{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r07-finite-directed-graph-0001","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-directed-graph"],"title_zh":"有限有向圖","stem_zh":"完整有限ordered-pair edge set與loop-free simple digraph convention。操作=digraph-qualification；資料={\"n\":5,\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,0]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, [[0, 1], [1, 2], [2, 3], [3, 4], [4, 0]], 是, 是, ordered_pairs_and_no_parallel_duplicate_edges)","explanation_zh":"Wave6第二層核對=(5, [[0, 1], [1, 2], [2, 3], [3, 4], [4, 0]], 是, 是, ordered_pairs_and_no_parallel_duplicate_edges)；依完整明示資料獨立重算；(u,v)與(v,u)不同，平行同向edge不重複且loop在主模型拒絕；out-degree與in-degree各自總和等於edge count；不沿用undirected degree或connectivity。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限有向圖"]},"variant":{"family_id":"qf-mod39-digraph-qualification","seed":202611087001,"parameters":{"n":5,"edges":[[0,1],[1,2],[2,3],[3,4],[4,0]],"domain":"Complete_finite_loopless_simple_digraph","operation":"digraph-qualification","semantic_template":"qf-mod39-digraph-qualification/{\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,0]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r07-finite-directed-graph-0002","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-directed-graph"],"title_zh":"有限有向圖","stem_zh":"完整有限ordered-pair edge set與loop-free simple digraph convention。操作=digraph-qualification；資料={\"n\":5,\"edges\":[[0,1],[0,2],[1,2],[2,3],[3,4],[4,0]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, [[0, 1], [0, 2], [1, 2], [2, 3], [3, 4], [4, 0]], 是, 是, ordered_pairs_and_no_parallel_duplicate_edges)","explanation_zh":"Wave6第二層核對=(5, [[0, 1], [0, 2], [1, 2], [2, 3], [3, 4], [4, 0]], 是, 是, ordered_pairs_and_no_parallel_duplicate_edges)；依完整明示資料獨立重算；(u,v)與(v,u)不同，平行同向edge不重複且loop在主模型拒絕；out-degree與in-degree各自總和等於edge count；不沿用undirected degree或connectivity。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限有向圖"]},"variant":{"family_id":"qf-mod39-digraph-qualification","seed":202611087002,"parameters":{"n":5,"edges":[[0,1],[0,2],[1,2],[2,3],[3,4],[4,0]],"domain":"Complete_finite_loopless_simple_digraph","operation":"digraph-qualification","semantic_template":"qf-mod39-digraph-qualification/{\"edges\":[[0,1],[0,2],[1,2],[2,3],[3,4],[4,0]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r07-finite-directed-graph-0003","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-directed-graph"],"title_zh":"有限有向圖","stem_zh":"完整有限ordered-pair edge set與loop-free simple digraph convention。操作=digraph-qualification；資料={\"n\":5,\"edges\":[[0,1],[1,2],[1,3],[2,3],[3,4],[4,0]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, [[0, 1], [1, 2], [1, 3], [2, 3], [3, 4], [4, 0]], 是, 是, ordered_pairs_and_no_parallel_duplicate_edges)","explanation_zh":"Wave6第二層核對=(5, [[0, 1], [1, 2], [1, 3], [2, 3], [3, 4], [4, 0]], 是, 是, ordered_pairs_and_no_parallel_duplicate_edges)；依完整明示資料獨立重算；(u,v)與(v,u)不同，平行同向edge不重複且loop在主模型拒絕；out-degree與in-degree各自總和等於edge count；不沿用undirected degree或connectivity。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限有向圖"]},"variant":{"family_id":"qf-mod39-digraph-qualification","seed":202611087003,"parameters":{"n":5,"edges":[[0,1],[1,2],[1,3],[2,3],[3,4],[4,0]],"domain":"Complete_finite_loopless_simple_digraph","operation":"digraph-qualification","semantic_template":"qf-mod39-digraph-qualification/{\"edges\":[[0,1],[1,2],[1,3],[2,3],[3,4],[4,0]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r07-finite-directed-graph-0004","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-directed-graph"],"title_zh":"有限有向圖","stem_zh":"完整有限ordered-pair edge set與loop-free simple digraph convention。操作=digraph-qualification；資料={\"n\":5,\"edges\":[[0,1],[0,2],[1,2],[1,3],[2,3],[3,4],[4,0]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, [[0, 1], [0, 2], [1, 2], [1, 3], [2, 3], [3, 4], [4, 0]], 是, 是, ordered_pairs_and_no_parallel_duplicate_edges)","explanation_zh":"Wave6第二層核對=(5, [[0, 1], [0, 2], [1, 2], [1, 3], [2, 3], [3, 4], [4, 0]], 是, 是, ordered_pairs_and_no_parallel_duplicate_edges)；依完整明示資料獨立重算；(u,v)與(v,u)不同，平行同向edge不重複且loop在主模型拒絕；out-degree與in-degree各自總和等於edge count；不沿用undirected degree或connectivity。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限有向圖"]},"variant":{"family_id":"qf-mod39-digraph-qualification","seed":202611087004,"parameters":{"n":5,"edges":[[0,1],[0,2],[1,2],[1,3],[2,3],[3,4],[4,0]],"domain":"Complete_finite_loopless_simple_digraph","operation":"digraph-qualification","semantic_template":"qf-mod39-digraph-qualification/{\"edges\":[[0,1],[0,2],[1,2],[1,3],[2,3],[3,4],[4,0]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r07-finite-directed-graph-0005","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-directed-graph"],"title_zh":"有限有向圖","stem_zh":"完整有限ordered-pair edge set與loop-free simple digraph convention。操作=digraph-qualification；資料={\"n\":5,\"edges\":[[0,1],[1,2],[2,3],[2,4],[3,4],[4,0]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, [[0, 1], [1, 2], [2, 3], [2, 4], [3, 4], [4, 0]], 是, 是, ordered_pairs_and_no_parallel_duplicate_edges)","explanation_zh":"Wave6第二層核對=(5, [[0, 1], [1, 2], [2, 3], [2, 4], [3, 4], [4, 0]], 是, 是, ordered_pairs_and_no_parallel_duplicate_edges)；依完整明示資料獨立重算；(u,v)與(v,u)不同，平行同向edge不重複且loop在主模型拒絕；out-degree與in-degree各自總和等於edge count；不沿用undirected degree或connectivity。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限有向圖"]},"variant":{"family_id":"qf-mod39-digraph-qualification","seed":202611087005,"parameters":{"n":5,"edges":[[0,1],[1,2],[2,3],[2,4],[3,4],[4,0]],"domain":"Complete_finite_loopless_simple_digraph","operation":"digraph-qualification","semantic_template":"qf-mod39-digraph-qualification/{\"edges\":[[0,1],[1,2],[2,3],[2,4],[3,4],[4,0]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r07-finite-directed-graph-0006","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-directed-graph"],"title_zh":"有限有向圖","stem_zh":"完整有限ordered-pair edge set與loop-free simple digraph convention。操作=digraph-qualification；資料={\"n\":5,\"edges\":[[0,1],[0,2],[1,2],[2,3],[2,4],[3,4],[4,0]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, [[0, 1], [0, 2], [1, 2], [2, 3], [2, 4], [3, 4], [4, 0]], 是, 是, ordered_pairs_and_no_parallel_duplicate_edges)","explanation_zh":"Wave6第二層核對=(5, [[0, 1], [0, 2], [1, 2], [2, 3], [2, 4], [3, 4], [4, 0]], 是, 是, ordered_pairs_and_no_parallel_duplicate_edges)；依完整明示資料獨立重算；(u,v)與(v,u)不同，平行同向edge不重複且loop在主模型拒絕；out-degree與in-degree各自總和等於edge count；不沿用undirected degree或connectivity。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限有向圖"]},"variant":{"family_id":"qf-mod39-digraph-qualification","seed":202611087006,"parameters":{"n":5,"edges":[[0,1],[0,2],[1,2],[2,3],[2,4],[3,4],[4,0]],"domain":"Complete_finite_loopless_simple_digraph","operation":"digraph-qualification","semantic_template":"qf-mod39-digraph-qualification/{\"edges\":[[0,1],[0,2],[1,2],[2,3],[2,4],[3,4],[4,0]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r07-finite-directed-graph-0007","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-directed-graph"],"title_zh":"有限有向圖","stem_zh":"完整有限ordered-pair edge set與loop-free simple digraph convention。操作=digraph-qualification；資料={\"n\":5,\"edges\":[[0,1],[1,2],[1,3],[2,3],[2,4],[3,4],[4,0]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, [[0, 1], [1, 2], [1, 3], [2, 3], [2, 4], [3, 4], [4, 0]], 是, 是, ordered_pairs_and_no_parallel_duplicate_edges)","explanation_zh":"Wave6第二層核對=(5, [[0, 1], [1, 2], [1, 3], [2, 3], [2, 4], [3, 4], [4, 0]], 是, 是, ordered_pairs_and_no_parallel_duplicate_edges)；依完整明示資料獨立重算；(u,v)與(v,u)不同，平行同向edge不重複且loop在主模型拒絕；out-degree與in-degree各自總和等於edge count；不沿用undirected degree或connectivity。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限有向圖"]},"variant":{"family_id":"qf-mod39-digraph-qualification","seed":202611087007,"parameters":{"n":5,"edges":[[0,1],[1,2],[1,3],[2,3],[2,4],[3,4],[4,0]],"domain":"Complete_finite_loopless_simple_digraph","operation":"digraph-qualification","semantic_template":"qf-mod39-digraph-qualification/{\"edges\":[[0,1],[1,2],[1,3],[2,3],[2,4],[3,4],[4,0]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r07-finite-directed-graph-0008","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-directed-graph"],"title_zh":"有限有向圖","stem_zh":"完整有限ordered-pair edge set與loop-free simple digraph convention。操作=digraph-qualification；資料={\"n\":5,\"edges\":[[0,1],[0,2],[1,2],[1,3],[2,3],[2,4],[3,4],[4,0]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, [[0, 1], [0, 2], [1, 2], [1, 3], [2, 3], [2, 4], [3, 4], [4, 0]], 是, 是, ordered_pairs_and_no_parallel_duplicate_edges)","explanation_zh":"Wave6第二層核對=(5, [[0, 1], [0, 2], [1, 2], [1, 3], [2, 3], [2, 4], [3, 4], [4, 0]], 是, 是, ordered_pairs_and_no_parallel_duplicate_edges)；依完整明示資料獨立重算；(u,v)與(v,u)不同，平行同向edge不重複且loop在主模型拒絕；out-degree與in-degree各自總和等於edge count；不沿用undirected degree或connectivity。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限有向圖"]},"variant":{"family_id":"qf-mod39-digraph-qualification","seed":202611087008,"parameters":{"n":5,"edges":[[0,1],[0,2],[1,2],[1,3],[2,3],[2,4],[3,4],[4,0]],"domain":"Complete_finite_loopless_simple_digraph","operation":"digraph-qualification","semantic_template":"qf-mod39-digraph-qualification/{\"edges\":[[0,1],[0,2],[1,2],[1,3],[2,3],[2,4],[3,4],[4,0]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r07-finite-directed-graph-0009","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-directed-graph"],"title_zh":"有限有向圖","stem_zh":"完整有限ordered-pair edge set與loop-free simple digraph convention。操作=digraph-qualification；資料={\"n\":5,\"edges\":[[0,1],[1,2],[2,3],[3,0],[3,4],[4,0]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, [[0, 1], [1, 2], [2, 3], [3, 0], [3, 4], [4, 0]], 是, 是, ordered_pairs_and_no_parallel_duplicate_edges)","explanation_zh":"Wave6第二層核對=(5, [[0, 1], [1, 2], [2, 3], [3, 0], [3, 4], [4, 0]], 是, 是, ordered_pairs_and_no_parallel_duplicate_edges)；依完整明示資料獨立重算；(u,v)與(v,u)不同，平行同向edge不重複且loop在主模型拒絕；out-degree與in-degree各自總和等於edge count；不沿用undirected degree或connectivity。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限有向圖"]},"variant":{"family_id":"qf-mod39-digraph-qualification","seed":202611087009,"parameters":{"n":5,"edges":[[0,1],[1,2],[2,3],[3,0],[3,4],[4,0]],"domain":"Complete_finite_loopless_simple_digraph","operation":"digraph-qualification","semantic_template":"qf-mod39-digraph-qualification/{\"edges\":[[0,1],[1,2],[2,3],[3,0],[3,4],[4,0]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r07-finite-directed-graph-0010","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-directed-graph"],"title_zh":"有限有向圖","stem_zh":"完整有限ordered-pair edge set與loop-free simple digraph convention。操作=digraph-qualification；資料={\"n\":5,\"edges\":[[0,1],[0,2],[1,2],[2,3],[3,0],[3,4],[4,0]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, [[0, 1], [0, 2], [1, 2], [2, 3], [3, 0], [3, 4], [4, 0]], 是, 是, ordered_pairs_and_no_parallel_duplicate_edges)","explanation_zh":"Wave6第二層核對=(5, [[0, 1], [0, 2], [1, 2], [2, 3], [3, 0], [3, 4], [4, 0]], 是, 是, ordered_pairs_and_no_parallel_duplicate_edges)；依完整明示資料獨立重算；(u,v)與(v,u)不同，平行同向edge不重複且loop在主模型拒絕；out-degree與in-degree各自總和等於edge count；不沿用undirected degree或connectivity。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限有向圖"]},"variant":{"family_id":"qf-mod39-digraph-qualification","seed":202611087010,"parameters":{"n":5,"edges":[[0,1],[0,2],[1,2],[2,3],[3,0],[3,4],[4,0]],"domain":"Complete_finite_loopless_simple_digraph","operation":"digraph-qualification","semantic_template":"qf-mod39-digraph-qualification/{\"edges\":[[0,1],[0,2],[1,2],[2,3],[3,0],[3,4],[4,0]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r07-finite-directed-graph-0011","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-directed-graph"],"title_zh":"有限有向圖","stem_zh":"完整有限ordered-pair edge set與loop-free simple digraph convention。操作=digraph-qualification；資料={\"n\":5,\"edges\":[[0,1],[1,2],[1,3],[2,3],[3,0],[3,4],[4,0]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, [[0, 1], [1, 2], [1, 3], [2, 3], [3, 0], [3, 4], [4, 0]], 是, 是, ordered_pairs_and_no_parallel_duplicate_edges)","explanation_zh":"Wave6第二層核對=(5, [[0, 1], [1, 2], [1, 3], [2, 3], [3, 0], [3, 4], [4, 0]], 是, 是, ordered_pairs_and_no_parallel_duplicate_edges)；依完整明示資料獨立重算；(u,v)與(v,u)不同，平行同向edge不重複且loop在主模型拒絕；out-degree與in-degree各自總和等於edge count；不沿用undirected degree或connectivity。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限有向圖"]},"variant":{"family_id":"qf-mod39-digraph-qualification","seed":202611087011,"parameters":{"n":5,"edges":[[0,1],[1,2],[1,3],[2,3],[3,0],[3,4],[4,0]],"domain":"Complete_finite_loopless_simple_digraph","operation":"digraph-qualification","semantic_template":"qf-mod39-digraph-qualification/{\"edges\":[[0,1],[1,2],[1,3],[2,3],[3,0],[3,4],[4,0]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r07-finite-directed-graph-0012","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-directed-graph"],"title_zh":"有限有向圖","stem_zh":"完整有限ordered-pair edge set與loop-free simple digraph convention。操作=digraph-qualification；資料={\"n\":5,\"edges\":[[0,1],[0,2],[1,2],[1,3],[2,3],[3,0],[3,4],[4,0]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, [[0, 1], [0, 2], [1, 2], [1, 3], [2, 3], [3, 0], [3, 4], [4, 0]], 是, 是, ordered_pairs_and_no_parallel_duplicate_edges)","explanation_zh":"Wave6第二層核對=(5, [[0, 1], [0, 2], [1, 2], [1, 3], [2, 3], [3, 0], [3, 4], [4, 0]], 是, 是, ordered_pairs_and_no_parallel_duplicate_edges)；依完整明示資料獨立重算；(u,v)與(v,u)不同，平行同向edge不重複且loop在主模型拒絕；out-degree與in-degree各自總和等於edge count；不沿用undirected degree或connectivity。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限有向圖"]},"variant":{"family_id":"qf-mod39-digraph-qualification","seed":202611087012,"parameters":{"n":5,"edges":[[0,1],[0,2],[1,2],[1,3],[2,3],[3,0],[3,4],[4,0]],"domain":"Complete_finite_loopless_simple_digraph","operation":"digraph-qualification","semantic_template":"qf-mod39-digraph-qualification/{\"edges\":[[0,1],[0,2],[1,2],[1,3],[2,3],[3,0],[3,4],[4,0]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r07-finite-directed-graph-0013","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-directed-graph"],"title_zh":"有限有向圖","stem_zh":"完整有限ordered-pair edge set與loop-free simple digraph convention。操作=digraph-qualification；資料={\"n\":5,\"edges\":[[0,1],[1,2],[2,3],[2,4],[3,0],[3,4],[4,0]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, [[0, 1], [1, 2], [2, 3], [2, 4], [3, 0], [3, 4], [4, 0]], 是, 是, ordered_pairs_and_no_parallel_duplicate_edges)","explanation_zh":"Wave6第二層核對=(5, [[0, 1], [1, 2], [2, 3], [2, 4], [3, 0], [3, 4], [4, 0]], 是, 是, ordered_pairs_and_no_parallel_duplicate_edges)；依完整明示資料獨立重算；(u,v)與(v,u)不同，平行同向edge不重複且loop在主模型拒絕；out-degree與in-degree各自總和等於edge count；不沿用undirected degree或connectivity。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限有向圖"]},"variant":{"family_id":"qf-mod39-digraph-qualification","seed":202611087013,"parameters":{"n":5,"edges":[[0,1],[1,2],[2,3],[2,4],[3,0],[3,4],[4,0]],"domain":"Complete_finite_loopless_simple_digraph","operation":"digraph-qualification","semantic_template":"qf-mod39-digraph-qualification/{\"edges\":[[0,1],[1,2],[2,3],[2,4],[3,0],[3,4],[4,0]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r07-finite-directed-graph-0014","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-directed-graph"],"title_zh":"有限有向圖","stem_zh":"完整有限ordered-pair edge set與loop-free simple digraph convention。操作=digraph-qualification；資料={\"n\":5,\"edges\":[[0,1],[0,2],[1,2],[2,3],[2,4],[3,0],[3,4],[4,0]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, [[0, 1], [0, 2], [1, 2], [2, 3], [2, 4], [3, 0], [3, 4], [4, 0]], 是, 是, ordered_pairs_and_no_parallel_duplicate_edges)","explanation_zh":"Wave6第二層核對=(5, [[0, 1], [0, 2], [1, 2], [2, 3], [2, 4], [3, 0], [3, 4], [4, 0]], 是, 是, ordered_pairs_and_no_parallel_duplicate_edges)；依完整明示資料獨立重算；(u,v)與(v,u)不同，平行同向edge不重複且loop在主模型拒絕；out-degree與in-degree各自總和等於edge count；不沿用undirected degree或connectivity。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限有向圖"]},"variant":{"family_id":"qf-mod39-digraph-qualification","seed":202611087014,"parameters":{"n":5,"edges":[[0,1],[0,2],[1,2],[2,3],[2,4],[3,0],[3,4],[4,0]],"domain":"Complete_finite_loopless_simple_digraph","operation":"digraph-qualification","semantic_template":"qf-mod39-digraph-qualification/{\"edges\":[[0,1],[0,2],[1,2],[2,3],[2,4],[3,0],[3,4],[4,0]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r07-finite-directed-graph-0015","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-directed-graph"],"title_zh":"有限有向圖","stem_zh":"完整有限ordered-pair edge set與loop-free simple digraph convention。操作=digraph-qualification；資料={\"n\":5,\"edges\":[[0,1],[1,2],[1,3],[2,3],[2,4],[3,0],[3,4],[4,0]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, [[0, 1], [1, 2], [1, 3], [2, 3], [2, 4], [3, 0], [3, 4], [4, 0]], 是, 是, ordered_pairs_and_no_parallel_duplicate_edges)","explanation_zh":"Wave6第二層核對=(5, [[0, 1], [1, 2], [1, 3], [2, 3], [2, 4], [3, 0], [3, 4], [4, 0]], 是, 是, ordered_pairs_and_no_parallel_duplicate_edges)；依完整明示資料獨立重算；(u,v)與(v,u)不同，平行同向edge不重複且loop在主模型拒絕；out-degree與in-degree各自總和等於edge count；不沿用undirected degree或connectivity。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限有向圖"]},"variant":{"family_id":"qf-mod39-digraph-qualification","seed":202611087015,"parameters":{"n":5,"edges":[[0,1],[1,2],[1,3],[2,3],[2,4],[3,0],[3,4],[4,0]],"domain":"Complete_finite_loopless_simple_digraph","operation":"digraph-qualification","semantic_template":"qf-mod39-digraph-qualification/{\"edges\":[[0,1],[1,2],[1,3],[2,3],[2,4],[3,0],[3,4],[4,0]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r07-finite-directed-graph-0016","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-directed-graph"],"title_zh":"有限有向圖","stem_zh":"完整有限ordered-pair edge set與loop-free simple digraph convention。操作=digraph-qualification；資料={\"n\":5,\"edges\":[[0,1],[0,2],[1,2],[1,3],[2,3],[2,4],[3,0],[3,4],[4,0]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, [[0, 1], [0, 2], [1, 2], [1, 3], [2, 3], [2, 4], [3, 0], [3, 4], [4, 0]], 是, 是, ordered_pairs_and_no_parallel_duplicate_edges)","explanation_zh":"Wave6第二層核對=(5, [[0, 1], [0, 2], [1, 2], [1, 3], [2, 3], [2, 4], [3, 0], [3, 4], [4, 0]], 是, 是, ordered_pairs_and_no_parallel_duplicate_edges)；依完整明示資料獨立重算；(u,v)與(v,u)不同，平行同向edge不重複且loop在主模型拒絕；out-degree與in-degree各自總和等於edge count；不沿用undirected degree或connectivity。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限有向圖"]},"variant":{"family_id":"qf-mod39-digraph-qualification","seed":202611087016,"parameters":{"n":5,"edges":[[0,1],[0,2],[1,2],[1,3],[2,3],[2,4],[3,0],[3,4],[4,0]],"domain":"Complete_finite_loopless_simple_digraph","operation":"digraph-qualification","semantic_template":"qf-mod39-digraph-qualification/{\"edges\":[[0,1],[0,2],[1,2],[1,3],[2,3],[2,4],[3,0],[3,4],[4,0]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r07-finite-directed-graph-0017","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-directed-graph"],"title_zh":"有限有向圖","stem_zh":"完整有限ordered-pair edge set與loop-free simple digraph convention。操作=digraph-qualification；資料={\"n\":5,\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,0],[4,1]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, [[0, 1], [1, 2], [2, 3], [3, 4], [4, 0], [4, 1]], 是, 是, ordered_pairs_and_no_parallel_duplicate_edges)","explanation_zh":"Wave6第二層核對=(5, [[0, 1], [1, 2], [2, 3], [3, 4], [4, 0], [4, 1]], 是, 是, ordered_pairs_and_no_parallel_duplicate_edges)；依完整明示資料獨立重算；(u,v)與(v,u)不同，平行同向edge不重複且loop在主模型拒絕；out-degree與in-degree各自總和等於edge count；不沿用undirected degree或connectivity。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限有向圖"]},"variant":{"family_id":"qf-mod39-digraph-qualification","seed":202611087017,"parameters":{"n":5,"edges":[[0,1],[1,2],[2,3],[3,4],[4,0],[4,1]],"domain":"Complete_finite_loopless_simple_digraph","operation":"digraph-qualification","semantic_template":"qf-mod39-digraph-qualification/{\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,0],[4,1]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r07-finite-directed-graph-0018","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-directed-graph"],"title_zh":"有限有向圖","stem_zh":"完整有限ordered-pair edge set與loop-free simple digraph convention。操作=digraph-qualification；資料={\"n\":5,\"edges\":[[0,1],[0,2],[1,2],[2,3],[3,4],[4,0],[4,1]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, [[0, 1], [0, 2], [1, 2], [2, 3], [3, 4], [4, 0], [4, 1]], 是, 是, ordered_pairs_and_no_parallel_duplicate_edges)","explanation_zh":"Wave6第二層核對=(5, [[0, 1], [0, 2], [1, 2], [2, 3], [3, 4], [4, 0], [4, 1]], 是, 是, ordered_pairs_and_no_parallel_duplicate_edges)；依完整明示資料獨立重算；(u,v)與(v,u)不同，平行同向edge不重複且loop在主模型拒絕；out-degree與in-degree各自總和等於edge count；不沿用undirected degree或connectivity。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限有向圖"]},"variant":{"family_id":"qf-mod39-digraph-qualification","seed":202611087018,"parameters":{"n":5,"edges":[[0,1],[0,2],[1,2],[2,3],[3,4],[4,0],[4,1]],"domain":"Complete_finite_loopless_simple_digraph","operation":"digraph-qualification","semantic_template":"qf-mod39-digraph-qualification/{\"edges\":[[0,1],[0,2],[1,2],[2,3],[3,4],[4,0],[4,1]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r07-finite-directed-graph-0019","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-directed-graph"],"title_zh":"有限有向圖","stem_zh":"完整有限ordered-pair edge set與loop-free simple digraph convention。操作=digraph-qualification；資料={\"n\":5,\"edges\":[[0,1],[1,2],[1,3],[2,3],[3,4],[4,0],[4,1]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, [[0, 1], [1, 2], [1, 3], [2, 3], [3, 4], [4, 0], [4, 1]], 是, 是, ordered_pairs_and_no_parallel_duplicate_edges)","explanation_zh":"Wave6第二層核對=(5, [[0, 1], [1, 2], [1, 3], [2, 3], [3, 4], [4, 0], [4, 1]], 是, 是, ordered_pairs_and_no_parallel_duplicate_edges)；依完整明示資料獨立重算；(u,v)與(v,u)不同，平行同向edge不重複且loop在主模型拒絕；out-degree與in-degree各自總和等於edge count；不沿用undirected degree或connectivity。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限有向圖"]},"variant":{"family_id":"qf-mod39-digraph-qualification","seed":202611087019,"parameters":{"n":5,"edges":[[0,1],[1,2],[1,3],[2,3],[3,4],[4,0],[4,1]],"domain":"Complete_finite_loopless_simple_digraph","operation":"digraph-qualification","semantic_template":"qf-mod39-digraph-qualification/{\"edges\":[[0,1],[1,2],[1,3],[2,3],[3,4],[4,0],[4,1]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r07-finite-directed-graph-0020","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-directed-graph"],"title_zh":"有限有向圖","stem_zh":"完整有限ordered-pair edge set與loop-free simple digraph convention。操作=digraph-qualification；資料={\"n\":5,\"edges\":[[0,1],[0,2],[1,2],[1,3],[2,3],[3,4],[4,0],[4,1]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, [[0, 1], [0, 2], [1, 2], [1, 3], [2, 3], [3, 4], [4, 0], [4, 1]], 是, 是, ordered_pairs_and_no_parallel_duplicate_edges)","explanation_zh":"Wave6第二層核對=(5, [[0, 1], [0, 2], [1, 2], [1, 3], [2, 3], [3, 4], [4, 0], [4, 1]], 是, 是, ordered_pairs_and_no_parallel_duplicate_edges)；依完整明示資料獨立重算；(u,v)與(v,u)不同，平行同向edge不重複且loop在主模型拒絕；out-degree與in-degree各自總和等於edge count；不沿用undirected degree或connectivity。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限有向圖"]},"variant":{"family_id":"qf-mod39-digraph-qualification","seed":202611087020,"parameters":{"n":5,"edges":[[0,1],[0,2],[1,2],[1,3],[2,3],[3,4],[4,0],[4,1]],"domain":"Complete_finite_loopless_simple_digraph","operation":"digraph-qualification","semantic_template":"qf-mod39-digraph-qualification/{\"edges\":[[0,1],[0,2],[1,2],[1,3],[2,3],[3,4],[4,0],[4,1]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r07-finite-directed-graph-0021","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-directed-graph"],"title_zh":"有限有向圖","stem_zh":"完整有限ordered-pair edge set與loop-free simple digraph convention。操作=digraph-qualification；資料={\"n\":5,\"edges\":[[0,1],[1,2],[2,3],[2,4],[3,4],[4,0],[4,1]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, [[0, 1], [1, 2], [2, 3], [2, 4], [3, 4], [4, 0], [4, 1]], 是, 是, ordered_pairs_and_no_parallel_duplicate_edges)","explanation_zh":"Wave6第二層核對=(5, [[0, 1], [1, 2], [2, 3], [2, 4], [3, 4], [4, 0], [4, 1]], 是, 是, ordered_pairs_and_no_parallel_duplicate_edges)；依完整明示資料獨立重算；(u,v)與(v,u)不同，平行同向edge不重複且loop在主模型拒絕；out-degree與in-degree各自總和等於edge count；不沿用undirected degree或connectivity。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限有向圖"]},"variant":{"family_id":"qf-mod39-digraph-qualification","seed":202611087021,"parameters":{"n":5,"edges":[[0,1],[1,2],[2,3],[2,4],[3,4],[4,0],[4,1]],"domain":"Complete_finite_loopless_simple_digraph","operation":"digraph-qualification","semantic_template":"qf-mod39-digraph-qualification/{\"edges\":[[0,1],[1,2],[2,3],[2,4],[3,4],[4,0],[4,1]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r07-finite-directed-graph-0022","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-directed-graph"],"title_zh":"有限有向圖","stem_zh":"完整有限ordered-pair edge set與loop-free simple digraph convention。操作=digraph-qualification；資料={\"n\":5,\"edges\":[[0,1],[0,2],[1,2],[2,3],[2,4],[3,4],[4,0],[4,1]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, [[0, 1], [0, 2], [1, 2], [2, 3], [2, 4], [3, 4], [4, 0], [4, 1]], 是, 是, ordered_pairs_and_no_parallel_duplicate_edges)","explanation_zh":"Wave6第二層核對=(5, [[0, 1], [0, 2], [1, 2], [2, 3], [2, 4], [3, 4], [4, 0], [4, 1]], 是, 是, ordered_pairs_and_no_parallel_duplicate_edges)；依完整明示資料獨立重算；(u,v)與(v,u)不同，平行同向edge不重複且loop在主模型拒絕；out-degree與in-degree各自總和等於edge count；不沿用undirected degree或connectivity。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限有向圖"]},"variant":{"family_id":"qf-mod39-digraph-qualification","seed":202611087022,"parameters":{"n":5,"edges":[[0,1],[0,2],[1,2],[2,3],[2,4],[3,4],[4,0],[4,1]],"domain":"Complete_finite_loopless_simple_digraph","operation":"digraph-qualification","semantic_template":"qf-mod39-digraph-qualification/{\"edges\":[[0,1],[0,2],[1,2],[2,3],[2,4],[3,4],[4,0],[4,1]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r07-finite-directed-graph-0023","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-directed-graph"],"title_zh":"有限有向圖","stem_zh":"完整有限ordered-pair edge set與loop-free simple digraph convention。操作=digraph-qualification；資料={\"n\":5,\"edges\":[[0,1],[1,2],[1,3],[2,3],[2,4],[3,4],[4,0],[4,1]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, [[0, 1], [1, 2], [1, 3], [2, 3], [2, 4], [3, 4], [4, 0], [4, 1]], 是, 是, ordered_pairs_and_no_parallel_duplicate_edges)","explanation_zh":"Wave6第二層核對=(5, [[0, 1], [1, 2], [1, 3], [2, 3], [2, 4], [3, 4], [4, 0], [4, 1]], 是, 是, ordered_pairs_and_no_parallel_duplicate_edges)；依完整明示資料獨立重算；(u,v)與(v,u)不同，平行同向edge不重複且loop在主模型拒絕；out-degree與in-degree各自總和等於edge count；不沿用undirected degree或connectivity。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限有向圖"]},"variant":{"family_id":"qf-mod39-digraph-qualification","seed":202611087023,"parameters":{"n":5,"edges":[[0,1],[1,2],[1,3],[2,3],[2,4],[3,4],[4,0],[4,1]],"domain":"Complete_finite_loopless_simple_digraph","operation":"digraph-qualification","semantic_template":"qf-mod39-digraph-qualification/{\"edges\":[[0,1],[1,2],[1,3],[2,3],[2,4],[3,4],[4,0],[4,1]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r07-finite-directed-graph-0024","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-directed-graph"],"title_zh":"有限有向圖","stem_zh":"完整有限ordered-pair edge set與loop-free simple digraph convention。操作=digraph-qualification；資料={\"n\":5,\"edges\":[[0,1],[0,2],[1,2],[1,3],[2,3],[2,4],[3,4],[4,0],[4,1]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, [[0, 1], [0, 2], [1, 2], [1, 3], [2, 3], [2, 4], [3, 4], [4, 0], [4, 1]], 是, 是, ordered_pairs_and_no_parallel_duplicate_edges)","explanation_zh":"Wave6第二層核對=(5, [[0, 1], [0, 2], [1, 2], [1, 3], [2, 3], [2, 4], [3, 4], [4, 0], [4, 1]], 是, 是, ordered_pairs_and_no_parallel_duplicate_edges)；依完整明示資料獨立重算；(u,v)與(v,u)不同，平行同向edge不重複且loop在主模型拒絕；out-degree與in-degree各自總和等於edge count；不沿用undirected degree或connectivity。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限有向圖"]},"variant":{"family_id":"qf-mod39-digraph-qualification","seed":202611087024,"parameters":{"n":5,"edges":[[0,1],[0,2],[1,2],[1,3],[2,3],[2,4],[3,4],[4,0],[4,1]],"domain":"Complete_finite_loopless_simple_digraph","operation":"digraph-qualification","semantic_template":"qf-mod39-digraph-qualification/{\"edges\":[[0,1],[0,2],[1,2],[1,3],[2,3],[2,4],[3,4],[4,0],[4,1]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r07-finite-directed-graph-0025","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-directed-graph"],"title_zh":"有限有向圖","stem_zh":"完整有限ordered-pair edge set與loop-free simple digraph convention。操作=digraph-qualification；資料={\"n\":5,\"edges\":[[0,1],[1,2],[2,3],[3,0],[3,4],[4,0],[4,1]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, [[0, 1], [1, 2], [2, 3], [3, 0], [3, 4], [4, 0], [4, 1]], 是, 是, ordered_pairs_and_no_parallel_duplicate_edges)","explanation_zh":"Wave6第二層核對=(5, [[0, 1], [1, 2], [2, 3], [3, 0], [3, 4], [4, 0], [4, 1]], 是, 是, ordered_pairs_and_no_parallel_duplicate_edges)；依完整明示資料獨立重算；(u,v)與(v,u)不同，平行同向edge不重複且loop在主模型拒絕；out-degree與in-degree各自總和等於edge count；不沿用undirected degree或connectivity。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限有向圖"]},"variant":{"family_id":"qf-mod39-digraph-qualification","seed":202611087025,"parameters":{"n":5,"edges":[[0,1],[1,2],[2,3],[3,0],[3,4],[4,0],[4,1]],"domain":"Complete_finite_loopless_simple_digraph","operation":"digraph-qualification","semantic_template":"qf-mod39-digraph-qualification/{\"edges\":[[0,1],[1,2],[2,3],[3,0],[3,4],[4,0],[4,1]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r07-finite-directed-graph-0026","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-directed-graph"],"title_zh":"有限有向圖","stem_zh":"完整有限ordered-pair edge set與loop-free simple digraph convention。操作=digraph-in-out-degree；資料={\"n\":5,\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,0]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([1, 1, 1, 1, 1], [1, 1, 1, 1, 1], 5, 5, 5, 是)","explanation_zh":"Wave6第二層核對=([1, 1, 1, 1, 1], [1, 1, 1, 1, 1], 5, 5, 5, 是)；依完整明示資料獨立重算；(u,v)與(v,u)不同，平行同向edge不重複且loop在主模型拒絕；out-degree與in-degree各自總和等於edge count；不沿用undirected degree或connectivity。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限有向圖"]},"variant":{"family_id":"qf-mod39-digraph-in-out-degree","seed":202611087026,"parameters":{"n":5,"edges":[[0,1],[1,2],[2,3],[3,4],[4,0]],"domain":"Complete_finite_loopless_simple_digraph","operation":"digraph-in-out-degree","semantic_template":"qf-mod39-digraph-in-out-degree/{\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,0]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r07-finite-directed-graph-0027","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-directed-graph"],"title_zh":"有限有向圖","stem_zh":"完整有限ordered-pair edge set與loop-free simple digraph convention。操作=digraph-in-out-degree；資料={\"n\":5,\"edges\":[[0,1],[0,2],[1,2],[2,3],[3,4],[4,0]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([2, 1, 1, 1, 1], [1, 1, 2, 1, 1], 6, 6, 6, 是)","explanation_zh":"Wave6第二層核對=([2, 1, 1, 1, 1], [1, 1, 2, 1, 1], 6, 6, 6, 是)；依完整明示資料獨立重算；(u,v)與(v,u)不同，平行同向edge不重複且loop在主模型拒絕；out-degree與in-degree各自總和等於edge count；不沿用undirected degree或connectivity。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限有向圖"]},"variant":{"family_id":"qf-mod39-digraph-in-out-degree","seed":202611087027,"parameters":{"n":5,"edges":[[0,1],[0,2],[1,2],[2,3],[3,4],[4,0]],"domain":"Complete_finite_loopless_simple_digraph","operation":"digraph-in-out-degree","semantic_template":"qf-mod39-digraph-in-out-degree/{\"edges\":[[0,1],[0,2],[1,2],[2,3],[3,4],[4,0]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r07-finite-directed-graph-0028","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-directed-graph"],"title_zh":"有限有向圖","stem_zh":"完整有限ordered-pair edge set與loop-free simple digraph convention。操作=digraph-in-out-degree；資料={\"n\":5,\"edges\":[[0,1],[1,2],[1,3],[2,3],[3,4],[4,0]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([1, 2, 1, 1, 1], [1, 1, 1, 2, 1], 6, 6, 6, 是)","explanation_zh":"Wave6第二層核對=([1, 2, 1, 1, 1], [1, 1, 1, 2, 1], 6, 6, 6, 是)；依完整明示資料獨立重算；(u,v)與(v,u)不同，平行同向edge不重複且loop在主模型拒絕；out-degree與in-degree各自總和等於edge count；不沿用undirected degree或connectivity。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限有向圖"]},"variant":{"family_id":"qf-mod39-digraph-in-out-degree","seed":202611087028,"parameters":{"n":5,"edges":[[0,1],[1,2],[1,3],[2,3],[3,4],[4,0]],"domain":"Complete_finite_loopless_simple_digraph","operation":"digraph-in-out-degree","semantic_template":"qf-mod39-digraph-in-out-degree/{\"edges\":[[0,1],[1,2],[1,3],[2,3],[3,4],[4,0]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r07-finite-directed-graph-0029","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-directed-graph"],"title_zh":"有限有向圖","stem_zh":"完整有限ordered-pair edge set與loop-free simple digraph convention。操作=digraph-in-out-degree；資料={\"n\":5,\"edges\":[[0,1],[0,2],[1,2],[1,3],[2,3],[3,4],[4,0]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([2, 2, 1, 1, 1], [1, 1, 2, 2, 1], 7, 7, 7, 是)","explanation_zh":"Wave6第二層核對=([2, 2, 1, 1, 1], [1, 1, 2, 2, 1], 7, 7, 7, 是)；依完整明示資料獨立重算；(u,v)與(v,u)不同，平行同向edge不重複且loop在主模型拒絕；out-degree與in-degree各自總和等於edge count；不沿用undirected degree或connectivity。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限有向圖"]},"variant":{"family_id":"qf-mod39-digraph-in-out-degree","seed":202611087029,"parameters":{"n":5,"edges":[[0,1],[0,2],[1,2],[1,3],[2,3],[3,4],[4,0]],"domain":"Complete_finite_loopless_simple_digraph","operation":"digraph-in-out-degree","semantic_template":"qf-mod39-digraph-in-out-degree/{\"edges\":[[0,1],[0,2],[1,2],[1,3],[2,3],[3,4],[4,0]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r07-finite-directed-graph-0030","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-directed-graph"],"title_zh":"有限有向圖","stem_zh":"完整有限ordered-pair edge set與loop-free simple digraph convention。操作=digraph-in-out-degree；資料={\"n\":5,\"edges\":[[0,1],[1,2],[2,3],[2,4],[3,4],[4,0]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([1, 1, 2, 1, 1], [1, 1, 1, 1, 2], 6, 6, 6, 是)","explanation_zh":"Wave6第二層核對=([1, 1, 2, 1, 1], [1, 1, 1, 1, 2], 6, 6, 6, 是)；依完整明示資料獨立重算；(u,v)與(v,u)不同，平行同向edge不重複且loop在主模型拒絕；out-degree與in-degree各自總和等於edge count；不沿用undirected degree或connectivity。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限有向圖"]},"variant":{"family_id":"qf-mod39-digraph-in-out-degree","seed":202611087030,"parameters":{"n":5,"edges":[[0,1],[1,2],[2,3],[2,4],[3,4],[4,0]],"domain":"Complete_finite_loopless_simple_digraph","operation":"digraph-in-out-degree","semantic_template":"qf-mod39-digraph-in-out-degree/{\"edges\":[[0,1],[1,2],[2,3],[2,4],[3,4],[4,0]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r07-finite-directed-graph-0031","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-directed-graph"],"title_zh":"有限有向圖","stem_zh":"完整有限ordered-pair edge set與loop-free simple digraph convention。操作=digraph-in-out-degree；資料={\"n\":5,\"edges\":[[0,1],[0,2],[1,2],[2,3],[2,4],[3,4],[4,0]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([2, 1, 2, 1, 1], [1, 1, 2, 1, 2], 7, 7, 7, 是)","explanation_zh":"Wave6第二層核對=([2, 1, 2, 1, 1], [1, 1, 2, 1, 2], 7, 7, 7, 是)；依完整明示資料獨立重算；(u,v)與(v,u)不同，平行同向edge不重複且loop在主模型拒絕；out-degree與in-degree各自總和等於edge count；不沿用undirected degree或connectivity。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限有向圖"]},"variant":{"family_id":"qf-mod39-digraph-in-out-degree","seed":202611087031,"parameters":{"n":5,"edges":[[0,1],[0,2],[1,2],[2,3],[2,4],[3,4],[4,0]],"domain":"Complete_finite_loopless_simple_digraph","operation":"digraph-in-out-degree","semantic_template":"qf-mod39-digraph-in-out-degree/{\"edges\":[[0,1],[0,2],[1,2],[2,3],[2,4],[3,4],[4,0]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r07-finite-directed-graph-0032","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-directed-graph"],"title_zh":"有限有向圖","stem_zh":"完整有限ordered-pair edge set與loop-free simple digraph convention。操作=digraph-in-out-degree；資料={\"n\":5,\"edges\":[[0,1],[1,2],[1,3],[2,3],[2,4],[3,4],[4,0]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([1, 2, 2, 1, 1], [1, 1, 1, 2, 2], 7, 7, 7, 是)","explanation_zh":"Wave6第二層核對=([1, 2, 2, 1, 1], [1, 1, 1, 2, 2], 7, 7, 7, 是)；依完整明示資料獨立重算；(u,v)與(v,u)不同，平行同向edge不重複且loop在主模型拒絕；out-degree與in-degree各自總和等於edge count；不沿用undirected degree或connectivity。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限有向圖"]},"variant":{"family_id":"qf-mod39-digraph-in-out-degree","seed":202611087032,"parameters":{"n":5,"edges":[[0,1],[1,2],[1,3],[2,3],[2,4],[3,4],[4,0]],"domain":"Complete_finite_loopless_simple_digraph","operation":"digraph-in-out-degree","semantic_template":"qf-mod39-digraph-in-out-degree/{\"edges\":[[0,1],[1,2],[1,3],[2,3],[2,4],[3,4],[4,0]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r07-finite-directed-graph-0033","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-directed-graph"],"title_zh":"有限有向圖","stem_zh":"完整有限ordered-pair edge set與loop-free simple digraph convention。操作=digraph-in-out-degree；資料={\"n\":5,\"edges\":[[0,1],[0,2],[1,2],[1,3],[2,3],[2,4],[3,4],[4,0]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([2, 2, 2, 1, 1], [1, 1, 2, 2, 2], 8, 8, 8, 是)","explanation_zh":"Wave6第二層核對=([2, 2, 2, 1, 1], [1, 1, 2, 2, 2], 8, 8, 8, 是)；依完整明示資料獨立重算；(u,v)與(v,u)不同，平行同向edge不重複且loop在主模型拒絕；out-degree與in-degree各自總和等於edge count；不沿用undirected degree或connectivity。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限有向圖"]},"variant":{"family_id":"qf-mod39-digraph-in-out-degree","seed":202611087033,"parameters":{"n":5,"edges":[[0,1],[0,2],[1,2],[1,3],[2,3],[2,4],[3,4],[4,0]],"domain":"Complete_finite_loopless_simple_digraph","operation":"digraph-in-out-degree","semantic_template":"qf-mod39-digraph-in-out-degree/{\"edges\":[[0,1],[0,2],[1,2],[1,3],[2,3],[2,4],[3,4],[4,0]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r07-finite-directed-graph-0034","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-directed-graph"],"title_zh":"有限有向圖","stem_zh":"完整有限ordered-pair edge set與loop-free simple digraph convention。操作=digraph-in-out-degree；資料={\"n\":5,\"edges\":[[0,1],[1,2],[2,3],[3,0],[3,4],[4,0]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([1, 1, 1, 2, 1], [2, 1, 1, 1, 1], 6, 6, 6, 是)","explanation_zh":"Wave6第二層核對=([1, 1, 1, 2, 1], [2, 1, 1, 1, 1], 6, 6, 6, 是)；依完整明示資料獨立重算；(u,v)與(v,u)不同，平行同向edge不重複且loop在主模型拒絕；out-degree與in-degree各自總和等於edge count；不沿用undirected degree或connectivity。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限有向圖"]},"variant":{"family_id":"qf-mod39-digraph-in-out-degree","seed":202611087034,"parameters":{"n":5,"edges":[[0,1],[1,2],[2,3],[3,0],[3,4],[4,0]],"domain":"Complete_finite_loopless_simple_digraph","operation":"digraph-in-out-degree","semantic_template":"qf-mod39-digraph-in-out-degree/{\"edges\":[[0,1],[1,2],[2,3],[3,0],[3,4],[4,0]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r07-finite-directed-graph-0035","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-directed-graph"],"title_zh":"有限有向圖","stem_zh":"完整有限ordered-pair edge set與loop-free simple digraph convention。操作=digraph-in-out-degree；資料={\"n\":5,\"edges\":[[0,1],[0,2],[1,2],[2,3],[3,0],[3,4],[4,0]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([2, 1, 1, 2, 1], [2, 1, 2, 1, 1], 7, 7, 7, 是)","explanation_zh":"Wave6第二層核對=([2, 1, 1, 2, 1], [2, 1, 2, 1, 1], 7, 7, 7, 是)；依完整明示資料獨立重算；(u,v)與(v,u)不同，平行同向edge不重複且loop在主模型拒絕；out-degree與in-degree各自總和等於edge count；不沿用undirected degree或connectivity。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限有向圖"]},"variant":{"family_id":"qf-mod39-digraph-in-out-degree","seed":202611087035,"parameters":{"n":5,"edges":[[0,1],[0,2],[1,2],[2,3],[3,0],[3,4],[4,0]],"domain":"Complete_finite_loopless_simple_digraph","operation":"digraph-in-out-degree","semantic_template":"qf-mod39-digraph-in-out-degree/{\"edges\":[[0,1],[0,2],[1,2],[2,3],[3,0],[3,4],[4,0]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r07-finite-directed-graph-0036","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-directed-graph"],"title_zh":"有限有向圖","stem_zh":"完整有限ordered-pair edge set與loop-free simple digraph convention。操作=digraph-in-out-degree；資料={\"n\":5,\"edges\":[[0,1],[1,2],[1,3],[2,3],[3,0],[3,4],[4,0]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([1, 2, 1, 2, 1], [2, 1, 1, 2, 1], 7, 7, 7, 是)","explanation_zh":"Wave6第二層核對=([1, 2, 1, 2, 1], [2, 1, 1, 2, 1], 7, 7, 7, 是)；依完整明示資料獨立重算；(u,v)與(v,u)不同，平行同向edge不重複且loop在主模型拒絕；out-degree與in-degree各自總和等於edge count；不沿用undirected degree或connectivity。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限有向圖"]},"variant":{"family_id":"qf-mod39-digraph-in-out-degree","seed":202611087036,"parameters":{"n":5,"edges":[[0,1],[1,2],[1,3],[2,3],[3,0],[3,4],[4,0]],"domain":"Complete_finite_loopless_simple_digraph","operation":"digraph-in-out-degree","semantic_template":"qf-mod39-digraph-in-out-degree/{\"edges\":[[0,1],[1,2],[1,3],[2,3],[3,0],[3,4],[4,0]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r07-finite-directed-graph-0037","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-directed-graph"],"title_zh":"有限有向圖","stem_zh":"完整有限ordered-pair edge set與loop-free simple digraph convention。操作=digraph-in-out-degree；資料={\"n\":5,\"edges\":[[0,1],[0,2],[1,2],[1,3],[2,3],[3,0],[3,4],[4,0]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([2, 2, 1, 2, 1], [2, 1, 2, 2, 1], 8, 8, 8, 是)","explanation_zh":"Wave6第二層核對=([2, 2, 1, 2, 1], [2, 1, 2, 2, 1], 8, 8, 8, 是)；依完整明示資料獨立重算；(u,v)與(v,u)不同，平行同向edge不重複且loop在主模型拒絕；out-degree與in-degree各自總和等於edge count；不沿用undirected degree或connectivity。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限有向圖"]},"variant":{"family_id":"qf-mod39-digraph-in-out-degree","seed":202611087037,"parameters":{"n":5,"edges":[[0,1],[0,2],[1,2],[1,3],[2,3],[3,0],[3,4],[4,0]],"domain":"Complete_finite_loopless_simple_digraph","operation":"digraph-in-out-degree","semantic_template":"qf-mod39-digraph-in-out-degree/{\"edges\":[[0,1],[0,2],[1,2],[1,3],[2,3],[3,0],[3,4],[4,0]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r07-finite-directed-graph-0038","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-directed-graph"],"title_zh":"有限有向圖","stem_zh":"完整有限ordered-pair edge set與loop-free simple digraph convention。操作=digraph-in-out-degree；資料={\"n\":5,\"edges\":[[0,1],[1,2],[2,3],[2,4],[3,0],[3,4],[4,0]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([1, 1, 2, 2, 1], [2, 1, 1, 1, 2], 7, 7, 7, 是)","explanation_zh":"Wave6第二層核對=([1, 1, 2, 2, 1], [2, 1, 1, 1, 2], 7, 7, 7, 是)；依完整明示資料獨立重算；(u,v)與(v,u)不同，平行同向edge不重複且loop在主模型拒絕；out-degree與in-degree各自總和等於edge count；不沿用undirected degree或connectivity。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限有向圖"]},"variant":{"family_id":"qf-mod39-digraph-in-out-degree","seed":202611087038,"parameters":{"n":5,"edges":[[0,1],[1,2],[2,3],[2,4],[3,0],[3,4],[4,0]],"domain":"Complete_finite_loopless_simple_digraph","operation":"digraph-in-out-degree","semantic_template":"qf-mod39-digraph-in-out-degree/{\"edges\":[[0,1],[1,2],[2,3],[2,4],[3,0],[3,4],[4,0]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r07-finite-directed-graph-0039","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-directed-graph"],"title_zh":"有限有向圖","stem_zh":"完整有限ordered-pair edge set與loop-free simple digraph convention。操作=digraph-in-out-degree；資料={\"n\":5,\"edges\":[[0,1],[0,2],[1,2],[2,3],[2,4],[3,0],[3,4],[4,0]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([2, 1, 2, 2, 1], [2, 1, 2, 1, 2], 8, 8, 8, 是)","explanation_zh":"Wave6第二層核對=([2, 1, 2, 2, 1], [2, 1, 2, 1, 2], 8, 8, 8, 是)；依完整明示資料獨立重算；(u,v)與(v,u)不同，平行同向edge不重複且loop在主模型拒絕；out-degree與in-degree各自總和等於edge count；不沿用undirected degree或connectivity。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限有向圖"]},"variant":{"family_id":"qf-mod39-digraph-in-out-degree","seed":202611087039,"parameters":{"n":5,"edges":[[0,1],[0,2],[1,2],[2,3],[2,4],[3,0],[3,4],[4,0]],"domain":"Complete_finite_loopless_simple_digraph","operation":"digraph-in-out-degree","semantic_template":"qf-mod39-digraph-in-out-degree/{\"edges\":[[0,1],[0,2],[1,2],[2,3],[2,4],[3,0],[3,4],[4,0]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r07-finite-directed-graph-0040","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-directed-graph"],"title_zh":"有限有向圖","stem_zh":"完整有限ordered-pair edge set與loop-free simple digraph convention。操作=digraph-in-out-degree；資料={\"n\":5,\"edges\":[[0,1],[1,2],[1,3],[2,3],[2,4],[3,0],[3,4],[4,0]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([1, 2, 2, 2, 1], [2, 1, 1, 2, 2], 8, 8, 8, 是)","explanation_zh":"Wave6第二層核對=([1, 2, 2, 2, 1], [2, 1, 1, 2, 2], 8, 8, 8, 是)；依完整明示資料獨立重算；(u,v)與(v,u)不同，平行同向edge不重複且loop在主模型拒絕；out-degree與in-degree各自總和等於edge count；不沿用undirected degree或connectivity。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限有向圖"]},"variant":{"family_id":"qf-mod39-digraph-in-out-degree","seed":202611087040,"parameters":{"n":5,"edges":[[0,1],[1,2],[1,3],[2,3],[2,4],[3,0],[3,4],[4,0]],"domain":"Complete_finite_loopless_simple_digraph","operation":"digraph-in-out-degree","semantic_template":"qf-mod39-digraph-in-out-degree/{\"edges\":[[0,1],[1,2],[1,3],[2,3],[2,4],[3,0],[3,4],[4,0]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r07-finite-directed-graph-0041","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-directed-graph"],"title_zh":"有限有向圖","stem_zh":"完整有限ordered-pair edge set與loop-free simple digraph convention。操作=digraph-in-out-degree；資料={\"n\":5,\"edges\":[[0,1],[0,2],[1,2],[1,3],[2,3],[2,4],[3,0],[3,4],[4,0]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([2, 2, 2, 2, 1], [2, 1, 2, 2, 2], 9, 9, 9, 是)","explanation_zh":"Wave6第二層核對=([2, 2, 2, 2, 1], [2, 1, 2, 2, 2], 9, 9, 9, 是)；依完整明示資料獨立重算；(u,v)與(v,u)不同，平行同向edge不重複且loop在主模型拒絕；out-degree與in-degree各自總和等於edge count；不沿用undirected degree或connectivity。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限有向圖"]},"variant":{"family_id":"qf-mod39-digraph-in-out-degree","seed":202611087041,"parameters":{"n":5,"edges":[[0,1],[0,2],[1,2],[1,3],[2,3],[2,4],[3,0],[3,4],[4,0]],"domain":"Complete_finite_loopless_simple_digraph","operation":"digraph-in-out-degree","semantic_template":"qf-mod39-digraph-in-out-degree/{\"edges\":[[0,1],[0,2],[1,2],[1,3],[2,3],[2,4],[3,0],[3,4],[4,0]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r07-finite-directed-graph-0042","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-directed-graph"],"title_zh":"有限有向圖","stem_zh":"完整有限ordered-pair edge set與loop-free simple digraph convention。操作=digraph-in-out-degree；資料={\"n\":5,\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,0],[4,1]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([1, 1, 1, 1, 2], [1, 2, 1, 1, 1], 6, 6, 6, 是)","explanation_zh":"Wave6第二層核對=([1, 1, 1, 1, 2], [1, 2, 1, 1, 1], 6, 6, 6, 是)；依完整明示資料獨立重算；(u,v)與(v,u)不同，平行同向edge不重複且loop在主模型拒絕；out-degree與in-degree各自總和等於edge count；不沿用undirected degree或connectivity。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限有向圖"]},"variant":{"family_id":"qf-mod39-digraph-in-out-degree","seed":202611087042,"parameters":{"n":5,"edges":[[0,1],[1,2],[2,3],[3,4],[4,0],[4,1]],"domain":"Complete_finite_loopless_simple_digraph","operation":"digraph-in-out-degree","semantic_template":"qf-mod39-digraph-in-out-degree/{\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,0],[4,1]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r07-finite-directed-graph-0043","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-directed-graph"],"title_zh":"有限有向圖","stem_zh":"完整有限ordered-pair edge set與loop-free simple digraph convention。操作=digraph-in-out-degree；資料={\"n\":5,\"edges\":[[0,1],[0,2],[1,2],[2,3],[3,4],[4,0],[4,1]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([2, 1, 1, 1, 2], [1, 2, 2, 1, 1], 7, 7, 7, 是)","explanation_zh":"Wave6第二層核對=([2, 1, 1, 1, 2], [1, 2, 2, 1, 1], 7, 7, 7, 是)；依完整明示資料獨立重算；(u,v)與(v,u)不同，平行同向edge不重複且loop在主模型拒絕；out-degree與in-degree各自總和等於edge count；不沿用undirected degree或connectivity。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限有向圖"]},"variant":{"family_id":"qf-mod39-digraph-in-out-degree","seed":202611087043,"parameters":{"n":5,"edges":[[0,1],[0,2],[1,2],[2,3],[3,4],[4,0],[4,1]],"domain":"Complete_finite_loopless_simple_digraph","operation":"digraph-in-out-degree","semantic_template":"qf-mod39-digraph-in-out-degree/{\"edges\":[[0,1],[0,2],[1,2],[2,3],[3,4],[4,0],[4,1]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r07-finite-directed-graph-0044","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-directed-graph"],"title_zh":"有限有向圖","stem_zh":"完整有限ordered-pair edge set與loop-free simple digraph convention。操作=digraph-in-out-degree；資料={\"n\":5,\"edges\":[[0,1],[1,2],[1,3],[2,3],[3,4],[4,0],[4,1]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([1, 2, 1, 1, 2], [1, 2, 1, 2, 1], 7, 7, 7, 是)","explanation_zh":"Wave6第二層核對=([1, 2, 1, 1, 2], [1, 2, 1, 2, 1], 7, 7, 7, 是)；依完整明示資料獨立重算；(u,v)與(v,u)不同，平行同向edge不重複且loop在主模型拒絕；out-degree與in-degree各自總和等於edge count；不沿用undirected degree或connectivity。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限有向圖"]},"variant":{"family_id":"qf-mod39-digraph-in-out-degree","seed":202611087044,"parameters":{"n":5,"edges":[[0,1],[1,2],[1,3],[2,3],[3,4],[4,0],[4,1]],"domain":"Complete_finite_loopless_simple_digraph","operation":"digraph-in-out-degree","semantic_template":"qf-mod39-digraph-in-out-degree/{\"edges\":[[0,1],[1,2],[1,3],[2,3],[3,4],[4,0],[4,1]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r07-finite-directed-graph-0045","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-directed-graph"],"title_zh":"有限有向圖","stem_zh":"完整有限ordered-pair edge set與loop-free simple digraph convention。操作=digraph-in-out-degree；資料={\"n\":5,\"edges\":[[0,1],[0,2],[1,2],[1,3],[2,3],[3,4],[4,0],[4,1]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([2, 2, 1, 1, 2], [1, 2, 2, 2, 1], 8, 8, 8, 是)","explanation_zh":"Wave6第二層核對=([2, 2, 1, 1, 2], [1, 2, 2, 2, 1], 8, 8, 8, 是)；依完整明示資料獨立重算；(u,v)與(v,u)不同，平行同向edge不重複且loop在主模型拒絕；out-degree與in-degree各自總和等於edge count；不沿用undirected degree或connectivity。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限有向圖"]},"variant":{"family_id":"qf-mod39-digraph-in-out-degree","seed":202611087045,"parameters":{"n":5,"edges":[[0,1],[0,2],[1,2],[1,3],[2,3],[3,4],[4,0],[4,1]],"domain":"Complete_finite_loopless_simple_digraph","operation":"digraph-in-out-degree","semantic_template":"qf-mod39-digraph-in-out-degree/{\"edges\":[[0,1],[0,2],[1,2],[1,3],[2,3],[3,4],[4,0],[4,1]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r07-finite-directed-graph-0046","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-directed-graph"],"title_zh":"有限有向圖","stem_zh":"完整有限ordered-pair edge set與loop-free simple digraph convention。操作=digraph-in-out-degree；資料={\"n\":5,\"edges\":[[0,1],[1,2],[2,3],[2,4],[3,4],[4,0],[4,1]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([1, 1, 2, 1, 2], [1, 2, 1, 1, 2], 7, 7, 7, 是)","explanation_zh":"Wave6第二層核對=([1, 1, 2, 1, 2], [1, 2, 1, 1, 2], 7, 7, 7, 是)；依完整明示資料獨立重算；(u,v)與(v,u)不同，平行同向edge不重複且loop在主模型拒絕；out-degree與in-degree各自總和等於edge count；不沿用undirected degree或connectivity。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限有向圖"]},"variant":{"family_id":"qf-mod39-digraph-in-out-degree","seed":202611087046,"parameters":{"n":5,"edges":[[0,1],[1,2],[2,3],[2,4],[3,4],[4,0],[4,1]],"domain":"Complete_finite_loopless_simple_digraph","operation":"digraph-in-out-degree","semantic_template":"qf-mod39-digraph-in-out-degree/{\"edges\":[[0,1],[1,2],[2,3],[2,4],[3,4],[4,0],[4,1]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r07-finite-directed-graph-0047","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-directed-graph"],"title_zh":"有限有向圖","stem_zh":"完整有限ordered-pair edge set與loop-free simple digraph convention。操作=digraph-in-out-degree；資料={\"n\":5,\"edges\":[[0,1],[0,2],[1,2],[2,3],[2,4],[3,4],[4,0],[4,1]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([2, 1, 2, 1, 2], [1, 2, 2, 1, 2], 8, 8, 8, 是)","explanation_zh":"Wave6第二層核對=([2, 1, 2, 1, 2], [1, 2, 2, 1, 2], 8, 8, 8, 是)；依完整明示資料獨立重算；(u,v)與(v,u)不同，平行同向edge不重複且loop在主模型拒絕；out-degree與in-degree各自總和等於edge count；不沿用undirected degree或connectivity。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限有向圖"]},"variant":{"family_id":"qf-mod39-digraph-in-out-degree","seed":202611087047,"parameters":{"n":5,"edges":[[0,1],[0,2],[1,2],[2,3],[2,4],[3,4],[4,0],[4,1]],"domain":"Complete_finite_loopless_simple_digraph","operation":"digraph-in-out-degree","semantic_template":"qf-mod39-digraph-in-out-degree/{\"edges\":[[0,1],[0,2],[1,2],[2,3],[2,4],[3,4],[4,0],[4,1]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r07-finite-directed-graph-0048","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-directed-graph"],"title_zh":"有限有向圖","stem_zh":"完整有限ordered-pair edge set與loop-free simple digraph convention。操作=digraph-in-out-degree；資料={\"n\":5,\"edges\":[[0,1],[1,2],[1,3],[2,3],[2,4],[3,4],[4,0],[4,1]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([1, 2, 2, 1, 2], [1, 2, 1, 2, 2], 8, 8, 8, 是)","explanation_zh":"Wave6第二層核對=([1, 2, 2, 1, 2], [1, 2, 1, 2, 2], 8, 8, 8, 是)；依完整明示資料獨立重算；(u,v)與(v,u)不同，平行同向edge不重複且loop在主模型拒絕；out-degree與in-degree各自總和等於edge count；不沿用undirected degree或connectivity。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限有向圖"]},"variant":{"family_id":"qf-mod39-digraph-in-out-degree","seed":202611087048,"parameters":{"n":5,"edges":[[0,1],[1,2],[1,3],[2,3],[2,4],[3,4],[4,0],[4,1]],"domain":"Complete_finite_loopless_simple_digraph","operation":"digraph-in-out-degree","semantic_template":"qf-mod39-digraph-in-out-degree/{\"edges\":[[0,1],[1,2],[1,3],[2,3],[2,4],[3,4],[4,0],[4,1]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r07-finite-directed-graph-0049","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-directed-graph"],"title_zh":"有限有向圖","stem_zh":"完整有限ordered-pair edge set與loop-free simple digraph convention。操作=digraph-in-out-degree；資料={\"n\":5,\"edges\":[[0,1],[0,2],[1,2],[1,3],[2,3],[2,4],[3,4],[4,0],[4,1]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([2, 2, 2, 1, 2], [1, 2, 2, 2, 2], 9, 9, 9, 是)","explanation_zh":"Wave6第二層核對=([2, 2, 2, 1, 2], [1, 2, 2, 2, 2], 9, 9, 9, 是)；依完整明示資料獨立重算；(u,v)與(v,u)不同，平行同向edge不重複且loop在主模型拒絕；out-degree與in-degree各自總和等於edge count；不沿用undirected degree或connectivity。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限有向圖"]},"variant":{"family_id":"qf-mod39-digraph-in-out-degree","seed":202611087049,"parameters":{"n":5,"edges":[[0,1],[0,2],[1,2],[1,3],[2,3],[2,4],[3,4],[4,0],[4,1]],"domain":"Complete_finite_loopless_simple_digraph","operation":"digraph-in-out-degree","semantic_template":"qf-mod39-digraph-in-out-degree/{\"edges\":[[0,1],[0,2],[1,2],[1,3],[2,3],[2,4],[3,4],[4,0],[4,1]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r07-finite-directed-graph-0050","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-directed-graph"],"title_zh":"有限有向圖","stem_zh":"完整有限ordered-pair edge set與loop-free simple digraph convention。操作=digraph-in-out-degree；資料={\"n\":5,\"edges\":[[0,1],[1,2],[2,3],[3,0],[3,4],[4,0],[4,1]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([1, 1, 1, 2, 2], [2, 2, 1, 1, 1], 7, 7, 7, 是)","explanation_zh":"Wave6第二層核對=([1, 1, 1, 2, 2], [2, 2, 1, 1, 1], 7, 7, 7, 是)；依完整明示資料獨立重算；(u,v)與(v,u)不同，平行同向edge不重複且loop在主模型拒絕；out-degree與in-degree各自總和等於edge count；不沿用undirected degree或connectivity。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限有向圖"]},"variant":{"family_id":"qf-mod39-digraph-in-out-degree","seed":202611087050,"parameters":{"n":5,"edges":[[0,1],[1,2],[2,3],[3,0],[3,4],[4,0],[4,1]],"domain":"Complete_finite_loopless_simple_digraph","operation":"digraph-in-out-degree","semantic_template":"qf-mod39-digraph-in-out-degree/{\"edges\":[[0,1],[1,2],[2,3],[3,0],[3,4],[4,0],[4,1]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r07-finite-directed-graph-0051","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-directed-graph"],"title_zh":"有限有向圖","stem_zh":"完整有限ordered-pair edge set與loop-free simple digraph convention。操作=digraph-directed-walk-audit；資料={\"n\":5,\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,0]],\"walk\":[0,1,2,3,4,0]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2, 3, 4, 0], [[0, 1, 是], [1, 2, 是], [2, 3, 是], [3, 4, 是], [4, 0, 是]], 是, edge_orientation_must_match)","explanation_zh":"Wave6第二層核對=([0, 1, 2, 3, 4, 0], [[0, 1, 是], [1, 2, 是], [2, 3, 是], [3, 4, 是], [4, 0, 是]], 是, edge_orientation_must_match)；依完整明示資料獨立重算；(u,v)與(v,u)不同，平行同向edge不重複且loop在主模型拒絕；out-degree與in-degree各自總和等於edge count；不沿用undirected degree或connectivity。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限有向圖"]},"variant":{"family_id":"qf-mod39-digraph-directed-walk-audit","seed":202611087051,"parameters":{"n":5,"edges":[[0,1],[1,2],[2,3],[3,4],[4,0]],"walk":[0,1,2,3,4,0],"domain":"Complete_finite_loopless_simple_digraph","operation":"digraph-directed-walk-audit","semantic_template":"qf-mod39-digraph-directed-walk-audit/{\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,0]],\"n\":5,\"walk\":[0,1,2,3,4,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r07-finite-directed-graph-0052","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-directed-graph"],"title_zh":"有限有向圖","stem_zh":"完整有限ordered-pair edge set與loop-free simple digraph convention。操作=digraph-directed-walk-audit；資料={\"n\":5,\"edges\":[[0,1],[0,2],[1,2],[2,3],[3,4],[4,0]],\"walk\":[0,1,2,3,4,0]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2, 3, 4, 0], [[0, 1, 是], [1, 2, 是], [2, 3, 是], [3, 4, 是], [4, 0, 是]], 是, edge_orientation_must_match)","explanation_zh":"Wave6第二層核對=([0, 1, 2, 3, 4, 0], [[0, 1, 是], [1, 2, 是], [2, 3, 是], [3, 4, 是], [4, 0, 是]], 是, edge_orientation_must_match)；依完整明示資料獨立重算；(u,v)與(v,u)不同，平行同向edge不重複且loop在主模型拒絕；out-degree與in-degree各自總和等於edge count；不沿用undirected degree或connectivity。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限有向圖"]},"variant":{"family_id":"qf-mod39-digraph-directed-walk-audit","seed":202611087052,"parameters":{"n":5,"edges":[[0,1],[0,2],[1,2],[2,3],[3,4],[4,0]],"walk":[0,1,2,3,4,0],"domain":"Complete_finite_loopless_simple_digraph","operation":"digraph-directed-walk-audit","semantic_template":"qf-mod39-digraph-directed-walk-audit/{\"edges\":[[0,1],[0,2],[1,2],[2,3],[3,4],[4,0]],\"n\":5,\"walk\":[0,1,2,3,4,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r07-finite-directed-graph-0053","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-directed-graph"],"title_zh":"有限有向圖","stem_zh":"完整有限ordered-pair edge set與loop-free simple digraph convention。操作=digraph-directed-walk-audit；資料={\"n\":5,\"edges\":[[0,1],[1,2],[1,3],[2,3],[3,4],[4,0]],\"walk\":[0,1,2,3,4,0]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2, 3, 4, 0], [[0, 1, 是], [1, 2, 是], [2, 3, 是], [3, 4, 是], [4, 0, 是]], 是, edge_orientation_must_match)","explanation_zh":"Wave6第二層核對=([0, 1, 2, 3, 4, 0], [[0, 1, 是], [1, 2, 是], [2, 3, 是], [3, 4, 是], [4, 0, 是]], 是, edge_orientation_must_match)；依完整明示資料獨立重算；(u,v)與(v,u)不同，平行同向edge不重複且loop在主模型拒絕；out-degree與in-degree各自總和等於edge count；不沿用undirected degree或connectivity。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限有向圖"]},"variant":{"family_id":"qf-mod39-digraph-directed-walk-audit","seed":202611087053,"parameters":{"n":5,"edges":[[0,1],[1,2],[1,3],[2,3],[3,4],[4,0]],"walk":[0,1,2,3,4,0],"domain":"Complete_finite_loopless_simple_digraph","operation":"digraph-directed-walk-audit","semantic_template":"qf-mod39-digraph-directed-walk-audit/{\"edges\":[[0,1],[1,2],[1,3],[2,3],[3,4],[4,0]],\"n\":5,\"walk\":[0,1,2,3,4,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r07-finite-directed-graph-0054","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-directed-graph"],"title_zh":"有限有向圖","stem_zh":"完整有限ordered-pair edge set與loop-free simple digraph convention。操作=digraph-directed-walk-audit；資料={\"n\":5,\"edges\":[[0,1],[0,2],[1,2],[1,3],[2,3],[3,4],[4,0]],\"walk\":[0,1,2,3,4,0]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2, 3, 4, 0], [[0, 1, 是], [1, 2, 是], [2, 3, 是], [3, 4, 是], [4, 0, 是]], 是, edge_orientation_must_match)","explanation_zh":"Wave6第二層核對=([0, 1, 2, 3, 4, 0], [[0, 1, 是], [1, 2, 是], [2, 3, 是], [3, 4, 是], [4, 0, 是]], 是, edge_orientation_must_match)；依完整明示資料獨立重算；(u,v)與(v,u)不同，平行同向edge不重複且loop在主模型拒絕；out-degree與in-degree各自總和等於edge count；不沿用undirected degree或connectivity。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限有向圖"]},"variant":{"family_id":"qf-mod39-digraph-directed-walk-audit","seed":202611087054,"parameters":{"n":5,"edges":[[0,1],[0,2],[1,2],[1,3],[2,3],[3,4],[4,0]],"walk":[0,1,2,3,4,0],"domain":"Complete_finite_loopless_simple_digraph","operation":"digraph-directed-walk-audit","semantic_template":"qf-mod39-digraph-directed-walk-audit/{\"edges\":[[0,1],[0,2],[1,2],[1,3],[2,3],[3,4],[4,0]],\"n\":5,\"walk\":[0,1,2,3,4,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r07-finite-directed-graph-0055","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-directed-graph"],"title_zh":"有限有向圖","stem_zh":"完整有限ordered-pair edge set與loop-free simple digraph convention。操作=digraph-directed-walk-audit；資料={\"n\":5,\"edges\":[[0,1],[1,2],[2,3],[2,4],[3,4],[4,0]],\"walk\":[0,1,2,3,4,0]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2, 3, 4, 0], [[0, 1, 是], [1, 2, 是], [2, 3, 是], [3, 4, 是], [4, 0, 是]], 是, edge_orientation_must_match)","explanation_zh":"Wave6第二層核對=([0, 1, 2, 3, 4, 0], [[0, 1, 是], [1, 2, 是], [2, 3, 是], [3, 4, 是], [4, 0, 是]], 是, edge_orientation_must_match)；依完整明示資料獨立重算；(u,v)與(v,u)不同，平行同向edge不重複且loop在主模型拒絕；out-degree與in-degree各自總和等於edge count；不沿用undirected degree或connectivity。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限有向圖"]},"variant":{"family_id":"qf-mod39-digraph-directed-walk-audit","seed":202611087055,"parameters":{"n":5,"edges":[[0,1],[1,2],[2,3],[2,4],[3,4],[4,0]],"walk":[0,1,2,3,4,0],"domain":"Complete_finite_loopless_simple_digraph","operation":"digraph-directed-walk-audit","semantic_template":"qf-mod39-digraph-directed-walk-audit/{\"edges\":[[0,1],[1,2],[2,3],[2,4],[3,4],[4,0]],\"n\":5,\"walk\":[0,1,2,3,4,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r07-finite-directed-graph-0056","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-directed-graph"],"title_zh":"有限有向圖","stem_zh":"完整有限ordered-pair edge set與loop-free simple digraph convention。操作=digraph-directed-walk-audit；資料={\"n\":5,\"edges\":[[0,1],[0,2],[1,2],[2,3],[2,4],[3,4],[4,0]],\"walk\":[0,1,2,3,4,0]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2, 3, 4, 0], [[0, 1, 是], [1, 2, 是], [2, 3, 是], [3, 4, 是], [4, 0, 是]], 是, edge_orientation_must_match)","explanation_zh":"Wave6第二層核對=([0, 1, 2, 3, 4, 0], [[0, 1, 是], [1, 2, 是], [2, 3, 是], [3, 4, 是], [4, 0, 是]], 是, edge_orientation_must_match)；依完整明示資料獨立重算；(u,v)與(v,u)不同，平行同向edge不重複且loop在主模型拒絕；out-degree與in-degree各自總和等於edge count；不沿用undirected degree或connectivity。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限有向圖"]},"variant":{"family_id":"qf-mod39-digraph-directed-walk-audit","seed":202611087056,"parameters":{"n":5,"edges":[[0,1],[0,2],[1,2],[2,3],[2,4],[3,4],[4,0]],"walk":[0,1,2,3,4,0],"domain":"Complete_finite_loopless_simple_digraph","operation":"digraph-directed-walk-audit","semantic_template":"qf-mod39-digraph-directed-walk-audit/{\"edges\":[[0,1],[0,2],[1,2],[2,3],[2,4],[3,4],[4,0]],\"n\":5,\"walk\":[0,1,2,3,4,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r07-finite-directed-graph-0057","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-directed-graph"],"title_zh":"有限有向圖","stem_zh":"完整有限ordered-pair edge set與loop-free simple digraph convention。操作=digraph-directed-walk-audit；資料={\"n\":5,\"edges\":[[0,1],[1,2],[1,3],[2,3],[2,4],[3,4],[4,0]],\"walk\":[0,1,2,3,4,0]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2, 3, 4, 0], [[0, 1, 是], [1, 2, 是], [2, 3, 是], [3, 4, 是], [4, 0, 是]], 是, edge_orientation_must_match)","explanation_zh":"Wave6第二層核對=([0, 1, 2, 3, 4, 0], [[0, 1, 是], [1, 2, 是], [2, 3, 是], [3, 4, 是], [4, 0, 是]], 是, edge_orientation_must_match)；依完整明示資料獨立重算；(u,v)與(v,u)不同，平行同向edge不重複且loop在主模型拒絕；out-degree與in-degree各自總和等於edge count；不沿用undirected degree或connectivity。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限有向圖"]},"variant":{"family_id":"qf-mod39-digraph-directed-walk-audit","seed":202611087057,"parameters":{"n":5,"edges":[[0,1],[1,2],[1,3],[2,3],[2,4],[3,4],[4,0]],"walk":[0,1,2,3,4,0],"domain":"Complete_finite_loopless_simple_digraph","operation":"digraph-directed-walk-audit","semantic_template":"qf-mod39-digraph-directed-walk-audit/{\"edges\":[[0,1],[1,2],[1,3],[2,3],[2,4],[3,4],[4,0]],\"n\":5,\"walk\":[0,1,2,3,4,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r07-finite-directed-graph-0058","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-directed-graph"],"title_zh":"有限有向圖","stem_zh":"完整有限ordered-pair edge set與loop-free simple digraph convention。操作=digraph-directed-walk-audit；資料={\"n\":5,\"edges\":[[0,1],[0,2],[1,2],[1,3],[2,3],[2,4],[3,4],[4,0]],\"walk\":[0,1,2,3,4,0]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2, 3, 4, 0], [[0, 1, 是], [1, 2, 是], [2, 3, 是], [3, 4, 是], [4, 0, 是]], 是, edge_orientation_must_match)","explanation_zh":"Wave6第二層核對=([0, 1, 2, 3, 4, 0], [[0, 1, 是], [1, 2, 是], [2, 3, 是], [3, 4, 是], [4, 0, 是]], 是, edge_orientation_must_match)；依完整明示資料獨立重算；(u,v)與(v,u)不同，平行同向edge不重複且loop在主模型拒絕；out-degree與in-degree各自總和等於edge count；不沿用undirected degree或connectivity。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限有向圖"]},"variant":{"family_id":"qf-mod39-digraph-directed-walk-audit","seed":202611087058,"parameters":{"n":5,"edges":[[0,1],[0,2],[1,2],[1,3],[2,3],[2,4],[3,4],[4,0]],"walk":[0,1,2,3,4,0],"domain":"Complete_finite_loopless_simple_digraph","operation":"digraph-directed-walk-audit","semantic_template":"qf-mod39-digraph-directed-walk-audit/{\"edges\":[[0,1],[0,2],[1,2],[1,3],[2,3],[2,4],[3,4],[4,0]],\"n\":5,\"walk\":[0,1,2,3,4,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r07-finite-directed-graph-0059","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-directed-graph"],"title_zh":"有限有向圖","stem_zh":"完整有限ordered-pair edge set與loop-free simple digraph convention。操作=digraph-directed-walk-audit；資料={\"n\":5,\"edges\":[[0,1],[1,2],[2,3],[3,0],[3,4],[4,0]],\"walk\":[0,1,2,3,4,0]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2, 3, 4, 0], [[0, 1, 是], [1, 2, 是], [2, 3, 是], [3, 4, 是], [4, 0, 是]], 是, edge_orientation_must_match)","explanation_zh":"Wave6第二層核對=([0, 1, 2, 3, 4, 0], [[0, 1, 是], [1, 2, 是], [2, 3, 是], [3, 4, 是], [4, 0, 是]], 是, edge_orientation_must_match)；依完整明示資料獨立重算；(u,v)與(v,u)不同，平行同向edge不重複且loop在主模型拒絕；out-degree與in-degree各自總和等於edge count；不沿用undirected degree或connectivity。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限有向圖"]},"variant":{"family_id":"qf-mod39-digraph-directed-walk-audit","seed":202611087059,"parameters":{"n":5,"edges":[[0,1],[1,2],[2,3],[3,0],[3,4],[4,0]],"walk":[0,1,2,3,4,0],"domain":"Complete_finite_loopless_simple_digraph","operation":"digraph-directed-walk-audit","semantic_template":"qf-mod39-digraph-directed-walk-audit/{\"edges\":[[0,1],[1,2],[2,3],[3,0],[3,4],[4,0]],\"n\":5,\"walk\":[0,1,2,3,4,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r07-finite-directed-graph-0060","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-directed-graph"],"title_zh":"有限有向圖","stem_zh":"完整有限ordered-pair edge set與loop-free simple digraph convention。操作=digraph-directed-walk-audit；資料={\"n\":5,\"edges\":[[0,1],[0,2],[1,2],[2,3],[3,0],[3,4],[4,0]],\"walk\":[0,1,2,3,4,0]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2, 3, 4, 0], [[0, 1, 是], [1, 2, 是], [2, 3, 是], [3, 4, 是], [4, 0, 是]], 是, edge_orientation_must_match)","explanation_zh":"Wave6第二層核對=([0, 1, 2, 3, 4, 0], [[0, 1, 是], [1, 2, 是], [2, 3, 是], [3, 4, 是], [4, 0, 是]], 是, edge_orientation_must_match)；依完整明示資料獨立重算；(u,v)與(v,u)不同，平行同向edge不重複且loop在主模型拒絕；out-degree與in-degree各自總和等於edge count；不沿用undirected degree或connectivity。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限有向圖"]},"variant":{"family_id":"qf-mod39-digraph-directed-walk-audit","seed":202611087060,"parameters":{"n":5,"edges":[[0,1],[0,2],[1,2],[2,3],[3,0],[3,4],[4,0]],"walk":[0,1,2,3,4,0],"domain":"Complete_finite_loopless_simple_digraph","operation":"digraph-directed-walk-audit","semantic_template":"qf-mod39-digraph-directed-walk-audit/{\"edges\":[[0,1],[0,2],[1,2],[2,3],[3,0],[3,4],[4,0]],\"n\":5,\"walk\":[0,1,2,3,4,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r07-finite-directed-graph-0061","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-directed-graph"],"title_zh":"有限有向圖","stem_zh":"完整有限ordered-pair edge set與loop-free simple digraph convention。操作=digraph-directed-walk-audit；資料={\"n\":5,\"edges\":[[0,1],[1,2],[1,3],[2,3],[3,0],[3,4],[4,0]],\"walk\":[0,1,2,3,4,0]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2, 3, 4, 0], [[0, 1, 是], [1, 2, 是], [2, 3, 是], [3, 4, 是], [4, 0, 是]], 是, edge_orientation_must_match)","explanation_zh":"Wave6第二層核對=([0, 1, 2, 3, 4, 0], [[0, 1, 是], [1, 2, 是], [2, 3, 是], [3, 4, 是], [4, 0, 是]], 是, edge_orientation_must_match)；依完整明示資料獨立重算；(u,v)與(v,u)不同，平行同向edge不重複且loop在主模型拒絕；out-degree與in-degree各自總和等於edge count；不沿用undirected degree或connectivity。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限有向圖"]},"variant":{"family_id":"qf-mod39-digraph-directed-walk-audit","seed":202611087061,"parameters":{"n":5,"edges":[[0,1],[1,2],[1,3],[2,3],[3,0],[3,4],[4,0]],"walk":[0,1,2,3,4,0],"domain":"Complete_finite_loopless_simple_digraph","operation":"digraph-directed-walk-audit","semantic_template":"qf-mod39-digraph-directed-walk-audit/{\"edges\":[[0,1],[1,2],[1,3],[2,3],[3,0],[3,4],[4,0]],\"n\":5,\"walk\":[0,1,2,3,4,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r07-finite-directed-graph-0062","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-directed-graph"],"title_zh":"有限有向圖","stem_zh":"完整有限ordered-pair edge set與loop-free simple digraph convention。操作=digraph-directed-walk-audit；資料={\"n\":5,\"edges\":[[0,1],[0,2],[1,2],[1,3],[2,3],[3,0],[3,4],[4,0]],\"walk\":[0,1,2,3,4,0]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2, 3, 4, 0], [[0, 1, 是], [1, 2, 是], [2, 3, 是], [3, 4, 是], [4, 0, 是]], 是, edge_orientation_must_match)","explanation_zh":"Wave6第二層核對=([0, 1, 2, 3, 4, 0], [[0, 1, 是], [1, 2, 是], [2, 3, 是], [3, 4, 是], [4, 0, 是]], 是, edge_orientation_must_match)；依完整明示資料獨立重算；(u,v)與(v,u)不同，平行同向edge不重複且loop在主模型拒絕；out-degree與in-degree各自總和等於edge count；不沿用undirected degree或connectivity。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限有向圖"]},"variant":{"family_id":"qf-mod39-digraph-directed-walk-audit","seed":202611087062,"parameters":{"n":5,"edges":[[0,1],[0,2],[1,2],[1,3],[2,3],[3,0],[3,4],[4,0]],"walk":[0,1,2,3,4,0],"domain":"Complete_finite_loopless_simple_digraph","operation":"digraph-directed-walk-audit","semantic_template":"qf-mod39-digraph-directed-walk-audit/{\"edges\":[[0,1],[0,2],[1,2],[1,3],[2,3],[3,0],[3,4],[4,0]],\"n\":5,\"walk\":[0,1,2,3,4,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r07-finite-directed-graph-0063","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-directed-graph"],"title_zh":"有限有向圖","stem_zh":"完整有限ordered-pair edge set與loop-free simple digraph convention。操作=digraph-directed-walk-audit；資料={\"n\":5,\"edges\":[[0,1],[1,2],[2,3],[2,4],[3,0],[3,4],[4,0]],\"walk\":[0,1,2,3,4,0]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2, 3, 4, 0], [[0, 1, 是], [1, 2, 是], [2, 3, 是], [3, 4, 是], [4, 0, 是]], 是, edge_orientation_must_match)","explanation_zh":"Wave6第二層核對=([0, 1, 2, 3, 4, 0], [[0, 1, 是], [1, 2, 是], [2, 3, 是], [3, 4, 是], [4, 0, 是]], 是, edge_orientation_must_match)；依完整明示資料獨立重算；(u,v)與(v,u)不同，平行同向edge不重複且loop在主模型拒絕；out-degree與in-degree各自總和等於edge count；不沿用undirected degree或connectivity。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限有向圖"]},"variant":{"family_id":"qf-mod39-digraph-directed-walk-audit","seed":202611087063,"parameters":{"n":5,"edges":[[0,1],[1,2],[2,3],[2,4],[3,0],[3,4],[4,0]],"walk":[0,1,2,3,4,0],"domain":"Complete_finite_loopless_simple_digraph","operation":"digraph-directed-walk-audit","semantic_template":"qf-mod39-digraph-directed-walk-audit/{\"edges\":[[0,1],[1,2],[2,3],[2,4],[3,0],[3,4],[4,0]],\"n\":5,\"walk\":[0,1,2,3,4,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r07-finite-directed-graph-0064","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-directed-graph"],"title_zh":"有限有向圖","stem_zh":"完整有限ordered-pair edge set與loop-free simple digraph convention。操作=digraph-directed-walk-audit；資料={\"n\":5,\"edges\":[[0,1],[0,2],[1,2],[2,3],[2,4],[3,0],[3,4],[4,0]],\"walk\":[0,1,2,3,4,0]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2, 3, 4, 0], [[0, 1, 是], [1, 2, 是], [2, 3, 是], [3, 4, 是], [4, 0, 是]], 是, edge_orientation_must_match)","explanation_zh":"Wave6第二層核對=([0, 1, 2, 3, 4, 0], [[0, 1, 是], [1, 2, 是], [2, 3, 是], [3, 4, 是], [4, 0, 是]], 是, edge_orientation_must_match)；依完整明示資料獨立重算；(u,v)與(v,u)不同，平行同向edge不重複且loop在主模型拒絕；out-degree與in-degree各自總和等於edge count；不沿用undirected degree或connectivity。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限有向圖"]},"variant":{"family_id":"qf-mod39-digraph-directed-walk-audit","seed":202611087064,"parameters":{"n":5,"edges":[[0,1],[0,2],[1,2],[2,3],[2,4],[3,0],[3,4],[4,0]],"walk":[0,1,2,3,4,0],"domain":"Complete_finite_loopless_simple_digraph","operation":"digraph-directed-walk-audit","semantic_template":"qf-mod39-digraph-directed-walk-audit/{\"edges\":[[0,1],[0,2],[1,2],[2,3],[2,4],[3,0],[3,4],[4,0]],\"n\":5,\"walk\":[0,1,2,3,4,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r07-finite-directed-graph-0065","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-directed-graph"],"title_zh":"有限有向圖","stem_zh":"完整有限ordered-pair edge set與loop-free simple digraph convention。操作=digraph-directed-walk-audit；資料={\"n\":5,\"edges\":[[0,1],[1,2],[1,3],[2,3],[2,4],[3,0],[3,4],[4,0]],\"walk\":[0,1,2,3,4,0]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2, 3, 4, 0], [[0, 1, 是], [1, 2, 是], [2, 3, 是], [3, 4, 是], [4, 0, 是]], 是, edge_orientation_must_match)","explanation_zh":"Wave6第二層核對=([0, 1, 2, 3, 4, 0], [[0, 1, 是], [1, 2, 是], [2, 3, 是], [3, 4, 是], [4, 0, 是]], 是, edge_orientation_must_match)；依完整明示資料獨立重算；(u,v)與(v,u)不同，平行同向edge不重複且loop在主模型拒絕；out-degree與in-degree各自總和等於edge count；不沿用undirected degree或connectivity。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限有向圖"]},"variant":{"family_id":"qf-mod39-digraph-directed-walk-audit","seed":202611087065,"parameters":{"n":5,"edges":[[0,1],[1,2],[1,3],[2,3],[2,4],[3,0],[3,4],[4,0]],"walk":[0,1,2,3,4,0],"domain":"Complete_finite_loopless_simple_digraph","operation":"digraph-directed-walk-audit","semantic_template":"qf-mod39-digraph-directed-walk-audit/{\"edges\":[[0,1],[1,2],[1,3],[2,3],[2,4],[3,0],[3,4],[4,0]],\"n\":5,\"walk\":[0,1,2,3,4,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r07-finite-directed-graph-0066","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-directed-graph"],"title_zh":"有限有向圖","stem_zh":"完整有限ordered-pair edge set與loop-free simple digraph convention。操作=digraph-directed-walk-audit；資料={\"n\":5,\"edges\":[[0,1],[0,2],[1,2],[1,3],[2,3],[2,4],[3,0],[3,4],[4,0]],\"walk\":[0,1,2,3,4,0]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2, 3, 4, 0], [[0, 1, 是], [1, 2, 是], [2, 3, 是], [3, 4, 是], [4, 0, 是]], 是, edge_orientation_must_match)","explanation_zh":"Wave6第二層核對=([0, 1, 2, 3, 4, 0], [[0, 1, 是], [1, 2, 是], [2, 3, 是], [3, 4, 是], [4, 0, 是]], 是, edge_orientation_must_match)；依完整明示資料獨立重算；(u,v)與(v,u)不同，平行同向edge不重複且loop在主模型拒絕；out-degree與in-degree各自總和等於edge count；不沿用undirected degree或connectivity。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限有向圖"]},"variant":{"family_id":"qf-mod39-digraph-directed-walk-audit","seed":202611087066,"parameters":{"n":5,"edges":[[0,1],[0,2],[1,2],[1,3],[2,3],[2,4],[3,0],[3,4],[4,0]],"walk":[0,1,2,3,4,0],"domain":"Complete_finite_loopless_simple_digraph","operation":"digraph-directed-walk-audit","semantic_template":"qf-mod39-digraph-directed-walk-audit/{\"edges\":[[0,1],[0,2],[1,2],[1,3],[2,3],[2,4],[3,0],[3,4],[4,0]],\"n\":5,\"walk\":[0,1,2,3,4,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r07-finite-directed-graph-0067","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-directed-graph"],"title_zh":"有限有向圖","stem_zh":"完整有限ordered-pair edge set與loop-free simple digraph convention。操作=digraph-directed-walk-audit；資料={\"n\":5,\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,0],[4,1]],\"walk\":[0,1,2,3,4,0]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2, 3, 4, 0], [[0, 1, 是], [1, 2, 是], [2, 3, 是], [3, 4, 是], [4, 0, 是]], 是, edge_orientation_must_match)","explanation_zh":"Wave6第二層核對=([0, 1, 2, 3, 4, 0], [[0, 1, 是], [1, 2, 是], [2, 3, 是], [3, 4, 是], [4, 0, 是]], 是, edge_orientation_must_match)；依完整明示資料獨立重算；(u,v)與(v,u)不同，平行同向edge不重複且loop在主模型拒絕；out-degree與in-degree各自總和等於edge count；不沿用undirected degree或connectivity。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限有向圖"]},"variant":{"family_id":"qf-mod39-digraph-directed-walk-audit","seed":202611087067,"parameters":{"n":5,"edges":[[0,1],[1,2],[2,3],[3,4],[4,0],[4,1]],"walk":[0,1,2,3,4,0],"domain":"Complete_finite_loopless_simple_digraph","operation":"digraph-directed-walk-audit","semantic_template":"qf-mod39-digraph-directed-walk-audit/{\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,0],[4,1]],\"n\":5,\"walk\":[0,1,2,3,4,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r07-finite-directed-graph-0068","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-directed-graph"],"title_zh":"有限有向圖","stem_zh":"完整有限ordered-pair edge set與loop-free simple digraph convention。操作=digraph-directed-walk-audit；資料={\"n\":5,\"edges\":[[0,1],[0,2],[1,2],[2,3],[3,4],[4,0],[4,1]],\"walk\":[0,1,2,3,4,0]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2, 3, 4, 0], [[0, 1, 是], [1, 2, 是], [2, 3, 是], [3, 4, 是], [4, 0, 是]], 是, edge_orientation_must_match)","explanation_zh":"Wave6第二層核對=([0, 1, 2, 3, 4, 0], [[0, 1, 是], [1, 2, 是], [2, 3, 是], [3, 4, 是], [4, 0, 是]], 是, edge_orientation_must_match)；依完整明示資料獨立重算；(u,v)與(v,u)不同，平行同向edge不重複且loop在主模型拒絕；out-degree與in-degree各自總和等於edge count；不沿用undirected degree或connectivity。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限有向圖"]},"variant":{"family_id":"qf-mod39-digraph-directed-walk-audit","seed":202611087068,"parameters":{"n":5,"edges":[[0,1],[0,2],[1,2],[2,3],[3,4],[4,0],[4,1]],"walk":[0,1,2,3,4,0],"domain":"Complete_finite_loopless_simple_digraph","operation":"digraph-directed-walk-audit","semantic_template":"qf-mod39-digraph-directed-walk-audit/{\"edges\":[[0,1],[0,2],[1,2],[2,3],[3,4],[4,0],[4,1]],\"n\":5,\"walk\":[0,1,2,3,4,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r07-finite-directed-graph-0069","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-directed-graph"],"title_zh":"有限有向圖","stem_zh":"完整有限ordered-pair edge set與loop-free simple digraph convention。操作=digraph-directed-walk-audit；資料={\"n\":5,\"edges\":[[0,1],[1,2],[1,3],[2,3],[3,4],[4,0],[4,1]],\"walk\":[0,1,2,3,4,0]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2, 3, 4, 0], [[0, 1, 是], [1, 2, 是], [2, 3, 是], [3, 4, 是], [4, 0, 是]], 是, edge_orientation_must_match)","explanation_zh":"Wave6第二層核對=([0, 1, 2, 3, 4, 0], [[0, 1, 是], [1, 2, 是], [2, 3, 是], [3, 4, 是], [4, 0, 是]], 是, edge_orientation_must_match)；依完整明示資料獨立重算；(u,v)與(v,u)不同，平行同向edge不重複且loop在主模型拒絕；out-degree與in-degree各自總和等於edge count；不沿用undirected degree或connectivity。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限有向圖"]},"variant":{"family_id":"qf-mod39-digraph-directed-walk-audit","seed":202611087069,"parameters":{"n":5,"edges":[[0,1],[1,2],[1,3],[2,3],[3,4],[4,0],[4,1]],"walk":[0,1,2,3,4,0],"domain":"Complete_finite_loopless_simple_digraph","operation":"digraph-directed-walk-audit","semantic_template":"qf-mod39-digraph-directed-walk-audit/{\"edges\":[[0,1],[1,2],[1,3],[2,3],[3,4],[4,0],[4,1]],\"n\":5,\"walk\":[0,1,2,3,4,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r07-finite-directed-graph-0070","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-directed-graph"],"title_zh":"有限有向圖","stem_zh":"完整有限ordered-pair edge set與loop-free simple digraph convention。操作=digraph-directed-walk-audit；資料={\"n\":5,\"edges\":[[0,1],[0,2],[1,2],[1,3],[2,3],[3,4],[4,0],[4,1]],\"walk\":[0,1,2,3,4,0]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2, 3, 4, 0], [[0, 1, 是], [1, 2, 是], [2, 3, 是], [3, 4, 是], [4, 0, 是]], 是, edge_orientation_must_match)","explanation_zh":"Wave6第二層核對=([0, 1, 2, 3, 4, 0], [[0, 1, 是], [1, 2, 是], [2, 3, 是], [3, 4, 是], [4, 0, 是]], 是, edge_orientation_must_match)；依完整明示資料獨立重算；(u,v)與(v,u)不同，平行同向edge不重複且loop在主模型拒絕；out-degree與in-degree各自總和等於edge count；不沿用undirected degree或connectivity。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限有向圖"]},"variant":{"family_id":"qf-mod39-digraph-directed-walk-audit","seed":202611087070,"parameters":{"n":5,"edges":[[0,1],[0,2],[1,2],[1,3],[2,3],[3,4],[4,0],[4,1]],"walk":[0,1,2,3,4,0],"domain":"Complete_finite_loopless_simple_digraph","operation":"digraph-directed-walk-audit","semantic_template":"qf-mod39-digraph-directed-walk-audit/{\"edges\":[[0,1],[0,2],[1,2],[1,3],[2,3],[3,4],[4,0],[4,1]],\"n\":5,\"walk\":[0,1,2,3,4,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r07-finite-directed-graph-0071","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-directed-graph"],"title_zh":"有限有向圖","stem_zh":"完整有限ordered-pair edge set與loop-free simple digraph convention。操作=digraph-directed-walk-audit；資料={\"n\":5,\"edges\":[[0,1],[1,2],[2,3],[2,4],[3,4],[4,0],[4,1]],\"walk\":[0,1,2,3,4,0]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2, 3, 4, 0], [[0, 1, 是], [1, 2, 是], [2, 3, 是], [3, 4, 是], [4, 0, 是]], 是, edge_orientation_must_match)","explanation_zh":"Wave6第二層核對=([0, 1, 2, 3, 4, 0], [[0, 1, 是], [1, 2, 是], [2, 3, 是], [3, 4, 是], [4, 0, 是]], 是, edge_orientation_must_match)；依完整明示資料獨立重算；(u,v)與(v,u)不同，平行同向edge不重複且loop在主模型拒絕；out-degree與in-degree各自總和等於edge count；不沿用undirected degree或connectivity。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限有向圖"]},"variant":{"family_id":"qf-mod39-digraph-directed-walk-audit","seed":202611087071,"parameters":{"n":5,"edges":[[0,1],[1,2],[2,3],[2,4],[3,4],[4,0],[4,1]],"walk":[0,1,2,3,4,0],"domain":"Complete_finite_loopless_simple_digraph","operation":"digraph-directed-walk-audit","semantic_template":"qf-mod39-digraph-directed-walk-audit/{\"edges\":[[0,1],[1,2],[2,3],[2,4],[3,4],[4,0],[4,1]],\"n\":5,\"walk\":[0,1,2,3,4,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r07-finite-directed-graph-0072","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-directed-graph"],"title_zh":"有限有向圖","stem_zh":"完整有限ordered-pair edge set與loop-free simple digraph convention。操作=digraph-directed-walk-audit；資料={\"n\":5,\"edges\":[[0,1],[0,2],[1,2],[2,3],[2,4],[3,4],[4,0],[4,1]],\"walk\":[0,1,2,3,4,0]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2, 3, 4, 0], [[0, 1, 是], [1, 2, 是], [2, 3, 是], [3, 4, 是], [4, 0, 是]], 是, edge_orientation_must_match)","explanation_zh":"Wave6第二層核對=([0, 1, 2, 3, 4, 0], [[0, 1, 是], [1, 2, 是], [2, 3, 是], [3, 4, 是], [4, 0, 是]], 是, edge_orientation_must_match)；依完整明示資料獨立重算；(u,v)與(v,u)不同，平行同向edge不重複且loop在主模型拒絕；out-degree與in-degree各自總和等於edge count；不沿用undirected degree或connectivity。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限有向圖"]},"variant":{"family_id":"qf-mod39-digraph-directed-walk-audit","seed":202611087072,"parameters":{"n":5,"edges":[[0,1],[0,2],[1,2],[2,3],[2,4],[3,4],[4,0],[4,1]],"walk":[0,1,2,3,4,0],"domain":"Complete_finite_loopless_simple_digraph","operation":"digraph-directed-walk-audit","semantic_template":"qf-mod39-digraph-directed-walk-audit/{\"edges\":[[0,1],[0,2],[1,2],[2,3],[2,4],[3,4],[4,0],[4,1]],\"n\":5,\"walk\":[0,1,2,3,4,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r07-finite-directed-graph-0073","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-directed-graph"],"title_zh":"有限有向圖","stem_zh":"完整有限ordered-pair edge set與loop-free simple digraph convention。操作=digraph-directed-walk-audit；資料={\"n\":5,\"edges\":[[0,1],[1,2],[1,3],[2,3],[2,4],[3,4],[4,0],[4,1]],\"walk\":[0,1,2,3,4,0]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2, 3, 4, 0], [[0, 1, 是], [1, 2, 是], [2, 3, 是], [3, 4, 是], [4, 0, 是]], 是, edge_orientation_must_match)","explanation_zh":"Wave6第二層核對=([0, 1, 2, 3, 4, 0], [[0, 1, 是], [1, 2, 是], [2, 3, 是], [3, 4, 是], [4, 0, 是]], 是, edge_orientation_must_match)；依完整明示資料獨立重算；(u,v)與(v,u)不同，平行同向edge不重複且loop在主模型拒絕；out-degree與in-degree各自總和等於edge count；不沿用undirected degree或connectivity。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限有向圖"]},"variant":{"family_id":"qf-mod39-digraph-directed-walk-audit","seed":202611087073,"parameters":{"n":5,"edges":[[0,1],[1,2],[1,3],[2,3],[2,4],[3,4],[4,0],[4,1]],"walk":[0,1,2,3,4,0],"domain":"Complete_finite_loopless_simple_digraph","operation":"digraph-directed-walk-audit","semantic_template":"qf-mod39-digraph-directed-walk-audit/{\"edges\":[[0,1],[1,2],[1,3],[2,3],[2,4],[3,4],[4,0],[4,1]],\"n\":5,\"walk\":[0,1,2,3,4,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r07-finite-directed-graph-0074","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-directed-graph"],"title_zh":"有限有向圖","stem_zh":"完整有限ordered-pair edge set與loop-free simple digraph convention。操作=digraph-directed-walk-audit；資料={\"n\":5,\"edges\":[[0,1],[0,2],[1,2],[1,3],[2,3],[2,4],[3,4],[4,0],[4,1]],\"walk\":[0,1,2,3,4,0]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2, 3, 4, 0], [[0, 1, 是], [1, 2, 是], [2, 3, 是], [3, 4, 是], [4, 0, 是]], 是, edge_orientation_must_match)","explanation_zh":"Wave6第二層核對=([0, 1, 2, 3, 4, 0], [[0, 1, 是], [1, 2, 是], [2, 3, 是], [3, 4, 是], [4, 0, 是]], 是, edge_orientation_must_match)；依完整明示資料獨立重算；(u,v)與(v,u)不同，平行同向edge不重複且loop在主模型拒絕；out-degree與in-degree各自總和等於edge count；不沿用undirected degree或connectivity。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限有向圖"]},"variant":{"family_id":"qf-mod39-digraph-directed-walk-audit","seed":202611087074,"parameters":{"n":5,"edges":[[0,1],[0,2],[1,2],[1,3],[2,3],[2,4],[3,4],[4,0],[4,1]],"walk":[0,1,2,3,4,0],"domain":"Complete_finite_loopless_simple_digraph","operation":"digraph-directed-walk-audit","semantic_template":"qf-mod39-digraph-directed-walk-audit/{\"edges\":[[0,1],[0,2],[1,2],[1,3],[2,3],[2,4],[3,4],[4,0],[4,1]],\"n\":5,\"walk\":[0,1,2,3,4,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r07-finite-directed-graph-0075","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-directed-graph"],"title_zh":"有限有向圖","stem_zh":"完整有限ordered-pair edge set與loop-free simple digraph convention。操作=digraph-directed-walk-audit；資料={\"n\":5,\"edges\":[[0,1],[1,2],[2,3],[3,0],[3,4],[4,0],[4,1]],\"walk\":[0,1,2,3,4,0]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2, 3, 4, 0], [[0, 1, 是], [1, 2, 是], [2, 3, 是], [3, 4, 是], [4, 0, 是]], 是, edge_orientation_must_match)","explanation_zh":"Wave6第二層核對=([0, 1, 2, 3, 4, 0], [[0, 1, 是], [1, 2, 是], [2, 3, 是], [3, 4, 是], [4, 0, 是]], 是, edge_orientation_must_match)；依完整明示資料獨立重算；(u,v)與(v,u)不同，平行同向edge不重複且loop在主模型拒絕；out-degree與in-degree各自總和等於edge count；不沿用undirected degree或connectivity。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限有向圖"]},"variant":{"family_id":"qf-mod39-digraph-directed-walk-audit","seed":202611087075,"parameters":{"n":5,"edges":[[0,1],[1,2],[2,3],[3,0],[3,4],[4,0],[4,1]],"walk":[0,1,2,3,4,0],"domain":"Complete_finite_loopless_simple_digraph","operation":"digraph-directed-walk-audit","semantic_template":"qf-mod39-digraph-directed-walk-audit/{\"edges\":[[0,1],[1,2],[2,3],[3,0],[3,4],[4,0],[4,1]],\"n\":5,\"walk\":[0,1,2,3,4,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r07-finite-directed-graph-0076","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-directed-graph"],"title_zh":"有限有向圖","stem_zh":"完整有限ordered-pair edge set與loop-free simple digraph convention。操作=digraph-loop-boundary；資料={\"n\":5,\"edges\":[[0,0],[0,1],[1,2],[2,3],[3,4],[4,0]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 0]], 否, [2, 1, 1, 1, 1], [2, 1, 1, 1, 1], loops_rejected_by_declared_loop_free_simple_digraph_convention)","explanation_zh":"Wave6第二層核對=([[0, 0]], 否, [2, 1, 1, 1, 1], [2, 1, 1, 1, 1], loops_rejected_by_declared_loop_free_simple_digraph_convention)；依完整明示資料獨立重算；(u,v)與(v,u)不同，平行同向edge不重複且loop在主模型拒絕；out-degree與in-degree各自總和等於edge count；不沿用undirected degree或connectivity。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限有向圖"]},"variant":{"family_id":"qf-mod39-digraph-loop-boundary","seed":202611087076,"parameters":{"n":5,"edges":[[0,0],[0,1],[1,2],[2,3],[3,4],[4,0]],"domain":"Complete_finite_loopless_simple_digraph","operation":"digraph-loop-boundary","semantic_template":"qf-mod39-digraph-loop-boundary/{\"edges\":[[0,0],[0,1],[1,2],[2,3],[3,4],[4,0]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r07-finite-directed-graph-0077","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-directed-graph"],"title_zh":"有限有向圖","stem_zh":"完整有限ordered-pair edge set與loop-free simple digraph convention。操作=digraph-loop-boundary；資料={\"n\":5,\"edges\":[[0,0],[0,1],[0,2],[1,2],[2,3],[3,4],[4,0]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 0]], 否, [3, 1, 1, 1, 1], [2, 1, 2, 1, 1], loops_rejected_by_declared_loop_free_simple_digraph_convention)","explanation_zh":"Wave6第二層核對=([[0, 0]], 否, [3, 1, 1, 1, 1], [2, 1, 2, 1, 1], loops_rejected_by_declared_loop_free_simple_digraph_convention)；依完整明示資料獨立重算；(u,v)與(v,u)不同，平行同向edge不重複且loop在主模型拒絕；out-degree與in-degree各自總和等於edge count；不沿用undirected degree或connectivity。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限有向圖"]},"variant":{"family_id":"qf-mod39-digraph-loop-boundary","seed":202611087077,"parameters":{"n":5,"edges":[[0,0],[0,1],[0,2],[1,2],[2,3],[3,4],[4,0]],"domain":"Complete_finite_loopless_simple_digraph","operation":"digraph-loop-boundary","semantic_template":"qf-mod39-digraph-loop-boundary/{\"edges\":[[0,0],[0,1],[0,2],[1,2],[2,3],[3,4],[4,0]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r07-finite-directed-graph-0078","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-directed-graph"],"title_zh":"有限有向圖","stem_zh":"完整有限ordered-pair edge set與loop-free simple digraph convention。操作=digraph-loop-boundary；資料={\"n\":5,\"edges\":[[0,0],[0,1],[1,2],[1,3],[2,3],[3,4],[4,0]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 0]], 否, [2, 2, 1, 1, 1], [2, 1, 1, 2, 1], loops_rejected_by_declared_loop_free_simple_digraph_convention)","explanation_zh":"Wave6第二層核對=([[0, 0]], 否, [2, 2, 1, 1, 1], [2, 1, 1, 2, 1], loops_rejected_by_declared_loop_free_simple_digraph_convention)；依完整明示資料獨立重算；(u,v)與(v,u)不同，平行同向edge不重複且loop在主模型拒絕；out-degree與in-degree各自總和等於edge count；不沿用undirected degree或connectivity。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限有向圖"]},"variant":{"family_id":"qf-mod39-digraph-loop-boundary","seed":202611087078,"parameters":{"n":5,"edges":[[0,0],[0,1],[1,2],[1,3],[2,3],[3,4],[4,0]],"domain":"Complete_finite_loopless_simple_digraph","operation":"digraph-loop-boundary","semantic_template":"qf-mod39-digraph-loop-boundary/{\"edges\":[[0,0],[0,1],[1,2],[1,3],[2,3],[3,4],[4,0]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r07-finite-directed-graph-0079","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-directed-graph"],"title_zh":"有限有向圖","stem_zh":"完整有限ordered-pair edge set與loop-free simple digraph convention。操作=digraph-loop-boundary；資料={\"n\":5,\"edges\":[[0,0],[0,1],[0,2],[1,2],[1,3],[2,3],[3,4],[4,0]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 0]], 否, [3, 2, 1, 1, 1], [2, 1, 2, 2, 1], loops_rejected_by_declared_loop_free_simple_digraph_convention)","explanation_zh":"Wave6第二層核對=([[0, 0]], 否, [3, 2, 1, 1, 1], [2, 1, 2, 2, 1], loops_rejected_by_declared_loop_free_simple_digraph_convention)；依完整明示資料獨立重算；(u,v)與(v,u)不同，平行同向edge不重複且loop在主模型拒絕；out-degree與in-degree各自總和等於edge count；不沿用undirected degree或connectivity。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限有向圖"]},"variant":{"family_id":"qf-mod39-digraph-loop-boundary","seed":202611087079,"parameters":{"n":5,"edges":[[0,0],[0,1],[0,2],[1,2],[1,3],[2,3],[3,4],[4,0]],"domain":"Complete_finite_loopless_simple_digraph","operation":"digraph-loop-boundary","semantic_template":"qf-mod39-digraph-loop-boundary/{\"edges\":[[0,0],[0,1],[0,2],[1,2],[1,3],[2,3],[3,4],[4,0]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r07-finite-directed-graph-0080","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-directed-graph"],"title_zh":"有限有向圖","stem_zh":"完整有限ordered-pair edge set與loop-free simple digraph convention。操作=digraph-loop-boundary；資料={\"n\":5,\"edges\":[[0,0],[0,1],[1,2],[2,3],[2,4],[3,4],[4,0]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 0]], 否, [2, 1, 2, 1, 1], [2, 1, 1, 1, 2], loops_rejected_by_declared_loop_free_simple_digraph_convention)","explanation_zh":"Wave6第二層核對=([[0, 0]], 否, [2, 1, 2, 1, 1], [2, 1, 1, 1, 2], loops_rejected_by_declared_loop_free_simple_digraph_convention)；依完整明示資料獨立重算；(u,v)與(v,u)不同，平行同向edge不重複且loop在主模型拒絕；out-degree與in-degree各自總和等於edge count；不沿用undirected degree或connectivity。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限有向圖"]},"variant":{"family_id":"qf-mod39-digraph-loop-boundary","seed":202611087080,"parameters":{"n":5,"edges":[[0,0],[0,1],[1,2],[2,3],[2,4],[3,4],[4,0]],"domain":"Complete_finite_loopless_simple_digraph","operation":"digraph-loop-boundary","semantic_template":"qf-mod39-digraph-loop-boundary/{\"edges\":[[0,0],[0,1],[1,2],[2,3],[2,4],[3,4],[4,0]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r07-finite-directed-graph-0081","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-directed-graph"],"title_zh":"有限有向圖","stem_zh":"完整有限ordered-pair edge set與loop-free simple digraph convention。操作=digraph-loop-boundary；資料={\"n\":5,\"edges\":[[0,0],[0,1],[0,2],[1,2],[2,3],[2,4],[3,4],[4,0]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 0]], 否, [3, 1, 2, 1, 1], [2, 1, 2, 1, 2], loops_rejected_by_declared_loop_free_simple_digraph_convention)","explanation_zh":"Wave6第二層核對=([[0, 0]], 否, [3, 1, 2, 1, 1], [2, 1, 2, 1, 2], loops_rejected_by_declared_loop_free_simple_digraph_convention)；依完整明示資料獨立重算；(u,v)與(v,u)不同，平行同向edge不重複且loop在主模型拒絕；out-degree與in-degree各自總和等於edge count；不沿用undirected degree或connectivity。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限有向圖"]},"variant":{"family_id":"qf-mod39-digraph-loop-boundary","seed":202611087081,"parameters":{"n":5,"edges":[[0,0],[0,1],[0,2],[1,2],[2,3],[2,4],[3,4],[4,0]],"domain":"Complete_finite_loopless_simple_digraph","operation":"digraph-loop-boundary","semantic_template":"qf-mod39-digraph-loop-boundary/{\"edges\":[[0,0],[0,1],[0,2],[1,2],[2,3],[2,4],[3,4],[4,0]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r07-finite-directed-graph-0082","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-directed-graph"],"title_zh":"有限有向圖","stem_zh":"完整有限ordered-pair edge set與loop-free simple digraph convention。操作=digraph-loop-boundary；資料={\"n\":5,\"edges\":[[0,0],[0,1],[1,2],[1,3],[2,3],[2,4],[3,4],[4,0]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 0]], 否, [2, 2, 2, 1, 1], [2, 1, 1, 2, 2], loops_rejected_by_declared_loop_free_simple_digraph_convention)","explanation_zh":"Wave6第二層核對=([[0, 0]], 否, [2, 2, 2, 1, 1], [2, 1, 1, 2, 2], loops_rejected_by_declared_loop_free_simple_digraph_convention)；依完整明示資料獨立重算；(u,v)與(v,u)不同，平行同向edge不重複且loop在主模型拒絕；out-degree與in-degree各自總和等於edge count；不沿用undirected degree或connectivity。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限有向圖"]},"variant":{"family_id":"qf-mod39-digraph-loop-boundary","seed":202611087082,"parameters":{"n":5,"edges":[[0,0],[0,1],[1,2],[1,3],[2,3],[2,4],[3,4],[4,0]],"domain":"Complete_finite_loopless_simple_digraph","operation":"digraph-loop-boundary","semantic_template":"qf-mod39-digraph-loop-boundary/{\"edges\":[[0,0],[0,1],[1,2],[1,3],[2,3],[2,4],[3,4],[4,0]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r07-finite-directed-graph-0083","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-directed-graph"],"title_zh":"有限有向圖","stem_zh":"完整有限ordered-pair edge set與loop-free simple digraph convention。操作=digraph-loop-boundary；資料={\"n\":5,\"edges\":[[0,0],[0,1],[0,2],[1,2],[1,3],[2,3],[2,4],[3,4],[4,0]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 0]], 否, [3, 2, 2, 1, 1], [2, 1, 2, 2, 2], loops_rejected_by_declared_loop_free_simple_digraph_convention)","explanation_zh":"Wave6第二層核對=([[0, 0]], 否, [3, 2, 2, 1, 1], [2, 1, 2, 2, 2], loops_rejected_by_declared_loop_free_simple_digraph_convention)；依完整明示資料獨立重算；(u,v)與(v,u)不同，平行同向edge不重複且loop在主模型拒絕；out-degree與in-degree各自總和等於edge count；不沿用undirected degree或connectivity。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限有向圖"]},"variant":{"family_id":"qf-mod39-digraph-loop-boundary","seed":202611087083,"parameters":{"n":5,"edges":[[0,0],[0,1],[0,2],[1,2],[1,3],[2,3],[2,4],[3,4],[4,0]],"domain":"Complete_finite_loopless_simple_digraph","operation":"digraph-loop-boundary","semantic_template":"qf-mod39-digraph-loop-boundary/{\"edges\":[[0,0],[0,1],[0,2],[1,2],[1,3],[2,3],[2,4],[3,4],[4,0]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r07-finite-directed-graph-0084","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-directed-graph"],"title_zh":"有限有向圖","stem_zh":"完整有限ordered-pair edge set與loop-free simple digraph convention。操作=digraph-loop-boundary；資料={\"n\":5,\"edges\":[[0,0],[0,1],[1,2],[2,3],[3,0],[3,4],[4,0]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 0]], 否, [2, 1, 1, 2, 1], [3, 1, 1, 1, 1], loops_rejected_by_declared_loop_free_simple_digraph_convention)","explanation_zh":"Wave6第二層核對=([[0, 0]], 否, [2, 1, 1, 2, 1], [3, 1, 1, 1, 1], loops_rejected_by_declared_loop_free_simple_digraph_convention)；依完整明示資料獨立重算；(u,v)與(v,u)不同，平行同向edge不重複且loop在主模型拒絕；out-degree與in-degree各自總和等於edge count；不沿用undirected degree或connectivity。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限有向圖"]},"variant":{"family_id":"qf-mod39-digraph-loop-boundary","seed":202611087084,"parameters":{"n":5,"edges":[[0,0],[0,1],[1,2],[2,3],[3,0],[3,4],[4,0]],"domain":"Complete_finite_loopless_simple_digraph","operation":"digraph-loop-boundary","semantic_template":"qf-mod39-digraph-loop-boundary/{\"edges\":[[0,0],[0,1],[1,2],[2,3],[3,0],[3,4],[4,0]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r07-finite-directed-graph-0085","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-directed-graph"],"title_zh":"有限有向圖","stem_zh":"完整有限ordered-pair edge set與loop-free simple digraph convention。操作=digraph-loop-boundary；資料={\"n\":5,\"edges\":[[0,0],[0,1],[0,2],[1,2],[2,3],[3,0],[3,4],[4,0]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 0]], 否, [3, 1, 1, 2, 1], [3, 1, 2, 1, 1], loops_rejected_by_declared_loop_free_simple_digraph_convention)","explanation_zh":"Wave6第二層核對=([[0, 0]], 否, [3, 1, 1, 2, 1], [3, 1, 2, 1, 1], loops_rejected_by_declared_loop_free_simple_digraph_convention)；依完整明示資料獨立重算；(u,v)與(v,u)不同，平行同向edge不重複且loop在主模型拒絕；out-degree與in-degree各自總和等於edge count；不沿用undirected degree或connectivity。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限有向圖"]},"variant":{"family_id":"qf-mod39-digraph-loop-boundary","seed":202611087085,"parameters":{"n":5,"edges":[[0,0],[0,1],[0,2],[1,2],[2,3],[3,0],[3,4],[4,0]],"domain":"Complete_finite_loopless_simple_digraph","operation":"digraph-loop-boundary","semantic_template":"qf-mod39-digraph-loop-boundary/{\"edges\":[[0,0],[0,1],[0,2],[1,2],[2,3],[3,0],[3,4],[4,0]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r07-finite-directed-graph-0086","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-directed-graph"],"title_zh":"有限有向圖","stem_zh":"完整有限ordered-pair edge set與loop-free simple digraph convention。操作=digraph-loop-boundary；資料={\"n\":5,\"edges\":[[0,0],[0,1],[1,2],[1,3],[2,3],[3,0],[3,4],[4,0]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 0]], 否, [2, 2, 1, 2, 1], [3, 1, 1, 2, 1], loops_rejected_by_declared_loop_free_simple_digraph_convention)","explanation_zh":"Wave6第二層核對=([[0, 0]], 否, [2, 2, 1, 2, 1], [3, 1, 1, 2, 1], loops_rejected_by_declared_loop_free_simple_digraph_convention)；依完整明示資料獨立重算；(u,v)與(v,u)不同，平行同向edge不重複且loop在主模型拒絕；out-degree與in-degree各自總和等於edge count；不沿用undirected degree或connectivity。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限有向圖"]},"variant":{"family_id":"qf-mod39-digraph-loop-boundary","seed":202611087086,"parameters":{"n":5,"edges":[[0,0],[0,1],[1,2],[1,3],[2,3],[3,0],[3,4],[4,0]],"domain":"Complete_finite_loopless_simple_digraph","operation":"digraph-loop-boundary","semantic_template":"qf-mod39-digraph-loop-boundary/{\"edges\":[[0,0],[0,1],[1,2],[1,3],[2,3],[3,0],[3,4],[4,0]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r07-finite-directed-graph-0087","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-directed-graph"],"title_zh":"有限有向圖","stem_zh":"完整有限ordered-pair edge set與loop-free simple digraph convention。操作=digraph-loop-boundary；資料={\"n\":5,\"edges\":[[0,0],[0,1],[0,2],[1,2],[1,3],[2,3],[3,0],[3,4],[4,0]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 0]], 否, [3, 2, 1, 2, 1], [3, 1, 2, 2, 1], loops_rejected_by_declared_loop_free_simple_digraph_convention)","explanation_zh":"Wave6第二層核對=([[0, 0]], 否, [3, 2, 1, 2, 1], [3, 1, 2, 2, 1], loops_rejected_by_declared_loop_free_simple_digraph_convention)；依完整明示資料獨立重算；(u,v)與(v,u)不同，平行同向edge不重複且loop在主模型拒絕；out-degree與in-degree各自總和等於edge count；不沿用undirected degree或connectivity。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限有向圖"]},"variant":{"family_id":"qf-mod39-digraph-loop-boundary","seed":202611087087,"parameters":{"n":5,"edges":[[0,0],[0,1],[0,2],[1,2],[1,3],[2,3],[3,0],[3,4],[4,0]],"domain":"Complete_finite_loopless_simple_digraph","operation":"digraph-loop-boundary","semantic_template":"qf-mod39-digraph-loop-boundary/{\"edges\":[[0,0],[0,1],[0,2],[1,2],[1,3],[2,3],[3,0],[3,4],[4,0]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r07-finite-directed-graph-0088","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-directed-graph"],"title_zh":"有限有向圖","stem_zh":"完整有限ordered-pair edge set與loop-free simple digraph convention。操作=digraph-loop-boundary；資料={\"n\":5,\"edges\":[[0,0],[0,1],[1,2],[2,3],[2,4],[3,0],[3,4],[4,0]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 0]], 否, [2, 1, 2, 2, 1], [3, 1, 1, 1, 2], loops_rejected_by_declared_loop_free_simple_digraph_convention)","explanation_zh":"Wave6第二層核對=([[0, 0]], 否, [2, 1, 2, 2, 1], [3, 1, 1, 1, 2], loops_rejected_by_declared_loop_free_simple_digraph_convention)；依完整明示資料獨立重算；(u,v)與(v,u)不同，平行同向edge不重複且loop在主模型拒絕；out-degree與in-degree各自總和等於edge count；不沿用undirected degree或connectivity。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限有向圖"]},"variant":{"family_id":"qf-mod39-digraph-loop-boundary","seed":202611087088,"parameters":{"n":5,"edges":[[0,0],[0,1],[1,2],[2,3],[2,4],[3,0],[3,4],[4,0]],"domain":"Complete_finite_loopless_simple_digraph","operation":"digraph-loop-boundary","semantic_template":"qf-mod39-digraph-loop-boundary/{\"edges\":[[0,0],[0,1],[1,2],[2,3],[2,4],[3,0],[3,4],[4,0]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r07-finite-directed-graph-0089","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-directed-graph"],"title_zh":"有限有向圖","stem_zh":"完整有限ordered-pair edge set與loop-free simple digraph convention。操作=digraph-loop-boundary；資料={\"n\":5,\"edges\":[[0,0],[0,1],[0,2],[1,2],[2,3],[2,4],[3,0],[3,4],[4,0]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 0]], 否, [3, 1, 2, 2, 1], [3, 1, 2, 1, 2], loops_rejected_by_declared_loop_free_simple_digraph_convention)","explanation_zh":"Wave6第二層核對=([[0, 0]], 否, [3, 1, 2, 2, 1], [3, 1, 2, 1, 2], loops_rejected_by_declared_loop_free_simple_digraph_convention)；依完整明示資料獨立重算；(u,v)與(v,u)不同，平行同向edge不重複且loop在主模型拒絕；out-degree與in-degree各自總和等於edge count；不沿用undirected degree或connectivity。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限有向圖"]},"variant":{"family_id":"qf-mod39-digraph-loop-boundary","seed":202611087089,"parameters":{"n":5,"edges":[[0,0],[0,1],[0,2],[1,2],[2,3],[2,4],[3,0],[3,4],[4,0]],"domain":"Complete_finite_loopless_simple_digraph","operation":"digraph-loop-boundary","semantic_template":"qf-mod39-digraph-loop-boundary/{\"edges\":[[0,0],[0,1],[0,2],[1,2],[2,3],[2,4],[3,0],[3,4],[4,0]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r07-finite-directed-graph-0090","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-directed-graph"],"title_zh":"有限有向圖","stem_zh":"完整有限ordered-pair edge set與loop-free simple digraph convention。操作=digraph-loop-boundary；資料={\"n\":5,\"edges\":[[0,0],[0,1],[1,2],[1,3],[2,3],[2,4],[3,0],[3,4],[4,0]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 0]], 否, [2, 2, 2, 2, 1], [3, 1, 1, 2, 2], loops_rejected_by_declared_loop_free_simple_digraph_convention)","explanation_zh":"Wave6第二層核對=([[0, 0]], 否, [2, 2, 2, 2, 1], [3, 1, 1, 2, 2], loops_rejected_by_declared_loop_free_simple_digraph_convention)；依完整明示資料獨立重算；(u,v)與(v,u)不同，平行同向edge不重複且loop在主模型拒絕；out-degree與in-degree各自總和等於edge count；不沿用undirected degree或connectivity。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限有向圖"]},"variant":{"family_id":"qf-mod39-digraph-loop-boundary","seed":202611087090,"parameters":{"n":5,"edges":[[0,0],[0,1],[1,2],[1,3],[2,3],[2,4],[3,0],[3,4],[4,0]],"domain":"Complete_finite_loopless_simple_digraph","operation":"digraph-loop-boundary","semantic_template":"qf-mod39-digraph-loop-boundary/{\"edges\":[[0,0],[0,1],[1,2],[1,3],[2,3],[2,4],[3,0],[3,4],[4,0]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r07-finite-directed-graph-0091","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-directed-graph"],"title_zh":"有限有向圖","stem_zh":"完整有限ordered-pair edge set與loop-free simple digraph convention。操作=digraph-loop-boundary；資料={\"n\":5,\"edges\":[[0,0],[0,1],[0,2],[1,2],[1,3],[2,3],[2,4],[3,0],[3,4],[4,0]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 0]], 否, [3, 2, 2, 2, 1], [3, 1, 2, 2, 2], loops_rejected_by_declared_loop_free_simple_digraph_convention)","explanation_zh":"Wave6第二層核對=([[0, 0]], 否, [3, 2, 2, 2, 1], [3, 1, 2, 2, 2], loops_rejected_by_declared_loop_free_simple_digraph_convention)；依完整明示資料獨立重算；(u,v)與(v,u)不同，平行同向edge不重複且loop在主模型拒絕；out-degree與in-degree各自總和等於edge count；不沿用undirected degree或connectivity。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限有向圖"]},"variant":{"family_id":"qf-mod39-digraph-loop-boundary","seed":202611087091,"parameters":{"n":5,"edges":[[0,0],[0,1],[0,2],[1,2],[1,3],[2,3],[2,4],[3,0],[3,4],[4,0]],"domain":"Complete_finite_loopless_simple_digraph","operation":"digraph-loop-boundary","semantic_template":"qf-mod39-digraph-loop-boundary/{\"edges\":[[0,0],[0,1],[0,2],[1,2],[1,3],[2,3],[2,4],[3,0],[3,4],[4,0]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r07-finite-directed-graph-0092","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-directed-graph"],"title_zh":"有限有向圖","stem_zh":"完整有限ordered-pair edge set與loop-free simple digraph convention。操作=digraph-loop-boundary；資料={\"n\":5,\"edges\":[[0,0],[0,1],[1,2],[2,3],[3,4],[4,0],[4,1]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 0]], 否, [2, 1, 1, 1, 2], [2, 2, 1, 1, 1], loops_rejected_by_declared_loop_free_simple_digraph_convention)","explanation_zh":"Wave6第二層核對=([[0, 0]], 否, [2, 1, 1, 1, 2], [2, 2, 1, 1, 1], loops_rejected_by_declared_loop_free_simple_digraph_convention)；依完整明示資料獨立重算；(u,v)與(v,u)不同，平行同向edge不重複且loop在主模型拒絕；out-degree與in-degree各自總和等於edge count；不沿用undirected degree或connectivity。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限有向圖"]},"variant":{"family_id":"qf-mod39-digraph-loop-boundary","seed":202611087092,"parameters":{"n":5,"edges":[[0,0],[0,1],[1,2],[2,3],[3,4],[4,0],[4,1]],"domain":"Complete_finite_loopless_simple_digraph","operation":"digraph-loop-boundary","semantic_template":"qf-mod39-digraph-loop-boundary/{\"edges\":[[0,0],[0,1],[1,2],[2,3],[3,4],[4,0],[4,1]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r07-finite-directed-graph-0093","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-directed-graph"],"title_zh":"有限有向圖","stem_zh":"完整有限ordered-pair edge set與loop-free simple digraph convention。操作=digraph-loop-boundary；資料={\"n\":5,\"edges\":[[0,0],[0,1],[0,2],[1,2],[2,3],[3,4],[4,0],[4,1]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 0]], 否, [3, 1, 1, 1, 2], [2, 2, 2, 1, 1], loops_rejected_by_declared_loop_free_simple_digraph_convention)","explanation_zh":"Wave6第二層核對=([[0, 0]], 否, [3, 1, 1, 1, 2], [2, 2, 2, 1, 1], loops_rejected_by_declared_loop_free_simple_digraph_convention)；依完整明示資料獨立重算；(u,v)與(v,u)不同，平行同向edge不重複且loop在主模型拒絕；out-degree與in-degree各自總和等於edge count；不沿用undirected degree或connectivity。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限有向圖"]},"variant":{"family_id":"qf-mod39-digraph-loop-boundary","seed":202611087093,"parameters":{"n":5,"edges":[[0,0],[0,1],[0,2],[1,2],[2,3],[3,4],[4,0],[4,1]],"domain":"Complete_finite_loopless_simple_digraph","operation":"digraph-loop-boundary","semantic_template":"qf-mod39-digraph-loop-boundary/{\"edges\":[[0,0],[0,1],[0,2],[1,2],[2,3],[3,4],[4,0],[4,1]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r07-finite-directed-graph-0094","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-directed-graph"],"title_zh":"有限有向圖","stem_zh":"完整有限ordered-pair edge set與loop-free simple digraph convention。操作=digraph-loop-boundary；資料={\"n\":5,\"edges\":[[0,0],[0,1],[1,2],[1,3],[2,3],[3,4],[4,0],[4,1]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 0]], 否, [2, 2, 1, 1, 2], [2, 2, 1, 2, 1], loops_rejected_by_declared_loop_free_simple_digraph_convention)","explanation_zh":"Wave6第二層核對=([[0, 0]], 否, [2, 2, 1, 1, 2], [2, 2, 1, 2, 1], loops_rejected_by_declared_loop_free_simple_digraph_convention)；依完整明示資料獨立重算；(u,v)與(v,u)不同，平行同向edge不重複且loop在主模型拒絕；out-degree與in-degree各自總和等於edge count；不沿用undirected degree或connectivity。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限有向圖"]},"variant":{"family_id":"qf-mod39-digraph-loop-boundary","seed":202611087094,"parameters":{"n":5,"edges":[[0,0],[0,1],[1,2],[1,3],[2,3],[3,4],[4,0],[4,1]],"domain":"Complete_finite_loopless_simple_digraph","operation":"digraph-loop-boundary","semantic_template":"qf-mod39-digraph-loop-boundary/{\"edges\":[[0,0],[0,1],[1,2],[1,3],[2,3],[3,4],[4,0],[4,1]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r07-finite-directed-graph-0095","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-directed-graph"],"title_zh":"有限有向圖","stem_zh":"完整有限ordered-pair edge set與loop-free simple digraph convention。操作=digraph-loop-boundary；資料={\"n\":5,\"edges\":[[0,0],[0,1],[0,2],[1,2],[1,3],[2,3],[3,4],[4,0],[4,1]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 0]], 否, [3, 2, 1, 1, 2], [2, 2, 2, 2, 1], loops_rejected_by_declared_loop_free_simple_digraph_convention)","explanation_zh":"Wave6第二層核對=([[0, 0]], 否, [3, 2, 1, 1, 2], [2, 2, 2, 2, 1], loops_rejected_by_declared_loop_free_simple_digraph_convention)；依完整明示資料獨立重算；(u,v)與(v,u)不同，平行同向edge不重複且loop在主模型拒絕；out-degree與in-degree各自總和等於edge count；不沿用undirected degree或connectivity。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限有向圖"]},"variant":{"family_id":"qf-mod39-digraph-loop-boundary","seed":202611087095,"parameters":{"n":5,"edges":[[0,0],[0,1],[0,2],[1,2],[1,3],[2,3],[3,4],[4,0],[4,1]],"domain":"Complete_finite_loopless_simple_digraph","operation":"digraph-loop-boundary","semantic_template":"qf-mod39-digraph-loop-boundary/{\"edges\":[[0,0],[0,1],[0,2],[1,2],[1,3],[2,3],[3,4],[4,0],[4,1]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r07-finite-directed-graph-0096","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-directed-graph"],"title_zh":"有限有向圖","stem_zh":"完整有限ordered-pair edge set與loop-free simple digraph convention。操作=digraph-loop-boundary；資料={\"n\":5,\"edges\":[[0,0],[0,1],[1,2],[2,3],[2,4],[3,4],[4,0],[4,1]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 0]], 否, [2, 1, 2, 1, 2], [2, 2, 1, 1, 2], loops_rejected_by_declared_loop_free_simple_digraph_convention)","explanation_zh":"Wave6第二層核對=([[0, 0]], 否, [2, 1, 2, 1, 2], [2, 2, 1, 1, 2], loops_rejected_by_declared_loop_free_simple_digraph_convention)；依完整明示資料獨立重算；(u,v)與(v,u)不同，平行同向edge不重複且loop在主模型拒絕；out-degree與in-degree各自總和等於edge count；不沿用undirected degree或connectivity。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限有向圖"]},"variant":{"family_id":"qf-mod39-digraph-loop-boundary","seed":202611087096,"parameters":{"n":5,"edges":[[0,0],[0,1],[1,2],[2,3],[2,4],[3,4],[4,0],[4,1]],"domain":"Complete_finite_loopless_simple_digraph","operation":"digraph-loop-boundary","semantic_template":"qf-mod39-digraph-loop-boundary/{\"edges\":[[0,0],[0,1],[1,2],[2,3],[2,4],[3,4],[4,0],[4,1]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r07-finite-directed-graph-0097","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-directed-graph"],"title_zh":"有限有向圖","stem_zh":"完整有限ordered-pair edge set與loop-free simple digraph convention。操作=digraph-loop-boundary；資料={\"n\":5,\"edges\":[[0,0],[0,1],[0,2],[1,2],[2,3],[2,4],[3,4],[4,0],[4,1]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 0]], 否, [3, 1, 2, 1, 2], [2, 2, 2, 1, 2], loops_rejected_by_declared_loop_free_simple_digraph_convention)","explanation_zh":"Wave6第二層核對=([[0, 0]], 否, [3, 1, 2, 1, 2], [2, 2, 2, 1, 2], loops_rejected_by_declared_loop_free_simple_digraph_convention)；依完整明示資料獨立重算；(u,v)與(v,u)不同，平行同向edge不重複且loop在主模型拒絕；out-degree與in-degree各自總和等於edge count；不沿用undirected degree或connectivity。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限有向圖"]},"variant":{"family_id":"qf-mod39-digraph-loop-boundary","seed":202611087097,"parameters":{"n":5,"edges":[[0,0],[0,1],[0,2],[1,2],[2,3],[2,4],[3,4],[4,0],[4,1]],"domain":"Complete_finite_loopless_simple_digraph","operation":"digraph-loop-boundary","semantic_template":"qf-mod39-digraph-loop-boundary/{\"edges\":[[0,0],[0,1],[0,2],[1,2],[2,3],[2,4],[3,4],[4,0],[4,1]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r07-finite-directed-graph-0098","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-directed-graph"],"title_zh":"有限有向圖","stem_zh":"完整有限ordered-pair edge set與loop-free simple digraph convention。操作=digraph-loop-boundary；資料={\"n\":5,\"edges\":[[0,0],[0,1],[1,2],[1,3],[2,3],[2,4],[3,4],[4,0],[4,1]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 0]], 否, [2, 2, 2, 1, 2], [2, 2, 1, 2, 2], loops_rejected_by_declared_loop_free_simple_digraph_convention)","explanation_zh":"Wave6第二層核對=([[0, 0]], 否, [2, 2, 2, 1, 2], [2, 2, 1, 2, 2], loops_rejected_by_declared_loop_free_simple_digraph_convention)；依完整明示資料獨立重算；(u,v)與(v,u)不同，平行同向edge不重複且loop在主模型拒絕；out-degree與in-degree各自總和等於edge count；不沿用undirected degree或connectivity。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限有向圖"]},"variant":{"family_id":"qf-mod39-digraph-loop-boundary","seed":202611087098,"parameters":{"n":5,"edges":[[0,0],[0,1],[1,2],[1,3],[2,3],[2,4],[3,4],[4,0],[4,1]],"domain":"Complete_finite_loopless_simple_digraph","operation":"digraph-loop-boundary","semantic_template":"qf-mod39-digraph-loop-boundary/{\"edges\":[[0,0],[0,1],[1,2],[1,3],[2,3],[2,4],[3,4],[4,0],[4,1]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r07-finite-directed-graph-0099","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-directed-graph"],"title_zh":"有限有向圖","stem_zh":"完整有限ordered-pair edge set與loop-free simple digraph convention。操作=digraph-loop-boundary；資料={\"n\":5,\"edges\":[[0,0],[0,1],[0,2],[1,2],[1,3],[2,3],[2,4],[3,4],[4,0],[4,1]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 0]], 否, [3, 2, 2, 1, 2], [2, 2, 2, 2, 2], loops_rejected_by_declared_loop_free_simple_digraph_convention)","explanation_zh":"Wave6第二層核對=([[0, 0]], 否, [3, 2, 2, 1, 2], [2, 2, 2, 2, 2], loops_rejected_by_declared_loop_free_simple_digraph_convention)；依完整明示資料獨立重算；(u,v)與(v,u)不同，平行同向edge不重複且loop在主模型拒絕；out-degree與in-degree各自總和等於edge count；不沿用undirected degree或connectivity。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限有向圖"]},"variant":{"family_id":"qf-mod39-digraph-loop-boundary","seed":202611087099,"parameters":{"n":5,"edges":[[0,0],[0,1],[0,2],[1,2],[1,3],[2,3],[2,4],[3,4],[4,0],[4,1]],"domain":"Complete_finite_loopless_simple_digraph","operation":"digraph-loop-boundary","semantic_template":"qf-mod39-digraph-loop-boundary/{\"edges\":[[0,0],[0,1],[0,2],[1,2],[1,3],[2,3],[2,4],[3,4],[4,0],[4,1]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r07-finite-directed-graph-0100","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-directed-graph"],"title_zh":"有限有向圖","stem_zh":"完整有限ordered-pair edge set與loop-free simple digraph convention。操作=digraph-loop-boundary；資料={\"n\":5,\"edges\":[[0,0],[0,1],[1,2],[2,3],[3,0],[3,4],[4,0],[4,1]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 0]], 否, [2, 1, 1, 2, 2], [3, 2, 1, 1, 1], loops_rejected_by_declared_loop_free_simple_digraph_convention)","explanation_zh":"Wave6第二層核對=([[0, 0]], 否, [2, 1, 1, 2, 2], [3, 2, 1, 1, 1], loops_rejected_by_declared_loop_free_simple_digraph_convention)；依完整明示資料獨立重算；(u,v)與(v,u)不同，平行同向edge不重複且loop在主模型拒絕；out-degree與in-degree各自總和等於edge count；不沿用undirected degree或connectivity。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限有向圖"]},"variant":{"family_id":"qf-mod39-digraph-loop-boundary","seed":202611087100,"parameters":{"n":5,"edges":[[0,0],[0,1],[1,2],[2,3],[3,0],[3,4],[4,0],[4,1]],"domain":"Complete_finite_loopless_simple_digraph","operation":"digraph-loop-boundary","semantic_template":"qf-mod39-digraph-loop-boundary/{\"edges\":[[0,0],[0,1],[1,2],[2,3],[3,0],[3,4],[4,0],[4,1]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}