{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r07-connected-component-0001","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-component"],"title_zh":"連通分量","stem_zh":"完整有限無向無loop簡單圖。操作=component-of-vertex；資料={\"n\":7,\"edges\":[[0,1],[0,2],[0,3],[0,4],[0,6],[1,2],[1,3],[1,5],[1,6],[2,3],[2,4]],\"vertex\":0}。請依 [vertex,component] 欄序回答。","answer":{"kind":"expression","value":"(0, [0, 1, 2, 3, 4, 5, 6])","explanation_zh":"Wave4第二層核對=(0, [0, 1, 2, 3, 4, 5, 6])；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；長度零path保證反身且孤立點為單點分量；完整可達性等價類不切換有向強弱連通。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["連通分量"]},"variant":{"family_id":"qf-mod28-component-of-vertex","seed":202610287001,"parameters":{"n":7,"edges":[[0,1],[0,2],[0,3],[0,4],[0,6],[1,2],[1,3],[1,5],[1,6],[2,3],[2,4]],"vertex":0,"domain":"Complete_finite_undirected_components","operation":"component-of-vertex","semantic_template":"qf-mod28-component-of-vertex/{\"edges\":[[0,1],[0,2],[0,3],[0,4],[0,6],[1,2],[1,3],[1,5],[1,6],[2,3],[2,4]],\"n\":7,\"vertex\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r07-connected-component-0002","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-component"],"title_zh":"連通分量","stem_zh":"完整有限無向無loop簡單圖。操作=component-of-vertex；資料={\"n\":7,\"edges\":[[0,2],[0,3],[0,4],[0,5],[1,2],[1,3],[1,4],[1,6],[2,3],[2,4],[2,5]],\"vertex\":1}。請依 [vertex,component] 欄序回答。","answer":{"kind":"expression","value":"(1, [0, 1, 2, 3, 4, 5, 6])","explanation_zh":"Wave4第二層核對=(1, [0, 1, 2, 3, 4, 5, 6])；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；長度零path保證反身且孤立點為單點分量；完整可達性等價類不切換有向強弱連通。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["連通分量"]},"variant":{"family_id":"qf-mod28-component-of-vertex","seed":202610287002,"parameters":{"n":7,"edges":[[0,2],[0,3],[0,4],[0,5],[1,2],[1,3],[1,4],[1,6],[2,3],[2,4],[2,5]],"vertex":1,"domain":"Complete_finite_undirected_components","operation":"component-of-vertex","semantic_template":"qf-mod28-component-of-vertex/{\"edges\":[[0,2],[0,3],[0,4],[0,5],[1,2],[1,3],[1,4],[1,6],[2,3],[2,4],[2,5]],\"n\":7,\"vertex\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r07-connected-component-0003","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-component"],"title_zh":"連通分量","stem_zh":"完整有限無向無loop簡單圖。操作=component-of-vertex；資料={\"n\":7,\"edges\":[[0,1],[0,3],[0,4],[1,2],[1,3],[1,6],[2,3],[2,4],[2,6]],\"vertex\":2}。請依 [vertex,component] 欄序回答。","answer":{"kind":"expression","value":"(2, [0, 1, 2, 3, 4, 6])","explanation_zh":"Wave4第二層核對=(2, [0, 1, 2, 3, 4, 6])；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；長度零path保證反身且孤立點為單點分量；完整可達性等價類不切換有向強弱連通。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["連通分量"]},"variant":{"family_id":"qf-mod28-component-of-vertex","seed":202610287003,"parameters":{"n":7,"edges":[[0,1],[0,3],[0,4],[1,2],[1,3],[1,6],[2,3],[2,4],[2,6]],"vertex":2,"domain":"Complete_finite_undirected_components","operation":"component-of-vertex","semantic_template":"qf-mod28-component-of-vertex/{\"edges\":[[0,1],[0,3],[0,4],[1,2],[1,3],[1,6],[2,3],[2,4],[2,6]],\"n\":7,\"vertex\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r07-connected-component-0004","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-component"],"title_zh":"連通分量","stem_zh":"完整有限無向無loop簡單圖。操作=component-of-vertex；資料={\"n\":7,\"edges\":[[0,3],[0,4],[0,5],[0,6],[1,3],[1,4],[1,5],[2,3],[2,4],[2,5],[2,6]],\"vertex\":3}。請依 [vertex,component] 欄序回答。","answer":{"kind":"expression","value":"(3, [0, 1, 2, 3, 4, 5, 6])","explanation_zh":"Wave4第二層核對=(3, [0, 1, 2, 3, 4, 5, 6])；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；長度零path保證反身且孤立點為單點分量；完整可達性等價類不切換有向強弱連通。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["連通分量"]},"variant":{"family_id":"qf-mod28-component-of-vertex","seed":202610287004,"parameters":{"n":7,"edges":[[0,3],[0,4],[0,5],[0,6],[1,3],[1,4],[1,5],[2,3],[2,4],[2,5],[2,6]],"vertex":3,"domain":"Complete_finite_undirected_components","operation":"component-of-vertex","semantic_template":"qf-mod28-component-of-vertex/{\"edges\":[[0,3],[0,4],[0,5],[0,6],[1,3],[1,4],[1,5],[2,3],[2,4],[2,5],[2,6]],\"n\":7,\"vertex\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r07-connected-component-0005","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-component"],"title_zh":"連通分量","stem_zh":"完整有限無向無loop簡單圖。操作=component-of-vertex；資料={\"n\":7,\"edges\":[[0,1],[0,2],[0,4],[0,6],[1,3],[1,5],[2,3],[2,4],[3,4]],\"vertex\":4}。請依 [vertex,component] 欄序回答。","answer":{"kind":"expression","value":"(4, [0, 1, 2, 3, 4, 5, 6])","explanation_zh":"Wave4第二層核對=(4, [0, 1, 2, 3, 4, 5, 6])；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；長度零path保證反身且孤立點為單點分量；完整可達性等價類不切換有向強弱連通。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["連通分量"]},"variant":{"family_id":"qf-mod28-component-of-vertex","seed":202610287005,"parameters":{"n":7,"edges":[[0,1],[0,2],[0,4],[0,6],[1,3],[1,5],[2,3],[2,4],[3,4]],"vertex":4,"domain":"Complete_finite_undirected_components","operation":"component-of-vertex","semantic_template":"qf-mod28-component-of-vertex/{\"edges\":[[0,1],[0,2],[0,4],[0,6],[1,3],[1,5],[2,3],[2,4],[3,4]],\"n\":7,\"vertex\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r07-connected-component-0006","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-component"],"title_zh":"連通分量","stem_zh":"完整有限無向無loop簡單圖。操作=component-of-vertex；資料={\"n\":7,\"edges\":[[0,2],[0,4],[0,5],[1,3],[1,4],[2,3],[2,4],[2,5],[3,4]],\"vertex\":5}。請依 [vertex,component] 欄序回答。","answer":{"kind":"expression","value":"(5, [0, 1, 2, 3, 4, 5])","explanation_zh":"Wave4第二層核對=(5, [0, 1, 2, 3, 4, 5])；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；長度零path保證反身且孤立點為單點分量；完整可達性等價類不切換有向強弱連通。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["連通分量"]},"variant":{"family_id":"qf-mod28-component-of-vertex","seed":202610287006,"parameters":{"n":7,"edges":[[0,2],[0,4],[0,5],[1,3],[1,4],[2,3],[2,4],[2,5],[3,4]],"vertex":5,"domain":"Complete_finite_undirected_components","operation":"component-of-vertex","semantic_template":"qf-mod28-component-of-vertex/{\"edges\":[[0,2],[0,4],[0,5],[1,3],[1,4],[2,3],[2,4],[2,5],[3,4]],\"n\":7,\"vertex\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r07-connected-component-0007","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-component"],"title_zh":"連通分量","stem_zh":"完整有限無向無loop簡單圖。操作=component-of-vertex；資料={\"n\":7,\"edges\":[[0,1],[0,4],[1,3],[2,3],[2,4],[2,6],[3,4]],\"vertex\":6}。請依 [vertex,component] 欄序回答。","answer":{"kind":"expression","value":"(6, [0, 1, 2, 3, 4, 6])","explanation_zh":"Wave4第二層核對=(6, [0, 1, 2, 3, 4, 6])；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；長度零path保證反身且孤立點為單點分量；完整可達性等價類不切換有向強弱連通。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["連通分量"]},"variant":{"family_id":"qf-mod28-component-of-vertex","seed":202610287007,"parameters":{"n":7,"edges":[[0,1],[0,4],[1,3],[2,3],[2,4],[2,6],[3,4]],"vertex":6,"domain":"Complete_finite_undirected_components","operation":"component-of-vertex","semantic_template":"qf-mod28-component-of-vertex/{\"edges\":[[0,1],[0,4],[1,3],[2,3],[2,4],[2,6],[3,4]],\"n\":7,\"vertex\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r07-connected-component-0008","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-component"],"title_zh":"連通分量","stem_zh":"完整有限無向無loop簡單圖。操作=component-of-vertex；資料={\"n\":7,\"edges\":[[0,4],[0,5],[0,6],[1,2],[1,4],[1,5],[1,6],[2,4],[2,5],[2,6],[3,4]],\"vertex\":0}。請依 [vertex,component] 欄序回答。","answer":{"kind":"expression","value":"(0, [0, 1, 2, 3, 4, 5, 6])","explanation_zh":"Wave4第二層核對=(0, [0, 1, 2, 3, 4, 5, 6])；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；長度零path保證反身且孤立點為單點分量；完整可達性等價類不切換有向強弱連通。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["連通分量"]},"variant":{"family_id":"qf-mod28-component-of-vertex","seed":202610287008,"parameters":{"n":7,"edges":[[0,4],[0,5],[0,6],[1,2],[1,4],[1,5],[1,6],[2,4],[2,5],[2,6],[3,4]],"vertex":0,"domain":"Complete_finite_undirected_components","operation":"component-of-vertex","semantic_template":"qf-mod28-component-of-vertex/{\"edges\":[[0,4],[0,5],[0,6],[1,2],[1,4],[1,5],[1,6],[2,4],[2,5],[2,6],[3,4]],\"n\":7,\"vertex\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r07-connected-component-0009","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-component"],"title_zh":"連通分量","stem_zh":"完整有限無向無loop簡單圖。操作=component-of-vertex；資料={\"n\":7,\"edges\":[[0,1],[0,2],[0,3],[0,6],[1,2],[1,5],[1,6],[2,4],[3,5]],\"vertex\":1}。請依 [vertex,component] 欄序回答。","answer":{"kind":"expression","value":"(1, [0, 1, 2, 3, 4, 5, 6])","explanation_zh":"Wave4第二層核對=(1, [0, 1, 2, 3, 4, 5, 6])；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；長度零path保證反身且孤立點為單點分量；完整可達性等價類不切換有向強弱連通。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["連通分量"]},"variant":{"family_id":"qf-mod28-component-of-vertex","seed":202610287009,"parameters":{"n":7,"edges":[[0,1],[0,2],[0,3],[0,6],[1,2],[1,5],[1,6],[2,4],[3,5]],"vertex":1,"domain":"Complete_finite_undirected_components","operation":"component-of-vertex","semantic_template":"qf-mod28-component-of-vertex/{\"edges\":[[0,1],[0,2],[0,3],[0,6],[1,2],[1,5],[1,6],[2,4],[3,5]],\"n\":7,\"vertex\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r07-connected-component-0010","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-component"],"title_zh":"連通分量","stem_zh":"完整有限無向無loop簡單圖。操作=component-of-vertex；資料={\"n\":7,\"edges\":[[0,2],[0,3],[0,5],[1,2],[1,4],[1,6],[2,4],[2,5],[3,5]],\"vertex\":2}。請依 [vertex,component] 欄序回答。","answer":{"kind":"expression","value":"(2, [0, 1, 2, 3, 4, 5, 6])","explanation_zh":"Wave4第二層核對=(2, [0, 1, 2, 3, 4, 5, 6])；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；長度零path保證反身且孤立點為單點分量；完整可達性等價類不切換有向強弱連通。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["連通分量"]},"variant":{"family_id":"qf-mod28-component-of-vertex","seed":202610287010,"parameters":{"n":7,"edges":[[0,2],[0,3],[0,5],[1,2],[1,4],[1,6],[2,4],[2,5],[3,5]],"vertex":2,"domain":"Complete_finite_undirected_components","operation":"component-of-vertex","semantic_template":"qf-mod28-component-of-vertex/{\"edges\":[[0,2],[0,3],[0,5],[1,2],[1,4],[1,6],[2,4],[2,5],[3,5]],\"n\":7,\"vertex\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r07-connected-component-0011","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-component"],"title_zh":"連通分量","stem_zh":"完整有限無向無loop簡單圖。操作=component-of-vertex；資料={\"n\":7,\"edges\":[[0,1],[0,3],[1,2],[1,6],[2,4],[2,6],[3,5]],\"vertex\":3}。請依 [vertex,component] 欄序回答。","answer":{"kind":"expression","value":"(3, [0, 1, 2, 3, 4, 5, 6])","explanation_zh":"Wave4第二層核對=(3, [0, 1, 2, 3, 4, 5, 6])；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；長度零path保證反身且孤立點為單點分量；完整可達性等價類不切換有向強弱連通。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["連通分量"]},"variant":{"family_id":"qf-mod28-component-of-vertex","seed":202610287011,"parameters":{"n":7,"edges":[[0,1],[0,3],[1,2],[1,6],[2,4],[2,6],[3,5]],"vertex":3,"domain":"Complete_finite_undirected_components","operation":"component-of-vertex","semantic_template":"qf-mod28-component-of-vertex/{\"edges\":[[0,1],[0,3],[1,2],[1,6],[2,4],[2,6],[3,5]],\"n\":7,\"vertex\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r07-connected-component-0012","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-component"],"title_zh":"連通分量","stem_zh":"完整有限無向無loop簡單圖。操作=component-of-vertex；資料={\"n\":7,\"edges\":[[0,3],[0,5],[0,6],[1,4],[1,5],[2,4],[2,5],[2,6],[3,5]],\"vertex\":4}。請依 [vertex,component] 欄序回答。","answer":{"kind":"expression","value":"(4, [0, 1, 2, 3, 4, 5, 6])","explanation_zh":"Wave4第二層核對=(4, [0, 1, 2, 3, 4, 5, 6])；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；長度零path保證反身且孤立點為單點分量；完整可達性等價類不切換有向強弱連通。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["連通分量"]},"variant":{"family_id":"qf-mod28-component-of-vertex","seed":202610287012,"parameters":{"n":7,"edges":[[0,3],[0,5],[0,6],[1,4],[1,5],[2,4],[2,5],[2,6],[3,5]],"vertex":4,"domain":"Complete_finite_undirected_components","operation":"component-of-vertex","semantic_template":"qf-mod28-component-of-vertex/{\"edges\":[[0,3],[0,5],[0,6],[1,4],[1,5],[2,4],[2,5],[2,6],[3,5]],\"n\":7,\"vertex\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r07-connected-component-0013","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-component"],"title_zh":"連通分量","stem_zh":"完整有限無向無loop簡單圖。操作=component-of-vertex；資料={\"n\":7,\"edges\":[[0,1],[0,2],[0,6],[1,5],[2,4],[3,4],[3,5]],\"vertex\":5}。請依 [vertex,component] 欄序回答。","answer":{"kind":"expression","value":"(5, [0, 1, 2, 3, 4, 5, 6])","explanation_zh":"Wave4第二層核對=(5, [0, 1, 2, 3, 4, 5, 6])；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；長度零path保證反身且孤立點為單點分量；完整可達性等價類不切換有向強弱連通。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["連通分量"]},"variant":{"family_id":"qf-mod28-component-of-vertex","seed":202610287013,"parameters":{"n":7,"edges":[[0,1],[0,2],[0,6],[1,5],[2,4],[3,4],[3,5]],"vertex":5,"domain":"Complete_finite_undirected_components","operation":"component-of-vertex","semantic_template":"qf-mod28-component-of-vertex/{\"edges\":[[0,1],[0,2],[0,6],[1,5],[2,4],[3,4],[3,5]],\"n\":7,\"vertex\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r07-connected-component-0014","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-component"],"title_zh":"連通分量","stem_zh":"完整有限無向無loop簡單圖。操作=component-of-vertex；資料={\"n\":7,\"edges\":[[0,2],[0,5],[1,4],[2,4],[2,5],[3,4],[3,5]],\"vertex\":6}。請依 [vertex,component] 欄序回答。","answer":{"kind":"expression","value":"(6, [6])","explanation_zh":"Wave4第二層核對=(6, [6])；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；長度零path保證反身且孤立點為單點分量；完整可達性等價類不切換有向強弱連通。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["連通分量"]},"variant":{"family_id":"qf-mod28-component-of-vertex","seed":202610287014,"parameters":{"n":7,"edges":[[0,2],[0,5],[1,4],[2,4],[2,5],[3,4],[3,5]],"vertex":6,"domain":"Complete_finite_undirected_components","operation":"component-of-vertex","semantic_template":"qf-mod28-component-of-vertex/{\"edges\":[[0,2],[0,5],[1,4],[2,4],[2,5],[3,4],[3,5]],\"n\":7,\"vertex\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r07-connected-component-0015","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-component"],"title_zh":"連通分量","stem_zh":"完整有限無向無loop簡單圖。操作=component-of-vertex；資料={\"n\":7,\"edges\":[[0,1],[2,4],[2,6],[3,4],[3,5]],\"vertex\":0}。請依 [vertex,component] 欄序回答。","answer":{"kind":"expression","value":"(0, [0, 1])","explanation_zh":"Wave4第二層核對=(0, [0, 1])；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；長度零path保證反身且孤立點為單點分量；完整可達性等價類不切換有向強弱連通。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["連通分量"]},"variant":{"family_id":"qf-mod28-component-of-vertex","seed":202610287015,"parameters":{"n":7,"edges":[[0,1],[2,4],[2,6],[3,4],[3,5]],"vertex":0,"domain":"Complete_finite_undirected_components","operation":"component-of-vertex","semantic_template":"qf-mod28-component-of-vertex/{\"edges\":[[0,1],[2,4],[2,6],[3,4],[3,5]],\"n\":7,\"vertex\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r07-connected-component-0016","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-component"],"title_zh":"連通分量","stem_zh":"完整有限無向無loop簡單圖。操作=component-of-vertex；資料={\"n\":7,\"edges\":[[0,5],[0,6],[1,2],[1,3],[1,5],[1,6],[2,3],[2,5],[2,6],[3,4],[3,5]],\"vertex\":1}。請依 [vertex,component] 欄序回答。","answer":{"kind":"expression","value":"(1, [0, 1, 2, 3, 4, 5, 6])","explanation_zh":"Wave4第二層核對=(1, [0, 1, 2, 3, 4, 5, 6])；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；長度零path保證反身且孤立點為單點分量；完整可達性等價類不切換有向強弱連通。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["連通分量"]},"variant":{"family_id":"qf-mod28-component-of-vertex","seed":202610287016,"parameters":{"n":7,"edges":[[0,5],[0,6],[1,2],[1,3],[1,5],[1,6],[2,3],[2,5],[2,6],[3,4],[3,5]],"vertex":1,"domain":"Complete_finite_undirected_components","operation":"component-of-vertex","semantic_template":"qf-mod28-component-of-vertex/{\"edges\":[[0,5],[0,6],[1,2],[1,3],[1,5],[1,6],[2,3],[2,5],[2,6],[3,4],[3,5]],\"n\":7,\"vertex\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r07-connected-component-0017","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-component"],"title_zh":"連通分量","stem_zh":"完整有限無向無loop簡單圖。操作=component-of-vertex；資料={\"n\":7,\"edges\":[[0,1],[0,2],[0,3],[0,4],[0,5],[1,2],[1,3],[1,4],[1,6],[2,3],[3,6]],\"vertex\":2}。請依 [vertex,component] 欄序回答。","answer":{"kind":"expression","value":"(2, [0, 1, 2, 3, 4, 5, 6])","explanation_zh":"Wave4第二層核對=(2, [0, 1, 2, 3, 4, 5, 6])；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；長度零path保證反身且孤立點為單點分量；完整可達性等價類不切換有向強弱連通。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["連通分量"]},"variant":{"family_id":"qf-mod28-component-of-vertex","seed":202610287017,"parameters":{"n":7,"edges":[[0,1],[0,2],[0,3],[0,4],[0,5],[1,2],[1,3],[1,4],[1,6],[2,3],[3,6]],"vertex":2,"domain":"Complete_finite_undirected_components","operation":"component-of-vertex","semantic_template":"qf-mod28-component-of-vertex/{\"edges\":[[0,1],[0,2],[0,3],[0,4],[0,5],[1,2],[1,3],[1,4],[1,6],[2,3],[3,6]],\"n\":7,\"vertex\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r07-connected-component-0018","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-component"],"title_zh":"連通分量","stem_zh":"完整有限無向無loop簡單圖。操作=component-of-vertex；資料={\"n\":7,\"edges\":[[0,2],[0,3],[0,4],[1,2],[1,3],[1,6],[2,3],[2,5],[3,6]],\"vertex\":3}。請依 [vertex,component] 欄序回答。","answer":{"kind":"expression","value":"(3, [0, 1, 2, 3, 4, 5, 6])","explanation_zh":"Wave4第二層核對=(3, [0, 1, 2, 3, 4, 5, 6])；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；長度零path保證反身且孤立點為單點分量；完整可達性等價類不切換有向強弱連通。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["連通分量"]},"variant":{"family_id":"qf-mod28-component-of-vertex","seed":202610287018,"parameters":{"n":7,"edges":[[0,2],[0,3],[0,4],[1,2],[1,3],[1,6],[2,3],[2,5],[3,6]],"vertex":3,"domain":"Complete_finite_undirected_components","operation":"component-of-vertex","semantic_template":"qf-mod28-component-of-vertex/{\"edges\":[[0,2],[0,3],[0,4],[1,2],[1,3],[1,6],[2,3],[2,5],[3,6]],\"n\":7,\"vertex\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r07-connected-component-0019","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-component"],"title_zh":"連通分量","stem_zh":"完整有限無向無loop簡單圖。操作=component-of-vertex；資料={\"n\":7,\"edges\":[[0,1],[0,3],[0,4],[0,5],[0,6],[1,3],[1,4],[1,5],[2,3],[2,6],[3,6]],\"vertex\":4}。請依 [vertex,component] 欄序回答。","answer":{"kind":"expression","value":"(4, [0, 1, 2, 3, 4, 5, 6])","explanation_zh":"Wave4第二層核對=(4, [0, 1, 2, 3, 4, 5, 6])；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；長度零path保證反身且孤立點為單點分量；完整可達性等價類不切換有向強弱連通。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["連通分量"]},"variant":{"family_id":"qf-mod28-component-of-vertex","seed":202610287019,"parameters":{"n":7,"edges":[[0,1],[0,3],[0,4],[0,5],[0,6],[1,3],[1,4],[1,5],[2,3],[2,6],[3,6]],"vertex":4,"domain":"Complete_finite_undirected_components","operation":"component-of-vertex","semantic_template":"qf-mod28-component-of-vertex/{\"edges\":[[0,1],[0,3],[0,4],[0,5],[0,6],[1,3],[1,4],[1,5],[2,3],[2,6],[3,6]],\"n\":7,\"vertex\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r07-connected-component-0020","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-component"],"title_zh":"連通分量","stem_zh":"完整有限無向無loop簡單圖。操作=component-of-vertex；資料={\"n\":7,\"edges\":[[0,3],[0,4],[0,6],[1,3],[1,5],[2,3],[2,5],[2,6],[3,6]],\"vertex\":5}。請依 [vertex,component] 欄序回答。","answer":{"kind":"expression","value":"(5, [0, 1, 2, 3, 4, 5, 6])","explanation_zh":"Wave4第二層核對=(5, [0, 1, 2, 3, 4, 5, 6])；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；長度零path保證反身且孤立點為單點分量；完整可達性等價類不切換有向強弱連通。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["連通分量"]},"variant":{"family_id":"qf-mod28-component-of-vertex","seed":202610287020,"parameters":{"n":7,"edges":[[0,3],[0,4],[0,6],[1,3],[1,5],[2,3],[2,5],[2,6],[3,6]],"vertex":5,"domain":"Complete_finite_undirected_components","operation":"component-of-vertex","semantic_template":"qf-mod28-component-of-vertex/{\"edges\":[[0,3],[0,4],[0,6],[1,3],[1,5],[2,3],[2,5],[2,6],[3,6]],\"n\":7,\"vertex\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r07-connected-component-0021","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-component"],"title_zh":"連通分量","stem_zh":"完整有限無向無loop簡單圖。操作=component-of-vertex；資料={\"n\":7,\"edges\":[[0,1],[0,2],[0,4],[0,5],[1,3],[1,4],[2,3],[3,4],[3,6]],\"vertex\":6}。請依 [vertex,component] 欄序回答。","answer":{"kind":"expression","value":"(6, [0, 1, 2, 3, 4, 5, 6])","explanation_zh":"Wave4第二層核對=(6, [0, 1, 2, 3, 4, 5, 6])；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；長度零path保證反身且孤立點為單點分量；完整可達性等價類不切換有向強弱連通。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["連通分量"]},"variant":{"family_id":"qf-mod28-component-of-vertex","seed":202610287021,"parameters":{"n":7,"edges":[[0,1],[0,2],[0,4],[0,5],[1,3],[1,4],[2,3],[3,4],[3,6]],"vertex":6,"domain":"Complete_finite_undirected_components","operation":"component-of-vertex","semantic_template":"qf-mod28-component-of-vertex/{\"edges\":[[0,1],[0,2],[0,4],[0,5],[1,3],[1,4],[2,3],[3,4],[3,6]],\"n\":7,\"vertex\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r07-connected-component-0022","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-component"],"title_zh":"連通分量","stem_zh":"完整有限無向無loop簡單圖。操作=component-of-vertex；資料={\"n\":7,\"edges\":[[0,2],[0,4],[1,3],[2,3],[2,5],[3,4],[3,6]],\"vertex\":0}。請依 [vertex,component] 欄序回答。","answer":{"kind":"expression","value":"(0, [0, 1, 2, 3, 4, 5, 6])","explanation_zh":"Wave4第二層核對=(0, [0, 1, 2, 3, 4, 5, 6])；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；長度零path保證反身且孤立點為單點分量；完整可達性等價類不切換有向強弱連通。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["連通分量"]},"variant":{"family_id":"qf-mod28-component-of-vertex","seed":202610287022,"parameters":{"n":7,"edges":[[0,2],[0,4],[1,3],[2,3],[2,5],[3,4],[3,6]],"vertex":0,"domain":"Complete_finite_undirected_components","operation":"component-of-vertex","semantic_template":"qf-mod28-component-of-vertex/{\"edges\":[[0,2],[0,4],[1,3],[2,3],[2,5],[3,4],[3,6]],\"n\":7,\"vertex\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r07-connected-component-0023","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-component"],"title_zh":"連通分量","stem_zh":"完整有限無向無loop簡單圖。操作=component-of-vertex；資料={\"n\":7,\"edges\":[[0,1],[0,4],[0,5],[0,6],[1,2],[1,4],[1,5],[1,6],[2,6],[3,4],[3,6]],\"vertex\":1}。請依 [vertex,component] 欄序回答。","answer":{"kind":"expression","value":"(1, [0, 1, 2, 3, 4, 5, 6])","explanation_zh":"Wave4第二層核對=(1, [0, 1, 2, 3, 4, 5, 6])；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；長度零path保證反身且孤立點為單點分量；完整可達性等價類不切換有向強弱連通。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["連通分量"]},"variant":{"family_id":"qf-mod28-component-of-vertex","seed":202610287023,"parameters":{"n":7,"edges":[[0,1],[0,4],[0,5],[0,6],[1,2],[1,4],[1,5],[1,6],[2,6],[3,4],[3,6]],"vertex":1,"domain":"Complete_finite_undirected_components","operation":"component-of-vertex","semantic_template":"qf-mod28-component-of-vertex/{\"edges\":[[0,1],[0,4],[0,5],[0,6],[1,2],[1,4],[1,5],[1,6],[2,6],[3,4],[3,6]],\"n\":7,\"vertex\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r07-connected-component-0024","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-component"],"title_zh":"連通分量","stem_zh":"完整有限無向無loop簡單圖。操作=component-of-vertex；資料={\"n\":7,\"edges\":[[0,4],[0,6],[1,2],[1,5],[1,6],[2,5],[2,6],[3,4],[3,6]],\"vertex\":2}。請依 [vertex,component] 欄序回答。","answer":{"kind":"expression","value":"(2, [0, 1, 2, 3, 4, 5, 6])","explanation_zh":"Wave4第二層核對=(2, [0, 1, 2, 3, 4, 5, 6])；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；長度零path保證反身且孤立點為單點分量；完整可達性等價類不切換有向強弱連通。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["連通分量"]},"variant":{"family_id":"qf-mod28-component-of-vertex","seed":202610287024,"parameters":{"n":7,"edges":[[0,4],[0,6],[1,2],[1,5],[1,6],[2,5],[2,6],[3,4],[3,6]],"vertex":2,"domain":"Complete_finite_undirected_components","operation":"component-of-vertex","semantic_template":"qf-mod28-component-of-vertex/{\"edges\":[[0,4],[0,6],[1,2],[1,5],[1,6],[2,5],[2,6],[3,4],[3,6]],\"n\":7,\"vertex\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r07-connected-component-0025","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-component"],"title_zh":"連通分量","stem_zh":"完整有限無向無loop簡單圖。操作=component-of-vertex；資料={\"n\":7,\"edges\":[[0,1],[0,2],[0,3],[0,5],[1,2],[1,4],[1,6],[3,5],[3,6]],\"vertex\":3}。請依 [vertex,component] 欄序回答。","answer":{"kind":"expression","value":"(3, [0, 1, 2, 3, 4, 5, 6])","explanation_zh":"Wave4第二層核對=(3, [0, 1, 2, 3, 4, 5, 6])；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；長度零path保證反身且孤立點為單點分量；完整可達性等價類不切換有向強弱連通。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["連通分量"]},"variant":{"family_id":"qf-mod28-component-of-vertex","seed":202610287025,"parameters":{"n":7,"edges":[[0,1],[0,2],[0,3],[0,5],[1,2],[1,4],[1,6],[3,5],[3,6]],"vertex":3,"domain":"Complete_finite_undirected_components","operation":"component-of-vertex","semantic_template":"qf-mod28-component-of-vertex/{\"edges\":[[0,1],[0,2],[0,3],[0,5],[1,2],[1,4],[1,6],[3,5],[3,6]],\"n\":7,\"vertex\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r07-connected-component-0026","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-component"],"title_zh":"連通分量","stem_zh":"完整有限無向無loop簡單圖。操作=component-all；資料={\"n\":7,\"edges\":[[0,1],[0,2],[0,3],[0,4],[0,6],[1,2],[1,3],[1,5],[1,6],[2,3],[2,4]]}。請依 [components,count] 欄序回答。","answer":{"kind":"expression","value":"([[0, 1, 2, 3, 4, 5, 6]], 1)","explanation_zh":"Wave4第二層核對=([[0, 1, 2, 3, 4, 5, 6]], 1)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；長度零path保證反身且孤立點為單點分量；完整可達性等價類不切換有向強弱連通。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["連通分量"]},"variant":{"family_id":"qf-mod28-component-all","seed":202610287026,"parameters":{"n":7,"edges":[[0,1],[0,2],[0,3],[0,4],[0,6],[1,2],[1,3],[1,5],[1,6],[2,3],[2,4]],"domain":"Complete_finite_undirected_components","operation":"component-all","semantic_template":"qf-mod28-component-all/{\"edges\":[[0,1],[0,2],[0,3],[0,4],[0,6],[1,2],[1,3],[1,5],[1,6],[2,3],[2,4]],\"n\":7}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r07-connected-component-0027","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-component"],"title_zh":"連通分量","stem_zh":"完整有限無向無loop簡單圖。操作=component-all；資料={\"n\":7,\"edges\":[[0,2],[0,3],[0,4],[0,5],[1,2],[1,3],[1,4],[1,6],[2,3],[2,4],[2,5]]}。請依 [components,count] 欄序回答。","answer":{"kind":"expression","value":"([[0, 1, 2, 3, 4, 5, 6]], 1)","explanation_zh":"Wave4第二層核對=([[0, 1, 2, 3, 4, 5, 6]], 1)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；長度零path保證反身且孤立點為單點分量；完整可達性等價類不切換有向強弱連通。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["連通分量"]},"variant":{"family_id":"qf-mod28-component-all","seed":202610287027,"parameters":{"n":7,"edges":[[0,2],[0,3],[0,4],[0,5],[1,2],[1,3],[1,4],[1,6],[2,3],[2,4],[2,5]],"domain":"Complete_finite_undirected_components","operation":"component-all","semantic_template":"qf-mod28-component-all/{\"edges\":[[0,2],[0,3],[0,4],[0,5],[1,2],[1,3],[1,4],[1,6],[2,3],[2,4],[2,5]],\"n\":7}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r07-connected-component-0028","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-component"],"title_zh":"連通分量","stem_zh":"完整有限無向無loop簡單圖。操作=component-all；資料={\"n\":7,\"edges\":[[0,1],[0,3],[0,4],[1,2],[1,3],[1,6],[2,3],[2,4],[2,6]]}。請依 [components,count] 欄序回答。","answer":{"kind":"expression","value":"([[0, 1, 2, 3, 4, 6], [5]], 2)","explanation_zh":"Wave4第二層核對=([[0, 1, 2, 3, 4, 6], [5]], 2)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；長度零path保證反身且孤立點為單點分量；完整可達性等價類不切換有向強弱連通。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["連通分量"]},"variant":{"family_id":"qf-mod28-component-all","seed":202610287028,"parameters":{"n":7,"edges":[[0,1],[0,3],[0,4],[1,2],[1,3],[1,6],[2,3],[2,4],[2,6]],"domain":"Complete_finite_undirected_components","operation":"component-all","semantic_template":"qf-mod28-component-all/{\"edges\":[[0,1],[0,3],[0,4],[1,2],[1,3],[1,6],[2,3],[2,4],[2,6]],\"n\":7}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r07-connected-component-0029","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-component"],"title_zh":"連通分量","stem_zh":"完整有限無向無loop簡單圖。操作=component-all；資料={\"n\":7,\"edges\":[[0,3],[0,4],[0,5],[0,6],[1,3],[1,4],[1,5],[2,3],[2,4],[2,5],[2,6]]}。請依 [components,count] 欄序回答。","answer":{"kind":"expression","value":"([[0, 1, 2, 3, 4, 5, 6]], 1)","explanation_zh":"Wave4第二層核對=([[0, 1, 2, 3, 4, 5, 6]], 1)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；長度零path保證反身且孤立點為單點分量；完整可達性等價類不切換有向強弱連通。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["連通分量"]},"variant":{"family_id":"qf-mod28-component-all","seed":202610287029,"parameters":{"n":7,"edges":[[0,3],[0,4],[0,5],[0,6],[1,3],[1,4],[1,5],[2,3],[2,4],[2,5],[2,6]],"domain":"Complete_finite_undirected_components","operation":"component-all","semantic_template":"qf-mod28-component-all/{\"edges\":[[0,3],[0,4],[0,5],[0,6],[1,3],[1,4],[1,5],[2,3],[2,4],[2,5],[2,6]],\"n\":7}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
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{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r07-connected-component-0044","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-component"],"title_zh":"連通分量","stem_zh":"完整有限無向無loop簡單圖。操作=component-all；資料={\"n\":7,\"edges\":[[0,1],[0,3],[0,4],[0,5],[0,6],[1,3],[1,4],[1,5],[2,3],[2,6],[3,6]]}。請依 [components,count] 欄序回答。","answer":{"kind":"expression","value":"([[0, 1, 2, 3, 4, 5, 6]], 1)","explanation_zh":"Wave4第二層核對=([[0, 1, 2, 3, 4, 5, 6]], 1)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；長度零path保證反身且孤立點為單點分量；完整可達性等價類不切換有向強弱連通。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["連通分量"]},"variant":{"family_id":"qf-mod28-component-all","seed":202610287044,"parameters":{"n":7,"edges":[[0,1],[0,3],[0,4],[0,5],[0,6],[1,3],[1,4],[1,5],[2,3],[2,6],[3,6]],"domain":"Complete_finite_undirected_components","operation":"component-all","semantic_template":"qf-mod28-component-all/{\"edges\":[[0,1],[0,3],[0,4],[0,5],[0,6],[1,3],[1,4],[1,5],[2,3],[2,6],[3,6]],\"n\":7}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
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{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r07-connected-component-0046","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-component"],"title_zh":"連通分量","stem_zh":"完整有限無向無loop簡單圖。操作=component-all；資料={\"n\":7,\"edges\":[[0,1],[0,2],[0,4],[0,5],[1,3],[1,4],[2,3],[3,4],[3,6]]}。請依 [components,count] 欄序回答。","answer":{"kind":"expression","value":"([[0, 1, 2, 3, 4, 5, 6]], 1)","explanation_zh":"Wave4第二層核對=([[0, 1, 2, 3, 4, 5, 6]], 1)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；長度零path保證反身且孤立點為單點分量；完整可達性等價類不切換有向強弱連通。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["連通分量"]},"variant":{"family_id":"qf-mod28-component-all","seed":202610287046,"parameters":{"n":7,"edges":[[0,1],[0,2],[0,4],[0,5],[1,3],[1,4],[2,3],[3,4],[3,6]],"domain":"Complete_finite_undirected_components","operation":"component-all","semantic_template":"qf-mod28-component-all/{\"edges\":[[0,1],[0,2],[0,4],[0,5],[1,3],[1,4],[2,3],[3,4],[3,6]],\"n\":7}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r07-connected-component-0047","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-component"],"title_zh":"連通分量","stem_zh":"完整有限無向無loop簡單圖。操作=component-all；資料={\"n\":7,\"edges\":[[0,2],[0,4],[1,3],[2,3],[2,5],[3,4],[3,6]]}。請依 [components,count] 欄序回答。","answer":{"kind":"expression","value":"([[0, 1, 2, 3, 4, 5, 6]], 1)","explanation_zh":"Wave4第二層核對=([[0, 1, 2, 3, 4, 5, 6]], 1)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；長度零path保證反身且孤立點為單點分量；完整可達性等價類不切換有向強弱連通。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["連通分量"]},"variant":{"family_id":"qf-mod28-component-all","seed":202610287047,"parameters":{"n":7,"edges":[[0,2],[0,4],[1,3],[2,3],[2,5],[3,4],[3,6]],"domain":"Complete_finite_undirected_components","operation":"component-all","semantic_template":"qf-mod28-component-all/{\"edges\":[[0,2],[0,4],[1,3],[2,3],[2,5],[3,4],[3,6]],\"n\":7}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r07-connected-component-0048","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-component"],"title_zh":"連通分量","stem_zh":"完整有限無向無loop簡單圖。操作=component-all；資料={\"n\":7,\"edges\":[[0,1],[0,4],[0,5],[0,6],[1,2],[1,4],[1,5],[1,6],[2,6],[3,4],[3,6]]}。請依 [components,count] 欄序回答。","answer":{"kind":"expression","value":"([[0, 1, 2, 3, 4, 5, 6]], 1)","explanation_zh":"Wave4第二層核對=([[0, 1, 2, 3, 4, 5, 6]], 1)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；長度零path保證反身且孤立點為單點分量；完整可達性等價類不切換有向強弱連通。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["連通分量"]},"variant":{"family_id":"qf-mod28-component-all","seed":202610287048,"parameters":{"n":7,"edges":[[0,1],[0,4],[0,5],[0,6],[1,2],[1,4],[1,5],[1,6],[2,6],[3,4],[3,6]],"domain":"Complete_finite_undirected_components","operation":"component-all","semantic_template":"qf-mod28-component-all/{\"edges\":[[0,1],[0,4],[0,5],[0,6],[1,2],[1,4],[1,5],[1,6],[2,6],[3,4],[3,6]],\"n\":7}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r07-connected-component-0049","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-component"],"title_zh":"連通分量","stem_zh":"完整有限無向無loop簡單圖。操作=component-all；資料={\"n\":7,\"edges\":[[0,4],[0,6],[1,2],[1,5],[1,6],[2,5],[2,6],[3,4],[3,6]]}。請依 [components,count] 欄序回答。","answer":{"kind":"expression","value":"([[0, 1, 2, 3, 4, 5, 6]], 1)","explanation_zh":"Wave4第二層核對=([[0, 1, 2, 3, 4, 5, 6]], 1)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；長度零path保證反身且孤立點為單點分量；完整可達性等價類不切換有向強弱連通。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["連通分量"]},"variant":{"family_id":"qf-mod28-component-all","seed":202610287049,"parameters":{"n":7,"edges":[[0,4],[0,6],[1,2],[1,5],[1,6],[2,5],[2,6],[3,4],[3,6]],"domain":"Complete_finite_undirected_components","operation":"component-all","semantic_template":"qf-mod28-component-all/{\"edges\":[[0,4],[0,6],[1,2],[1,5],[1,6],[2,5],[2,6],[3,4],[3,6]],\"n\":7}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r07-connected-component-0050","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-component"],"title_zh":"連通分量","stem_zh":"完整有限無向無loop簡單圖。操作=component-all；資料={\"n\":7,\"edges\":[[0,1],[0,2],[0,3],[0,5],[1,2],[1,4],[1,6],[3,5],[3,6]]}。請依 [components,count] 欄序回答。","answer":{"kind":"expression","value":"([[0, 1, 2, 3, 4, 5, 6]], 1)","explanation_zh":"Wave4第二層核對=([[0, 1, 2, 3, 4, 5, 6]], 1)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；長度零path保證反身且孤立點為單點分量；完整可達性等價類不切換有向強弱連通。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["連通分量"]},"variant":{"family_id":"qf-mod28-component-all","seed":202610287050,"parameters":{"n":7,"edges":[[0,1],[0,2],[0,3],[0,5],[1,2],[1,4],[1,6],[3,5],[3,6]],"domain":"Complete_finite_undirected_components","operation":"component-all","semantic_template":"qf-mod28-component-all/{\"edges\":[[0,1],[0,2],[0,3],[0,5],[1,2],[1,4],[1,6],[3,5],[3,6]],\"n\":7}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r07-connected-component-0051","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-component"],"title_zh":"連通分量","stem_zh":"完整有限無向無loop簡單圖。操作=component-same-audit；資料={\"n\":7,\"edges\":[[0,1],[0,2],[0,3],[0,4],[0,6],[1,2],[1,3],[1,5],[1,6],[2,3],[2,4]],\"left\":0,\"right\":1}。請依 [left,right,left_component,same_component] 欄序回答。","answer":{"kind":"expression","value":"(0, 1, [0, 1, 2, 3, 4, 5, 6], 是)","explanation_zh":"Wave4第二層核對=(0, 1, [0, 1, 2, 3, 4, 5, 6], 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；長度零path保證反身且孤立點為單點分量；完整可達性等價類不切換有向強弱連通。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["連通分量"]},"variant":{"family_id":"qf-mod28-component-same-audit","seed":202610287051,"parameters":{"n":7,"edges":[[0,1],[0,2],[0,3],[0,4],[0,6],[1,2],[1,3],[1,5],[1,6],[2,3],[2,4]],"left":0,"right":1,"domain":"Complete_finite_undirected_components","operation":"component-same-audit","semantic_template":"qf-mod28-component-same-audit/{\"edges\":[[0,1],[0,2],[0,3],[0,4],[0,6],[1,2],[1,3],[1,5],[1,6],[2,3],[2,4]],\"left\":0,\"n\":7,\"right\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r07-connected-component-0052","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-component"],"title_zh":"連通分量","stem_zh":"完整有限無向無loop簡單圖。操作=component-same-audit；資料={\"n\":7,\"edges\":[[0,2],[0,3],[0,4],[0,5],[1,2],[1,3],[1,4],[1,6],[2,3],[2,4],[2,5]],\"left\":1,\"right\":4}。請依 [left,right,left_component,same_component] 欄序回答。","answer":{"kind":"expression","value":"(1, 4, [0, 1, 2, 3, 4, 5, 6], 是)","explanation_zh":"Wave4第二層核對=(1, 4, [0, 1, 2, 3, 4, 5, 6], 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；長度零path保證反身且孤立點為單點分量；完整可達性等價類不切換有向強弱連通。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["連通分量"]},"variant":{"family_id":"qf-mod28-component-same-audit","seed":202610287052,"parameters":{"n":7,"edges":[[0,2],[0,3],[0,4],[0,5],[1,2],[1,3],[1,4],[1,6],[2,3],[2,4],[2,5]],"left":1,"right":4,"domain":"Complete_finite_undirected_components","operation":"component-same-audit","semantic_template":"qf-mod28-component-same-audit/{\"edges\":[[0,2],[0,3],[0,4],[0,5],[1,2],[1,3],[1,4],[1,6],[2,3],[2,4],[2,5]],\"left\":1,\"n\":7,\"right\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r07-connected-component-0053","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-component"],"title_zh":"連通分量","stem_zh":"完整有限無向無loop簡單圖。操作=component-same-audit；資料={\"n\":7,\"edges\":[[0,1],[0,3],[0,4],[1,2],[1,3],[1,6],[2,3],[2,4],[2,6]],\"left\":2,\"right\":0}。請依 [left,right,left_component,same_component] 欄序回答。","answer":{"kind":"expression","value":"(2, 0, [0, 1, 2, 3, 4, 6], 是)","explanation_zh":"Wave4第二層核對=(2, 0, [0, 1, 2, 3, 4, 6], 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；長度零path保證反身且孤立點為單點分量；完整可達性等價類不切換有向強弱連通。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["連通分量"]},"variant":{"family_id":"qf-mod28-component-same-audit","seed":202610287053,"parameters":{"n":7,"edges":[[0,1],[0,3],[0,4],[1,2],[1,3],[1,6],[2,3],[2,4],[2,6]],"left":2,"right":0,"domain":"Complete_finite_undirected_components","operation":"component-same-audit","semantic_template":"qf-mod28-component-same-audit/{\"edges\":[[0,1],[0,3],[0,4],[1,2],[1,3],[1,6],[2,3],[2,4],[2,6]],\"left\":2,\"n\":7,\"right\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r07-connected-component-0054","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-component"],"title_zh":"連通分量","stem_zh":"完整有限無向無loop簡單圖。操作=component-same-audit；資料={\"n\":7,\"edges\":[[0,3],[0,4],[0,5],[0,6],[1,3],[1,4],[1,5],[2,3],[2,4],[2,5],[2,6]],\"left\":3,\"right\":3}。請依 [left,right,left_component,same_component] 欄序回答。","answer":{"kind":"expression","value":"(3, 3, [0, 1, 2, 3, 4, 5, 6], 是)","explanation_zh":"Wave4第二層核對=(3, 3, [0, 1, 2, 3, 4, 5, 6], 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；長度零path保證反身且孤立點為單點分量；完整可達性等價類不切換有向強弱連通。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["連通分量"]},"variant":{"family_id":"qf-mod28-component-same-audit","seed":202610287054,"parameters":{"n":7,"edges":[[0,3],[0,4],[0,5],[0,6],[1,3],[1,4],[1,5],[2,3],[2,4],[2,5],[2,6]],"left":3,"right":3,"domain":"Complete_finite_undirected_components","operation":"component-same-audit","semantic_template":"qf-mod28-component-same-audit/{\"edges\":[[0,3],[0,4],[0,5],[0,6],[1,3],[1,4],[1,5],[2,3],[2,4],[2,5],[2,6]],\"left\":3,\"n\":7,\"right\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r07-connected-component-0055","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-component"],"title_zh":"連通分量","stem_zh":"完整有限無向無loop簡單圖。操作=component-same-audit；資料={\"n\":7,\"edges\":[[0,1],[0,2],[0,4],[0,6],[1,3],[1,5],[2,3],[2,4],[3,4]],\"left\":4,\"right\":6}。請依 [left,right,left_component,same_component] 欄序回答。","answer":{"kind":"expression","value":"(4, 6, [0, 1, 2, 3, 4, 5, 6], 是)","explanation_zh":"Wave4第二層核對=(4, 6, [0, 1, 2, 3, 4, 5, 6], 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；長度零path保證反身且孤立點為單點分量；完整可達性等價類不切換有向強弱連通。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["連通分量"]},"variant":{"family_id":"qf-mod28-component-same-audit","seed":202610287055,"parameters":{"n":7,"edges":[[0,1],[0,2],[0,4],[0,6],[1,3],[1,5],[2,3],[2,4],[3,4]],"left":4,"right":6,"domain":"Complete_finite_undirected_components","operation":"component-same-audit","semantic_template":"qf-mod28-component-same-audit/{\"edges\":[[0,1],[0,2],[0,4],[0,6],[1,3],[1,5],[2,3],[2,4],[3,4]],\"left\":4,\"n\":7,\"right\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r07-connected-component-0056","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-component"],"title_zh":"連通分量","stem_zh":"完整有限無向無loop簡單圖。操作=component-same-audit；資料={\"n\":7,\"edges\":[[0,2],[0,4],[0,5],[1,3],[1,4],[2,3],[2,4],[2,5],[3,4]],\"left\":5,\"right\":2}。請依 [left,right,left_component,same_component] 欄序回答。","answer":{"kind":"expression","value":"(5, 2, [0, 1, 2, 3, 4, 5], 是)","explanation_zh":"Wave4第二層核對=(5, 2, [0, 1, 2, 3, 4, 5], 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；長度零path保證反身且孤立點為單點分量；完整可達性等價類不切換有向強弱連通。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["連通分量"]},"variant":{"family_id":"qf-mod28-component-same-audit","seed":202610287056,"parameters":{"n":7,"edges":[[0,2],[0,4],[0,5],[1,3],[1,4],[2,3],[2,4],[2,5],[3,4]],"left":5,"right":2,"domain":"Complete_finite_undirected_components","operation":"component-same-audit","semantic_template":"qf-mod28-component-same-audit/{\"edges\":[[0,2],[0,4],[0,5],[1,3],[1,4],[2,3],[2,4],[2,5],[3,4]],\"left\":5,\"n\":7,\"right\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r07-connected-component-0057","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-component"],"title_zh":"連通分量","stem_zh":"完整有限無向無loop簡單圖。操作=component-same-audit；資料={\"n\":7,\"edges\":[[0,1],[0,4],[1,3],[2,3],[2,4],[2,6],[3,4]],\"left\":6,\"right\":5}。請依 [left,right,left_component,same_component] 欄序回答。","answer":{"kind":"expression","value":"(6, 5, [0, 1, 2, 3, 4, 6], 否)","explanation_zh":"Wave4第二層核對=(6, 5, [0, 1, 2, 3, 4, 6], 否)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；長度零path保證反身且孤立點為單點分量；完整可達性等價類不切換有向強弱連通。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["連通分量"]},"variant":{"family_id":"qf-mod28-component-same-audit","seed":202610287057,"parameters":{"n":7,"edges":[[0,1],[0,4],[1,3],[2,3],[2,4],[2,6],[3,4]],"left":6,"right":5,"domain":"Complete_finite_undirected_components","operation":"component-same-audit","semantic_template":"qf-mod28-component-same-audit/{\"edges\":[[0,1],[0,4],[1,3],[2,3],[2,4],[2,6],[3,4]],\"left\":6,\"n\":7,\"right\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r07-connected-component-0058","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-component"],"title_zh":"連通分量","stem_zh":"完整有限無向無loop簡單圖。操作=component-same-audit；資料={\"n\":7,\"edges\":[[0,4],[0,5],[0,6],[1,2],[1,4],[1,5],[1,6],[2,4],[2,5],[2,6],[3,4]],\"left\":0,\"right\":1}。請依 [left,right,left_component,same_component] 欄序回答。","answer":{"kind":"expression","value":"(0, 1, [0, 1, 2, 3, 4, 5, 6], 是)","explanation_zh":"Wave4第二層核對=(0, 1, [0, 1, 2, 3, 4, 5, 6], 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；長度零path保證反身且孤立點為單點分量；完整可達性等價類不切換有向強弱連通。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["連通分量"]},"variant":{"family_id":"qf-mod28-component-same-audit","seed":202610287058,"parameters":{"n":7,"edges":[[0,4],[0,5],[0,6],[1,2],[1,4],[1,5],[1,6],[2,4],[2,5],[2,6],[3,4]],"left":0,"right":1,"domain":"Complete_finite_undirected_components","operation":"component-same-audit","semantic_template":"qf-mod28-component-same-audit/{\"edges\":[[0,4],[0,5],[0,6],[1,2],[1,4],[1,5],[1,6],[2,4],[2,5],[2,6],[3,4]],\"left\":0,\"n\":7,\"right\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r07-connected-component-0059","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-component"],"title_zh":"連通分量","stem_zh":"完整有限無向無loop簡單圖。操作=component-same-audit；資料={\"n\":7,\"edges\":[[0,1],[0,2],[0,3],[0,6],[1,2],[1,5],[1,6],[2,4],[3,5]],\"left\":1,\"right\":4}。請依 [left,right,left_component,same_component] 欄序回答。","answer":{"kind":"expression","value":"(1, 4, [0, 1, 2, 3, 4, 5, 6], 是)","explanation_zh":"Wave4第二層核對=(1, 4, [0, 1, 2, 3, 4, 5, 6], 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；長度零path保證反身且孤立點為單點分量；完整可達性等價類不切換有向強弱連通。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["連通分量"]},"variant":{"family_id":"qf-mod28-component-same-audit","seed":202610287059,"parameters":{"n":7,"edges":[[0,1],[0,2],[0,3],[0,6],[1,2],[1,5],[1,6],[2,4],[3,5]],"left":1,"right":4,"domain":"Complete_finite_undirected_components","operation":"component-same-audit","semantic_template":"qf-mod28-component-same-audit/{\"edges\":[[0,1],[0,2],[0,3],[0,6],[1,2],[1,5],[1,6],[2,4],[3,5]],\"left\":1,\"n\":7,\"right\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r07-connected-component-0060","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-component"],"title_zh":"連通分量","stem_zh":"完整有限無向無loop簡單圖。操作=component-same-audit；資料={\"n\":7,\"edges\":[[0,2],[0,3],[0,5],[1,2],[1,4],[1,6],[2,4],[2,5],[3,5]],\"left\":2,\"right\":0}。請依 [left,right,left_component,same_component] 欄序回答。","answer":{"kind":"expression","value":"(2, 0, [0, 1, 2, 3, 4, 5, 6], 是)","explanation_zh":"Wave4第二層核對=(2, 0, [0, 1, 2, 3, 4, 5, 6], 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；長度零path保證反身且孤立點為單點分量；完整可達性等價類不切換有向強弱連通。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["連通分量"]},"variant":{"family_id":"qf-mod28-component-same-audit","seed":202610287060,"parameters":{"n":7,"edges":[[0,2],[0,3],[0,5],[1,2],[1,4],[1,6],[2,4],[2,5],[3,5]],"left":2,"right":0,"domain":"Complete_finite_undirected_components","operation":"component-same-audit","semantic_template":"qf-mod28-component-same-audit/{\"edges\":[[0,2],[0,3],[0,5],[1,2],[1,4],[1,6],[2,4],[2,5],[3,5]],\"left\":2,\"n\":7,\"right\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r07-connected-component-0061","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-component"],"title_zh":"連通分量","stem_zh":"完整有限無向無loop簡單圖。操作=component-same-audit；資料={\"n\":7,\"edges\":[[0,1],[0,3],[1,2],[1,6],[2,4],[2,6],[3,5]],\"left\":3,\"right\":3}。請依 [left,right,left_component,same_component] 欄序回答。","answer":{"kind":"expression","value":"(3, 3, [0, 1, 2, 3, 4, 5, 6], 是)","explanation_zh":"Wave4第二層核對=(3, 3, [0, 1, 2, 3, 4, 5, 6], 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；長度零path保證反身且孤立點為單點分量；完整可達性等價類不切換有向強弱連通。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["連通分量"]},"variant":{"family_id":"qf-mod28-component-same-audit","seed":202610287061,"parameters":{"n":7,"edges":[[0,1],[0,3],[1,2],[1,6],[2,4],[2,6],[3,5]],"left":3,"right":3,"domain":"Complete_finite_undirected_components","operation":"component-same-audit","semantic_template":"qf-mod28-component-same-audit/{\"edges\":[[0,1],[0,3],[1,2],[1,6],[2,4],[2,6],[3,5]],\"left\":3,\"n\":7,\"right\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r07-connected-component-0062","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-component"],"title_zh":"連通分量","stem_zh":"完整有限無向無loop簡單圖。操作=component-same-audit；資料={\"n\":7,\"edges\":[[0,3],[0,5],[0,6],[1,4],[1,5],[2,4],[2,5],[2,6],[3,5]],\"left\":4,\"right\":6}。請依 [left,right,left_component,same_component] 欄序回答。","answer":{"kind":"expression","value":"(4, 6, [0, 1, 2, 3, 4, 5, 6], 是)","explanation_zh":"Wave4第二層核對=(4, 6, [0, 1, 2, 3, 4, 5, 6], 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；長度零path保證反身且孤立點為單點分量；完整可達性等價類不切換有向強弱連通。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["連通分量"]},"variant":{"family_id":"qf-mod28-component-same-audit","seed":202610287062,"parameters":{"n":7,"edges":[[0,3],[0,5],[0,6],[1,4],[1,5],[2,4],[2,5],[2,6],[3,5]],"left":4,"right":6,"domain":"Complete_finite_undirected_components","operation":"component-same-audit","semantic_template":"qf-mod28-component-same-audit/{\"edges\":[[0,3],[0,5],[0,6],[1,4],[1,5],[2,4],[2,5],[2,6],[3,5]],\"left\":4,\"n\":7,\"right\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r07-connected-component-0063","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-component"],"title_zh":"連通分量","stem_zh":"完整有限無向無loop簡單圖。操作=component-same-audit；資料={\"n\":7,\"edges\":[[0,1],[0,2],[0,6],[1,5],[2,4],[3,4],[3,5]],\"left\":5,\"right\":2}。請依 [left,right,left_component,same_component] 欄序回答。","answer":{"kind":"expression","value":"(5, 2, [0, 1, 2, 3, 4, 5, 6], 是)","explanation_zh":"Wave4第二層核對=(5, 2, [0, 1, 2, 3, 4, 5, 6], 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；長度零path保證反身且孤立點為單點分量；完整可達性等價類不切換有向強弱連通。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["連通分量"]},"variant":{"family_id":"qf-mod28-component-same-audit","seed":202610287063,"parameters":{"n":7,"edges":[[0,1],[0,2],[0,6],[1,5],[2,4],[3,4],[3,5]],"left":5,"right":2,"domain":"Complete_finite_undirected_components","operation":"component-same-audit","semantic_template":"qf-mod28-component-same-audit/{\"edges\":[[0,1],[0,2],[0,6],[1,5],[2,4],[3,4],[3,5]],\"left\":5,\"n\":7,\"right\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r07-connected-component-0064","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-component"],"title_zh":"連通分量","stem_zh":"完整有限無向無loop簡單圖。操作=component-same-audit；資料={\"n\":7,\"edges\":[[0,2],[0,5],[1,4],[2,4],[2,5],[3,4],[3,5]],\"left\":6,\"right\":5}。請依 [left,right,left_component,same_component] 欄序回答。","answer":{"kind":"expression","value":"(6, 5, [6], 否)","explanation_zh":"Wave4第二層核對=(6, 5, [6], 否)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；長度零path保證反身且孤立點為單點分量；完整可達性等價類不切換有向強弱連通。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["連通分量"]},"variant":{"family_id":"qf-mod28-component-same-audit","seed":202610287064,"parameters":{"n":7,"edges":[[0,2],[0,5],[1,4],[2,4],[2,5],[3,4],[3,5]],"left":6,"right":5,"domain":"Complete_finite_undirected_components","operation":"component-same-audit","semantic_template":"qf-mod28-component-same-audit/{\"edges\":[[0,2],[0,5],[1,4],[2,4],[2,5],[3,4],[3,5]],\"left\":6,\"n\":7,\"right\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r07-connected-component-0065","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-component"],"title_zh":"連通分量","stem_zh":"完整有限無向無loop簡單圖。操作=component-same-audit；資料={\"n\":7,\"edges\":[[0,1],[2,4],[2,6],[3,4],[3,5]],\"left\":0,\"right\":1}。請依 [left,right,left_component,same_component] 欄序回答。","answer":{"kind":"expression","value":"(0, 1, [0, 1], 是)","explanation_zh":"Wave4第二層核對=(0, 1, [0, 1], 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；長度零path保證反身且孤立點為單點分量；完整可達性等價類不切換有向強弱連通。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["連通分量"]},"variant":{"family_id":"qf-mod28-component-same-audit","seed":202610287065,"parameters":{"n":7,"edges":[[0,1],[2,4],[2,6],[3,4],[3,5]],"left":0,"right":1,"domain":"Complete_finite_undirected_components","operation":"component-same-audit","semantic_template":"qf-mod28-component-same-audit/{\"edges\":[[0,1],[2,4],[2,6],[3,4],[3,5]],\"left\":0,\"n\":7,\"right\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r07-connected-component-0066","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-component"],"title_zh":"連通分量","stem_zh":"完整有限無向無loop簡單圖。操作=component-same-audit；資料={\"n\":7,\"edges\":[[0,5],[0,6],[1,2],[1,3],[1,5],[1,6],[2,3],[2,5],[2,6],[3,4],[3,5]],\"left\":1,\"right\":4}。請依 [left,right,left_component,same_component] 欄序回答。","answer":{"kind":"expression","value":"(1, 4, [0, 1, 2, 3, 4, 5, 6], 是)","explanation_zh":"Wave4第二層核對=(1, 4, [0, 1, 2, 3, 4, 5, 6], 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；長度零path保證反身且孤立點為單點分量；完整可達性等價類不切換有向強弱連通。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["連通分量"]},"variant":{"family_id":"qf-mod28-component-same-audit","seed":202610287066,"parameters":{"n":7,"edges":[[0,5],[0,6],[1,2],[1,3],[1,5],[1,6],[2,3],[2,5],[2,6],[3,4],[3,5]],"left":1,"right":4,"domain":"Complete_finite_undirected_components","operation":"component-same-audit","semantic_template":"qf-mod28-component-same-audit/{\"edges\":[[0,5],[0,6],[1,2],[1,3],[1,5],[1,6],[2,3],[2,5],[2,6],[3,4],[3,5]],\"left\":1,\"n\":7,\"right\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r07-connected-component-0067","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-component"],"title_zh":"連通分量","stem_zh":"完整有限無向無loop簡單圖。操作=component-same-audit；資料={\"n\":7,\"edges\":[[0,1],[0,2],[0,3],[0,4],[0,5],[1,2],[1,3],[1,4],[1,6],[2,3],[3,6]],\"left\":2,\"right\":0}。請依 [left,right,left_component,same_component] 欄序回答。","answer":{"kind":"expression","value":"(2, 0, [0, 1, 2, 3, 4, 5, 6], 是)","explanation_zh":"Wave4第二層核對=(2, 0, [0, 1, 2, 3, 4, 5, 6], 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；長度零path保證反身且孤立點為單點分量；完整可達性等價類不切換有向強弱連通。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["連通分量"]},"variant":{"family_id":"qf-mod28-component-same-audit","seed":202610287067,"parameters":{"n":7,"edges":[[0,1],[0,2],[0,3],[0,4],[0,5],[1,2],[1,3],[1,4],[1,6],[2,3],[3,6]],"left":2,"right":0,"domain":"Complete_finite_undirected_components","operation":"component-same-audit","semantic_template":"qf-mod28-component-same-audit/{\"edges\":[[0,1],[0,2],[0,3],[0,4],[0,5],[1,2],[1,3],[1,4],[1,6],[2,3],[3,6]],\"left\":2,\"n\":7,\"right\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r07-connected-component-0068","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-component"],"title_zh":"連通分量","stem_zh":"完整有限無向無loop簡單圖。操作=component-same-audit；資料={\"n\":7,\"edges\":[[0,2],[0,3],[0,4],[1,2],[1,3],[1,6],[2,3],[2,5],[3,6]],\"left\":3,\"right\":3}。請依 [left,right,left_component,same_component] 欄序回答。","answer":{"kind":"expression","value":"(3, 3, [0, 1, 2, 3, 4, 5, 6], 是)","explanation_zh":"Wave4第二層核對=(3, 3, [0, 1, 2, 3, 4, 5, 6], 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；長度零path保證反身且孤立點為單點分量；完整可達性等價類不切換有向強弱連通。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["連通分量"]},"variant":{"family_id":"qf-mod28-component-same-audit","seed":202610287068,"parameters":{"n":7,"edges":[[0,2],[0,3],[0,4],[1,2],[1,3],[1,6],[2,3],[2,5],[3,6]],"left":3,"right":3,"domain":"Complete_finite_undirected_components","operation":"component-same-audit","semantic_template":"qf-mod28-component-same-audit/{\"edges\":[[0,2],[0,3],[0,4],[1,2],[1,3],[1,6],[2,3],[2,5],[3,6]],\"left\":3,\"n\":7,\"right\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r07-connected-component-0069","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-component"],"title_zh":"連通分量","stem_zh":"完整有限無向無loop簡單圖。操作=component-same-audit；資料={\"n\":7,\"edges\":[[0,1],[0,3],[0,4],[0,5],[0,6],[1,3],[1,4],[1,5],[2,3],[2,6],[3,6]],\"left\":4,\"right\":6}。請依 [left,right,left_component,same_component] 欄序回答。","answer":{"kind":"expression","value":"(4, 6, [0, 1, 2, 3, 4, 5, 6], 是)","explanation_zh":"Wave4第二層核對=(4, 6, [0, 1, 2, 3, 4, 5, 6], 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；長度零path保證反身且孤立點為單點分量；完整可達性等價類不切換有向強弱連通。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["連通分量"]},"variant":{"family_id":"qf-mod28-component-same-audit","seed":202610287069,"parameters":{"n":7,"edges":[[0,1],[0,3],[0,4],[0,5],[0,6],[1,3],[1,4],[1,5],[2,3],[2,6],[3,6]],"left":4,"right":6,"domain":"Complete_finite_undirected_components","operation":"component-same-audit","semantic_template":"qf-mod28-component-same-audit/{\"edges\":[[0,1],[0,3],[0,4],[0,5],[0,6],[1,3],[1,4],[1,5],[2,3],[2,6],[3,6]],\"left\":4,\"n\":7,\"right\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r07-connected-component-0070","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-component"],"title_zh":"連通分量","stem_zh":"完整有限無向無loop簡單圖。操作=component-same-audit；資料={\"n\":7,\"edges\":[[0,3],[0,4],[0,6],[1,3],[1,5],[2,3],[2,5],[2,6],[3,6]],\"left\":5,\"right\":2}。請依 [left,right,left_component,same_component] 欄序回答。","answer":{"kind":"expression","value":"(5, 2, [0, 1, 2, 3, 4, 5, 6], 是)","explanation_zh":"Wave4第二層核對=(5, 2, [0, 1, 2, 3, 4, 5, 6], 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；長度零path保證反身且孤立點為單點分量；完整可達性等價類不切換有向強弱連通。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["連通分量"]},"variant":{"family_id":"qf-mod28-component-same-audit","seed":202610287070,"parameters":{"n":7,"edges":[[0,3],[0,4],[0,6],[1,3],[1,5],[2,3],[2,5],[2,6],[3,6]],"left":5,"right":2,"domain":"Complete_finite_undirected_components","operation":"component-same-audit","semantic_template":"qf-mod28-component-same-audit/{\"edges\":[[0,3],[0,4],[0,6],[1,3],[1,5],[2,3],[2,5],[2,6],[3,6]],\"left\":5,\"n\":7,\"right\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r07-connected-component-0071","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-component"],"title_zh":"連通分量","stem_zh":"完整有限無向無loop簡單圖。操作=component-same-audit；資料={\"n\":7,\"edges\":[[0,1],[0,2],[0,4],[0,5],[1,3],[1,4],[2,3],[3,4],[3,6]],\"left\":6,\"right\":5}。請依 [left,right,left_component,same_component] 欄序回答。","answer":{"kind":"expression","value":"(6, 5, [0, 1, 2, 3, 4, 5, 6], 是)","explanation_zh":"Wave4第二層核對=(6, 5, [0, 1, 2, 3, 4, 5, 6], 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；長度零path保證反身且孤立點為單點分量；完整可達性等價類不切換有向強弱連通。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["連通分量"]},"variant":{"family_id":"qf-mod28-component-same-audit","seed":202610287071,"parameters":{"n":7,"edges":[[0,1],[0,2],[0,4],[0,5],[1,3],[1,4],[2,3],[3,4],[3,6]],"left":6,"right":5,"domain":"Complete_finite_undirected_components","operation":"component-same-audit","semantic_template":"qf-mod28-component-same-audit/{\"edges\":[[0,1],[0,2],[0,4],[0,5],[1,3],[1,4],[2,3],[3,4],[3,6]],\"left\":6,\"n\":7,\"right\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r07-connected-component-0072","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-component"],"title_zh":"連通分量","stem_zh":"完整有限無向無loop簡單圖。操作=component-same-audit；資料={\"n\":7,\"edges\":[[0,2],[0,4],[1,3],[2,3],[2,5],[3,4],[3,6]],\"left\":0,\"right\":1}。請依 [left,right,left_component,same_component] 欄序回答。","answer":{"kind":"expression","value":"(0, 1, [0, 1, 2, 3, 4, 5, 6], 是)","explanation_zh":"Wave4第二層核對=(0, 1, [0, 1, 2, 3, 4, 5, 6], 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；長度零path保證反身且孤立點為單點分量；完整可達性等價類不切換有向強弱連通。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["連通分量"]},"variant":{"family_id":"qf-mod28-component-same-audit","seed":202610287072,"parameters":{"n":7,"edges":[[0,2],[0,4],[1,3],[2,3],[2,5],[3,4],[3,6]],"left":0,"right":1,"domain":"Complete_finite_undirected_components","operation":"component-same-audit","semantic_template":"qf-mod28-component-same-audit/{\"edges\":[[0,2],[0,4],[1,3],[2,3],[2,5],[3,4],[3,6]],\"left\":0,\"n\":7,\"right\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r07-connected-component-0073","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-component"],"title_zh":"連通分量","stem_zh":"完整有限無向無loop簡單圖。操作=component-same-audit；資料={\"n\":7,\"edges\":[[0,1],[0,4],[0,5],[0,6],[1,2],[1,4],[1,5],[1,6],[2,6],[3,4],[3,6]],\"left\":1,\"right\":4}。請依 [left,right,left_component,same_component] 欄序回答。","answer":{"kind":"expression","value":"(1, 4, [0, 1, 2, 3, 4, 5, 6], 是)","explanation_zh":"Wave4第二層核對=(1, 4, [0, 1, 2, 3, 4, 5, 6], 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；長度零path保證反身且孤立點為單點分量；完整可達性等價類不切換有向強弱連通。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["連通分量"]},"variant":{"family_id":"qf-mod28-component-same-audit","seed":202610287073,"parameters":{"n":7,"edges":[[0,1],[0,4],[0,5],[0,6],[1,2],[1,4],[1,5],[1,6],[2,6],[3,4],[3,6]],"left":1,"right":4,"domain":"Complete_finite_undirected_components","operation":"component-same-audit","semantic_template":"qf-mod28-component-same-audit/{\"edges\":[[0,1],[0,4],[0,5],[0,6],[1,2],[1,4],[1,5],[1,6],[2,6],[3,4],[3,6]],\"left\":1,\"n\":7,\"right\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r07-connected-component-0074","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-component"],"title_zh":"連通分量","stem_zh":"完整有限無向無loop簡單圖。操作=component-same-audit；資料={\"n\":7,\"edges\":[[0,4],[0,6],[1,2],[1,5],[1,6],[2,5],[2,6],[3,4],[3,6]],\"left\":2,\"right\":0}。請依 [left,right,left_component,same_component] 欄序回答。","answer":{"kind":"expression","value":"(2, 0, [0, 1, 2, 3, 4, 5, 6], 是)","explanation_zh":"Wave4第二層核對=(2, 0, [0, 1, 2, 3, 4, 5, 6], 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；長度零path保證反身且孤立點為單點分量；完整可達性等價類不切換有向強弱連通。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["連通分量"]},"variant":{"family_id":"qf-mod28-component-same-audit","seed":202610287074,"parameters":{"n":7,"edges":[[0,4],[0,6],[1,2],[1,5],[1,6],[2,5],[2,6],[3,4],[3,6]],"left":2,"right":0,"domain":"Complete_finite_undirected_components","operation":"component-same-audit","semantic_template":"qf-mod28-component-same-audit/{\"edges\":[[0,4],[0,6],[1,2],[1,5],[1,6],[2,5],[2,6],[3,4],[3,6]],\"left\":2,\"n\":7,\"right\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r07-connected-component-0075","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-component"],"title_zh":"連通分量","stem_zh":"完整有限無向無loop簡單圖。操作=component-same-audit；資料={\"n\":7,\"edges\":[[0,1],[0,2],[0,3],[0,5],[1,2],[1,4],[1,6],[3,5],[3,6]],\"left\":3,\"right\":3}。請依 [left,right,left_component,same_component] 欄序回答。","answer":{"kind":"expression","value":"(3, 3, [0, 1, 2, 3, 4, 5, 6], 是)","explanation_zh":"Wave4第二層核對=(3, 3, [0, 1, 2, 3, 4, 5, 6], 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；長度零path保證反身且孤立點為單點分量；完整可達性等價類不切換有向強弱連通。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["連通分量"]},"variant":{"family_id":"qf-mod28-component-same-audit","seed":202610287075,"parameters":{"n":7,"edges":[[0,1],[0,2],[0,3],[0,5],[1,2],[1,4],[1,6],[3,5],[3,6]],"left":3,"right":3,"domain":"Complete_finite_undirected_components","operation":"component-same-audit","semantic_template":"qf-mod28-component-same-audit/{\"edges\":[[0,1],[0,2],[0,3],[0,5],[1,2],[1,4],[1,6],[3,5],[3,6]],\"left\":3,\"n\":7,\"right\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r07-connected-component-0076","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-component"],"title_zh":"連通分量","stem_zh":"完整有限無向無loop簡單圖。操作=component-partition；資料={\"n\":7,\"edges\":[[0,1],[0,2],[0,3],[0,4],[0,6],[1,2],[1,3],[1,5],[1,6],[2,3],[2,4]]}。請依 [components,covers_vertices,disjoint,singletons,boundary] 欄序回答。","answer":{"kind":"expression","value":"([[0, 1, 2, 3, 4, 5, 6]], 是, 是, [], length_zero_paths_include_every_vertex)","explanation_zh":"Wave4第二層核對=([[0, 1, 2, 3, 4, 5, 6]], 是, 是, [], length_zero_paths_include_every_vertex)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；長度零path保證反身且孤立點為單點分量；完整可達性等價類不切換有向強弱連通。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["連通分量"]},"variant":{"family_id":"qf-mod28-component-partition","seed":202610287076,"parameters":{"n":7,"edges":[[0,1],[0,2],[0,3],[0,4],[0,6],[1,2],[1,3],[1,5],[1,6],[2,3],[2,4]],"domain":"Complete_finite_undirected_components","operation":"component-partition","semantic_template":"qf-mod28-component-partition/{\"edges\":[[0,1],[0,2],[0,3],[0,4],[0,6],[1,2],[1,3],[1,5],[1,6],[2,3],[2,4]],\"n\":7}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r07-connected-component-0077","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-component"],"title_zh":"連通分量","stem_zh":"完整有限無向無loop簡單圖。操作=component-partition；資料={\"n\":7,\"edges\":[[0,2],[0,3],[0,4],[0,5],[1,2],[1,3],[1,4],[1,6],[2,3],[2,4],[2,5]]}。請依 [components,covers_vertices,disjoint,singletons,boundary] 欄序回答。","answer":{"kind":"expression","value":"([[0, 1, 2, 3, 4, 5, 6]], 是, 是, [], length_zero_paths_include_every_vertex)","explanation_zh":"Wave4第二層核對=([[0, 1, 2, 3, 4, 5, 6]], 是, 是, [], length_zero_paths_include_every_vertex)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；長度零path保證反身且孤立點為單點分量；完整可達性等價類不切換有向強弱連通。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["連通分量"]},"variant":{"family_id":"qf-mod28-component-partition","seed":202610287077,"parameters":{"n":7,"edges":[[0,2],[0,3],[0,4],[0,5],[1,2],[1,3],[1,4],[1,6],[2,3],[2,4],[2,5]],"domain":"Complete_finite_undirected_components","operation":"component-partition","semantic_template":"qf-mod28-component-partition/{\"edges\":[[0,2],[0,3],[0,4],[0,5],[1,2],[1,3],[1,4],[1,6],[2,3],[2,4],[2,5]],\"n\":7}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r07-connected-component-0078","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-component"],"title_zh":"連通分量","stem_zh":"完整有限無向無loop簡單圖。操作=component-partition；資料={\"n\":7,\"edges\":[[0,1],[0,3],[0,4],[1,2],[1,3],[1,6],[2,3],[2,4],[2,6]]}。請依 [components,covers_vertices,disjoint,singletons,boundary] 欄序回答。","answer":{"kind":"expression","value":"([[0, 1, 2, 3, 4, 6], [5]], 是, 是, [[5]], length_zero_paths_include_every_vertex)","explanation_zh":"Wave4第二層核對=([[0, 1, 2, 3, 4, 6], [5]], 是, 是, [[5]], length_zero_paths_include_every_vertex)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；長度零path保證反身且孤立點為單點分量；完整可達性等價類不切換有向強弱連通。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["連通分量"]},"variant":{"family_id":"qf-mod28-component-partition","seed":202610287078,"parameters":{"n":7,"edges":[[0,1],[0,3],[0,4],[1,2],[1,3],[1,6],[2,3],[2,4],[2,6]],"domain":"Complete_finite_undirected_components","operation":"component-partition","semantic_template":"qf-mod28-component-partition/{\"edges\":[[0,1],[0,3],[0,4],[1,2],[1,3],[1,6],[2,3],[2,4],[2,6]],\"n\":7}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r07-connected-component-0079","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-component"],"title_zh":"連通分量","stem_zh":"完整有限無向無loop簡單圖。操作=component-partition；資料={\"n\":7,\"edges\":[[0,3],[0,4],[0,5],[0,6],[1,3],[1,4],[1,5],[2,3],[2,4],[2,5],[2,6]]}。請依 [components,covers_vertices,disjoint,singletons,boundary] 欄序回答。","answer":{"kind":"expression","value":"([[0, 1, 2, 3, 4, 5, 6]], 是, 是, [], length_zero_paths_include_every_vertex)","explanation_zh":"Wave4第二層核對=([[0, 1, 2, 3, 4, 5, 6]], 是, 是, [], length_zero_paths_include_every_vertex)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；長度零path保證反身且孤立點為單點分量；完整可達性等價類不切換有向強弱連通。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["連通分量"]},"variant":{"family_id":"qf-mod28-component-partition","seed":202610287079,"parameters":{"n":7,"edges":[[0,3],[0,4],[0,5],[0,6],[1,3],[1,4],[1,5],[2,3],[2,4],[2,5],[2,6]],"domain":"Complete_finite_undirected_components","operation":"component-partition","semantic_template":"qf-mod28-component-partition/{\"edges\":[[0,3],[0,4],[0,5],[0,6],[1,3],[1,4],[1,5],[2,3],[2,4],[2,5],[2,6]],\"n\":7}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r07-connected-component-0080","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-component"],"title_zh":"連通分量","stem_zh":"完整有限無向無loop簡單圖。操作=component-partition；資料={\"n\":7,\"edges\":[[0,1],[0,2],[0,4],[0,6],[1,3],[1,5],[2,3],[2,4],[3,4]]}。請依 [components,covers_vertices,disjoint,singletons,boundary] 欄序回答。","answer":{"kind":"expression","value":"([[0, 1, 2, 3, 4, 5, 6]], 是, 是, [], length_zero_paths_include_every_vertex)","explanation_zh":"Wave4第二層核對=([[0, 1, 2, 3, 4, 5, 6]], 是, 是, [], length_zero_paths_include_every_vertex)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；長度零path保證反身且孤立點為單點分量；完整可達性等價類不切換有向強弱連通。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["連通分量"]},"variant":{"family_id":"qf-mod28-component-partition","seed":202610287080,"parameters":{"n":7,"edges":[[0,1],[0,2],[0,4],[0,6],[1,3],[1,5],[2,3],[2,4],[3,4]],"domain":"Complete_finite_undirected_components","operation":"component-partition","semantic_template":"qf-mod28-component-partition/{\"edges\":[[0,1],[0,2],[0,4],[0,6],[1,3],[1,5],[2,3],[2,4],[3,4]],\"n\":7}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r07-connected-component-0081","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-component"],"title_zh":"連通分量","stem_zh":"完整有限無向無loop簡單圖。操作=component-partition；資料={\"n\":7,\"edges\":[[0,2],[0,4],[0,5],[1,3],[1,4],[2,3],[2,4],[2,5],[3,4]]}。請依 [components,covers_vertices,disjoint,singletons,boundary] 欄序回答。","answer":{"kind":"expression","value":"([[0, 1, 2, 3, 4, 5], [6]], 是, 是, [[6]], length_zero_paths_include_every_vertex)","explanation_zh":"Wave4第二層核對=([[0, 1, 2, 3, 4, 5], [6]], 是, 是, [[6]], length_zero_paths_include_every_vertex)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；長度零path保證反身且孤立點為單點分量；完整可達性等價類不切換有向強弱連通。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["連通分量"]},"variant":{"family_id":"qf-mod28-component-partition","seed":202610287081,"parameters":{"n":7,"edges":[[0,2],[0,4],[0,5],[1,3],[1,4],[2,3],[2,4],[2,5],[3,4]],"domain":"Complete_finite_undirected_components","operation":"component-partition","semantic_template":"qf-mod28-component-partition/{\"edges\":[[0,2],[0,4],[0,5],[1,3],[1,4],[2,3],[2,4],[2,5],[3,4]],\"n\":7}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r07-connected-component-0082","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-component"],"title_zh":"連通分量","stem_zh":"完整有限無向無loop簡單圖。操作=component-partition；資料={\"n\":7,\"edges\":[[0,1],[0,4],[1,3],[2,3],[2,4],[2,6],[3,4]]}。請依 [components,covers_vertices,disjoint,singletons,boundary] 欄序回答。","answer":{"kind":"expression","value":"([[0, 1, 2, 3, 4, 6], [5]], 是, 是, [[5]], length_zero_paths_include_every_vertex)","explanation_zh":"Wave4第二層核對=([[0, 1, 2, 3, 4, 6], [5]], 是, 是, [[5]], length_zero_paths_include_every_vertex)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；長度零path保證反身且孤立點為單點分量；完整可達性等價類不切換有向強弱連通。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["連通分量"]},"variant":{"family_id":"qf-mod28-component-partition","seed":202610287082,"parameters":{"n":7,"edges":[[0,1],[0,4],[1,3],[2,3],[2,4],[2,6],[3,4]],"domain":"Complete_finite_undirected_components","operation":"component-partition","semantic_template":"qf-mod28-component-partition/{\"edges\":[[0,1],[0,4],[1,3],[2,3],[2,4],[2,6],[3,4]],\"n\":7}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r07-connected-component-0083","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-component"],"title_zh":"連通分量","stem_zh":"完整有限無向無loop簡單圖。操作=component-partition；資料={\"n\":7,\"edges\":[[0,4],[0,5],[0,6],[1,2],[1,4],[1,5],[1,6],[2,4],[2,5],[2,6],[3,4]]}。請依 [components,covers_vertices,disjoint,singletons,boundary] 欄序回答。","answer":{"kind":"expression","value":"([[0, 1, 2, 3, 4, 5, 6]], 是, 是, [], length_zero_paths_include_every_vertex)","explanation_zh":"Wave4第二層核對=([[0, 1, 2, 3, 4, 5, 6]], 是, 是, [], length_zero_paths_include_every_vertex)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；長度零path保證反身且孤立點為單點分量；完整可達性等價類不切換有向強弱連通。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["連通分量"]},"variant":{"family_id":"qf-mod28-component-partition","seed":202610287083,"parameters":{"n":7,"edges":[[0,4],[0,5],[0,6],[1,2],[1,4],[1,5],[1,6],[2,4],[2,5],[2,6],[3,4]],"domain":"Complete_finite_undirected_components","operation":"component-partition","semantic_template":"qf-mod28-component-partition/{\"edges\":[[0,4],[0,5],[0,6],[1,2],[1,4],[1,5],[1,6],[2,4],[2,5],[2,6],[3,4]],\"n\":7}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r07-connected-component-0084","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-component"],"title_zh":"連通分量","stem_zh":"完整有限無向無loop簡單圖。操作=component-partition；資料={\"n\":7,\"edges\":[[0,1],[0,2],[0,3],[0,6],[1,2],[1,5],[1,6],[2,4],[3,5]]}。請依 [components,covers_vertices,disjoint,singletons,boundary] 欄序回答。","answer":{"kind":"expression","value":"([[0, 1, 2, 3, 4, 5, 6]], 是, 是, [], length_zero_paths_include_every_vertex)","explanation_zh":"Wave4第二層核對=([[0, 1, 2, 3, 4, 5, 6]], 是, 是, [], length_zero_paths_include_every_vertex)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；長度零path保證反身且孤立點為單點分量；完整可達性等價類不切換有向強弱連通。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["連通分量"]},"variant":{"family_id":"qf-mod28-component-partition","seed":202610287084,"parameters":{"n":7,"edges":[[0,1],[0,2],[0,3],[0,6],[1,2],[1,5],[1,6],[2,4],[3,5]],"domain":"Complete_finite_undirected_components","operation":"component-partition","semantic_template":"qf-mod28-component-partition/{\"edges\":[[0,1],[0,2],[0,3],[0,6],[1,2],[1,5],[1,6],[2,4],[3,5]],\"n\":7}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r07-connected-component-0085","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-component"],"title_zh":"連通分量","stem_zh":"完整有限無向無loop簡單圖。操作=component-partition；資料={\"n\":7,\"edges\":[[0,2],[0,3],[0,5],[1,2],[1,4],[1,6],[2,4],[2,5],[3,5]]}。請依 [components,covers_vertices,disjoint,singletons,boundary] 欄序回答。","answer":{"kind":"expression","value":"([[0, 1, 2, 3, 4, 5, 6]], 是, 是, [], length_zero_paths_include_every_vertex)","explanation_zh":"Wave4第二層核對=([[0, 1, 2, 3, 4, 5, 6]], 是, 是, [], length_zero_paths_include_every_vertex)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；長度零path保證反身且孤立點為單點分量；完整可達性等價類不切換有向強弱連通。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["連通分量"]},"variant":{"family_id":"qf-mod28-component-partition","seed":202610287085,"parameters":{"n":7,"edges":[[0,2],[0,3],[0,5],[1,2],[1,4],[1,6],[2,4],[2,5],[3,5]],"domain":"Complete_finite_undirected_components","operation":"component-partition","semantic_template":"qf-mod28-component-partition/{\"edges\":[[0,2],[0,3],[0,5],[1,2],[1,4],[1,6],[2,4],[2,5],[3,5]],\"n\":7}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r07-connected-component-0086","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-component"],"title_zh":"連通分量","stem_zh":"完整有限無向無loop簡單圖。操作=component-partition；資料={\"n\":7,\"edges\":[[0,1],[0,3],[1,2],[1,6],[2,4],[2,6],[3,5]]}。請依 [components,covers_vertices,disjoint,singletons,boundary] 欄序回答。","answer":{"kind":"expression","value":"([[0, 1, 2, 3, 4, 5, 6]], 是, 是, [], length_zero_paths_include_every_vertex)","explanation_zh":"Wave4第二層核對=([[0, 1, 2, 3, 4, 5, 6]], 是, 是, [], length_zero_paths_include_every_vertex)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；長度零path保證反身且孤立點為單點分量；完整可達性等價類不切換有向強弱連通。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["連通分量"]},"variant":{"family_id":"qf-mod28-component-partition","seed":202610287086,"parameters":{"n":7,"edges":[[0,1],[0,3],[1,2],[1,6],[2,4],[2,6],[3,5]],"domain":"Complete_finite_undirected_components","operation":"component-partition","semantic_template":"qf-mod28-component-partition/{\"edges\":[[0,1],[0,3],[1,2],[1,6],[2,4],[2,6],[3,5]],\"n\":7}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r07-connected-component-0087","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-component"],"title_zh":"連通分量","stem_zh":"完整有限無向無loop簡單圖。操作=component-partition；資料={\"n\":7,\"edges\":[[0,3],[0,5],[0,6],[1,4],[1,5],[2,4],[2,5],[2,6],[3,5]]}。請依 [components,covers_vertices,disjoint,singletons,boundary] 欄序回答。","answer":{"kind":"expression","value":"([[0, 1, 2, 3, 4, 5, 6]], 是, 是, [], length_zero_paths_include_every_vertex)","explanation_zh":"Wave4第二層核對=([[0, 1, 2, 3, 4, 5, 6]], 是, 是, [], length_zero_paths_include_every_vertex)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；長度零path保證反身且孤立點為單點分量；完整可達性等價類不切換有向強弱連通。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["連通分量"]},"variant":{"family_id":"qf-mod28-component-partition","seed":202610287087,"parameters":{"n":7,"edges":[[0,3],[0,5],[0,6],[1,4],[1,5],[2,4],[2,5],[2,6],[3,5]],"domain":"Complete_finite_undirected_components","operation":"component-partition","semantic_template":"qf-mod28-component-partition/{\"edges\":[[0,3],[0,5],[0,6],[1,4],[1,5],[2,4],[2,5],[2,6],[3,5]],\"n\":7}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r07-connected-component-0088","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-component"],"title_zh":"連通分量","stem_zh":"完整有限無向無loop簡單圖。操作=component-partition；資料={\"n\":7,\"edges\":[[0,1],[0,2],[0,6],[1,5],[2,4],[3,4],[3,5]]}。請依 [components,covers_vertices,disjoint,singletons,boundary] 欄序回答。","answer":{"kind":"expression","value":"([[0, 1, 2, 3, 4, 5, 6]], 是, 是, [], length_zero_paths_include_every_vertex)","explanation_zh":"Wave4第二層核對=([[0, 1, 2, 3, 4, 5, 6]], 是, 是, [], length_zero_paths_include_every_vertex)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；長度零path保證反身且孤立點為單點分量；完整可達性等價類不切換有向強弱連通。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["連通分量"]},"variant":{"family_id":"qf-mod28-component-partition","seed":202610287088,"parameters":{"n":7,"edges":[[0,1],[0,2],[0,6],[1,5],[2,4],[3,4],[3,5]],"domain":"Complete_finite_undirected_components","operation":"component-partition","semantic_template":"qf-mod28-component-partition/{\"edges\":[[0,1],[0,2],[0,6],[1,5],[2,4],[3,4],[3,5]],\"n\":7}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r07-connected-component-0089","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-component"],"title_zh":"連通分量","stem_zh":"完整有限無向無loop簡單圖。操作=component-partition；資料={\"n\":7,\"edges\":[[0,2],[0,5],[1,4],[2,4],[2,5],[3,4],[3,5]]}。請依 [components,covers_vertices,disjoint,singletons,boundary] 欄序回答。","answer":{"kind":"expression","value":"([[0, 1, 2, 3, 4, 5], [6]], 是, 是, [[6]], length_zero_paths_include_every_vertex)","explanation_zh":"Wave4第二層核對=([[0, 1, 2, 3, 4, 5], [6]], 是, 是, [[6]], length_zero_paths_include_every_vertex)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；長度零path保證反身且孤立點為單點分量；完整可達性等價類不切換有向強弱連通。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["連通分量"]},"variant":{"family_id":"qf-mod28-component-partition","seed":202610287089,"parameters":{"n":7,"edges":[[0,2],[0,5],[1,4],[2,4],[2,5],[3,4],[3,5]],"domain":"Complete_finite_undirected_components","operation":"component-partition","semantic_template":"qf-mod28-component-partition/{\"edges\":[[0,2],[0,5],[1,4],[2,4],[2,5],[3,4],[3,5]],\"n\":7}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r07-connected-component-0090","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-component"],"title_zh":"連通分量","stem_zh":"完整有限無向無loop簡單圖。操作=component-partition；資料={\"n\":7,\"edges\":[[0,1],[2,4],[2,6],[3,4],[3,5]]}。請依 [components,covers_vertices,disjoint,singletons,boundary] 欄序回答。","answer":{"kind":"expression","value":"([[0, 1], [2, 3, 4, 5, 6]], 是, 是, [], length_zero_paths_include_every_vertex)","explanation_zh":"Wave4第二層核對=([[0, 1], [2, 3, 4, 5, 6]], 是, 是, [], length_zero_paths_include_every_vertex)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；長度零path保證反身且孤立點為單點分量；完整可達性等價類不切換有向強弱連通。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["連通分量"]},"variant":{"family_id":"qf-mod28-component-partition","seed":202610287090,"parameters":{"n":7,"edges":[[0,1],[2,4],[2,6],[3,4],[3,5]],"domain":"Complete_finite_undirected_components","operation":"component-partition","semantic_template":"qf-mod28-component-partition/{\"edges\":[[0,1],[2,4],[2,6],[3,4],[3,5]],\"n\":7}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r07-connected-component-0091","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-component"],"title_zh":"連通分量","stem_zh":"完整有限無向無loop簡單圖。操作=component-partition；資料={\"n\":7,\"edges\":[[0,5],[0,6],[1,2],[1,3],[1,5],[1,6],[2,3],[2,5],[2,6],[3,4],[3,5]]}。請依 [components,covers_vertices,disjoint,singletons,boundary] 欄序回答。","answer":{"kind":"expression","value":"([[0, 1, 2, 3, 4, 5, 6]], 是, 是, [], length_zero_paths_include_every_vertex)","explanation_zh":"Wave4第二層核對=([[0, 1, 2, 3, 4, 5, 6]], 是, 是, [], length_zero_paths_include_every_vertex)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；長度零path保證反身且孤立點為單點分量；完整可達性等價類不切換有向強弱連通。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["連通分量"]},"variant":{"family_id":"qf-mod28-component-partition","seed":202610287091,"parameters":{"n":7,"edges":[[0,5],[0,6],[1,2],[1,3],[1,5],[1,6],[2,3],[2,5],[2,6],[3,4],[3,5]],"domain":"Complete_finite_undirected_components","operation":"component-partition","semantic_template":"qf-mod28-component-partition/{\"edges\":[[0,5],[0,6],[1,2],[1,3],[1,5],[1,6],[2,3],[2,5],[2,6],[3,4],[3,5]],\"n\":7}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r07-connected-component-0092","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-component"],"title_zh":"連通分量","stem_zh":"完整有限無向無loop簡單圖。操作=component-partition；資料={\"n\":7,\"edges\":[[0,1],[0,2],[0,3],[0,4],[0,5],[1,2],[1,3],[1,4],[1,6],[2,3],[3,6]]}。請依 [components,covers_vertices,disjoint,singletons,boundary] 欄序回答。","answer":{"kind":"expression","value":"([[0, 1, 2, 3, 4, 5, 6]], 是, 是, [], length_zero_paths_include_every_vertex)","explanation_zh":"Wave4第二層核對=([[0, 1, 2, 3, 4, 5, 6]], 是, 是, [], length_zero_paths_include_every_vertex)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；長度零path保證反身且孤立點為單點分量；完整可達性等價類不切換有向強弱連通。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["連通分量"]},"variant":{"family_id":"qf-mod28-component-partition","seed":202610287092,"parameters":{"n":7,"edges":[[0,1],[0,2],[0,3],[0,4],[0,5],[1,2],[1,3],[1,4],[1,6],[2,3],[3,6]],"domain":"Complete_finite_undirected_components","operation":"component-partition","semantic_template":"qf-mod28-component-partition/{\"edges\":[[0,1],[0,2],[0,3],[0,4],[0,5],[1,2],[1,3],[1,4],[1,6],[2,3],[3,6]],\"n\":7}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r07-connected-component-0093","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-component"],"title_zh":"連通分量","stem_zh":"完整有限無向無loop簡單圖。操作=component-partition；資料={\"n\":7,\"edges\":[[0,2],[0,3],[0,4],[1,2],[1,3],[1,6],[2,3],[2,5],[3,6]]}。請依 [components,covers_vertices,disjoint,singletons,boundary] 欄序回答。","answer":{"kind":"expression","value":"([[0, 1, 2, 3, 4, 5, 6]], 是, 是, [], length_zero_paths_include_every_vertex)","explanation_zh":"Wave4第二層核對=([[0, 1, 2, 3, 4, 5, 6]], 是, 是, [], length_zero_paths_include_every_vertex)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；長度零path保證反身且孤立點為單點分量；完整可達性等價類不切換有向強弱連通。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["連通分量"]},"variant":{"family_id":"qf-mod28-component-partition","seed":202610287093,"parameters":{"n":7,"edges":[[0,2],[0,3],[0,4],[1,2],[1,3],[1,6],[2,3],[2,5],[3,6]],"domain":"Complete_finite_undirected_components","operation":"component-partition","semantic_template":"qf-mod28-component-partition/{\"edges\":[[0,2],[0,3],[0,4],[1,2],[1,3],[1,6],[2,3],[2,5],[3,6]],\"n\":7}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r07-connected-component-0094","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-component"],"title_zh":"連通分量","stem_zh":"完整有限無向無loop簡單圖。操作=component-partition；資料={\"n\":7,\"edges\":[[0,1],[0,3],[0,4],[0,5],[0,6],[1,3],[1,4],[1,5],[2,3],[2,6],[3,6]]}。請依 [components,covers_vertices,disjoint,singletons,boundary] 欄序回答。","answer":{"kind":"expression","value":"([[0, 1, 2, 3, 4, 5, 6]], 是, 是, [], length_zero_paths_include_every_vertex)","explanation_zh":"Wave4第二層核對=([[0, 1, 2, 3, 4, 5, 6]], 是, 是, [], length_zero_paths_include_every_vertex)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；長度零path保證反身且孤立點為單點分量；完整可達性等價類不切換有向強弱連通。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["連通分量"]},"variant":{"family_id":"qf-mod28-component-partition","seed":202610287094,"parameters":{"n":7,"edges":[[0,1],[0,3],[0,4],[0,5],[0,6],[1,3],[1,4],[1,5],[2,3],[2,6],[3,6]],"domain":"Complete_finite_undirected_components","operation":"component-partition","semantic_template":"qf-mod28-component-partition/{\"edges\":[[0,1],[0,3],[0,4],[0,5],[0,6],[1,3],[1,4],[1,5],[2,3],[2,6],[3,6]],\"n\":7}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r07-connected-component-0095","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-component"],"title_zh":"連通分量","stem_zh":"完整有限無向無loop簡單圖。操作=component-partition；資料={\"n\":7,\"edges\":[[0,3],[0,4],[0,6],[1,3],[1,5],[2,3],[2,5],[2,6],[3,6]]}。請依 [components,covers_vertices,disjoint,singletons,boundary] 欄序回答。","answer":{"kind":"expression","value":"([[0, 1, 2, 3, 4, 5, 6]], 是, 是, [], length_zero_paths_include_every_vertex)","explanation_zh":"Wave4第二層核對=([[0, 1, 2, 3, 4, 5, 6]], 是, 是, [], length_zero_paths_include_every_vertex)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；長度零path保證反身且孤立點為單點分量；完整可達性等價類不切換有向強弱連通。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["連通分量"]},"variant":{"family_id":"qf-mod28-component-partition","seed":202610287095,"parameters":{"n":7,"edges":[[0,3],[0,4],[0,6],[1,3],[1,5],[2,3],[2,5],[2,6],[3,6]],"domain":"Complete_finite_undirected_components","operation":"component-partition","semantic_template":"qf-mod28-component-partition/{\"edges\":[[0,3],[0,4],[0,6],[1,3],[1,5],[2,3],[2,5],[2,6],[3,6]],\"n\":7}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r07-connected-component-0096","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-component"],"title_zh":"連通分量","stem_zh":"完整有限無向無loop簡單圖。操作=component-partition；資料={\"n\":7,\"edges\":[[0,1],[0,2],[0,4],[0,5],[1,3],[1,4],[2,3],[3,4],[3,6]]}。請依 [components,covers_vertices,disjoint,singletons,boundary] 欄序回答。","answer":{"kind":"expression","value":"([[0, 1, 2, 3, 4, 5, 6]], 是, 是, [], length_zero_paths_include_every_vertex)","explanation_zh":"Wave4第二層核對=([[0, 1, 2, 3, 4, 5, 6]], 是, 是, [], length_zero_paths_include_every_vertex)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；長度零path保證反身且孤立點為單點分量；完整可達性等價類不切換有向強弱連通。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["連通分量"]},"variant":{"family_id":"qf-mod28-component-partition","seed":202610287096,"parameters":{"n":7,"edges":[[0,1],[0,2],[0,4],[0,5],[1,3],[1,4],[2,3],[3,4],[3,6]],"domain":"Complete_finite_undirected_components","operation":"component-partition","semantic_template":"qf-mod28-component-partition/{\"edges\":[[0,1],[0,2],[0,4],[0,5],[1,3],[1,4],[2,3],[3,4],[3,6]],\"n\":7}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r07-connected-component-0097","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-component"],"title_zh":"連通分量","stem_zh":"完整有限無向無loop簡單圖。操作=component-partition；資料={\"n\":7,\"edges\":[[0,2],[0,4],[1,3],[2,3],[2,5],[3,4],[3,6]]}。請依 [components,covers_vertices,disjoint,singletons,boundary] 欄序回答。","answer":{"kind":"expression","value":"([[0, 1, 2, 3, 4, 5, 6]], 是, 是, [], length_zero_paths_include_every_vertex)","explanation_zh":"Wave4第二層核對=([[0, 1, 2, 3, 4, 5, 6]], 是, 是, [], length_zero_paths_include_every_vertex)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；長度零path保證反身且孤立點為單點分量；完整可達性等價類不切換有向強弱連通。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["連通分量"]},"variant":{"family_id":"qf-mod28-component-partition","seed":202610287097,"parameters":{"n":7,"edges":[[0,2],[0,4],[1,3],[2,3],[2,5],[3,4],[3,6]],"domain":"Complete_finite_undirected_components","operation":"component-partition","semantic_template":"qf-mod28-component-partition/{\"edges\":[[0,2],[0,4],[1,3],[2,3],[2,5],[3,4],[3,6]],\"n\":7}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r07-connected-component-0098","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-component"],"title_zh":"連通分量","stem_zh":"完整有限無向無loop簡單圖。操作=component-partition；資料={\"n\":7,\"edges\":[[0,1],[0,4],[0,5],[0,6],[1,2],[1,4],[1,5],[1,6],[2,6],[3,4],[3,6]]}。請依 [components,covers_vertices,disjoint,singletons,boundary] 欄序回答。","answer":{"kind":"expression","value":"([[0, 1, 2, 3, 4, 5, 6]], 是, 是, [], length_zero_paths_include_every_vertex)","explanation_zh":"Wave4第二層核對=([[0, 1, 2, 3, 4, 5, 6]], 是, 是, [], length_zero_paths_include_every_vertex)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；長度零path保證反身且孤立點為單點分量；完整可達性等價類不切換有向強弱連通。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["連通分量"]},"variant":{"family_id":"qf-mod28-component-partition","seed":202610287098,"parameters":{"n":7,"edges":[[0,1],[0,4],[0,5],[0,6],[1,2],[1,4],[1,5],[1,6],[2,6],[3,4],[3,6]],"domain":"Complete_finite_undirected_components","operation":"component-partition","semantic_template":"qf-mod28-component-partition/{\"edges\":[[0,1],[0,4],[0,5],[0,6],[1,2],[1,4],[1,5],[1,6],[2,6],[3,4],[3,6]],\"n\":7}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r07-connected-component-0099","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-component"],"title_zh":"連通分量","stem_zh":"完整有限無向無loop簡單圖。操作=component-partition；資料={\"n\":7,\"edges\":[[0,4],[0,6],[1,2],[1,5],[1,6],[2,5],[2,6],[3,4],[3,6]]}。請依 [components,covers_vertices,disjoint,singletons,boundary] 欄序回答。","answer":{"kind":"expression","value":"([[0, 1, 2, 3, 4, 5, 6]], 是, 是, [], length_zero_paths_include_every_vertex)","explanation_zh":"Wave4第二層核對=([[0, 1, 2, 3, 4, 5, 6]], 是, 是, [], length_zero_paths_include_every_vertex)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；長度零path保證反身且孤立點為單點分量；完整可達性等價類不切換有向強弱連通。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["連通分量"]},"variant":{"family_id":"qf-mod28-component-partition","seed":202610287099,"parameters":{"n":7,"edges":[[0,4],[0,6],[1,2],[1,5],[1,6],[2,5],[2,6],[3,4],[3,6]],"domain":"Complete_finite_undirected_components","operation":"component-partition","semantic_template":"qf-mod28-component-partition/{\"edges\":[[0,4],[0,6],[1,2],[1,5],[1,6],[2,5],[2,6],[3,4],[3,6]],\"n\":7}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r07-connected-component-0100","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-component"],"title_zh":"連通分量","stem_zh":"完整有限無向無loop簡單圖。操作=component-partition；資料={\"n\":7,\"edges\":[[0,1],[0,2],[0,3],[0,5],[1,2],[1,4],[1,6],[3,5],[3,6]]}。請依 [components,covers_vertices,disjoint,singletons,boundary] 欄序回答。","answer":{"kind":"expression","value":"([[0, 1, 2, 3, 4, 5, 6]], 是, 是, [], length_zero_paths_include_every_vertex)","explanation_zh":"Wave4第二層核對=([[0, 1, 2, 3, 4, 5, 6]], 是, 是, [], length_zero_paths_include_every_vertex)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；長度零path保證反身且孤立點為單點分量；完整可達性等價類不切換有向強弱連通。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["連通分量"]},"variant":{"family_id":"qf-mod28-component-partition","seed":202610287100,"parameters":{"n":7,"edges":[[0,1],[0,2],[0,3],[0,5],[1,2],[1,4],[1,6],[3,5],[3,6]],"domain":"Complete_finite_undirected_components","operation":"component-partition","semantic_template":"qf-mod28-component-partition/{\"edges\":[[0,1],[0,2],[0,3],[0,5],[1,2],[1,4],[1,6],[3,5],[3,6]],\"n\":7}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}