], 44, 110, 是], [[a21, t21], 66, 110, 是], [[s21, a21], -22, 110, 否], [[t21, a21], 22, 110, 是]], 否, [[s21, a21], -22, 110, 否], every_edge_requires_zero_less_equal_flow_less_equal_capacity)","explanation_zh":"Wave6第七層核對=(s21, t21, [[[a21, s21], 110, 44], [[a21, t21], 110, 66], [[s21, a21], 110, -22], [[t21, a21], 110, 22]], [[[a21, s21], 44, 110, 是], [[a21, t21], 66, 110, 是], [[s21, a21], -22, 110, 否], [[t21, a21], 22, 110, 是]], 否, [[s21, a21], -22, 110, 否], every_edge_requires_zero_less_equal_flow_less_equal_capacity)；依完整明示資料獨立重算；antiparallel edges明示，zero capacity與disconnected vertices允許；flow value由source與sink交叉計算；feasible不等於maximum，不提前宣稱min-cut equality或augmenting-path optimality。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限流網路"]},"variant":{"family_id":"qf-mod44-flow-capacity-audit","seed":202611137047,"parameters":{"vertices":["s21","a21","t21","u21"],"edges":[["s21","a21"],["a21","s21"],["a21","t21"],["t21","a21"]],"source":"s21","sink":"t21","capacities":[110,110,110,110],"flows":[-22,44,66,22],"domain":"Exact_finite_flow_network","operation":"flow-capacity-audit","semantic_template":"qf-mod44-flow-capacity-audit/{\"capacities\":[[110,1],[110,1],[110,1],[110,1]],\"edges\":[[\"a21\",\"s21\"],[\"a21\",\"t21\"],[\"s21\",\"a21\"],[\"t21\",\"a21\"]],\"flows\":[[44,1],[66,1],[-22,1],[22,1]],\"sink\":\"t21\",\"source\":\"s21\",\"vertices\":[\"a21\",\"s21\",\"t21\",\"u21\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、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-20261113-r07-finite-flow-network-0048","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-flow-network"],"title_zh":"有限流網路","stem_zh":"finite loop-free directed network、逐edge capacity與逐internal-vertex conservation。操作=flow-capacity-audit；資料={\"vertices\":[\"s22\",\"a22\",\"t22\",\"u22\"],\"edges\":[[\"s22\",\"a22\"],[\"a22\",\"s22\"],[\"a22\",\"t22\"],[\"t22\",\"a22\"]],\"source\":\"s22\",\"sink\":\"t22\",\"capacities\":[115,115,115,115],\"flows\":[138,46,69,23]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(s22, t22, [[[a22, s22], 115, 46], [[a22, t22], 115, 69], [[s22, a22], 115, 138], [[t22, a22], 115, 23]], [[[a22, s22], 46, 115, 是], [[a22, t22], 69, 115, 是], [[s22, a22], 138, 115, 否], [[t22, a22], 23, 115, 是]], 否, [[s22, a22], 138, 115, 否], every_edge_requires_zero_less_equal_flow_less_equal_capacity)","explanation_zh":"Wave6第七層核對=(s22, t22, [[[a22, s22], 115, 46], [[a22, t22], 115, 69], [[s22, a22], 115, 138], [[t22, a22], 115, 23]], [[[a22, s22], 46, 115, 是], [[a22, t22], 69, 115, 是], [[s22, a22], 138, 115, 否], [[t22, a22], 23, 115, 是]], 否, [[s22, a22], 138, 115, 否], every_edge_requires_zero_less_equal_flow_less_equal_capacity)；依完整明示資料獨立重算；antiparallel edges明示，zero capacity與disconnected vertices允許；flow value由source與sink交叉計算；feasible不等於maximum，不提前宣稱min-cut equality或augmenting-path optimality。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限流網路"]},"variant":{"family_id":"qf-mod44-flow-capacity-audit","seed":202611137048,"parameters":{"vertices":["s22","a22","t22","u22"],"edges":[["s22","a22"],["a22","s22"],["a22","t22"],["t22","a22"]],"source":"s22","sink":"t22","capacities":[115,115,115,115],"flows":[138,46,69,23],"domain":"Exact_finite_flow_network","operation":"flow-capacity-audit","semantic_template":"qf-mod44-flow-capacity-audit/{\"capacities\":[[115,1],[115,1],[115,1],[115,1]],\"edges\":[[\"a22\",\"s22\"],[\"a22\",\"t22\"],[\"s22\",\"a22\"],[\"t22\",\"a22\"]],\"flows\":[[46,1],[69,1],[138,1],[23,1]],\"sink\":\"t22\",\"source\":\"s22\",\"vertices\":[\"a22\",\"s22\",\"t22\",\"u22\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、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-20261113-r07-finite-flow-network-0049","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-flow-network"],"title_zh":"有限流網路","stem_zh":"finite loop-free directed network、逐edge capacity與逐internal-vertex conservation。操作=flow-capacity-audit；資料={\"vertices\":[\"s23\",\"a23\",\"t23\",\"u23\"],\"edges\":[[\"s23\",\"a23\"],[\"a23\",\"s23\"],[\"a23\",\"t23\"],[\"t23\",\"a23\"]],\"source\":\"s23\",\"sink\":\"t23\",\"capacities\":[120,120,120,120],\"flows\":[96,48,72,24]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(s23, t23, [[[a23, s23], 120, 48], [[a23, t23], 120, 72], [[s23, a23], 120, 96], [[t23, a23], 120, 24]], [[[a23, s23], 48, 120, 是], [[a23, t23], 72, 120, 是], [[s23, a23], 96, 120, 是], [[t23, a23], 24, 120, 是]], 是, null, every_edge_requires_zero_less_equal_flow_less_equal_capacity)","explanation_zh":"Wave6第七層核對=(s23, t23, [[[a23, s23], 120, 48], [[a23, t23], 120, 72], [[s23, a23], 120, 96], [[t23, a23], 120, 24]], [[[a23, s23], 48, 120, 是], [[a23, t23], 72, 120, 是], [[s23, a23], 96, 120, 是], [[t23, a23], 24, 120, 是]], 是, null, every_edge_requires_zero_less_equal_flow_less_equal_capacity)；依完整明示資料獨立重算；antiparallel edges明示，zero capacity與disconnected vertices允許；flow value由source與sink交叉計算；feasible不等於maximum，不提前宣稱min-cut equality或augmenting-path optimality。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限流網路"]},"variant":{"family_id":"qf-mod44-flow-capacity-audit","seed":202611137049,"parameters":{"vertices":["s23","a23","t23","u23"],"edges":[["s23","a23"],["a23","s23"],["a23","t23"],["t23","a23"]],"source":"s23","sink":"t23","capacities":[120,120,120,120],"flows":[96,48,72,24],"domain":"Exact_finite_flow_network","operation":"flow-capacity-audit","semantic_template":"qf-mod44-flow-capacity-audit/{\"capacities\":[[120,1],[120,1],[120,1],[120,1]],\"edges\":[[\"a23\",\"s23\"],[\"a23\",\"t23\"],[\"s23\",\"a23\"],[\"t23\",\"a23\"]],\"flows\":[[48,1],[72,1],[96,1],[24,1]],\"sink\":\"t23\",\"source\":\"s23\",\"vertices\":[\"a23\",\"s23\",\"t23\",\"u23\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、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-20261113-r07-finite-flow-network-0050","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-flow-network"],"title_zh":"有限流網路","stem_zh":"finite loop-free directed network、逐edge capacity與逐internal-vertex conservation。操作=flow-capacity-audit；資料={\"vertices\":[\"s24\",\"a24\",\"t24\",\"u24\"],\"edges\":[[\"s24\",\"a24\"],[\"a24\",\"s24\"],[\"a24\",\"t24\"],[\"t24\",\"a24\"]],\"source\":\"s24\",\"sink\":\"t24\",\"capacities\":[125,125,125,125],\"flows\":[-25,50,75,25]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(s24, t24, [[[a24, s24], 125, 50], [[a24, t24], 125, 75], [[s24, a24], 125, -25], [[t24, a24], 125, 25]], [[[a24, s24], 50, 125, 是], [[a24, t24], 75, 125, 是], [[s24, a24], -25, 125, 否], [[t24, a24], 25, 125, 是]], 否, [[s24, a24], -25, 125, 否], every_edge_requires_zero_less_equal_flow_less_equal_capacity)","explanation_zh":"Wave6第七層核對=(s24, t24, [[[a24, s24], 125, 50], [[a24, t24], 125, 75], [[s24, a24], 125, -25], [[t24, a24], 125, 25]], [[[a24, s24], 50, 125, 是], [[a24, t24], 75, 125, 是], [[s24, a24], -25, 125, 否], [[t24, a24], 25, 125, 是]], 否, [[s24, a24], -25, 125, 否], every_edge_requires_zero_less_equal_flow_less_equal_capacity)；依完整明示資料獨立重算；antiparallel edges明示，zero capacity與disconnected vertices允許；flow value由source與sink交叉計算；feasible不等於maximum，不提前宣稱min-cut equality或augmenting-path optimality。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限流網路"]},"variant":{"family_id":"qf-mod44-flow-capacity-audit","seed":202611137050,"parameters":{"vertices":["s24","a24","t24","u24"],"edges":[["s24","a24"],["a24","s24"],["a24","t24"],["t24","a24"]],"source":"s24","sink":"t24","capacities":[125,125,125,125],"flows":[-25,50,75,25],"domain":"Exact_finite_flow_network","operation":"flow-capacity-audit","semantic_template":"qf-mod44-flow-capacity-audit/{\"capacities\":[[125,1],[125,1],[125,1],[125,1]],\"edges\":[[\"a24\",\"s24\"],[\"a24\",\"t24\"],[\"s24\",\"a24\"],[\"t24\",\"a24\"]],\"flows\":[[50,1],[75,1],[-25,1],[25,1]],\"sink\":\"t24\",\"source\":\"s24\",\"vertices\":[\"a24\",\"s24\",\"t24\",\"u24\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、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-20261113-r07-finite-flow-network-0051","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-flow-network"],"title_zh":"有限流網路","stem_zh":"finite loop-free directed network、逐edge capacity與逐internal-vertex conservation。操作=flow-conservation-audit；資料={\"vertices\":[\"s0\",\"a0\",\"t0\",\"u0\"],\"edges\":[[\"s0\",\"a0\"],[\"a0\",\"s0\"],[\"a0\",\"t0\"],[\"t0\",\"a0\"]],\"source\":\"s0\",\"sink\":\"t0\",\"capacities\":[5,5,5,5],\"flows\":[4,2,3,1]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(s0, t0, [[[a0, s0], 5, 2], [[a0, t0], 5, 3], [[s0, a0], 5, 4], [[t0, a0], 5, 1]], [[a0, 5, 5, 是], [u0, 0, 0, 是]], 是, null, every_internal_vertex_requires_complete_inflow_equals_outflow)","explanation_zh":"Wave6第七層核對=(s0, t0, [[[a0, s0], 5, 2], [[a0, t0], 5, 3], [[s0, a0], 5, 4], [[t0, a0], 5, 1]], [[a0, 5, 5, 是], [u0, 0, 0, 是]], 是, null, every_internal_vertex_requires_complete_inflow_equals_outflow)；依完整明示資料獨立重算；antiparallel edges明示，zero capacity與disconnected vertices允許；flow value由source與sink交叉計算；feasible不等於maximum，不提前宣稱min-cut equality或augmenting-path optimality。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限流網路"]},"variant":{"family_id":"qf-mod44-flow-conservation-audit","seed":202611137051,"parameters":{"vertices":["s0","a0","t0","u0"],"edges":[["s0","a0"],["a0","s0"],["a0","t0"],["t0","a0"]],"source":"s0","sink":"t0","capacities":[5,5,5,5],"flows":[4,2,3,1],"domain":"Exact_finite_flow_network","operation":"flow-conservation-audit","semantic_template":"qf-mod44-flow-conservation-audit/{\"capacities\":[[5,1],[5,1],[5,1],[5,1]],\"edges\":[[\"a0\",\"s0\"],[\"a0\",\"t0\"],[\"s0\",\"a0\"],[\"t0\",\"a0\"]],\"flows\":[[2,1],[3,1],[4,1],[1,1]],\"sink\":\"t0\",\"source\":\"s0\",\"vertices\":[\"a0\",\"s0\",\"t0\",\"u0\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、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-20261113-r07-finite-flow-network-0052","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-flow-network"],"title_zh":"有限流網路","stem_zh":"finite loop-free directed network、逐edge capacity與逐internal-vertex conservation。操作=flow-conservation-audit；資料={\"vertices\":[\"s1\",\"a1\",\"t1\",\"u1\"],\"edges\":[[\"s1\",\"a1\"],[\"a1\",\"s1\"],[\"a1\",\"t1\"],[\"t1\",\"a1\"]],\"source\":\"s1\",\"sink\":\"t1\",\"capacities\":[10,10,10,10],\"flows\":[8,4,4,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(s1, t1, [[[a1, s1], 10, 4], [[a1, t1], 10, 4], [[s1, a1], 10, 8], [[t1, a1], 10, 2]], [[a1, 10, 8, 否], [u1, 0, 0, 是]], 否, [a1, 10, 8, 否], every_internal_vertex_requires_complete_inflow_equals_outflow)","explanation_zh":"Wave6第七層核對=(s1, t1, [[[a1, s1], 10, 4], [[a1, t1], 10, 4], [[s1, a1], 10, 8], [[t1, a1], 10, 2]], [[a1, 10, 8, 否], [u1, 0, 0, 是]], 否, [a1, 10, 8, 否], every_internal_vertex_requires_complete_inflow_equals_outflow)；依完整明示資料獨立重算；antiparallel edges明示，zero capacity與disconnected vertices允許；flow value由source與sink交叉計算；feasible不等於maximum，不提前宣稱min-cut equality或augmenting-path optimality。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限流網路"]},"variant":{"family_id":"qf-mod44-flow-conservation-audit","seed":202611137052,"parameters":{"vertices":["s1","a1","t1","u1"],"edges":[["s1","a1"],["a1","s1"],["a1","t1"],["t1","a1"]],"source":"s1","sink":"t1","capacities":[10,10,10,10],"flows":[8,4,4,2],"domain":"Exact_finite_flow_network","operation":"flow-conservation-audit","semantic_template":"qf-mod44-flow-conservation-audit/{\"capacities\":[[10,1],[10,1],[10,1],[10,1]],\"edges\":[[\"a1\",\"s1\"],[\"a1\",\"t1\"],[\"s1\",\"a1\"],[\"t1\",\"a1\"]],\"flows\":[[4,1],[4,1],[8,1],[2,1]],\"sink\":\"t1\",\"source\":\"s1\",\"vertices\":[\"a1\",\"s1\",\"t1\",\"u1\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、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-20261113-r07-finite-flow-network-0053","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-flow-network"],"title_zh":"有限流網路","stem_zh":"finite loop-free directed network、逐edge capacity與逐internal-vertex conservation。操作=flow-conservation-audit；資料={\"vertices\":[\"s2\",\"a2\",\"t2\",\"u2\"],\"edges\":[[\"s2\",\"a2\"],[\"a2\",\"s2\"],[\"a2\",\"t2\"],[\"t2\",\"a2\"]],\"source\":\"s2\",\"sink\":\"t2\",\"capacities\":[15,15,15,15],\"flows\":[12,6,9,3]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(s2, t2, [[[a2, s2], 15, 6], [[a2, t2], 15, 9], [[s2, a2], 15, 12], [[t2, a2], 15, 3]], [[a2, 15, 15, 是], [u2, 0, 0, 是]], 是, null, every_internal_vertex_requires_complete_inflow_equals_outflow)","explanation_zh":"Wave6第七層核對=(s2, t2, [[[a2, s2], 15, 6], [[a2, t2], 15, 9], [[s2, a2], 15, 12], [[t2, a2], 15, 3]], [[a2, 15, 15, 是], [u2, 0, 0, 是]], 是, null, every_internal_vertex_requires_complete_inflow_equals_outflow)；依完整明示資料獨立重算；antiparallel edges明示，zero capacity與disconnected vertices允許；flow value由source與sink交叉計算；feasible不等於maximum，不提前宣稱min-cut equality或augmenting-path optimality。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限流網路"]},"variant":{"family_id":"qf-mod44-flow-conservation-audit","seed":202611137053,"parameters":{"vertices":["s2","a2","t2","u2"],"edges":[["s2","a2"],["a2","s2"],["a2","t2"],["t2","a2"]],"source":"s2","sink":"t2","capacities":[15,15,15,15],"flows":[12,6,9,3],"domain":"Exact_finite_flow_network","operation":"flow-conservation-audit","semantic_template":"qf-mod44-flow-conservation-audit/{\"capacities\":[[15,1],[15,1],[15,1],[15,1]],\"edges\":[[\"a2\",\"s2\"],[\"a2\",\"t2\"],[\"s2\",\"a2\"],[\"t2\",\"a2\"]],\"flows\":[[6,1],[9,1],[12,1],[3,1]],\"sink\":\"t2\",\"source\":\"s2\",\"vertices\":[\"a2\",\"s2\",\"t2\",\"u2\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、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-20261113-r07-finite-flow-network-0054","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-flow-network"],"title_zh":"有限流網路","stem_zh":"finite loop-free directed network、逐edge capacity與逐internal-vertex conservation。操作=flow-conservation-audit；資料={\"vertices\":[\"s3\",\"a3\",\"t3\",\"u3\"],\"edges\":[[\"s3\",\"a3\"],[\"a3\",\"s3\"],[\"a3\",\"t3\"],[\"t3\",\"a3\"]],\"source\":\"s3\",\"sink\":\"t3\",\"capacities\":[20,20,20,20],\"flows\":[16,8,8,4]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(s3, t3, [[[a3, s3], 20, 8], [[a3, t3], 20, 8], [[s3, a3], 20, 16], [[t3, a3], 20, 4]], [[a3, 20, 16, 否], [u3, 0, 0, 是]], 否, [a3, 20, 16, 否], every_internal_vertex_requires_complete_inflow_equals_outflow)","explanation_zh":"Wave6第七層核對=(s3, t3, [[[a3, s3], 20, 8], [[a3, t3], 20, 8], [[s3, a3], 20, 16], [[t3, a3], 20, 4]], [[a3, 20, 16, 否], [u3, 0, 0, 是]], 否, [a3, 20, 16, 否], every_internal_vertex_requires_complete_inflow_equals_outflow)；依完整明示資料獨立重算；antiparallel edges明示，zero capacity與disconnected vertices允許；flow value由source與sink交叉計算；feasible不等於maximum，不提前宣稱min-cut equality或augmenting-path optimality。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限流網路"]},"variant":{"family_id":"qf-mod44-flow-conservation-audit","seed":202611137054,"parameters":{"vertices":["s3","a3","t3","u3"],"edges":[["s3","a3"],["a3","s3"],["a3","t3"],["t3","a3"]],"source":"s3","sink":"t3","capacities":[20,20,20,20],"flows":[16,8,8,4],"domain":"Exact_finite_flow_network","operation":"flow-conservation-audit","semantic_template":"qf-mod44-flow-conservation-audit/{\"capacities\":[[20,1],[20,1],[20,1],[20,1]],\"edges\":[[\"a3\",\"s3\"],[\"a3\",\"t3\"],[\"s3\",\"a3\"],[\"t3\",\"a3\"]],\"flows\":[[8,1],[8,1],[16,1],[4,1]],\"sink\":\"t3\",\"source\":\"s3\",\"vertices\":[\"a3\",\"s3\",\"t3\",\"u3\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、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-20261113-r07-finite-flow-network-0055","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-flow-network"],"title_zh":"有限流網路","stem_zh":"finite loop-free directed network、逐edge capacity與逐internal-vertex conservation。操作=flow-conservation-audit；資料={\"vertices\":[\"s4\",\"a4\",\"t4\",\"u4\"],\"edges\":[[\"s4\",\"a4\"],[\"a4\",\"s4\"],[\"a4\",\"t4\"],[\"t4\",\"a4\"]],\"source\":\"s4\",\"sink\":\"t4\",\"capacities\":[25,25,25,25],\"flows\":[20,10,15,5]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(s4, t4, [[[a4, s4], 25, 10], [[a4, t4], 25, 15], [[s4, a4], 25, 20], [[t4, a4], 25, 5]], [[a4, 25, 25, 是], [u4, 0, 0, 是]], 是, null, every_internal_vertex_requires_complete_inflow_equals_outflow)","explanation_zh":"Wave6第七層核對=(s4, t4, [[[a4, s4], 25, 10], [[a4, t4], 25, 15], [[s4, a4], 25, 20], [[t4, a4], 25, 5]], [[a4, 25, 25, 是], [u4, 0, 0, 是]], 是, null, every_internal_vertex_requires_complete_inflow_equals_outflow)；依完整明示資料獨立重算；antiparallel edges明示，zero capacity與disconnected vertices允許；flow value由source與sink交叉計算；feasible不等於maximum，不提前宣稱min-cut equality或augmenting-path optimality。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限流網路"]},"variant":{"family_id":"qf-mod44-flow-conservation-audit","seed":202611137055,"parameters":{"vertices":["s4","a4","t4","u4"],"edges":[["s4","a4"],["a4","s4"],["a4","t4"],["t4","a4"]],"source":"s4","sink":"t4","capacities":[25,25,25,25],"flows":[20,10,15,5],"domain":"Exact_finite_flow_network","operation":"flow-conservation-audit","semantic_template":"qf-mod44-flow-conservation-audit/{\"capacities\":[[25,1],[25,1],[25,1],[25,1]],\"edges\":[[\"a4\",\"s4\"],[\"a4\",\"t4\"],[\"s4\",\"a4\"],[\"t4\",\"a4\"]],\"flows\":[[10,1],[15,1],[20,1],[5,1]],\"sink\":\"t4\",\"source\":\"s4\",\"vertices\":[\"a4\",\"s4\",\"t4\",\"u4\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、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-20261113-r07-finite-flow-network-0056","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-flow-network"],"title_zh":"有限流網路","stem_zh":"finite loop-free directed network、逐edge capacity與逐internal-vertex conservation。操作=flow-conservation-audit；資料={\"vertices\":[\"s5\",\"a5\",\"t5\",\"u5\"],\"edges\":[[\"s5\",\"a5\"],[\"a5\",\"s5\"],[\"a5\",\"t5\"],[\"t5\",\"a5\"]],\"source\":\"s5\",\"sink\":\"t5\",\"capacities\":[30,30,30,30],\"flows\":[24,12,12,6]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(s5, t5, [[[a5, s5], 30, 12], [[a5, t5], 30, 12], [[s5, a5], 30, 24], [[t5, a5], 30, 6]], [[a5, 30, 24, 否], [u5, 0, 0, 是]], 否, [a5, 30, 24, 否], every_internal_vertex_requires_complete_inflow_equals_outflow)","explanation_zh":"Wave6第七層核對=(s5, t5, [[[a5, s5], 30, 12], [[a5, t5], 30, 12], [[s5, a5], 30, 24], [[t5, a5], 30, 6]], [[a5, 30, 24, 否], [u5, 0, 0, 是]], 否, [a5, 30, 24, 否], every_internal_vertex_requires_complete_inflow_equals_outflow)；依完整明示資料獨立重算；antiparallel edges明示，zero capacity與disconnected vertices允許；flow value由source與sink交叉計算；feasible不等於maximum，不提前宣稱min-cut equality或augmenting-path optimality。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限流網路"]},"variant":{"family_id":"qf-mod44-flow-conservation-audit","seed":202611137056,"parameters":{"vertices":["s5","a5","t5","u5"],"edges":[["s5","a5"],["a5","s5"],["a5","t5"],["t5","a5"]],"source":"s5","sink":"t5","capacities":[30,30,30,30],"flows":[24,12,12,6],"domain":"Exact_finite_flow_network","operation":"flow-conservation-audit","semantic_template":"qf-mod44-flow-conservation-audit/{\"capacities\":[[30,1],[30,1],[30,1],[30,1]],\"edges\":[[\"a5\",\"s5\"],[\"a5\",\"t5\"],[\"s5\",\"a5\"],[\"t5\",\"a5\"]],\"flows\":[[12,1],[12,1],[24,1],[6,1]],\"sink\":\"t5\",\"source\":\"s5\",\"vertices\":[\"a5\",\"s5\",\"t5\",\"u5\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、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-20261113-r07-finite-flow-network-0057","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-flow-network"],"title_zh":"有限流網路","stem_zh":"finite loop-free directed network、逐edge capacity與逐internal-vertex conservation。操作=flow-conservation-audit；資料={\"vertices\":[\"s6\",\"a6\",\"t6\",\"u6\"],\"edges\":[[\"s6\",\"a6\"],[\"a6\",\"s6\"],[\"a6\",\"t6\"],[\"t6\",\"a6\"]],\"source\":\"s6\",\"sink\":\"t6\",\"capacities\":[35,35,35,35],\"flows\":[28,14,21,7]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(s6, t6, [[[a6, s6], 35, 14], [[a6, t6], 35, 21], [[s6, a6], 35, 28], [[t6, a6], 35, 7]], [[a6, 35, 35, 是], [u6, 0, 0, 是]], 是, null, every_internal_vertex_requires_complete_inflow_equals_outflow)","explanation_zh":"Wave6第七層核對=(s6, t6, [[[a6, s6], 35, 14], [[a6, t6], 35, 21], [[s6, a6], 35, 28], [[t6, a6], 35, 7]], [[a6, 35, 35, 是], [u6, 0, 0, 是]], 是, null, every_internal_vertex_requires_complete_inflow_equals_outflow)；依完整明示資料獨立重算；antiparallel edges明示，zero capacity與disconnected vertices允許；flow value由source與sink交叉計算；feasible不等於maximum，不提前宣稱min-cut equality或augmenting-path optimality。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限流網路"]},"variant":{"family_id":"qf-mod44-flow-conservation-audit","seed":202611137057,"parameters":{"vertices":["s6","a6","t6","u6"],"edges":[["s6","a6"],["a6","s6"],["a6","t6"],["t6","a6"]],"source":"s6","sink":"t6","capacities":[35,35,35,35],"flows":[28,14,21,7],"domain":"Exact_finite_flow_network","operation":"flow-conservation-audit","semantic_template":"qf-mod44-flow-conservation-audit/{\"capacities\":[[35,1],[35,1],[35,1],[35,1]],\"edges\":[[\"a6\",\"s6\"],[\"a6\",\"t6\"],[\"s6\",\"a6\"],[\"t6\",\"a6\"]],\"flows\":[[14,1],[21,1],[28,1],[7,1]],\"sink\":\"t6\",\"source\":\"s6\",\"vertices\":[\"a6\",\"s6\",\"t6\",\"u6\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、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-20261113-r07-finite-flow-network-0058","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-flow-network"],"title_zh":"有限流網路","stem_zh":"finite loop-free directed network、逐edge capacity與逐internal-vertex conservation。操作=flow-conservation-audit；資料={\"vertices\":[\"s7\",\"a7\",\"t7\",\"u7\"],\"edges\":[[\"s7\",\"a7\"],[\"a7\",\"s7\"],[\"a7\",\"t7\"],[\"t7\",\"a7\"]],\"source\":\"s7\",\"sink\":\"t7\",\"capacities\":[40,40,40,40],\"flows\":[32,16,16,8]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(s7, t7, [[[a7, s7], 40, 16], [[a7, t7], 40, 16], [[s7, a7], 40, 32], [[t7, a7], 40, 8]], [[a7, 40, 32, 否], [u7, 0, 0, 是]], 否, [a7, 40, 32, 否], every_internal_vertex_requires_complete_inflow_equals_outflow)","explanation_zh":"Wave6第七層核對=(s7, t7, [[[a7, s7], 40, 16], [[a7, t7], 40, 16], [[s7, a7], 40, 32], [[t7, a7], 40, 8]], [[a7, 40, 32, 否], [u7, 0, 0, 是]], 否, [a7, 40, 32, 否], every_internal_vertex_requires_complete_inflow_equals_outflow)；依完整明示資料獨立重算；antiparallel edges明示，zero capacity與disconnected vertices允許；flow value由source與sink交叉計算；feasible不等於maximum，不提前宣稱min-cut equality或augmenting-path optimality。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限流網路"]},"variant":{"family_id":"qf-mod44-flow-conservation-audit","seed":202611137058,"parameters":{"vertices":["s7","a7","t7","u7"],"edges":[["s7","a7"],["a7","s7"],["a7","t7"],["t7","a7"]],"source":"s7","sink":"t7","capacities":[40,40,40,40],"flows":[32,16,16,8],"domain":"Exact_finite_flow_network","operation":"flow-conservation-audit","semantic_template":"qf-mod44-flow-conservation-audit/{\"capacities\":[[40,1],[40,1],[40,1],[40,1]],\"edges\":[[\"a7\",\"s7\"],[\"a7\",\"t7\"],[\"s7\",\"a7\"],[\"t7\",\"a7\"]],\"flows\":[[16,1],[16,1],[32,1],[8,1]],\"sink\":\"t7\",\"source\":\"s7\",\"vertices\":[\"a7\",\"s7\",\"t7\",\"u7\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、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-20261113-r07-finite-flow-network-0059","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-flow-network"],"title_zh":"有限流網路","stem_zh":"finite loop-free directed network、逐edge capacity與逐internal-vertex conservation。操作=flow-conservation-audit；資料={\"vertices\":[\"s8\",\"a8\",\"t8\",\"u8\"],\"edges\":[[\"s8\",\"a8\"],[\"a8\",\"s8\"],[\"a8\",\"t8\"],[\"t8\",\"a8\"]],\"source\":\"s8\",\"sink\":\"t8\",\"capacities\":[45,45,45,45],\"flows\":[36,18,27,9]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(s8, t8, [[[a8, s8], 45, 18], [[a8, t8], 45, 27], [[s8, a8], 45, 36], [[t8, a8], 45, 9]], [[a8, 45, 45, 是], [u8, 0, 0, 是]], 是, null, every_internal_vertex_requires_complete_inflow_equals_outflow)","explanation_zh":"Wave6第七層核對=(s8, t8, [[[a8, s8], 45, 18], [[a8, t8], 45, 27], [[s8, a8], 45, 36], [[t8, a8], 45, 9]], [[a8, 45, 45, 是], [u8, 0, 0, 是]], 是, null, every_internal_vertex_requires_complete_inflow_equals_outflow)；依完整明示資料獨立重算；antiparallel edges明示，zero capacity與disconnected vertices允許；flow value由source與sink交叉計算；feasible不等於maximum，不提前宣稱min-cut equality或augmenting-path optimality。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限流網路"]},"variant":{"family_id":"qf-mod44-flow-conservation-audit","seed":202611137059,"parameters":{"vertices":["s8","a8","t8","u8"],"edges":[["s8","a8"],["a8","s8"],["a8","t8"],["t8","a8"]],"source":"s8","sink":"t8","capacities":[45,45,45,45],"flows":[36,18,27,9],"domain":"Exact_finite_flow_network","operation":"flow-conservation-audit","semantic_template":"qf-mod44-flow-conservation-audit/{\"capacities\":[[45,1],[45,1],[45,1],[45,1]],\"edges\":[[\"a8\",\"s8\"],[\"a8\",\"t8\"],[\"s8\",\"a8\"],[\"t8\",\"a8\"]],\"flows\":[[18,1],[27,1],[36,1],[9,1]],\"sink\":\"t8\",\"source\":\"s8\",\"vertices\":[\"a8\",\"s8\",\"t8\",\"u8\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、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-20261113-r07-finite-flow-network-0060","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-flow-network"],"title_zh":"有限流網路","stem_zh":"finite loop-free directed network、逐edge capacity與逐internal-vertex conservation。操作=flow-conservation-audit；資料={\"vertices\":[\"s9\",\"a9\",\"t9\",\"u9\"],\"edges\":[[\"s9\",\"a9\"],[\"a9\",\"s9\"],[\"a9\",\"t9\"],[\"t9\",\"a9\"]],\"source\":\"s9\",\"sink\":\"t9\",\"capacities\":[50,50,50,50],\"flows\":[40,20,20,10]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(s9, t9, [[[a9, s9], 50, 20], [[a9, t9], 50, 20], [[s9, a9], 50, 40], [[t9, a9], 50, 10]], [[a9, 50, 40, 否], [u9, 0, 0, 是]], 否, [a9, 50, 40, 否], every_internal_vertex_requires_complete_inflow_equals_outflow)","explanation_zh":"Wave6第七層核對=(s9, t9, [[[a9, s9], 50, 20], [[a9, t9], 50, 20], [[s9, a9], 50, 40], [[t9, a9], 50, 10]], [[a9, 50, 40, 否], [u9, 0, 0, 是]], 否, [a9, 50, 40, 否], every_internal_vertex_requires_complete_inflow_equals_outflow)；依完整明示資料獨立重算；antiparallel edges明示，zero capacity與disconnected vertices允許；flow value由source與sink交叉計算；feasible不等於maximum，不提前宣稱min-cut equality或augmenting-path optimality。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限流網路"]},"variant":{"family_id":"qf-mod44-flow-conservation-audit","seed":202611137060,"parameters":{"vertices":["s9","a9","t9","u9"],"edges":[["s9","a9"],["a9","s9"],["a9","t9"],["t9","a9"]],"source":"s9","sink":"t9","capacities":[50,50,50,50],"flows":[40,20,20,10],"domain":"Exact_finite_flow_network","operation":"flow-conservation-audit","semantic_template":"qf-mod44-flow-conservation-audit/{\"capacities\":[[50,1],[50,1],[50,1],[50,1]],\"edges\":[[\"a9\",\"s9\"],[\"a9\",\"t9\"],[\"s9\",\"a9\"],[\"t9\",\"a9\"]],\"flows\":[[20,1],[20,1],[40,1],[10,1]],\"sink\":\"t9\",\"source\":\"s9\",\"vertices\":[\"a9\",\"s9\",\"t9\",\"u9\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、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-20261113-r07-finite-flow-network-0061","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-flow-network"],"title_zh":"有限流網路","stem_zh":"finite loop-free directed network、逐edge capacity與逐internal-vertex conservation。操作=flow-conservation-audit；資料={\"vertices\":[\"s10\",\"a10\",\"t10\",\"u10\"],\"edges\":[[\"s10\",\"a10\"],[\"a10\",\"s10\"],[\"a10\",\"t10\"],[\"t10\",\"a10\"]],\"source\":\"s10\",\"sink\":\"t10\",\"capacities\":[55,55,55,55],\"flows\":[44,22,33,11]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(s10, t10, [[[a10, s10], 55, 22], [[a10, t10], 55, 33], [[s10, a10], 55, 44], [[t10, a10], 55, 11]], [[a10, 55, 55, 是], [u10, 0, 0, 是]], 是, null, every_internal_vertex_requires_complete_inflow_equals_outflow)","explanation_zh":"Wave6第七層核對=(s10, t10, [[[a10, s10], 55, 22], [[a10, t10], 55, 33], [[s10, a10], 55, 44], [[t10, a10], 55, 11]], [[a10, 55, 55, 是], [u10, 0, 0, 是]], 是, null, every_internal_vertex_requires_complete_inflow_equals_outflow)；依完整明示資料獨立重算；antiparallel edges明示，zero capacity與disconnected vertices允許；flow value由source與sink交叉計算；feasible不等於maximum，不提前宣稱min-cut equality或augmenting-path optimality。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限流網路"]},"variant":{"family_id":"qf-mod44-flow-conservation-audit","seed":202611137061,"parameters":{"vertices":["s10","a10","t10","u10"],"edges":[["s10","a10"],["a10","s10"],["a10","t10"],["t10","a10"]],"source":"s10","sink":"t10","capacities":[55,55,55,55],"flows":[44,22,33,11],"domain":"Exact_finite_flow_network","operation":"flow-conservation-audit","semantic_template":"qf-mod44-flow-conservation-audit/{\"capacities\":[[55,1],[55,1],[55,1],[55,1]],\"edges\":[[\"a10\",\"s10\"],[\"a10\",\"t10\"],[\"s10\",\"a10\"],[\"t10\",\"a10\"]],\"flows\":[[22,1],[33,1],[44,1],[11,1]],\"sink\":\"t10\",\"source\":\"s10\",\"vertices\":[\"a10\",\"s10\",\"t10\",\"u10\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、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-20261113-r07-finite-flow-network-0062","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-flow-network"],"title_zh":"有限流網路","stem_zh":"finite loop-free directed network、逐edge capacity與逐internal-vertex conservation。操作=flow-conservation-audit；資料={\"vertices\":[\"s11\",\"a11\",\"t11\",\"u11\"],\"edges\":[[\"s11\",\"a11\"],[\"a11\",\"s11\"],[\"a11\",\"t11\"],[\"t11\",\"a11\"]],\"source\":\"s11\",\"sink\":\"t11\",\"capacities\":[60,60,60,60],\"flows\":[48,24,24,12]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(s11, t11, [[[a11, s11], 60, 24], [[a11, t11], 60, 24], [[s11, a11], 60, 48], [[t11, a11], 60, 12]], [[a11, 60, 48, 否], [u11, 0, 0, 是]], 否, [a11, 60, 48, 否], every_internal_vertex_requires_complete_inflow_equals_outflow)","explanation_zh":"Wave6第七層核對=(s11, t11, [[[a11, s11], 60, 24], [[a11, t11], 60, 24], [[s11, a11], 60, 48], [[t11, a11], 60, 12]], [[a11, 60, 48, 否], [u11, 0, 0, 是]], 否, [a11, 60, 48, 否], every_internal_vertex_requires_complete_inflow_equals_outflow)；依完整明示資料獨立重算；antiparallel edges明示，zero capacity與disconnected vertices允許；flow value由source與sink交叉計算；feasible不等於maximum，不提前宣稱min-cut equality或augmenting-path optimality。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限流網路"]},"variant":{"family_id":"qf-mod44-flow-conservation-audit","seed":202611137062,"parameters":{"vertices":["s11","a11","t11","u11"],"edges":[["s11","a11"],["a11","s11"],["a11","t11"],["t11","a11"]],"source":"s11","sink":"t11","capacities":[60,60,60,60],"flows":[48,24,24,12],"domain":"Exact_finite_flow_network","operation":"flow-conservation-audit","semantic_template":"qf-mod44-flow-conservation-audit/{\"capacities\":[[60,1],[60,1],[60,1],[60,1]],\"edges\":[[\"a11\",\"s11\"],[\"a11\",\"t11\"],[\"s11\",\"a11\"],[\"t11\",\"a11\"]],\"flows\":[[24,1],[24,1],[48,1],[12,1]],\"sink\":\"t11\",\"source\":\"s11\",\"vertices\":[\"a11\",\"s11\",\"t11\",\"u11\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、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-20261113-r07-finite-flow-network-0063","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-flow-network"],"title_zh":"有限流網路","stem_zh":"finite loop-free directed network、逐edge capacity與逐internal-vertex conservation。操作=flow-conservation-audit；資料={\"vertices\":[\"s12\",\"a12\",\"t12\",\"u12\"],\"edges\":[[\"s12\",\"a12\"],[\"a12\",\"s12\"],[\"a12\",\"t12\"],[\"t12\",\"a12\"]],\"source\":\"s12\",\"sink\":\"t12\",\"capacities\":[65,65,65,65],\"flows\":[52,26,39,13]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(s12, t12, [[[a12, s12], 65, 26], [[a12, t12], 65, 39], [[s12, a12], 65, 52], [[t12, a12], 65, 13]], [[a12, 65, 65, 是], [u12, 0, 0, 是]], 是, null, every_internal_vertex_requires_complete_inflow_equals_outflow)","explanation_zh":"Wave6第七層核對=(s12, t12, [[[a12, s12], 65, 26], [[a12, t12], 65, 39], [[s12, a12], 65, 52], [[t12, a12], 65, 13]], [[a12, 65, 65, 是], [u12, 0, 0, 是]], 是, null, every_internal_vertex_requires_complete_inflow_equals_outflow)；依完整明示資料獨立重算；antiparallel edges明示，zero capacity與disconnected vertices允許；flow value由source與sink交叉計算；feasible不等於maximum，不提前宣稱min-cut equality或augmenting-path optimality。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限流網路"]},"variant":{"family_id":"qf-mod44-flow-conservation-audit","seed":202611137063,"parameters":{"vertices":["s12","a12","t12","u12"],"edges":[["s12","a12"],["a12","s12"],["a12","t12"],["t12","a12"]],"source":"s12","sink":"t12","capacities":[65,65,65,65],"flows":[52,26,39,13],"domain":"Exact_finite_flow_network","operation":"flow-conservation-audit","semantic_template":"qf-mod44-flow-conservation-audit/{\"capacities\":[[65,1],[65,1],[65,1],[65,1]],\"edges\":[[\"a12\",\"s12\"],[\"a12\",\"t12\"],[\"s12\",\"a12\"],[\"t12\",\"a12\"]],\"flows\":[[26,1],[39,1],[52,1],[13,1]],\"sink\":\"t12\",\"source\":\"s12\",\"vertices\":[\"a12\",\"s12\",\"t12\",\"u12\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、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-20261113-r07-finite-flow-network-0064","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-flow-network"],"title_zh":"有限流網路","stem_zh":"finite loop-free directed network、逐edge capacity與逐internal-vertex conservation。操作=flow-conservation-audit；資料={\"vertices\":[\"s13\",\"a13\",\"t13\",\"u13\"],\"edges\":[[\"s13\",\"a13\"],[\"a13\",\"s13\"],[\"a13\",\"t13\"],[\"t13\",\"a13\"]],\"source\":\"s13\",\"sink\":\"t13\",\"capacities\":[70,70,70,70],\"flows\":[56,28,28,14]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(s13, t13, [[[a13, s13], 70, 28], [[a13, t13], 70, 28], [[s13, a13], 70, 56], [[t13, a13], 70, 14]], [[a13, 70, 56, 否], [u13, 0, 0, 是]], 否, [a13, 70, 56, 否], every_internal_vertex_requires_complete_inflow_equals_outflow)","explanation_zh":"Wave6第七層核對=(s13, t13, [[[a13, s13], 70, 28], [[a13, t13], 70, 28], [[s13, a13], 70, 56], [[t13, a13], 70, 14]], [[a13, 70, 56, 否], [u13, 0, 0, 是]], 否, [a13, 70, 56, 否], every_internal_vertex_requires_complete_inflow_equals_outflow)；依完整明示資料獨立重算；antiparallel edges明示，zero capacity與disconnected vertices允許；flow value由source與sink交叉計算；feasible不等於maximum，不提前宣稱min-cut equality或augmenting-path optimality。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限流網路"]},"variant":{"family_id":"qf-mod44-flow-conservation-audit","seed":202611137064,"parameters":{"vertices":["s13","a13","t13","u13"],"edges":[["s13","a13"],["a13","s13"],["a13","t13"],["t13","a13"]],"source":"s13","sink":"t13","capacities":[70,70,70,70],"flows":[56,28,28,14],"domain":"Exact_finite_flow_network","operation":"flow-conservation-audit","semantic_template":"qf-mod44-flow-conservation-audit/{\"capacities\":[[70,1],[70,1],[70,1],[70,1]],\"edges\":[[\"a13\",\"s13\"],[\"a13\",\"t13\"],[\"s13\",\"a13\"],[\"t13\",\"a13\"]],\"flows\":[[28,1],[28,1],[56,1],[14,1]],\"sink\":\"t13\",\"source\":\"s13\",\"vertices\":[\"a13\",\"s13\",\"t13\",\"u13\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、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-20261113-r07-finite-flow-network-0065","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-flow-network"],"title_zh":"有限流網路","stem_zh":"finite loop-free directed network、逐edge capacity與逐internal-vertex conservation。操作=flow-conservation-audit；資料={\"vertices\":[\"s14\",\"a14\",\"t14\",\"u14\"],\"edges\":[[\"s14\",\"a14\"],[\"a14\",\"s14\"],[\"a14\",\"t14\"],[\"t14\",\"a14\"]],\"source\":\"s14\",\"sink\":\"t14\",\"capacities\":[75,75,75,75],\"flows\":[60,30,45,15]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(s14, t14, [[[a14, s14], 75, 30], [[a14, t14], 75, 45], [[s14, a14], 75, 60], [[t14, a14], 75, 15]], [[a14, 75, 75, 是], [u14, 0, 0, 是]], 是, null, every_internal_vertex_requires_complete_inflow_equals_outflow)","explanation_zh":"Wave6第七層核對=(s14, t14, [[[a14, s14], 75, 30], [[a14, t14], 75, 45], [[s14, a14], 75, 60], [[t14, a14], 75, 15]], [[a14, 75, 75, 是], [u14, 0, 0, 是]], 是, null, every_internal_vertex_requires_complete_inflow_equals_outflow)；依完整明示資料獨立重算；antiparallel edges明示，zero capacity與disconnected vertices允許；flow value由source與sink交叉計算；feasible不等於maximum，不提前宣稱min-cut equality或augmenting-path optimality。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限流網路"]},"variant":{"family_id":"qf-mod44-flow-conservation-audit","seed":202611137065,"parameters":{"vertices":["s14","a14","t14","u14"],"edges":[["s14","a14"],["a14","s14"],["a14","t14"],["t14","a14"]],"source":"s14","sink":"t14","capacities":[75,75,75,75],"flows":[60,30,45,15],"domain":"Exact_finite_flow_network","operation":"flow-conservation-audit","semantic_template":"qf-mod44-flow-conservation-audit/{\"capacities\":[[75,1],[75,1],[75,1],[75,1]],\"edges\":[[\"a14\",\"s14\"],[\"a14\",\"t14\"],[\"s14\",\"a14\"],[\"t14\",\"a14\"]],\"flows\":[[30,1],[45,1],[60,1],[15,1]],\"sink\":\"t14\",\"source\":\"s14\",\"vertices\":[\"a14\",\"s14\",\"t14\",\"u14\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、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-20261113-r07-finite-flow-network-0066","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-flow-network"],"title_zh":"有限流網路","stem_zh":"finite loop-free directed network、逐edge capacity與逐internal-vertex conservation。操作=flow-conservation-audit；資料={\"vertices\":[\"s15\",\"a15\",\"t15\",\"u15\"],\"edges\":[[\"s15\",\"a15\"],[\"a15\",\"s15\"],[\"a15\",\"t15\"],[\"t15\",\"a15\"]],\"source\":\"s15\",\"sink\":\"t15\",\"capacities\":[80,80,80,80],\"flows\":[64,32,32,16]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(s15, t15, [[[a15, s15], 80, 32], [[a15, t15], 80, 32], [[s15, a15], 80, 64], [[t15, a15], 80, 16]], [[a15, 80, 64, 否], [u15, 0, 0, 是]], 否, [a15, 80, 64, 否], every_internal_vertex_requires_complete_inflow_equals_outflow)","explanation_zh":"Wave6第七層核對=(s15, t15, [[[a15, s15], 80, 32], [[a15, t15], 80, 32], [[s15, a15], 80, 64], [[t15, a15], 80, 16]], [[a15, 80, 64, 否], [u15, 0, 0, 是]], 否, [a15, 80, 64, 否], every_internal_vertex_requires_complete_inflow_equals_outflow)；依完整明示資料獨立重算；antiparallel edges明示，zero capacity與disconnected vertices允許；flow value由source與sink交叉計算；feasible不等於maximum，不提前宣稱min-cut equality或augmenting-path optimality。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限流網路"]},"variant":{"family_id":"qf-mod44-flow-conservation-audit","seed":202611137066,"parameters":{"vertices":["s15","a15","t15","u15"],"edges":[["s15","a15"],["a15","s15"],["a15","t15"],["t15","a15"]],"source":"s15","sink":"t15","capacities":[80,80,80,80],"flows":[64,32,32,16],"domain":"Exact_finite_flow_network","operation":"flow-conservation-audit","semantic_template":"qf-mod44-flow-conservation-audit/{\"capacities\":[[80,1],[80,1],[80,1],[80,1]],\"edges\":[[\"a15\",\"s15\"],[\"a15\",\"t15\"],[\"s15\",\"a15\"],[\"t15\",\"a15\"]],\"flows\":[[32,1],[32,1],[64,1],[16,1]],\"sink\":\"t15\",\"source\":\"s15\",\"vertices\":[\"a15\",\"s15\",\"t15\",\"u15\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、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-20261113-r07-finite-flow-network-0067","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-flow-network"],"title_zh":"有限流網路","stem_zh":"finite loop-free directed network、逐edge capacity與逐internal-vertex conservation。操作=flow-conservation-audit；資料={\"vertices\":[\"s16\",\"a16\",\"t16\",\"u16\"],\"edges\":[[\"s16\",\"a16\"],[\"a16\",\"s16\"],[\"a16\",\"t16\"],[\"t16\",\"a16\"]],\"source\":\"s16\",\"sink\":\"t16\",\"capacities\":[85,85,85,85],\"flows\":[68,34,51,17]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(s16, t16, [[[a16, s16], 85, 34], [[a16, t16], 85, 51], [[s16, a16], 85, 68], [[t16, a16], 85, 17]], [[a16, 85, 85, 是], [u16, 0, 0, 是]], 是, null, every_internal_vertex_requires_complete_inflow_equals_outflow)","explanation_zh":"Wave6第七層核對=(s16, t16, [[[a16, s16], 85, 34], [[a16, t16], 85, 51], [[s16, a16], 85, 68], [[t16, a16], 85, 17]], [[a16, 85, 85, 是], [u16, 0, 0, 是]], 是, null, every_internal_vertex_requires_complete_inflow_equals_outflow)；依完整明示資料獨立重算；antiparallel edges明示，zero capacity與disconnected vertices允許；flow value由source與sink交叉計算；feasible不等於maximum，不提前宣稱min-cut equality或augmenting-path optimality。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限流網路"]},"variant":{"family_id":"qf-mod44-flow-conservation-audit","seed":202611137067,"parameters":{"vertices":["s16","a16","t16","u16"],"edges":[["s16","a16"],["a16","s16"],["a16","t16"],["t16","a16"]],"source":"s16","sink":"t16","capacities":[85,85,85,85],"flows":[68,34,51,17],"domain":"Exact_finite_flow_network","operation":"flow-conservation-audit","semantic_template":"qf-mod44-flow-conservation-audit/{\"capacities\":[[85,1],[85,1],[85,1],[85,1]],\"edges\":[[\"a16\",\"s16\"],[\"a16\",\"t16\"],[\"s16\",\"a16\"],[\"t16\",\"a16\"]],\"flows\":[[34,1],[51,1],[68,1],[17,1]],\"sink\":\"t16\",\"source\":\"s16\",\"vertices\":[\"a16\",\"s16\",\"t16\",\"u16\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、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-20261113-r07-finite-flow-network-0068","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-flow-network"],"title_zh":"有限流網路","stem_zh":"finite loop-free directed network、逐edge capacity與逐internal-vertex conservation。操作=flow-conservation-audit；資料={\"vertices\":[\"s17\",\"a17\",\"t17\",\"u17\"],\"edges\":[[\"s17\",\"a17\"],[\"a17\",\"s17\"],[\"a17\",\"t17\"],[\"t17\",\"a17\"]],\"source\":\"s17\",\"sink\":\"t17\",\"capacities\":[90,90,90,90],\"flows\":[72,36,36,18]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(s17, t17, [[[a17, s17], 90, 36], [[a17, t17], 90, 36], [[s17, a17], 90, 72], [[t17, a17], 90, 18]], [[a17, 90, 72, 否], [u17, 0, 0, 是]], 否, [a17, 90, 72, 否], every_internal_vertex_requires_complete_inflow_equals_outflow)","explanation_zh":"Wave6第七層核對=(s17, t17, [[[a17, s17], 90, 36], [[a17, t17], 90, 36], [[s17, a17], 90, 72], [[t17, a17], 90, 18]], [[a17, 90, 72, 否], [u17, 0, 0, 是]], 否, [a17, 90, 72, 否], every_internal_vertex_requires_complete_inflow_equals_outflow)；依完整明示資料獨立重算；antiparallel edges明示，zero capacity與disconnected vertices允許；flow value由source與sink交叉計算；feasible不等於maximum，不提前宣稱min-cut equality或augmenting-path optimality。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限流網路"]},"variant":{"family_id":"qf-mod44-flow-conservation-audit","seed":202611137068,"parameters":{"vertices":["s17","a17","t17","u17"],"edges":[["s17","a17"],["a17","s17"],["a17","t17"],["t17","a17"]],"source":"s17","sink":"t17","capacities":[90,90,90,90],"flows":[72,36,36,18],"domain":"Exact_finite_flow_network","operation":"flow-conservation-audit","semantic_template":"qf-mod44-flow-conservation-audit/{\"capacities\":[[90,1],[90,1],[90,1],[90,1]],\"edges\":[[\"a17\",\"s17\"],[\"a17\",\"t17\"],[\"s17\",\"a17\"],[\"t17\",\"a17\"]],\"flows\":[[36,1],[36,1],[72,1],[18,1]],\"sink\":\"t17\",\"source\":\"s17\",\"vertices\":[\"a17\",\"s17\",\"t17\",\"u17\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、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-20261113-r07-finite-flow-network-0069","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-flow-network"],"title_zh":"有限流網路","stem_zh":"finite loop-free directed network、逐edge capacity與逐internal-vertex conservation。操作=flow-conservation-audit；資料={\"vertices\":[\"s18\",\"a18\",\"t18\",\"u18\"],\"edges\":[[\"s18\",\"a18\"],[\"a18\",\"s18\"],[\"a18\",\"t18\"],[\"t18\",\"a18\"]],\"source\":\"s18\",\"sink\":\"t18\",\"capacities\":[95,95,95,95],\"flows\":[76,38,57,19]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(s18, t18, [[[a18, s18], 95, 38], [[a18, t18], 95, 57], [[s18, a18], 95, 76], [[t18, a18], 95, 19]], [[a18, 95, 95, 是], [u18, 0, 0, 是]], 是, null, every_internal_vertex_requires_complete_inflow_equals_outflow)","explanation_zh":"Wave6第七層核對=(s18, t18, [[[a18, s18], 95, 38], [[a18, t18], 95, 57], [[s18, a18], 95, 76], [[t18, a18], 95, 19]], [[a18, 95, 95, 是], [u18, 0, 0, 是]], 是, null, every_internal_vertex_requires_complete_inflow_equals_outflow)；依完整明示資料獨立重算；antiparallel edges明示，zero capacity與disconnected vertices允許；flow value由source與sink交叉計算；feasible不等於maximum，不提前宣稱min-cut equality或augmenting-path optimality。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限流網路"]},"variant":{"family_id":"qf-mod44-flow-conservation-audit","seed":202611137069,"parameters":{"vertices":["s18","a18","t18","u18"],"edges":[["s18","a18"],["a18","s18"],["a18","t18"],["t18","a18"]],"source":"s18","sink":"t18","capacities":[95,95,95,95],"flows":[76,38,57,19],"domain":"Exact_finite_flow_network","operation":"flow-conservation-audit","semantic_template":"qf-mod44-flow-conservation-audit/{\"capacities\":[[95,1],[95,1],[95,1],[95,1]],\"edges\":[[\"a18\",\"s18\"],[\"a18\",\"t18\"],[\"s18\",\"a18\"],[\"t18\",\"a18\"]],\"flows\":[[38,1],[57,1],[76,1],[19,1]],\"sink\":\"t18\",\"source\":\"s18\",\"vertices\":[\"a18\",\"s18\",\"t18\",\"u18\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、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-20261113-r07-finite-flow-network-0070","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-flow-network"],"title_zh":"有限流網路","stem_zh":"finite loop-free directed network、逐edge capacity與逐internal-vertex conservation。操作=flow-conservation-audit；資料={\"vertices\":[\"s19\",\"a19\",\"t19\",\"u19\"],\"edges\":[[\"s19\",\"a19\"],[\"a19\",\"s19\"],[\"a19\",\"t19\"],[\"t19\",\"a19\"]],\"source\":\"s19\",\"sink\":\"t19\",\"capacities\":[100,100,100,100],\"flows\":[80,40,40,20]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(s19, t19, [[[a19, s19], 100, 40], [[a19, t19], 100, 40], [[s19, a19], 100, 80], [[t19, a19], 100, 20]], [[a19, 100, 80, 否], [u19, 0, 0, 是]], 否, [a19, 100, 80, 否], every_internal_vertex_requires_complete_inflow_equals_outflow)","explanation_zh":"Wave6第七層核對=(s19, t19, [[[a19, s19], 100, 40], [[a19, t19], 100, 40], [[s19, a19], 100, 80], [[t19, a19], 100, 20]], [[a19, 100, 80, 否], [u19, 0, 0, 是]], 否, [a19, 100, 80, 否], every_internal_vertex_requires_complete_inflow_equals_outflow)；依完整明示資料獨立重算；antiparallel edges明示，zero capacity與disconnected vertices允許；flow value由source與sink交叉計算；feasible不等於maximum，不提前宣稱min-cut equality或augmenting-path optimality。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限流網路"]},"variant":{"family_id":"qf-mod44-flow-conservation-audit","seed":202611137070,"parameters":{"vertices":["s19","a19","t19","u19"],"edges":[["s19","a19"],["a19","s19"],["a19","t19"],["t19","a19"]],"source":"s19","sink":"t19","capacities":[100,100,100,100],"flows":[80,40,40,20],"domain":"Exact_finite_flow_network","operation":"flow-conservation-audit","semantic_template":"qf-mod44-flow-conservation-audit/{\"capacities\":[[100,1],[100,1],[100,1],[100,1]],\"edges\":[[\"a19\",\"s19\"],[\"a19\",\"t19\"],[\"s19\",\"a19\"],[\"t19\",\"a19\"]],\"flows\":[[40,1],[40,1],[80,1],[20,1]],\"sink\":\"t19\",\"source\":\"s19\",\"vertices\":[\"a19\",\"s19\",\"t19\",\"u19\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、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-20261113-r07-finite-flow-network-0071","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-flow-network"],"title_zh":"有限流網路","stem_zh":"finite loop-free directed network、逐edge capacity與逐internal-vertex conservation。操作=flow-conservation-audit；資料={\"vertices\":[\"s20\",\"a20\",\"t20\",\"u20\"],\"edges\":[[\"s20\",\"a20\"],[\"a20\",\"s20\"],[\"a20\",\"t20\"],[\"t20\",\"a20\"]],\"source\":\"s20\",\"sink\":\"t20\",\"capacities\":[105,105,105,105],\"flows\":[84,42,63,21]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(s20, t20, [[[a20, s20], 105, 42], [[a20, t20], 105, 63], [[s20, a20], 105, 84], [[t20, a20], 105, 21]], [[a20, 105, 105, 是], [u20, 0, 0, 是]], 是, null, every_internal_vertex_requires_complete_inflow_equals_outflow)","explanation_zh":"Wave6第七層核對=(s20, t20, [[[a20, s20], 105, 42], [[a20, t20], 105, 63], [[s20, a20], 105, 84], [[t20, a20], 105, 21]], [[a20, 105, 105, 是], [u20, 0, 0, 是]], 是, null, every_internal_vertex_requires_complete_inflow_equals_outflow)；依完整明示資料獨立重算；antiparallel edges明示，zero capacity與disconnected vertices允許；flow value由source與sink交叉計算；feasible不等於maximum，不提前宣稱min-cut equality或augmenting-path optimality。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限流網路"]},"variant":{"family_id":"qf-mod44-flow-conservation-audit","seed":202611137071,"parameters":{"vertices":["s20","a20","t20","u20"],"edges":[["s20","a20"],["a20","s20"],["a20","t20"],["t20","a20"]],"source":"s20","sink":"t20","capacities":[105,105,105,105],"flows":[84,42,63,21],"domain":"Exact_finite_flow_network","operation":"flow-conservation-audit","semantic_template":"qf-mod44-flow-conservation-audit/{\"capacities\":[[105,1],[105,1],[105,1],[105,1]],\"edges\":[[\"a20\",\"s20\"],[\"a20\",\"t20\"],[\"s20\",\"a20\"],[\"t20\",\"a20\"]],\"flows\":[[42,1],[63,1],[84,1],[21,1]],\"sink\":\"t20\",\"source\":\"s20\",\"vertices\":[\"a20\",\"s20\",\"t20\",\"u20\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、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-20261113-r07-finite-flow-network-0072","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-flow-network"],"title_zh":"有限流網路","stem_zh":"finite loop-free directed network、逐edge capacity與逐internal-vertex conservation。操作=flow-conservation-audit；資料={\"vertices\":[\"s21\",\"a21\",\"t21\",\"u21\"],\"edges\":[[\"s21\",\"a21\"],[\"a21\",\"s21\"],[\"a21\",\"t21\"],[\"t21\",\"a21\"]],\"source\":\"s21\",\"sink\":\"t21\",\"capacities\":[110,110,110,110],\"flows\":[88,44,44,22]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(s21, t21, [[[a21, s21], 110, 44], [[a21, t21], 110, 44], [[s21, a21], 110, 88], [[t21, a21], 110, 22]], [[a21, 110, 88, 否], [u21, 0, 0, 是]], 否, [a21, 110, 88, 否], every_internal_vertex_requires_complete_inflow_equals_outflow)","explanation_zh":"Wave6第七層核對=(s21, t21, [[[a21, s21], 110, 44], [[a21, t21], 110, 44], [[s21, a21], 110, 88], [[t21, a21], 110, 22]], [[a21, 110, 88, 否], [u21, 0, 0, 是]], 否, [a21, 110, 88, 否], every_internal_vertex_requires_complete_inflow_equals_outflow)；依完整明示資料獨立重算；antiparallel edges明示，zero capacity與disconnected vertices允許；flow value由source與sink交叉計算；feasible不等於maximum，不提前宣稱min-cut equality或augmenting-path optimality。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限流網路"]},"variant":{"family_id":"qf-mod44-flow-conservation-audit","seed":202611137072,"parameters":{"vertices":["s21","a21","t21","u21"],"edges":[["s21","a21"],["a21","s21"],["a21","t21"],["t21","a21"]],"source":"s21","sink":"t21","capacities":[110,110,110,110],"flows":[88,44,44,22],"domain":"Exact_finite_flow_network","operation":"flow-conservation-audit","semantic_template":"qf-mod44-flow-conservation-audit/{\"capacities\":[[110,1],[110,1],[110,1],[110,1]],\"edges\":[[\"a21\",\"s21\"],[\"a21\",\"t21\"],[\"s21\",\"a21\"],[\"t21\",\"a21\"]],\"flows\":[[44,1],[44,1],[88,1],[22,1]],\"sink\":\"t21\",\"source\":\"s21\",\"vertices\":[\"a21\",\"s21\",\"t21\",\"u21\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、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-20261113-r07-finite-flow-network-0073","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-flow-network"],"title_zh":"有限流網路","stem_zh":"finite loop-free directed network、逐edge capacity與逐internal-vertex conservation。操作=flow-conservation-audit；資料={\"vertices\":[\"s22\",\"a22\",\"t22\",\"u22\"],\"edges\":[[\"s22\",\"a22\"],[\"a22\",\"s22\"],[\"a22\",\"t22\"],[\"t22\",\"a22\"]],\"source\":\"s22\",\"sink\":\"t22\",\"capacities\":[115,115,115,115],\"flows\":[92,46,69,23]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(s22, t22, [[[a22, s22], 115, 46], [[a22, t22], 115, 69], [[s22, a22], 115, 92], [[t22, a22], 115, 23]], [[a22, 115, 115, 是], [u22, 0, 0, 是]], 是, null, every_internal_vertex_requires_complete_inflow_equals_outflow)","explanation_zh":"Wave6第七層核對=(s22, t22, [[[a22, s22], 115, 46], [[a22, t22], 115, 69], [[s22, a22], 115, 92], [[t22, a22], 115, 23]], [[a22, 115, 115, 是], [u22, 0, 0, 是]], 是, null, every_internal_vertex_requires_complete_inflow_equals_outflow)；依完整明示資料獨立重算；antiparallel edges明示，zero capacity與disconnected vertices允許；flow value由source與sink交叉計算；feasible不等於maximum，不提前宣稱min-cut equality或augmenting-path optimality。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限流網路"]},"variant":{"family_id":"qf-mod44-flow-conservation-audit","seed":202611137073,"parameters":{"vertices":["s22","a22","t22","u22"],"edges":[["s22","a22"],["a22","s22"],["a22","t22"],["t22","a22"]],"source":"s22","sink":"t22","capacities":[115,115,115,115],"flows":[92,46,69,23],"domain":"Exact_finite_flow_network","operation":"flow-conservation-audit","semantic_template":"qf-mod44-flow-conservation-audit/{\"capacities\":[[115,1],[115,1],[115,1],[115,1]],\"edges\":[[\"a22\",\"s22\"],[\"a22\",\"t22\"],[\"s22\",\"a22\"],[\"t22\",\"a22\"]],\"flows\":[[46,1],[69,1],[92,1],[23,1]],\"sink\":\"t22\",\"source\":\"s22\",\"vertices\":[\"a22\",\"s22\",\"t22\",\"u22\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、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-20261113-r07-finite-flow-network-0074","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-flow-network"],"title_zh":"有限流網路","stem_zh":"finite loop-free directed network、逐edge capacity與逐internal-vertex conservation。操作=flow-conservation-audit；資料={\"vertices\":[\"s23\",\"a23\",\"t23\",\"u23\"],\"edges\":[[\"s23\",\"a23\"],[\"a23\",\"s23\"],[\"a23\",\"t23\"],[\"t23\",\"a23\"]],\"source\":\"s23\",\"sink\":\"t23\",\"capacities\":[120,120,120,120],\"flows\":[96,48,48,24]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(s23, t23, [[[a23, s23], 120, 48], [[a23, t23], 120, 48], [[s23, a23], 120, 96], [[t23, a23], 120, 24]], [[a23, 120, 96, 否], [u23, 0, 0, 是]], 否, [a23, 120, 96, 否], every_internal_vertex_requires_complete_inflow_equals_outflow)","explanation_zh":"Wave6第七層核對=(s23, t23, [[[a23, s23], 120, 48], [[a23, t23], 120, 48], [[s23, a23], 120, 96], [[t23, a23], 120, 24]], [[a23, 120, 96, 否], [u23, 0, 0, 是]], 否, [a23, 120, 96, 否], every_internal_vertex_requires_complete_inflow_equals_outflow)；依完整明示資料獨立重算；antiparallel edges明示，zero capacity與disconnected vertices允許；flow value由source與sink交叉計算；feasible不等於maximum，不提前宣稱min-cut equality或augmenting-path optimality。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限流網路"]},"variant":{"family_id":"qf-mod44-flow-conservation-audit","seed":202611137074,"parameters":{"vertices":["s23","a23","t23","u23"],"edges":[["s23","a23"],["a23","s23"],["a23","t23"],["t23","a23"]],"source":"s23","sink":"t23","capacities":[120,120,120,120],"flows":[96,48,48,24],"domain":"Exact_finite_flow_network","operation":"flow-conservation-audit","semantic_template":"qf-mod44-flow-conservation-audit/{\"capacities\":[[120,1],[120,1],[120,1],[120,1]],\"edges\":[[\"a23\",\"s23\"],[\"a23\",\"t23\"],[\"s23\",\"a23\"],[\"t23\",\"a23\"]],\"flows\":[[48,1],[48,1],[96,1],[24,1]],\"sink\":\"t23\",\"source\":\"s23\",\"vertices\":[\"a23\",\"s23\",\"t23\",\"u23\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、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-20261113-r07-finite-flow-network-0075","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-flow-network"],"title_zh":"有限流網路","stem_zh":"finite loop-free directed network、逐edge capacity與逐internal-vertex conservation。操作=flow-conservation-audit；資料={\"vertices\":[\"s24\",\"a24\",\"t24\",\"u24\"],\"edges\":[[\"s24\",\"a24\"],[\"a24\",\"s24\"],[\"a24\",\"t24\"],[\"t24\",\"a24\"]],\"source\":\"s24\",\"sink\":\"t24\",\"capacities\":[125,125,125,125],\"flows\":[100,50,75,25]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(s24, t24, [[[a24, s24], 125, 50], [[a24, t24], 125, 75], [[s24, a24], 125, 100], [[t24, a24], 125, 25]], [[a24, 125, 125, 是], [u24, 0, 0, 是]], 是, null, every_internal_vertex_requires_complete_inflow_equals_outflow)","explanation_zh":"Wave6第七層核對=(s24, t24, [[[a24, s24], 125, 50], [[a24, t24], 125, 75], [[s24, a24], 125, 100], [[t24, a24], 125, 25]], [[a24, 125, 125, 是], [u24, 0, 0, 是]], 是, null, every_internal_vertex_requires_complete_inflow_equals_outflow)；依完整明示資料獨立重算；antiparallel edges明示，zero capacity與disconnected vertices允許；flow value由source與sink交叉計算；feasible不等於maximum，不提前宣稱min-cut equality或augmenting-path optimality。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限流網路"]},"variant":{"family_id":"qf-mod44-flow-conservation-audit","seed":202611137075,"parameters":{"vertices":["s24","a24","t24","u24"],"edges":[["s24","a24"],["a24","s24"],["a24","t24"],["t24","a24"]],"source":"s24","sink":"t24","capacities":[125,125,125,125],"flows":[100,50,75,25],"domain":"Exact_finite_flow_network","operation":"flow-conservation-audit","semantic_template":"qf-mod44-flow-conservation-audit/{\"capacities\":[[125,1],[125,1],[125,1],[125,1]],\"edges\":[[\"a24\",\"s24\"],[\"a24\",\"t24\"],[\"s24\",\"a24\"],[\"t24\",\"a24\"]],\"flows\":[[50,1],[75,1],[100,1],[25,1]],\"sink\":\"t24\",\"source\":\"s24\",\"vertices\":[\"a24\",\"s24\",\"t24\",\"u24\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、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-20261113-r07-finite-flow-network-0076","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-flow-network"],"title_zh":"有限流網路","stem_zh":"finite loop-free directed network、逐edge capacity與逐internal-vertex conservation。操作=flow-value-cross-check；資料={\"vertices\":[\"s0\",\"a0\",\"t0\",\"u0\"],\"edges\":[[\"s0\",\"a0\"],[\"a0\",\"s0\"],[\"a0\",\"t0\"],[\"t0\",\"a0\"]],\"source\":\"s0\",\"sink\":\"t0\",\"capacities\":[5,5,5,5],\"flows\":[4,2,3,1]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(s0, t0, [[[a0, s0], 5, 2], [[a0, t0], 5, 3], [[s0, a0], 5, 4], [[t0, a0], 5, 1]], 2, 2, 是, 是, 是, 是, 否, feasible_flow_value_cross_check_does_not_claim_maximality_or_min_cut_equality)","explanation_zh":"Wave6第七層核對=(s0, t0, [[[a0, s0], 5, 2], [[a0, t0], 5, 3], [[s0, a0], 5, 4], [[t0, a0], 5, 1]], 2, 2, 是, 是, 是, 是, 否, feasible_flow_value_cross_check_does_not_claim_maximality_or_min_cut_equality)；依完整明示資料獨立重算；antiparallel edges明示，zero capacity與disconnected vertices允許；flow value由source與sink交叉計算；feasible不等於maximum，不提前宣稱min-cut equality或augmenting-path optimality。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限流網路"]},"variant":{"family_id":"qf-mod44-flow-value-cross-check","seed":202611137076,"parameters":{"vertices":["s0","a0","t0","u0"],"edges":[["s0","a0"],["a0","s0"],["a0","t0"],["t0","a0"]],"source":"s0","sink":"t0","capacities":[5,5,5,5],"flows":[4,2,3,1],"domain":"Exact_finite_flow_network","operation":"flow-value-cross-check","semantic_template":"qf-mod44-flow-value-cross-check/{\"capacities\":[[5,1],[5,1],[5,1],[5,1]],\"edges\":[[\"a0\",\"s0\"],[\"a0\",\"t0\"],[\"s0\",\"a0\"],[\"t0\",\"a0\"]],\"flows\":[[2,1],[3,1],[4,1],[1,1]],\"sink\":\"t0\",\"source\":\"s0\",\"vertices\":[\"a0\",\"s0\",\"t0\",\"u0\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、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-20261113-r07-finite-flow-network-0077","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-flow-network"],"title_zh":"有限流網路","stem_zh":"finite loop-free directed network、逐edge capacity與逐internal-vertex conservation。操作=flow-value-cross-check；資料={\"vertices\":[\"s1\",\"a1\",\"t1\",\"u1\"],\"edges\":[[\"s1\",\"a1\"],[\"a1\",\"s1\"],[\"a1\",\"t1\"],[\"t1\",\"a1\"]],\"source\":\"s1\",\"sink\":\"t1\",\"capacities\":[10,10,10,10],\"flows\":[8,4,6,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(s1, t1, [[[a1, s1], 10, 4], [[a1, t1], 10, 6], [[s1, a1], 10, 8], [[t1, a1], 10, 2]], 4, 4, 是, 是, 是, 是, 否, feasible_flow_value_cross_check_does_not_claim_maximality_or_min_cut_equality)","explanation_zh":"Wave6第七層核對=(s1, t1, [[[a1, s1], 10, 4], [[a1, t1], 10, 6], [[s1, a1], 10, 8], [[t1, a1], 10, 2]], 4, 4, 是, 是, 是, 是, 否, feasible_flow_value_cross_check_does_not_claim_maximality_or_min_cut_equality)；依完整明示資料獨立重算；antiparallel edges明示，zero capacity與disconnected vertices允許；flow value由source與sink交叉計算；feasible不等於maximum，不提前宣稱min-cut equality或augmenting-path optimality。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限流網路"]},"variant":{"family_id":"qf-mod44-flow-value-cross-check","seed":202611137077,"parameters":{"vertices":["s1","a1","t1","u1"],"edges":[["s1","a1"],["a1","s1"],["a1","t1"],["t1","a1"]],"source":"s1","sink":"t1","capacities":[10,10,10,10],"flows":[8,4,6,2],"domain":"Exact_finite_flow_network","operation":"flow-value-cross-check","semantic_template":"qf-mod44-flow-value-cross-check/{\"capacities\":[[10,1],[10,1],[10,1],[10,1]],\"edges\":[[\"a1\",\"s1\"],[\"a1\",\"t1\"],[\"s1\",\"a1\"],[\"t1\",\"a1\"]],\"flows\":[[4,1],[6,1],[8,1],[2,1]],\"sink\":\"t1\",\"source\":\"s1\",\"vertices\":[\"a1\",\"s1\",\"t1\",\"u1\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、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-20261113-r07-finite-flow-network-0078","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-flow-network"],"title_zh":"有限流網路","stem_zh":"finite loop-free directed network、逐edge capacity與逐internal-vertex conservation。操作=flow-value-cross-check；資料={\"vertices\":[\"s2\",\"a2\",\"t2\",\"u2\"],\"edges\":[[\"s2\",\"a2\"],[\"a2\",\"s2\"],[\"a2\",\"t2\"],[\"t2\",\"a2\"]],\"source\":\"s2\",\"sink\":\"t2\",\"capacities\":[15,15,15,15],\"flows\":[12,6,9,3]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(s2, t2, [[[a2, s2], 15, 6], [[a2, t2], 15, 9], [[s2, a2], 15, 12], [[t2, a2], 15, 3]], 6, 6, 是, 是, 是, 是, 否, feasible_flow_value_cross_check_does_not_claim_maximality_or_min_cut_equality)","explanation_zh":"Wave6第七層核對=(s2, t2, [[[a2, s2], 15, 6], [[a2, t2], 15, 9], [[s2, a2], 15, 12], [[t2, a2], 15, 3]], 6, 6, 是, 是, 是, 是, 否, feasible_flow_value_cross_check_does_not_claim_maximality_or_min_cut_equality)；依完整明示資料獨立重算；antiparallel edges明示，zero capacity與disconnected vertices允許；flow value由source與sink交叉計算；feasible不等於maximum，不提前宣稱min-cut equality或augmenting-path optimality。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限流網路"]},"variant":{"family_id":"qf-mod44-flow-value-cross-check","seed":202611137078,"parameters":{"vertices":["s2","a2","t2","u2"],"edges":[["s2","a2"],["a2","s2"],["a2","t2"],["t2","a2"]],"source":"s2","sink":"t2","capacities":[15,15,15,15],"flows":[12,6,9,3],"domain":"Exact_finite_flow_network","operation":"flow-value-cross-check","semantic_template":"qf-mod44-flow-value-cross-check/{\"capacities\":[[15,1],[15,1],[15,1],[15,1]],\"edges\":[[\"a2\",\"s2\"],[\"a2\",\"t2\"],[\"s2\",\"a2\"],[\"t2\",\"a2\"]],\"flows\":[[6,1],[9,1],[12,1],[3,1]],\"sink\":\"t2\",\"source\":\"s2\",\"vertices\":[\"a2\",\"s2\",\"t2\",\"u2\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、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-20261113-r07-finite-flow-network-0079","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-flow-network"],"title_zh":"有限流網路","stem_zh":"finite loop-free directed network、逐edge capacity與逐internal-vertex conservation。操作=flow-value-cross-check；資料={\"vertices\":[\"s3\",\"a3\",\"t3\",\"u3\"],\"edges\":[[\"s3\",\"a3\"],[\"a3\",\"s3\"],[\"a3\",\"t3\"],[\"t3\",\"a3\"]],\"source\":\"s3\",\"sink\":\"t3\",\"capacities\":[20,20,20,20],\"flows\":[16,8,12,4]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(s3, t3, [[[a3, s3], 20, 8], [[a3, t3], 20, 12], [[s3, a3], 20, 16], [[t3, a3], 20, 4]], 8, 8, 是, 是, 是, 是, 否, feasible_flow_value_cross_check_does_not_claim_maximality_or_min_cut_equality)","explanation_zh":"Wave6第七層核對=(s3, t3, [[[a3, s3], 20, 8], [[a3, t3], 20, 12], [[s3, a3], 20, 16], [[t3, a3], 20, 4]], 8, 8, 是, 是, 是, 是, 否, feasible_flow_value_cross_check_does_not_claim_maximality_or_min_cut_equality)；依完整明示資料獨立重算；antiparallel edges明示，zero capacity與disconnected vertices允許；flow value由source與sink交叉計算；feasible不等於maximum，不提前宣稱min-cut equality或augmenting-path optimality。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限流網路"]},"variant":{"family_id":"qf-mod44-flow-value-cross-check","seed":202611137079,"parameters":{"vertices":["s3","a3","t3","u3"],"edges":[["s3","a3"],["a3","s3"],["a3","t3"],["t3","a3"]],"source":"s3","sink":"t3","capacities":[20,20,20,20],"flows":[16,8,12,4],"domain":"Exact_finite_flow_network","operation":"flow-value-cross-check","semantic_template":"qf-mod44-flow-value-cross-check/{\"capacities\":[[20,1],[20,1],[20,1],[20,1]],\"edges\":[[\"a3\",\"s3\"],[\"a3\",\"t3\"],[\"s3\",\"a3\"],[\"t3\",\"a3\"]],\"flows\":[[8,1],[12,1],[16,1],[4,1]],\"sink\":\"t3\",\"source\":\"s3\",\"vertices\":[\"a3\",\"s3\",\"t3\",\"u3\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、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-20261113-r07-finite-flow-network-0080","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-flow-network"],"title_zh":"有限流網路","stem_zh":"finite loop-free directed network、逐edge capacity與逐internal-vertex conservation。操作=flow-value-cross-check；資料={\"vertices\":[\"s4\",\"a4\",\"t4\",\"u4\"],\"edges\":[[\"s4\",\"a4\"],[\"a4\",\"s4\"],[\"a4\",\"t4\"],[\"t4\",\"a4\"]],\"source\":\"s4\",\"sink\":\"t4\",\"capacities\":[25,25,25,25],\"flows\":[20,10,15,5]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(s4, t4, [[[a4, s4], 25, 10], [[a4, t4], 25, 15], [[s4, a4], 25, 20], [[t4, a4], 25, 5]], 10, 10, 是, 是, 是, 是, 否, feasible_flow_value_cross_check_does_not_claim_maximality_or_min_cut_equality)","explanation_zh":"Wave6第七層核對=(s4, t4, [[[a4, s4], 25, 10], [[a4, t4], 25, 15], [[s4, a4], 25, 20], [[t4, a4], 25, 5]], 10, 10, 是, 是, 是, 是, 否, feasible_flow_value_cross_check_does_not_claim_maximality_or_min_cut_equality)；依完整明示資料獨立重算；antiparallel edges明示，zero capacity與disconnected vertices允許；flow value由source與sink交叉計算；feasible不等於maximum，不提前宣稱min-cut equality或augmenting-path optimality。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限流網路"]},"variant":{"family_id":"qf-mod44-flow-value-cross-check","seed":202611137080,"parameters":{"vertices":["s4","a4","t4","u4"],"edges":[["s4","a4"],["a4","s4"],["a4","t4"],["t4","a4"]],"source":"s4","sink":"t4","capacities":[25,25,25,25],"flows":[20,10,15,5],"domain":"Exact_finite_flow_network","operation":"flow-value-cross-check","semantic_template":"qf-mod44-flow-value-cross-check/{\"capacities\":[[25,1],[25,1],[25,1],[25,1]],\"edges\":[[\"a4\",\"s4\"],[\"a4\",\"t4\"],[\"s4\",\"a4\"],[\"t4\",\"a4\"]],\"flows\":[[10,1],[15,1],[20,1],[5,1]],\"sink\":\"t4\",\"source\":\"s4\",\"vertices\":[\"a4\",\"s4\",\"t4\",\"u4\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、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-20261113-r07-finite-flow-network-0081","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-flow-network"],"title_zh":"有限流網路","stem_zh":"finite loop-free directed network、逐edge capacity與逐internal-vertex conservation。操作=flow-value-cross-check；資料={\"vertices\":[\"s5\",\"a5\",\"t5\",\"u5\"],\"edges\":[[\"s5\",\"a5\"],[\"a5\",\"s5\"],[\"a5\",\"t5\"],[\"t5\",\"a5\"]],\"source\":\"s5\",\"sink\":\"t5\",\"capacities\":[30,30,30,30],\"flows\":[24,12,18,6]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(s5, t5, [[[a5, s5], 30, 12], [[a5, t5], 30, 18], [[s5, a5], 30, 24], [[t5, a5], 30, 6]], 12, 12, 是, 是, 是, 是, 否, feasible_flow_value_cross_check_does_not_claim_maximality_or_min_cut_equality)","explanation_zh":"Wave6第七層核對=(s5, t5, [[[a5, s5], 30, 12], [[a5, t5], 30, 18], [[s5, a5], 30, 24], [[t5, a5], 30, 6]], 12, 12, 是, 是, 是, 是, 否, feasible_flow_value_cross_check_does_not_claim_maximality_or_min_cut_equality)；依完整明示資料獨立重算；antiparallel edges明示，zero capacity與disconnected vertices允許；flow value由source與sink交叉計算；feasible不等於maximum，不提前宣稱min-cut equality或augmenting-path optimality。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限流網路"]},"variant":{"family_id":"qf-mod44-flow-value-cross-check","seed":202611137081,"parameters":{"vertices":["s5","a5","t5","u5"],"edges":[["s5","a5"],["a5","s5"],["a5","t5"],["t5","a5"]],"source":"s5","sink":"t5","capacities":[30,30,30,30],"flows":[24,12,18,6],"domain":"Exact_finite_flow_network","operation":"flow-value-cross-check","semantic_template":"qf-mod44-flow-value-cross-check/{\"capacities\":[[30,1],[30,1],[30,1],[30,1]],\"edges\":[[\"a5\",\"s5\"],[\"a5\",\"t5\"],[\"s5\",\"a5\"],[\"t5\",\"a5\"]],\"flows\":[[12,1],[18,1],[24,1],[6,1]],\"sink\":\"t5\",\"source\":\"s5\",\"vertices\":[\"a5\",\"s5\",\"t5\",\"u5\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、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-20261113-r07-finite-flow-network-0082","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-flow-network"],"title_zh":"有限流網路","stem_zh":"finite loop-free directed network、逐edge capacity與逐internal-vertex conservation。操作=flow-value-cross-check；資料={\"vertices\":[\"s6\",\"a6\",\"t6\",\"u6\"],\"edges\":[[\"s6\",\"a6\"],[\"a6\",\"s6\"],[\"a6\",\"t6\"],[\"t6\",\"a6\"]],\"source\":\"s6\",\"sink\":\"t6\",\"capacities\":[35,35,35,35],\"flows\":[28,14,21,7]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(s6, t6, [[[a6, s6], 35, 14], [[a6, t6], 35, 21], [[s6, a6], 35, 28], [[t6, a6], 35, 7]], 14, 14, 是, 是, 是, 是, 否, feasible_flow_value_cross_check_does_not_claim_maximality_or_min_cut_equality)","explanation_zh":"Wave6第七層核對=(s6, t6, [[[a6, s6], 35, 14], [[a6, t6], 35, 21], [[s6, a6], 35, 28], [[t6, a6], 35, 7]], 14, 14, 是, 是, 是, 是, 否, feasible_flow_value_cross_check_does_not_claim_maximality_or_min_cut_equality)；依完整明示資料獨立重算；antiparallel edges明示，zero capacity與disconnected vertices允許；flow value由source與sink交叉計算；feasible不等於maximum，不提前宣稱min-cut equality或augmenting-path optimality。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限流網路"]},"variant":{"family_id":"qf-mod44-flow-value-cross-check","seed":202611137082,"parameters":{"vertices":["s6","a6","t6","u6"],"edges":[["s6","a6"],["a6","s6"],["a6","t6"],["t6","a6"]],"source":"s6","sink":"t6","capacities":[35,35,35,35],"flows":[28,14,21,7],"domain":"Exact_finite_flow_network","operation":"flow-value-cross-check","semantic_template":"qf-mod44-flow-value-cross-check/{\"capacities\":[[35,1],[35,1],[35,1],[35,1]],\"edges\":[[\"a6\",\"s6\"],[\"a6\",\"t6\"],[\"s6\",\"a6\"],[\"t6\",\"a6\"]],\"flows\":[[14,1],[21,1],[28,1],[7,1]],\"sink\":\"t6\",\"source\":\"s6\",\"vertices\":[\"a6\",\"s6\",\"t6\",\"u6\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、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-20261113-r07-finite-flow-network-0083","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-flow-network"],"title_zh":"有限流網路","stem_zh":"finite loop-free directed network、逐edge capacity與逐internal-vertex conservation。操作=flow-value-cross-check；資料={\"vertices\":[\"s7\",\"a7\",\"t7\",\"u7\"],\"edges\":[[\"s7\",\"a7\"],[\"a7\",\"s7\"],[\"a7\",\"t7\"],[\"t7\",\"a7\"]],\"source\":\"s7\",\"sink\":\"t7\",\"capacities\":[40,40,40,40],\"flows\":[32,16,24,8]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(s7, t7, [[[a7, s7], 40, 16], [[a7, t7], 40, 24], [[s7, a7], 40, 32], [[t7, a7], 40, 8]], 16, 16, 是, 是, 是, 是, 否, feasible_flow_value_cross_check_does_not_claim_maximality_or_min_cut_equality)","explanation_zh":"Wave6第七層核對=(s7, t7, [[[a7, s7], 40, 16], [[a7, t7], 40, 24], [[s7, a7], 40, 32], [[t7, a7], 40, 8]], 16, 16, 是, 是, 是, 是, 否, feasible_flow_value_cross_check_does_not_claim_maximality_or_min_cut_equality)；依完整明示資料獨立重算；antiparallel edges明示，zero capacity與disconnected vertices允許；flow value由source與sink交叉計算；feasible不等於maximum，不提前宣稱min-cut equality或augmenting-path optimality。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限流網路"]},"variant":{"family_id":"qf-mod44-flow-value-cross-check","seed":202611137083,"parameters":{"vertices":["s7","a7","t7","u7"],"edges":[["s7","a7"],["a7","s7"],["a7","t7"],["t7","a7"]],"source":"s7","sink":"t7","capacities":[40,40,40,40],"flows":[32,16,24,8],"domain":"Exact_finite_flow_network","operation":"flow-value-cross-check","semantic_template":"qf-mod44-flow-value-cross-check/{\"capacities\":[[40,1],[40,1],[40,1],[40,1]],\"edges\":[[\"a7\",\"s7\"],[\"a7\",\"t7\"],[\"s7\",\"a7\"],[\"t7\",\"a7\"]],\"flows\":[[16,1],[24,1],[32,1],[8,1]],\"sink\":\"t7\",\"source\":\"s7\",\"vertices\":[\"a7\",\"s7\",\"t7\",\"u7\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、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-20261113-r07-finite-flow-network-0084","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-flow-network"],"title_zh":"有限流網路","stem_zh":"finite loop-free directed network、逐edge capacity與逐internal-vertex conservation。操作=flow-value-cross-check；資料={\"vertices\":[\"s8\",\"a8\",\"t8\",\"u8\"],\"edges\":[[\"s8\",\"a8\"],[\"a8\",\"s8\"],[\"a8\",\"t8\"],[\"t8\",\"a8\"]],\"source\":\"s8\",\"sink\":\"t8\",\"capacities\":[45,45,45,45],\"flows\":[36,18,27,9]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(s8, t8, [[[a8, s8], 45, 18], [[a8, t8], 45, 27], [[s8, a8], 45, 36], [[t8, a8], 45, 9]], 18, 18, 是, 是, 是, 是, 否, feasible_flow_value_cross_check_does_not_claim_maximality_or_min_cut_equality)","explanation_zh":"Wave6第七層核對=(s8, t8, [[[a8, s8], 45, 18], [[a8, t8], 45, 27], [[s8, a8], 45, 36], [[t8, a8], 45, 9]], 18, 18, 是, 是, 是, 是, 否, feasible_flow_value_cross_check_does_not_claim_maximality_or_min_cut_equality)；依完整明示資料獨立重算；antiparallel edges明示，zero capacity與disconnected vertices允許；flow value由source與sink交叉計算；feasible不等於maximum，不提前宣稱min-cut equality或augmenting-path optimality。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限流網路"]},"variant":{"family_id":"qf-mod44-flow-value-cross-check","seed":202611137084,"parameters":{"vertices":["s8","a8","t8","u8"],"edges":[["s8","a8"],["a8","s8"],["a8","t8"],["t8","a8"]],"source":"s8","sink":"t8","capacities":[45,45,45,45],"flows":[36,18,27,9],"domain":"Exact_finite_flow_network","operation":"flow-value-cross-check","semantic_template":"qf-mod44-flow-value-cross-check/{\"capacities\":[[45,1],[45,1],[45,1],[45,1]],\"edges\":[[\"a8\",\"s8\"],[\"a8\",\"t8\"],[\"s8\",\"a8\"],[\"t8\",\"a8\"]],\"flows\":[[18,1],[27,1],[36,1],[9,1]],\"sink\":\"t8\",\"source\":\"s8\",\"vertices\":[\"a8\",\"s8\",\"t8\",\"u8\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、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-20261113-r07-finite-flow-network-0085","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-flow-network"],"title_zh":"有限流網路","stem_zh":"finite loop-free directed network、逐edge capacity與逐internal-vertex conservation。操作=flow-value-cross-check；資料={\"vertices\":[\"s9\",\"a9\",\"t9\",\"u9\"],\"edges\":[[\"s9\",\"a9\"],[\"a9\",\"s9\"],[\"a9\",\"t9\"],[\"t9\",\"a9\"]],\"source\":\"s9\",\"sink\":\"t9\",\"capacities\":[50,50,50,50],\"flows\":[40,20,30,10]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(s9, t9, [[[a9, s9], 50, 20], [[a9, t9], 50, 30], [[s9, a9], 50, 40], [[t9, a9], 50, 10]], 20, 20, 是, 是, 是, 是, 否, feasible_flow_value_cross_check_does_not_claim_maximality_or_min_cut_equality)","explanation_zh":"Wave6第七層核對=(s9, t9, [[[a9, s9], 50, 20], [[a9, t9], 50, 30], [[s9, a9], 50, 40], [[t9, a9], 50, 10]], 20, 20, 是, 是, 是, 是, 否, feasible_flow_value_cross_check_does_not_claim_maximality_or_min_cut_equality)；依完整明示資料獨立重算；antiparallel edges明示，zero capacity與disconnected vertices允許；flow value由source與sink交叉計算；feasible不等於maximum，不提前宣稱min-cut equality或augmenting-path optimality。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限流網路"]},"variant":{"family_id":"qf-mod44-flow-value-cross-check","seed":202611137085,"parameters":{"vertices":["s9","a9","t9","u9"],"edges":[["s9","a9"],["a9","s9"],["a9","t9"],["t9","a9"]],"source":"s9","sink":"t9","capacities":[50,50,50,50],"flows":[40,20,30,10],"domain":"Exact_finite_flow_network","operation":"flow-value-cross-check","semantic_template":"qf-mod44-flow-value-cross-check/{\"capacities\":[[50,1],[50,1],[50,1],[50,1]],\"edges\":[[\"a9\",\"s9\"],[\"a9\",\"t9\"],[\"s9\",\"a9\"],[\"t9\",\"a9\"]],\"flows\":[[20,1],[30,1],[40,1],[10,1]],\"sink\":\"t9\",\"source\":\"s9\",\"vertices\":[\"a9\",\"s9\",\"t9\",\"u9\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、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-20261113-r07-finite-flow-network-0086","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-flow-network"],"title_zh":"有限流網路","stem_zh":"finite loop-free directed network、逐edge capacity與逐internal-vertex conservation。操作=flow-value-cross-check；資料={\"vertices\":[\"s10\",\"a10\",\"t10\",\"u10\"],\"edges\":[[\"s10\",\"a10\"],[\"a10\",\"s10\"],[\"a10\",\"t10\"],[\"t10\",\"a10\"]],\"source\":\"s10\",\"sink\":\"t10\",\"capacities\":[55,55,55,55],\"flows\":[44,22,33,11]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(s10, t10, [[[a10, s10], 55, 22], [[a10, t10], 55, 33], [[s10, a10], 55, 44], [[t10, a10], 55, 11]], 22, 22, 是, 是, 是, 是, 否, feasible_flow_value_cross_check_does_not_claim_maximality_or_min_cut_equality)","explanation_zh":"Wave6第七層核對=(s10, t10, [[[a10, s10], 55, 22], [[a10, t10], 55, 33], [[s10, a10], 55, 44], [[t10, a10], 55, 11]], 22, 22, 是, 是, 是, 是, 否, feasible_flow_value_cross_check_does_not_claim_maximality_or_min_cut_equality)；依完整明示資料獨立重算；antiparallel edges明示，zero capacity與disconnected vertices允許；flow value由source與sink交叉計算；feasible不等於maximum，不提前宣稱min-cut equality或augmenting-path optimality。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限流網路"]},"variant":{"family_id":"qf-mod44-flow-value-cross-check","seed":202611137086,"parameters":{"vertices":["s10","a10","t10","u10"],"edges":[["s10","a10"],["a10","s10"],["a10","t10"],["t10","a10"]],"source":"s10","sink":"t10","capacities":[55,55,55,55],"flows":[44,22,33,11],"domain":"Exact_finite_flow_network","operation":"flow-value-cross-check","semantic_template":"qf-mod44-flow-value-cross-check/{\"capacities\":[[55,1],[55,1],[55,1],[55,1]],\"edges\":[[\"a10\",\"s10\"],[\"a10\",\"t10\"],[\"s10\",\"a10\"],[\"t10\",\"a10\"]],\"flows\":[[22,1],[33,1],[44,1],[11,1]],\"sink\":\"t10\",\"source\":\"s10\",\"vertices\":[\"a10\",\"s10\",\"t10\",\"u10\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、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-20261113-r07-finite-flow-network-0087","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-flow-network"],"title_zh":"有限流網路","stem_zh":"finite loop-free directed network、逐edge capacity與逐internal-vertex conservation。操作=flow-value-cross-check；資料={\"vertices\":[\"s11\",\"a11\",\"t11\",\"u11\"],\"edges\":[[\"s11\",\"a11\"],[\"a11\",\"s11\"],[\"a11\",\"t11\"],[\"t11\",\"a11\"]],\"source\":\"s11\",\"sink\":\"t11\",\"capacities\":[60,60,60,60],\"flows\":[48,24,36,12]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(s11, t11, [[[a11, s11], 60, 24], [[a11, t11], 60, 36], [[s11, a11], 60, 48], [[t11, a11], 60, 12]], 24, 24, 是, 是, 是, 是, 否, feasible_flow_value_cross_check_does_not_claim_maximality_or_min_cut_equality)","explanation_zh":"Wave6第七層核對=(s11, t11, [[[a11, s11], 60, 24], [[a11, t11], 60, 36], [[s11, a11], 60, 48], [[t11, a11], 60, 12]], 24, 24, 是, 是, 是, 是, 否, feasible_flow_value_cross_check_does_not_claim_maximality_or_min_cut_equality)；依完整明示資料獨立重算；antiparallel edges明示，zero capacity與disconnected vertices允許；flow value由source與sink交叉計算；feasible不等於maximum，不提前宣稱min-cut equality或augmenting-path optimality。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限流網路"]},"variant":{"family_id":"qf-mod44-flow-value-cross-check","seed":202611137087,"parameters":{"vertices":["s11","a11","t11","u11"],"edges":[["s11","a11"],["a11","s11"],["a11","t11"],["t11","a11"]],"source":"s11","sink":"t11","capacities":[60,60,60,60],"flows":[48,24,36,12],"domain":"Exact_finite_flow_network","operation":"flow-value-cross-check","semantic_template":"qf-mod44-flow-value-cross-check/{\"capacities\":[[60,1],[60,1],[60,1],[60,1]],\"edges\":[[\"a11\",\"s11\"],[\"a11\",\"t11\"],[\"s11\",\"a11\"],[\"t11\",\"a11\"]],\"flows\":[[24,1],[36,1],[48,1],[12,1]],\"sink\":\"t11\",\"source\":\"s11\",\"vertices\":[\"a11\",\"s11\",\"t11\",\"u11\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、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-20261113-r07-finite-flow-network-0088","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-flow-network"],"title_zh":"有限流網路","stem_zh":"finite loop-free directed network、逐edge capacity與逐internal-vertex conservation。操作=flow-value-cross-check；資料={\"vertices\":[\"s12\",\"a12\",\"t12\",\"u12\"],\"edges\":[[\"s12\",\"a12\"],[\"a12\",\"s12\"],[\"a12\",\"t12\"],[\"t12\",\"a12\"]],\"source\":\"s12\",\"sink\":\"t12\",\"capacities\":[65,65,65,65],\"flows\":[52,26,39,13]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(s12, t12, [[[a12, s12], 65, 26], [[a12, t12], 65, 39], [[s12, a12], 65, 52], [[t12, a12], 65, 13]], 26, 26, 是, 是, 是, 是, 否, feasible_flow_value_cross_check_does_not_claim_maximality_or_min_cut_equality)","explanation_zh":"Wave6第七層核對=(s12, t12, [[[a12, s12], 65, 26], [[a12, t12], 65, 39], [[s12, a12], 65, 52], [[t12, a12], 65, 13]], 26, 26, 是, 是, 是, 是, 否, feasible_flow_value_cross_check_does_not_claim_maximality_or_min_cut_equality)；依完整明示資料獨立重算；antiparallel edges明示，zero capacity與disconnected vertices允許；flow value由source與sink交叉計算；feasible不等於maximum，不提前宣稱min-cut equality或augmenting-path optimality。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限流網路"]},"variant":{"family_id":"qf-mod44-flow-value-cross-check","seed":202611137088,"parameters":{"vertices":["s12","a12","t12","u12"],"edges":[["s12","a12"],["a12","s12"],["a12","t12"],["t12","a12"]],"source":"s12","sink":"t12","capacities":[65,65,65,65],"flows":[52,26,39,13],"domain":"Exact_finite_flow_network","operation":"flow-value-cross-check","semantic_template":"qf-mod44-flow-value-cross-check/{\"capacities\":[[65,1],[65,1],[65,1],[65,1]],\"edges\":[[\"a12\",\"s12\"],[\"a12\",\"t12\"],[\"s12\",\"a12\"],[\"t12\",\"a12\"]],\"flows\":[[26,1],[39,1],[52,1],[13,1]],\"sink\":\"t12\",\"source\":\"s12\",\"vertices\":[\"a12\",\"s12\",\"t12\",\"u12\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、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-20261113-r07-finite-flow-network-0089","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-flow-network"],"title_zh":"有限流網路","stem_zh":"finite loop-free directed network、逐edge capacity與逐internal-vertex conservation。操作=flow-value-cross-check；資料={\"vertices\":[\"s13\",\"a13\",\"t13\",\"u13\"],\"edges\":[[\"s13\",\"a13\"],[\"a13\",\"s13\"],[\"a13\",\"t13\"],[\"t13\",\"a13\"]],\"source\":\"s13\",\"sink\":\"t13\",\"capacities\":[70,70,70,70],\"flows\":[56,28,42,14]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(s13, t13, [[[a13, s13], 70, 28], [[a13, t13], 70, 42], [[s13, a13], 70, 56], [[t13, a13], 70, 14]], 28, 28, 是, 是, 是, 是, 否, feasible_flow_value_cross_check_does_not_claim_maximality_or_min_cut_equality)","explanation_zh":"Wave6第七層核對=(s13, t13, [[[a13, s13], 70, 28], [[a13, t13], 70, 42], [[s13, a13], 70, 56], [[t13, a13], 70, 14]], 28, 28, 是, 是, 是, 是, 否, feasible_flow_value_cross_check_does_not_claim_maximality_or_min_cut_equality)；依完整明示資料獨立重算；antiparallel edges明示，zero capacity與disconnected vertices允許；flow value由source與sink交叉計算；feasible不等於maximum，不提前宣稱min-cut equality或augmenting-path optimality。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限流網路"]},"variant":{"family_id":"qf-mod44-flow-value-cross-check","seed":202611137089,"parameters":{"vertices":["s13","a13","t13","u13"],"edges":[["s13","a13"],["a13","s13"],["a13","t13"],["t13","a13"]],"source":"s13","sink":"t13","capacities":[70,70,70,70],"flows":[56,28,42,14],"domain":"Exact_finite_flow_network","operation":"flow-value-cross-check","semantic_template":"qf-mod44-flow-value-cross-check/{\"capacities\":[[70,1],[70,1],[70,1],[70,1]],\"edges\":[[\"a13\",\"s13\"],[\"a13\",\"t13\"],[\"s13\",\"a13\"],[\"t13\",\"a13\"]],\"flows\":[[28,1],[42,1],[56,1],[14,1]],\"sink\":\"t13\",\"source\":\"s13\",\"vertices\":[\"a13\",\"s13\",\"t13\",\"u13\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、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-20261113-r07-finite-flow-network-0090","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-flow-network"],"title_zh":"有限流網路","stem_zh":"finite loop-free directed network、逐edge capacity與逐internal-vertex conservation。操作=flow-value-cross-check；資料={\"vertices\":[\"s14\",\"a14\",\"t14\",\"u14\"],\"edges\":[[\"s14\",\"a14\"],[\"a14\",\"s14\"],[\"a14\",\"t14\"],[\"t14\",\"a14\"]],\"source\":\"s14\",\"sink\":\"t14\",\"capacities\":[75,75,75,75],\"flows\":[60,30,45,15]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(s14, t14, [[[a14, s14], 75, 30], [[a14, t14], 75, 45], [[s14, a14], 75, 60], [[t14, a14], 75, 15]], 30, 30, 是, 是, 是, 是, 否, feasible_flow_value_cross_check_does_not_claim_maximality_or_min_cut_equality)","explanation_zh":"Wave6第七層核對=(s14, t14, [[[a14, s14], 75, 30], [[a14, t14], 75, 45], [[s14, a14], 75, 60], [[t14, a14], 75, 15]], 30, 30, 是, 是, 是, 是, 否, feasible_flow_value_cross_check_does_not_claim_maximality_or_min_cut_equality)；依完整明示資料獨立重算；antiparallel edges明示，zero capacity與disconnected vertices允許；flow value由source與sink交叉計算；feasible不等於maximum，不提前宣稱min-cut equality或augmenting-path optimality。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限流網路"]},"variant":{"family_id":"qf-mod44-flow-value-cross-check","seed":202611137090,"parameters":{"vertices":["s14","a14","t14","u14"],"edges":[["s14","a14"],["a14","s14"],["a14","t14"],["t14","a14"]],"source":"s14","sink":"t14","capacities":[75,75,75,75],"flows":[60,30,45,15],"domain":"Exact_finite_flow_network","operation":"flow-value-cross-check","semantic_template":"qf-mod44-flow-value-cross-check/{\"capacities\":[[75,1],[75,1],[75,1],[75,1]],\"edges\":[[\"a14\",\"s14\"],[\"a14\",\"t14\"],[\"s14\",\"a14\"],[\"t14\",\"a14\"]],\"flows\":[[30,1],[45,1],[60,1],[15,1]],\"sink\":\"t14\",\"source\":\"s14\",\"vertices\":[\"a14\",\"s14\",\"t14\",\"u14\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、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-20261113-r07-finite-flow-network-0091","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-flow-network"],"title_zh":"有限流網路","stem_zh":"finite loop-free directed network、逐edge capacity與逐internal-vertex conservation。操作=flow-value-cross-check；資料={\"vertices\":[\"s15\",\"a15\",\"t15\",\"u15\"],\"edges\":[[\"s15\",\"a15\"],[\"a15\",\"s15\"],[\"a15\",\"t15\"],[\"t15\",\"a15\"]],\"source\":\"s15\",\"sink\":\"t15\",\"capacities\":[80,80,80,80],\"flows\":[64,32,48,16]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(s15, t15, [[[a15, s15], 80, 32], [[a15, t15], 80, 48], [[s15, a15], 80, 64], [[t15, a15], 80, 16]], 32, 32, 是, 是, 是, 是, 否, feasible_flow_value_cross_check_does_not_claim_maximality_or_min_cut_equality)","explanation_zh":"Wave6第七層核對=(s15, t15, [[[a15, s15], 80, 32], [[a15, t15], 80, 48], [[s15, a15], 80, 64], [[t15, a15], 80, 16]], 32, 32, 是, 是, 是, 是, 否, feasible_flow_value_cross_check_does_not_claim_maximality_or_min_cut_equality)；依完整明示資料獨立重算；antiparallel edges明示，zero capacity與disconnected vertices允許；flow value由source與sink交叉計算；feasible不等於maximum，不提前宣稱min-cut equality或augmenting-path optimality。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限流網路"]},"variant":{"family_id":"qf-mod44-flow-value-cross-check","seed":202611137091,"parameters":{"vertices":["s15","a15","t15","u15"],"edges":[["s15","a15"],["a15","s15"],["a15","t15"],["t15","a15"]],"source":"s15","sink":"t15","capacities":[80,80,80,80],"flows":[64,32,48,16],"domain":"Exact_finite_flow_network","operation":"flow-value-cross-check","semantic_template":"qf-mod44-flow-value-cross-check/{\"capacities\":[[80,1],[80,1],[80,1],[80,1]],\"edges\":[[\"a15\",\"s15\"],[\"a15\",\"t15\"],[\"s15\",\"a15\"],[\"t15\",\"a15\"]],\"flows\":[[32,1],[48,1],[64,1],[16,1]],\"sink\":\"t15\",\"source\":\"s15\",\"vertices\":[\"a15\",\"s15\",\"t15\",\"u15\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、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-20261113-r07-finite-flow-network-0092","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-flow-network"],"title_zh":"有限流網路","stem_zh":"finite loop-free directed network、逐edge capacity與逐internal-vertex conservation。操作=flow-value-cross-check；資料={\"vertices\":[\"s16\",\"a16\",\"t16\",\"u16\"],\"edges\":[[\"s16\",\"a16\"],[\"a16\",\"s16\"],[\"a16\",\"t16\"],[\"t16\",\"a16\"]],\"source\":\"s16\",\"sink\":\"t16\",\"capacities\":[85,85,85,85],\"flows\":[68,34,51,17]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(s16, t16, [[[a16, s16], 85, 34], [[a16, t16], 85, 51], [[s16, a16], 85, 68], [[t16, a16], 85, 17]], 34, 34, 是, 是, 是, 是, 否, feasible_flow_value_cross_check_does_not_claim_maximality_or_min_cut_equality)","explanation_zh":"Wave6第七層核對=(s16, t16, [[[a16, s16], 85, 34], [[a16, t16], 85, 51], [[s16, a16], 85, 68], [[t16, a16], 85, 17]], 34, 34, 是, 是, 是, 是, 否, feasible_flow_value_cross_check_does_not_claim_maximality_or_min_cut_equality)；依完整明示資料獨立重算；antiparallel edges明示，zero capacity與disconnected vertices允許；flow value由source與sink交叉計算；feasible不等於maximum，不提前宣稱min-cut equality或augmenting-path optimality。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限流網路"]},"variant":{"family_id":"qf-mod44-flow-value-cross-check","seed":202611137092,"parameters":{"vertices":["s16","a16","t16","u16"],"edges":[["s16","a16"],["a16","s16"],["a16","t16"],["t16","a16"]],"source":"s16","sink":"t16","capacities":[85,85,85,85],"flows":[68,34,51,17],"domain":"Exact_finite_flow_network","operation":"flow-value-cross-check","semantic_template":"qf-mod44-flow-value-cross-check/{\"capacities\":[[85,1],[85,1],[85,1],[85,1]],\"edges\":[[\"a16\",\"s16\"],[\"a16\",\"t16\"],[\"s16\",\"a16\"],[\"t16\",\"a16\"]],\"flows\":[[34,1],[51,1],[68,1],[17,1]],\"sink\":\"t16\",\"source\":\"s16\",\"vertices\":[\"a16\",\"s16\",\"t16\",\"u16\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、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-20261113-r07-finite-flow-network-0093","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-flow-network"],"title_zh":"有限流網路","stem_zh":"finite loop-free directed network、逐edge capacity與逐internal-vertex conservation。操作=flow-value-cross-check；資料={\"vertices\":[\"s17\",\"a17\",\"t17\",\"u17\"],\"edges\":[[\"s17\",\"a17\"],[\"a17\",\"s17\"],[\"a17\",\"t17\"],[\"t17\",\"a17\"]],\"source\":\"s17\",\"sink\":\"t17\",\"capacities\":[90,90,90,90],\"flows\":[72,36,54,18]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(s17, t17, [[[a17, s17], 90, 36], [[a17, t17], 90, 54], [[s17, a17], 90, 72], [[t17, a17], 90, 18]], 36, 36, 是, 是, 是, 是, 否, feasible_flow_value_cross_check_does_not_claim_maximality_or_min_cut_equality)","explanation_zh":"Wave6第七層核對=(s17, t17, [[[a17, s17], 90, 36], [[a17, t17], 90, 54], [[s17, a17], 90, 72], [[t17, a17], 90, 18]], 36, 36, 是, 是, 是, 是, 否, feasible_flow_value_cross_check_does_not_claim_maximality_or_min_cut_equality)；依完整明示資料獨立重算；antiparallel edges明示，zero capacity與disconnected vertices允許；flow value由source與sink交叉計算；feasible不等於maximum，不提前宣稱min-cut equality或augmenting-path optimality。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限流網路"]},"variant":{"family_id":"qf-mod44-flow-value-cross-check","seed":202611137093,"parameters":{"vertices":["s17","a17","t17","u17"],"edges":[["s17","a17"],["a17","s17"],["a17","t17"],["t17","a17"]],"source":"s17","sink":"t17","capacities":[90,90,90,90],"flows":[72,36,54,18],"domain":"Exact_finite_flow_network","operation":"flow-value-cross-check","semantic_template":"qf-mod44-flow-value-cross-check/{\"capacities\":[[90,1],[90,1],[90,1],[90,1]],\"edges\":[[\"a17\",\"s17\"],[\"a17\",\"t17\"],[\"s17\",\"a17\"],[\"t17\",\"a17\"]],\"flows\":[[36,1],[54,1],[72,1],[18,1]],\"sink\":\"t17\",\"source\":\"s17\",\"vertices\":[\"a17\",\"s17\",\"t17\",\"u17\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、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-20261113-r07-finite-flow-network-0094","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-flow-network"],"title_zh":"有限流網路","stem_zh":"finite loop-free directed network、逐edge capacity與逐internal-vertex conservation。操作=flow-value-cross-check；資料={\"vertices\":[\"s18\",\"a18\",\"t18\",\"u18\"],\"edges\":[[\"s18\",\"a18\"],[\"a18\",\"s18\"],[\"a18\",\"t18\"],[\"t18\",\"a18\"]],\"source\":\"s18\",\"sink\":\"t18\",\"capacities\":[95,95,95,95],\"flows\":[76,38,57,19]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(s18, t18, [[[a18, s18], 95, 38], [[a18, t18], 95, 57], [[s18, a18], 95, 76], [[t18, a18], 95, 19]], 38, 38, 是, 是, 是, 是, 否, feasible_flow_value_cross_check_does_not_claim_maximality_or_min_cut_equality)","explanation_zh":"Wave6第七層核對=(s18, t18, [[[a18, s18], 95, 38], [[a18, t18], 95, 57], [[s18, a18], 95, 76], [[t18, a18], 95, 19]], 38, 38, 是, 是, 是, 是, 否, feasible_flow_value_cross_check_does_not_claim_maximality_or_min_cut_equality)；依完整明示資料獨立重算；antiparallel edges明示，zero capacity與disconnected vertices允許；flow value由source與sink交叉計算；feasible不等於maximum，不提前宣稱min-cut equality或augmenting-path optimality。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限流網路"]},"variant":{"family_id":"qf-mod44-flow-value-cross-check","seed":202611137094,"parameters":{"vertices":["s18","a18","t18","u18"],"edges":[["s18","a18"],["a18","s18"],["a18","t18"],["t18","a18"]],"source":"s18","sink":"t18","capacities":[95,95,95,95],"flows":[76,38,57,19],"domain":"Exact_finite_flow_network","operation":"flow-value-cross-check","semantic_template":"qf-mod44-flow-value-cross-check/{\"capacities\":[[95,1],[95,1],[95,1],[95,1]],\"edges\":[[\"a18\",\"s18\"],[\"a18\",\"t18\"],[\"s18\",\"a18\"],[\"t18\",\"a18\"]],\"flows\":[[38,1],[57,1],[76,1],[19,1]],\"sink\":\"t18\",\"source\":\"s18\",\"vertices\":[\"a18\",\"s18\",\"t18\",\"u18\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、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-20261113-r07-finite-flow-network-0095","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-flow-network"],"title_zh":"有限流網路","stem_zh":"finite loop-free directed network、逐edge capacity與逐internal-vertex conservation。操作=flow-value-cross-check；資料={\"vertices\":[\"s19\",\"a19\",\"t19\",\"u19\"],\"edges\":[[\"s19\",\"a19\"],[\"a19\",\"s19\"],[\"a19\",\"t19\"],[\"t19\",\"a19\"]],\"source\":\"s19\",\"sink\":\"t19\",\"capacities\":[100,100,100,100],\"flows\":[80,40,60,20]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(s19, t19, [[[a19, s19], 100, 40], [[a19, t19], 100, 60], [[s19, a19], 100, 80], [[t19, a19], 100, 20]], 40, 40, 是, 是, 是, 是, 否, feasible_flow_value_cross_check_does_not_claim_maximality_or_min_cut_equality)","explanation_zh":"Wave6第七層核對=(s19, t19, [[[a19, s19], 100, 40], [[a19, t19], 100, 60], [[s19, a19], 100, 80], [[t19, a19], 100, 20]], 40, 40, 是, 是, 是, 是, 否, feasible_flow_value_cross_check_does_not_claim_maximality_or_min_cut_equality)；依完整明示資料獨立重算；antiparallel edges明示，zero capacity與disconnected vertices允許；flow value由source與sink交叉計算；feasible不等於maximum，不提前宣稱min-cut equality或augmenting-path optimality。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限流網路"]},"variant":{"family_id":"qf-mod44-flow-value-cross-check","seed":202611137095,"parameters":{"vertices":["s19","a19","t19","u19"],"edges":[["s19","a19"],["a19","s19"],["a19","t19"],["t19","a19"]],"source":"s19","sink":"t19","capacities":[100,100,100,100],"flows":[80,40,60,20],"domain":"Exact_finite_flow_network","operation":"flow-value-cross-check","semantic_template":"qf-mod44-flow-value-cross-check/{\"capacities\":[[100,1],[100,1],[100,1],[100,1]],\"edges\":[[\"a19\",\"s19\"],[\"a19\",\"t19\"],[\"s19\",\"a19\"],[\"t19\",\"a19\"]],\"flows\":[[40,1],[60,1],[80,1],[20,1]],\"sink\":\"t19\",\"source\":\"s19\",\"vertices\":[\"a19\",\"s19\",\"t19\",\"u19\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、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-20261113-r07-finite-flow-network-0096","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-flow-network"],"title_zh":"有限流網路","stem_zh":"finite loop-free directed network、逐edge capacity與逐internal-vertex conservation。操作=flow-value-cross-check；資料={\"vertices\":[\"s20\",\"a20\",\"t20\",\"u20\"],\"edges\":[[\"s20\",\"a20\"],[\"a20\",\"s20\"],[\"a20\",\"t20\"],[\"t20\",\"a20\"]],\"source\":\"s20\",\"sink\":\"t20\",\"capacities\":[105,105,105,105],\"flows\":[84,42,63,21]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(s20, t20, [[[a20, s20], 105, 42], [[a20, t20], 105, 63], [[s20, a20], 105, 84], [[t20, a20], 105, 21]], 42, 42, 是, 是, 是, 是, 否, feasible_flow_value_cross_check_does_not_claim_maximality_or_min_cut_equality)","explanation_zh":"Wave6第七層核對=(s20, t20, [[[a20, s20], 105, 42], [[a20, t20], 105, 63], [[s20, a20], 105, 84], [[t20, a20], 105, 21]], 42, 42, 是, 是, 是, 是, 否, feasible_flow_value_cross_check_does_not_claim_maximality_or_min_cut_equality)；依完整明示資料獨立重算；antiparallel edges明示，zero capacity與disconnected vertices允許；flow value由source與sink交叉計算；feasible不等於maximum，不提前宣稱min-cut equality或augmenting-path optimality。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限流網路"]},"variant":{"family_id":"qf-mod44-flow-value-cross-check","seed":202611137096,"parameters":{"vertices":["s20","a20","t20","u20"],"edges":[["s20","a20"],["a20","s20"],["a20","t20"],["t20","a20"]],"source":"s20","sink":"t20","capacities":[105,105,105,105],"flows":[84,42,63,21],"domain":"Exact_finite_flow_network","operation":"flow-value-cross-check","semantic_template":"qf-mod44-flow-value-cross-check/{\"capacities\":[[105,1],[105,1],[105,1],[105,1]],\"edges\":[[\"a20\",\"s20\"],[\"a20\",\"t20\"],[\"s20\",\"a20\"],[\"t20\",\"a20\"]],\"flows\":[[42,1],[63,1],[84,1],[21,1]],\"sink\":\"t20\",\"source\":\"s20\",\"vertices\":[\"a20\",\"s20\",\"t20\",\"u20\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、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-20261113-r07-finite-flow-network-0097","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-flow-network"],"title_zh":"有限流網路","stem_zh":"finite loop-free directed network、逐edge capacity與逐internal-vertex conservation。操作=flow-value-cross-check；資料={\"vertices\":[\"s21\",\"a21\",\"t21\",\"u21\"],\"edges\":[[\"s21\",\"a21\"],[\"a21\",\"s21\"],[\"a21\",\"t21\"],[\"t21\",\"a21\"]],\"source\":\"s21\",\"sink\":\"t21\",\"capacities\":[110,110,110,110],\"flows\":[88,44,66,22]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(s21, t21, [[[a21, s21], 110, 44], [[a21, t21], 110, 66], [[s21, a21], 110, 88], [[t21, a21], 110, 22]], 44, 44, 是, 是, 是, 是, 否, feasible_flow_value_cross_check_does_not_claim_maximality_or_min_cut_equality)","explanation_zh":"Wave6第七層核對=(s21, t21, [[[a21, s21], 110, 44], [[a21, t21], 110, 66], [[s21, a21], 110, 88], [[t21, a21], 110, 22]], 44, 44, 是, 是, 是, 是, 否, feasible_flow_value_cross_check_does_not_claim_maximality_or_min_cut_equality)；依完整明示資料獨立重算；antiparallel edges明示，zero capacity與disconnected vertices允許；flow value由source與sink交叉計算；feasible不等於maximum，不提前宣稱min-cut equality或augmenting-path optimality。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限流網路"]},"variant":{"family_id":"qf-mod44-flow-value-cross-check","seed":202611137097,"parameters":{"vertices":["s21","a21","t21","u21"],"edges":[["s21","a21"],["a21","s21"],["a21","t21"],["t21","a21"]],"source":"s21","sink":"t21","capacities":[110,110,110,110],"flows":[88,44,66,22],"domain":"Exact_finite_flow_network","operation":"flow-value-cross-check","semantic_template":"qf-mod44-flow-value-cross-check/{\"capacities\":[[110,1],[110,1],[110,1],[110,1]],\"edges\":[[\"a21\",\"s21\"],[\"a21\",\"t21\"],[\"s21\",\"a21\"],[\"t21\",\"a21\"]],\"flows\":[[44,1],[66,1],[88,1],[22,1]],\"sink\":\"t21\",\"source\":\"s21\",\"vertices\":[\"a21\",\"s21\",\"t21\",\"u21\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、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-20261113-r07-finite-flow-network-0098","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-flow-network"],"title_zh":"有限流網路","stem_zh":"finite loop-free directed network、逐edge capacity與逐internal-vertex conservation。操作=flow-value-cross-check；資料={\"vertices\":[\"s22\",\"a22\",\"t22\",\"u22\"],\"edges\":[[\"s22\",\"a22\"],[\"a22\",\"s22\"],[\"a22\",\"t22\"],[\"t22\",\"a22\"]],\"source\":\"s22\",\"sink\":\"t22\",\"capacities\":[115,115,115,115],\"flows\":[92,46,69,23]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(s22, t22, [[[a22, s22], 115, 46], [[a22, t22], 115, 69], [[s22, a22], 115, 92], [[t22, a22], 115, 23]], 46, 46, 是, 是, 是, 是, 否, feasible_flow_value_cross_check_does_not_claim_maximality_or_min_cut_equality)","explanation_zh":"Wave6第七層核對=(s22, t22, [[[a22, s22], 115, 46], [[a22, t22], 115, 69], [[s22, a22], 115, 92], [[t22, a22], 115, 23]], 46, 46, 是, 是, 是, 是, 否, feasible_flow_value_cross_check_does_not_claim_maximality_or_min_cut_equality)；依完整明示資料獨立重算；antiparallel edges明示，zero capacity與disconnected vertices允許；flow value由source與sink交叉計算；feasible不等於maximum，不提前宣稱min-cut equality或augmenting-path optimality。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限流網路"]},"variant":{"family_id":"qf-mod44-flow-value-cross-check","seed":202611137098,"parameters":{"vertices":["s22","a22","t22","u22"],"edges":[["s22","a22"],["a22","s22"],["a22","t22"],["t22","a22"]],"source":"s22","sink":"t22","capacities":[115,115,115,115],"flows":[92,46,69,23],"domain":"Exact_finite_flow_network","operation":"flow-value-cross-check","semantic_template":"qf-mod44-flow-value-cross-check/{\"capacities\":[[115,1],[115,1],[115,1],[115,1]],\"edges\":[[\"a22\",\"s22\"],[\"a22\",\"t22\"],[\"s22\",\"a22\"],[\"t22\",\"a22\"]],\"flows\":[[46,1],[69,1],[92,1],[23,1]],\"sink\":\"t22\",\"source\":\"s22\",\"vertices\":[\"a22\",\"s22\",\"t22\",\"u22\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、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-20261113-r07-finite-flow-network-0099","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-flow-network"],"title_zh":"有限流網路","stem_zh":"finite loop-free directed network、逐edge capacity與逐internal-vertex conservation。操作=flow-value-cross-check；資料={\"vertices\":[\"s23\",\"a23\",\"t23\",\"u23\"],\"edges\":[[\"s23\",\"a23\"],[\"a23\",\"s23\"],[\"a23\",\"t23\"],[\"t23\",\"a23\"]],\"source\":\"s23\",\"sink\":\"t23\",\"capacities\":[120,120,120,120],\"flows\":[96,48,72,24]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(s23, t23, [[[a23, s23], 120, 48], [[a23, t23], 120, 72], [[s23, a23], 120, 96], [[t23, a23], 120, 24]], 48, 48, 是, 是, 是, 是, 否, feasible_flow_value_cross_check_does_not_claim_maximality_or_min_cut_equality)","explanation_zh":"Wave6第七層核對=(s23, t23, [[[a23, s23], 120, 48], [[a23, t23], 120, 72], [[s23, a23], 120, 96], [[t23, a23], 120, 24]], 48, 48, 是, 是, 是, 是, 否, feasible_flow_value_cross_check_does_not_claim_maximality_or_min_cut_equality)；依完整明示資料獨立重算；antiparallel edges明示，zero capacity與disconnected vertices允許；flow value由source與sink交叉計算；feasible不等於maximum，不提前宣稱min-cut equality或augmenting-path optimality。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限流網路"]},"variant":{"family_id":"qf-mod44-flow-value-cross-check","seed":202611137099,"parameters":{"vertices":["s23","a23","t23","u23"],"edges":[["s23","a23"],["a23","s23"],["a23","t23"],["t23","a23"]],"source":"s23","sink":"t23","capacities":[120,120,120,120],"flows":[96,48,72,24],"domain":"Exact_finite_flow_network","operation":"flow-value-cross-check","semantic_template":"qf-mod44-flow-value-cross-check/{\"capacities\":[[120,1],[120,1],[120,1],[120,1]],\"edges\":[[\"a23\",\"s23\"],[\"a23\",\"t23\"],[\"s23\",\"a23\"],[\"t23\",\"a23\"]],\"flows\":[[48,1],[72,1],[96,1],[24,1]],\"sink\":\"t23\",\"source\":\"s23\",\"vertices\":[\"a23\",\"s23\",\"t23\",\"u23\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、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-20261113-r07-finite-flow-network-0100","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-flow-network"],"title_zh":"有限流網路","stem_zh":"finite loop-free directed network、逐edge capacity與逐internal-vertex conservation。操作=flow-value-cross-check；資料={\"vertices\":[\"s24\",\"a24\",\"t24\",\"u24\"],\"edges\":[[\"s24\",\"a24\"],[\"a24\",\"s24\"],[\"a24\",\"t24\"],[\"t24\",\"a24\"]],\"source\":\"s24\",\"sink\":\"t24\",\"capacities\":[125,125,125,125],\"flows\":[100,50,75,25]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(s24, t24, [[[a24, s24], 125, 50], [[a24, t24], 125, 75], [[s24, a24], 125, 100], [[t24, a24], 125, 25]], 50, 50, 是, 是, 是, 是, 否, feasible_flow_value_cross_check_does_not_claim_maximality_or_min_cut_equality)","explanation_zh":"Wave6第七層核對=(s24, t24, [[[a24, s24], 125, 50], [[a24, t24], 125, 75], [[s24, a24], 125, 100], [[t24, a24], 125, 25]], 50, 50, 是, 是, 是, 是, 否, feasible_flow_value_cross_check_does_not_claim_maximality_or_min_cut_equality)；依完整明示資料獨立重算；antiparallel edges明示，zero capacity與disconnected vertices允許；flow value由source與sink交叉計算；feasible不等於maximum，不提前宣稱min-cut equality或augmenting-path optimality。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限流網路"]},"variant":{"family_id":"qf-mod44-flow-value-cross-check","seed":202611137100,"parameters":{"vertices":["s24","a24","t24","u24"],"edges":[["s24","a24"],["a24","s24"],["a24","t24"],["t24","a24"]],"source":"s24","sink":"t24","capacities":[125,125,125,125],"flows":[100,50,75,25],"domain":"Exact_finite_flow_network","operation":"flow-value-cross-check","semantic_template":"qf-mod44-flow-value-cross-check/{\"capacities\":[[125,1],[125,1],[125,1],[125,1]],\"edges\":[[\"a24\",\"s24\"],[\"a24\",\"t24\"],[\"s24\",\"a24\"],[\"t24\",\"a24\"]],\"flows\":[[50,1],[75,1],[100,1],[25,1]],\"sink\":\"t24\",\"source\":\"s24\",\"vertices\":[\"a24\",\"s24\",\"t24\",\"u24\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}