{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261029-r07-equivalence-class-0001","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","language":"zh-Hant","source_mko_ids":["mko-equivalence-class"],"title_zh":"等價類","stem_zh":"同一完整載體上的已認證等價關係。操作=eqclass-of；資料={\"n\":6,\"pairs\":[[0,0],[0,1],[0,2],[0,3],[0,4],[0,5],[1,0],[1,1],[1,2],[1,3],[1,4],[1,5],[2,0],[2,1],[2,2],[2,3],[2,4],[2,5],[3,0],[3,1],[3,2],[3,3],[3,4],[3,5],[4,0],[4,1],[4,2],[4,3],[4,4],[4,5],[5,0],[5,1],[5,2],[5,3],[5,4],[5,5]],\"element\":0}。請依 [element,class] 欄序回答。","answer":{"kind":"expression","value":"(0, [0, 1, 2, 3, 4, 5])","explanation_zh":"Wave4第三層核對=(0, [0, 1, 2, 3, 4, 5])；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；class精確等於全部相關元素且代表元不變；任兩classes相等或不交且不從抽樣relation分群。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["等價類"]},"variant":{"family_id":"qf-mod29-eqclass-of","seed":202610297001,"parameters":{"n":6,"pairs":[[0,0],[0,1],[0,2],[0,3],[0,4],[0,5],[1,0],[1,1],[1,2],[1,3],[1,4],[1,5],[2,0],[2,1],[2,2],[2,3],[2,4],[2,5],[3,0],[3,1],[3,2],[3,3],[3,4],[3,5],[4,0],[4,1],[4,2],[4,3],[4,4],[4,5],[5,0],[5,1],[5,2],[5,3],[5,4],[5,5]],"element":0,"domain":"Complete_finite_equivalence_class","operation":"eqclass-of","semantic_template":"qf-mod29-eqclass-of/{\"element\":0,\"n\":6,\"pairs\":[[0,0],[0,1],[0,2],[0,3],[0,4],[0,5],[1,0],[1,1],[1,2],[1,3],[1,4],[1,5],[2,0],[2,1],[2,2],[2,3],[2,4],[2,5],[3,0],[3,1],[3,2],[3,3],[3,4],[3,5],[4,0],[4,1],[4,2],[4,3],[4,4],[4,5],[5,0],[5,1],[5,2],[5,3],[5,4],[5,5]]}"}},"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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r07-equivalence-class-0002","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","language":"zh-Hant","source_mko_ids":["mko-equivalence-class"],"title_zh":"等價類","stem_zh":"同一完整載體上的已認證等價關係。操作=eqclass-of；資料={\"n\":6,\"pairs\":[[0,0],[1,1],[1,2],[1,3],[1,4],[1,5],[2,1],[2,2],[2,3],[2,4],[2,5],[3,1],[3,2],[3,3],[3,4],[3,5],[4,1],[4,2],[4,3],[4,4],[4,5],[5,1],[5,2],[5,3],[5,4],[5,5]],\"element\":1}。請依 [element,class] 欄序回答。","answer":{"kind":"expression","value":"(1, [1, 2, 3, 4, 5])","explanation_zh":"Wave4第三層核對=(1, [1, 2, 3, 4, 5])；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；class精確等於全部相關元素且代表元不變；任兩classes相等或不交且不從抽樣relation分群。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["等價類"]},"variant":{"family_id":"qf-mod29-eqclass-of","seed":202610297002,"parameters":{"n":6,"pairs":[[0,0],[1,1],[1,2],[1,3],[1,4],[1,5],[2,1],[2,2],[2,3],[2,4],[2,5],[3,1],[3,2],[3,3],[3,4],[3,5],[4,1],[4,2],[4,3],[4,4],[4,5],[5,1],[5,2],[5,3],[5,4],[5,5]],"element":1,"domain":"Complete_finite_equivalence_class","operation":"eqclass-of","semantic_template":"qf-mod29-eqclass-of/{\"element\":1,\"n\":6,\"pairs\":[[0,0],[1,1],[1,2],[1,3],[1,4],[1,5],[2,1],[2,2],[2,3],[2,4],[2,5],[3,1],[3,2],[3,3],[3,4],[3,5],[4,1],[4,2],[4,3],[4,4],[4,5],[5,1],[5,2],[5,3],[5,4],[5,5]]}"}},"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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r07-equivalence-class-0003","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","language":"zh-Hant","source_mko_ids":["mko-equivalence-class"],"title_zh":"等價類","stem_zh":"同一完整載體上的已認證等價關係。操作=eqclass-of；資料={\"n\":6,\"pairs\":[[0,0],[0,2],[0,3],[0,4],[0,5],[1,1],[2,0],[2,2],[2,3],[2,4],[2,5],[3,0],[3,2],[3,3],[3,4],[3,5],[4,0],[4,2],[4,3],[4,4],[4,5],[5,0],[5,2],[5,3],[5,4],[5,5]],\"element\":2}。請依 [element,class] 欄序回答。","answer":{"kind":"expression","value":"(2, [0, 2, 3, 4, 5])","explanation_zh":"Wave4第三層核對=(2, [0, 2, 3, 4, 5])；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；class精確等於全部相關元素且代表元不變；任兩classes相等或不交且不從抽樣relation分群。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["等價類"]},"variant":{"family_id":"qf-mod29-eqclass-of","seed":202610297003,"parameters":{"n":6,"pairs":[[0,0],[0,2],[0,3],[0,4],[0,5],[1,1],[2,0],[2,2],[2,3],[2,4],[2,5],[3,0],[3,2],[3,3],[3,4],[3,5],[4,0],[4,2],[4,3],[4,4],[4,5],[5,0],[5,2],[5,3],[5,4],[5,5]],"element":2,"domain":"Complete_finite_equivalence_class","operation":"eqclass-of","semantic_template":"qf-mod29-eqclass-of/{\"element\":2,\"n\":6,\"pairs\":[[0,0],[0,2],[0,3],[0,4],[0,5],[1,1],[2,0],[2,2],[2,3],[2,4],[2,5],[3,0],[3,2],[3,3],[3,4],[3,5],[4,0],[4,2],[4,3],[4,4],[4,5],[5,0],[5,2],[5,3],[5,4],[5,5]]}"}},"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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r07-equivalence-class-0004","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","language":"zh-Hant","source_mko_ids":["mko-equivalence-class"],"title_zh":"等價類","stem_zh":"同一完整載體上的已認證等價關係。操作=eqclass-of；資料={\"n\":6,\"pairs\":[[0,0],[0,1],[1,0],[1,1],[2,2],[2,3],[2,4],[2,5],[3,2],[3,3],[3,4],[3,5],[4,2],[4,3],[4,4],[4,5],[5,2],[5,3],[5,4],[5,5]],\"element\":3}。請依 [element,class] 欄序回答。","answer":{"kind":"expression","value":"(3, [2, 3, 4, 5])","explanation_zh":"Wave4第三層核對=(3, [2, 3, 4, 5])；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；class精確等於全部相關元素且代表元不變；任兩classes相等或不交且不從抽樣relation分群。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["等價類"]},"variant":{"family_id":"qf-mod29-eqclass-of","seed":202610297004,"parameters":{"n":6,"pairs":[[0,0],[0,1],[1,0],[1,1],[2,2],[2,3],[2,4],[2,5],[3,2],[3,3],[3,4],[3,5],[4,2],[4,3],[4,4],[4,5],[5,2],[5,3],[5,4],[5,5]],"element":3,"domain":"Complete_finite_equivalence_class","operation":"eqclass-of","semantic_template":"qf-mod29-eqclass-of/{\"element\":3,\"n\":6,\"pairs\":[[0,0],[0,1],[1,0],[1,1],[2,2],[2,3],[2,4],[2,5],[3,2],[3,3],[3,4],[3,5],[4,2],[4,3],[4,4],[4,5],[5,2],[5,3],[5,4],[5,5]]}"}},"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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r07-equivalence-class-0005","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","language":"zh-Hant","source_mko_ids":["mko-equivalence-class"],"title_zh":"等價類","stem_zh":"同一完整載體上的已認證等價關係。操作=eqclass-of；資料={\"n\":6,\"pairs\":[[0,0],[1,1],[2,2],[2,3],[2,4],[2,5],[3,2],[3,3],[3,4],[3,5],[4,2],[4,3],[4,4],[4,5],[5,2],[5,3],[5,4],[5,5]],\"element\":4}。請依 [element,class] 欄序回答。","answer":{"kind":"expression","value":"(4, [2, 3, 4, 5])","explanation_zh":"Wave4第三層核對=(4, [2, 3, 4, 5])；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；class精確等於全部相關元素且代表元不變；任兩classes相等或不交且不從抽樣relation分群。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["等價類"]},"variant":{"family_id":"qf-mod29-eqclass-of","seed":202610297005,"parameters":{"n":6,"pairs":[[0,0],[1,1],[2,2],[2,3],[2,4],[2,5],[3,2],[3,3],[3,4],[3,5],[4,2],[4,3],[4,4],[4,5],[5,2],[5,3],[5,4],[5,5]],"element":4,"domain":"Complete_finite_equivalence_class","operation":"eqclass-of","semantic_template":"qf-mod29-eqclass-of/{\"element\":4,\"n\":6,\"pairs\":[[0,0],[1,1],[2,2],[2,3],[2,4],[2,5],[3,2],[3,3],[3,4],[3,5],[4,2],[4,3],[4,4],[4,5],[5,2],[5,3],[5,4],[5,5]]}"}},"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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r07-equivalence-class-0006","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","language":"zh-Hant","source_mko_ids":["mko-equivalence-class"],"title_zh":"等價類","stem_zh":"同一完整載體上的已認證等價關係。操作=eqclass-of；資料={\"n\":6,\"pairs\":[[0,0],[0,1],[0,3],[0,4],[0,5],[1,0],[1,1],[1,3],[1,4],[1,5],[2,2],[3,0],[3,1],[3,3],[3,4],[3,5],[4,0],[4,1],[4,3],[4,4],[4,5],[5,0],[5,1],[5,3],[5,4],[5,5]],\"element\":5}。請依 [element,class] 欄序回答。","answer":{"kind":"expression","value":"(5, [0, 1, 3, 4, 5])","explanation_zh":"Wave4第三層核對=(5, [0, 1, 3, 4, 5])；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；class精確等於全部相關元素且代表元不變；任兩classes相等或不交且不從抽樣relation分群。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["等價類"]},"variant":{"family_id":"qf-mod29-eqclass-of","seed":202610297006,"parameters":{"n":6,"pairs":[[0,0],[0,1],[0,3],[0,4],[0,5],[1,0],[1,1],[1,3],[1,4],[1,5],[2,2],[3,0],[3,1],[3,3],[3,4],[3,5],[4,0],[4,1],[4,3],[4,4],[4,5],[5,0],[5,1],[5,3],[5,4],[5,5]],"element":5,"domain":"Complete_finite_equivalence_class","operation":"eqclass-of","semantic_template":"qf-mod29-eqclass-of/{\"element\":5,\"n\":6,\"pairs\":[[0,0],[0,1],[0,3],[0,4],[0,5],[1,0],[1,1],[1,3],[1,4],[1,5],[2,2],[3,0],[3,1],[3,3],[3,4],[3,5],[4,0],[4,1],[4,3],[4,4],[4,5],[5,0],[5,1],[5,3],[5,4],[5,5]]}"}},"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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r07-equivalence-class-0007","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","language":"zh-Hant","source_mko_ids":["mko-equivalence-class"],"title_zh":"等價類","stem_zh":"同一完整載體上的已認證等價關係。操作=eqclass-of；資料={\"n\":6,\"pairs\":[[0,0],[0,2],[1,1],[1,3],[1,4],[1,5],[2,0],[2,2],[3,1],[3,3],[3,4],[3,5],[4,1],[4,3],[4,4],[4,5],[5,1],[5,3],[5,4],[5,5]],\"element\":0}。請依 [element,class] 欄序回答。","answer":{"kind":"expression","value":"(0, [0, 2])","explanation_zh":"Wave4第三層核對=(0, [0, 2])；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；class精確等於全部相關元素且代表元不變；任兩classes相等或不交且不從抽樣relation分群。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["等價類"]},"variant":{"family_id":"qf-mod29-eqclass-of","seed":202610297007,"parameters":{"n":6,"pairs":[[0,0],[0,2],[1,1],[1,3],[1,4],[1,5],[2,0],[2,2],[3,1],[3,3],[3,4],[3,5],[4,1],[4,3],[4,4],[4,5],[5,1],[5,3],[5,4],[5,5]],"element":0,"domain":"Complete_finite_equivalence_class","operation":"eqclass-of","semantic_template":"qf-mod29-eqclass-of/{\"element\":0,\"n\":6,\"pairs\":[[0,0],[0,2],[1,1],[1,3],[1,4],[1,5],[2,0],[2,2],[3,1],[3,3],[3,4],[3,5],[4,1],[4,3],[4,4],[4,5],[5,1],[5,3],[5,4],[5,5]]}"}},"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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r07-equivalence-class-0008","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","language":"zh-Hant","source_mko_ids":["mko-equivalence-class"],"title_zh":"等價類","stem_zh":"同一完整載體上的已認證等價關係。操作=eqclass-of；資料={\"n\":6,\"pairs\":[[0,0],[1,1],[1,3],[1,4],[1,5],[2,2],[3,1],[3,3],[3,4],[3,5],[4,1],[4,3],[4,4],[4,5],[5,1],[5,3],[5,4],[5,5]],\"element\":1}。請依 [element,class] 欄序回答。","answer":{"kind":"expression","value":"(1, [1, 3, 4, 5])","explanation_zh":"Wave4第三層核對=(1, [1, 3, 4, 5])；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；class精確等於全部相關元素且代表元不變；任兩classes相等或不交且不從抽樣relation分群。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["等價類"]},"variant":{"family_id":"qf-mod29-eqclass-of","seed":202610297008,"parameters":{"n":6,"pairs":[[0,0],[1,1],[1,3],[1,4],[1,5],[2,2],[3,1],[3,3],[3,4],[3,5],[4,1],[4,3],[4,4],[4,5],[5,1],[5,3],[5,4],[5,5]],"element":1,"domain":"Complete_finite_equivalence_class","operation":"eqclass-of","semantic_template":"qf-mod29-eqclass-of/{\"element\":1,\"n\":6,\"pairs\":[[0,0],[1,1],[1,3],[1,4],[1,5],[2,2],[3,1],[3,3],[3,4],[3,5],[4,1],[4,3],[4,4],[4,5],[5,1],[5,3],[5,4],[5,5]]}"}},"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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r07-equivalence-class-0009","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","language":"zh-Hant","source_mko_ids":["mko-equivalence-class"],"title_zh":"等價類","stem_zh":"同一完整載體上的已認證等價關係。操作=eqclass-of；資料={\"n\":6,\"pairs\":[[0,0],[0,3],[0,4],[0,5],[1,1],[1,2],[2,1],[2,2],[3,0],[3,3],[3,4],[3,5],[4,0],[4,3],[4,4],[4,5],[5,0],[5,3],[5,4],[5,5]],\"element\":2}。請依 [element,class] 欄序回答。","answer":{"kind":"expression","value":"(2, [1, 2])","explanation_zh":"Wave4第三層核對=(2, [1, 2])；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；class精確等於全部相關元素且代表元不變；任兩classes相等或不交且不從抽樣relation分群。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["等價類"]},"variant":{"family_id":"qf-mod29-eqclass-of","seed":202610297009,"parameters":{"n":6,"pairs":[[0,0],[0,3],[0,4],[0,5],[1,1],[1,2],[2,1],[2,2],[3,0],[3,3],[3,4],[3,5],[4,0],[4,3],[4,4],[4,5],[5,0],[5,3],[5,4],[5,5]],"element":2,"domain":"Complete_finite_equivalence_class","operation":"eqclass-of","semantic_template":"qf-mod29-eqclass-of/{\"element\":2,\"n\":6,\"pairs\":[[0,0],[0,3],[0,4],[0,5],[1,1],[1,2],[2,1],[2,2],[3,0],[3,3],[3,4],[3,5],[4,0],[4,3],[4,4],[4,5],[5,0],[5,3],[5,4],[5,5]]}"}},"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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r07-equivalence-class-0010","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","language":"zh-Hant","source_mko_ids":["mko-equivalence-class"],"title_zh":"等價類","stem_zh":"同一完整載體上的已認證等價關係。操作=eqclass-of；資料={\"n\":6,\"pairs\":[[0,0],[0,1],[0,2],[1,0],[1,1],[1,2],[2,0],[2,1],[2,2],[3,3],[3,4],[3,5],[4,3],[4,4],[4,5],[5,3],[5,4],[5,5]],\"element\":3}。請依 [element,class] 欄序回答。","answer":{"kind":"expression","value":"(3, [3, 4, 5])","explanation_zh":"Wave4第三層核對=(3, [3, 4, 5])；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；class精確等於全部相關元素且代表元不變；任兩classes相等或不交且不從抽樣relation分群。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["等價類"]},"variant":{"family_id":"qf-mod29-eqclass-of","seed":202610297010,"parameters":{"n":6,"pairs":[[0,0],[0,1],[0,2],[1,0],[1,1],[1,2],[2,0],[2,1],[2,2],[3,3],[3,4],[3,5],[4,3],[4,4],[4,5],[5,3],[5,4],[5,5]],"element":3,"domain":"Complete_finite_equivalence_class","operation":"eqclass-of","semantic_template":"qf-mod29-eqclass-of/{\"element\":3,\"n\":6,\"pairs\":[[0,0],[0,1],[0,2],[1,0],[1,1],[1,2],[2,0],[2,1],[2,2],[3,3],[3,4],[3,5],[4,3],[4,4],[4,5],[5,3],[5,4],[5,5]]}"}},"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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r07-equivalence-class-0011","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","language":"zh-Hant","source_mko_ids":["mko-equivalence-class"],"title_zh":"等價類","stem_zh":"同一完整載體上的已認證等價關係。操作=eqclass-of；資料={\"n\":6,\"pairs\":[[0,0],[1,1],[1,2],[2,1],[2,2],[3,3],[3,4],[3,5],[4,3],[4,4],[4,5],[5,3],[5,4],[5,5]],\"element\":4}。請依 [element,class] 欄序回答。","answer":{"kind":"expression","value":"(4, [3, 4, 5])","explanation_zh":"Wave4第三層核對=(4, [3, 4, 5])；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；class精確等於全部相關元素且代表元不變；任兩classes相等或不交且不從抽樣relation分群。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["等價類"]},"variant":{"family_id":"qf-mod29-eqclass-of","seed":202610297011,"parameters":{"n":6,"pairs":[[0,0],[1,1],[1,2],[2,1],[2,2],[3,3],[3,4],[3,5],[4,3],[4,4],[4,5],[5,3],[5,4],[5,5]],"element":4,"domain":"Complete_finite_equivalence_class","operation":"eqclass-of","semantic_template":"qf-mod29-eqclass-of/{\"element\":4,\"n\":6,\"pairs\":[[0,0],[1,1],[1,2],[2,1],[2,2],[3,3],[3,4],[3,5],[4,3],[4,4],[4,5],[5,3],[5,4],[5,5]]}"}},"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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r07-equivalence-class-0012","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","language":"zh-Hant","source_mko_ids":["mko-equivalence-class"],"title_zh":"等價類","stem_zh":"同一完整載體上的已認證等價關係。操作=eqclass-of；資料={\"n\":6,\"pairs\":[[0,0],[0,3],[0,4],[0,5],[1,1],[2,2],[3,0],[3,3],[3,4],[3,5],[4,0],[4,3],[4,4],[4,5],[5,0],[5,3],[5,4],[5,5]],\"element\":5}。請依 [element,class] 欄序回答。","answer":{"kind":"expression","value":"(5, [0, 3, 4, 5])","explanation_zh":"Wave4第三層核對=(5, [0, 3, 4, 5])；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；class精確等於全部相關元素且代表元不變；任兩classes相等或不交且不從抽樣relation分群。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["等價類"]},"variant":{"family_id":"qf-mod29-eqclass-of","seed":202610297012,"parameters":{"n":6,"pairs":[[0,0],[0,3],[0,4],[0,5],[1,1],[2,2],[3,0],[3,3],[3,4],[3,5],[4,0],[4,3],[4,4],[4,5],[5,0],[5,3],[5,4],[5,5]],"element":5,"domain":"Complete_finite_equivalence_class","operation":"eqclass-of","semantic_template":"qf-mod29-eqclass-of/{\"element\":5,\"n\":6,\"pairs\":[[0,0],[0,3],[0,4],[0,5],[1,1],[2,2],[3,0],[3,3],[3,4],[3,5],[4,0],[4,3],[4,4],[4,5],[5,0],[5,3],[5,4],[5,5]]}"}},"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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r07-equivalence-class-0013","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","language":"zh-Hant","source_mko_ids":["mko-equivalence-class"],"title_zh":"等價類","stem_zh":"同一完整載體上的已認證等價關係。操作=eqclass-of；資料={\"n\":6,\"pairs\":[[0,0],[0,2],[1,1],[2,0],[2,2],[3,3],[3,4],[3,5],[4,3],[4,4],[4,5],[5,3],[5,4],[5,5]],\"element\":0}。請依 [element,class] 欄序回答。","answer":{"kind":"expression","value":"(0, [0, 2])","explanation_zh":"Wave4第三層核對=(0, [0, 2])；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；class精確等於全部相關元素且代表元不變；任兩classes相等或不交且不從抽樣relation分群。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["等價類"]},"variant":{"family_id":"qf-mod29-eqclass-of","seed":202610297013,"parameters":{"n":6,"pairs":[[0,0],[0,2],[1,1],[2,0],[2,2],[3,3],[3,4],[3,5],[4,3],[4,4],[4,5],[5,3],[5,4],[5,5]],"element":0,"domain":"Complete_finite_equivalence_class","operation":"eqclass-of","semantic_template":"qf-mod29-eqclass-of/{\"element\":0,\"n\":6,\"pairs\":[[0,0],[0,2],[1,1],[2,0],[2,2],[3,3],[3,4],[3,5],[4,3],[4,4],[4,5],[5,3],[5,4],[5,5]]}"}},"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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r07-equivalence-class-0014","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","language":"zh-Hant","source_mko_ids":["mko-equivalence-class"],"title_zh":"等價類","stem_zh":"同一完整載體上的已認證等價關係。操作=eqclass-of；資料={\"n\":6,\"pairs\":[[0,0],[0,1],[1,0],[1,1],[2,2],[3,3],[3,4],[3,5],[4,3],[4,4],[4,5],[5,3],[5,4],[5,5]],\"element\":1}。請依 [element,class] 欄序回答。","answer":{"kind":"expression","value":"(1, [0, 1])","explanation_zh":"Wave4第三層核對=(1, [0, 1])；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；class精確等於全部相關元素且代表元不變；任兩classes相等或不交且不從抽樣relation分群。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["等價類"]},"variant":{"family_id":"qf-mod29-eqclass-of","seed":202610297014,"parameters":{"n":6,"pairs":[[0,0],[0,1],[1,0],[1,1],[2,2],[3,3],[3,4],[3,5],[4,3],[4,4],[4,5],[5,3],[5,4],[5,5]],"element":1,"domain":"Complete_finite_equivalence_class","operation":"eqclass-of","semantic_template":"qf-mod29-eqclass-of/{\"element\":1,\"n\":6,\"pairs\":[[0,0],[0,1],[1,0],[1,1],[2,2],[3,3],[3,4],[3,5],[4,3],[4,4],[4,5],[5,3],[5,4],[5,5]]}"}},"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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r07-equivalence-class-0015","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","language":"zh-Hant","source_mko_ids":["mko-equivalence-class"],"title_zh":"等價類","stem_zh":"同一完整載體上的已認證等價關係。操作=eqclass-of；資料={\"n\":6,\"pairs\":[[0,0],[0,1],[0,2],[0,4],[0,5],[1,0],[1,1],[1,2],[1,4],[1,5],[2,0],[2,1],[2,2],[2,4],[2,5],[3,3],[4,0],[4,1],[4,2],[4,4],[4,5],[5,0],[5,1],[5,2],[5,4],[5,5]],\"element\":2}。請依 [element,class] 欄序回答。","answer":{"kind":"expression","value":"(2, [0, 1, 2, 4, 5])","explanation_zh":"Wave4第三層核對=(2, [0, 1, 2, 4, 5])；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；class精確等於全部相關元素且代表元不變；任兩classes相等或不交且不從抽樣relation分群。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["等價類"]},"variant":{"family_id":"qf-mod29-eqclass-of","seed":202610297015,"parameters":{"n":6,"pairs":[[0,0],[0,1],[0,2],[0,4],[0,5],[1,0],[1,1],[1,2],[1,4],[1,5],[2,0],[2,1],[2,2],[2,4],[2,5],[3,3],[4,0],[4,1],[4,2],[4,4],[4,5],[5,0],[5,1],[5,2],[5,4],[5,5]],"element":2,"domain":"Complete_finite_equivalence_class","operation":"eqclass-of","semantic_template":"qf-mod29-eqclass-of/{\"element\":2,\"n\":6,\"pairs\":[[0,0],[0,1],[0,2],[0,4],[0,5],[1,0],[1,1],[1,2],[1,4],[1,5],[2,0],[2,1],[2,2],[2,4],[2,5],[3,3],[4,0],[4,1],[4,2],[4,4],[4,5],[5,0],[5,1],[5,2],[5,4],[5,5]]}"}},"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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r07-equivalence-class-0016","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","language":"zh-Hant","source_mko_ids":["mko-equivalence-class"],"title_zh":"等價類","stem_zh":"同一完整載體上的已認證等價關係。操作=eqclass-of；資料={\"n\":6,\"pairs\":[[0,0],[0,3],[1,1],[1,2],[1,4],[1,5],[2,1],[2,2],[2,4],[2,5],[3,0],[3,3],[4,1],[4,2],[4,4],[4,5],[5,1],[5,2],[5,4],[5,5]],\"element\":3}。請依 [element,class] 欄序回答。","answer":{"kind":"expression","value":"(3, [0, 3])","explanation_zh":"Wave4第三層核對=(3, [0, 3])；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；class精確等於全部相關元素且代表元不變；任兩classes相等或不交且不從抽樣relation分群。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["等價類"]},"variant":{"family_id":"qf-mod29-eqclass-of","seed":202610297016,"parameters":{"n":6,"pairs":[[0,0],[0,3],[1,1],[1,2],[1,4],[1,5],[2,1],[2,2],[2,4],[2,5],[3,0],[3,3],[4,1],[4,2],[4,4],[4,5],[5,1],[5,2],[5,4],[5,5]],"element":3,"domain":"Complete_finite_equivalence_class","operation":"eqclass-of","semantic_template":"qf-mod29-eqclass-of/{\"element\":3,\"n\":6,\"pairs\":[[0,0],[0,3],[1,1],[1,2],[1,4],[1,5],[2,1],[2,2],[2,4],[2,5],[3,0],[3,3],[4,1],[4,2],[4,4],[4,5],[5,1],[5,2],[5,4],[5,5]]}"}},"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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r07-equivalence-class-0017","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","language":"zh-Hant","source_mko_ids":["mko-equivalence-class"],"title_zh":"等價類","stem_zh":"同一完整載體上的已認證等價關係。操作=eqclass-of；資料={\"n\":6,\"pairs\":[[0,0],[1,1],[1,2],[1,4],[1,5],[2,1],[2,2],[2,4],[2,5],[3,3],[4,1],[4,2],[4,4],[4,5],[5,1],[5,2],[5,4],[5,5]],\"element\":4}。請依 [element,class] 欄序回答。","answer":{"kind":"expression","value":"(4, [1, 2, 4, 5])","explanation_zh":"Wave4第三層核對=(4, [1, 2, 4, 5])；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；class精確等於全部相關元素且代表元不變；任兩classes相等或不交且不從抽樣relation分群。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["等價類"]},"variant":{"family_id":"qf-mod29-eqclass-of","seed":202610297017,"parameters":{"n":6,"pairs":[[0,0],[1,1],[1,2],[1,4],[1,5],[2,1],[2,2],[2,4],[2,5],[3,3],[4,1],[4,2],[4,4],[4,5],[5,1],[5,2],[5,4],[5,5]],"element":4,"domain":"Complete_finite_equivalence_class","operation":"eqclass-of","semantic_template":"qf-mod29-eqclass-of/{\"element\":4,\"n\":6,\"pairs\":[[0,0],[1,1],[1,2],[1,4],[1,5],[2,1],[2,2],[2,4],[2,5],[3,3],[4,1],[4,2],[4,4],[4,5],[5,1],[5,2],[5,4],[5,5]]}"}},"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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r07-equivalence-class-0018","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","language":"zh-Hant","source_mko_ids":["mko-equivalence-class"],"title_zh":"等價類","stem_zh":"同一完整載體上的已認證等價關係。操作=eqclass-of；資料={\"n\":6,\"pairs\":[[0,0],[0,2],[0,4],[0,5],[1,1],[1,3],[2,0],[2,2],[2,4],[2,5],[3,1],[3,3],[4,0],[4,2],[4,4],[4,5],[5,0],[5,2],[5,4],[5,5]],\"element\":5}。請依 [element,class] 欄序回答。","answer":{"kind":"expression","value":"(5, [0, 2, 4, 5])","explanation_zh":"Wave4第三層核對=(5, [0, 2, 4, 5])；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；class精確等於全部相關元素且代表元不變；任兩classes相等或不交且不從抽樣relation分群。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["等價類"]},"variant":{"family_id":"qf-mod29-eqclass-of","seed":202610297018,"parameters":{"n":6,"pairs":[[0,0],[0,2],[0,4],[0,5],[1,1],[1,3],[2,0],[2,2],[2,4],[2,5],[3,1],[3,3],[4,0],[4,2],[4,4],[4,5],[5,0],[5,2],[5,4],[5,5]],"element":5,"domain":"Complete_finite_equivalence_class","operation":"eqclass-of","semantic_template":"qf-mod29-eqclass-of/{\"element\":5,\"n\":6,\"pairs\":[[0,0],[0,2],[0,4],[0,5],[1,1],[1,3],[2,0],[2,2],[2,4],[2,5],[3,1],[3,3],[4,0],[4,2],[4,4],[4,5],[5,0],[5,2],[5,4],[5,5]]}"}},"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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r07-equivalence-class-0019","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","language":"zh-Hant","source_mko_ids":["mko-equivalence-class"],"title_zh":"等價類","stem_zh":"同一完整載體上的已認證等價關係。操作=eqclass-of；資料={\"n\":6,\"pairs\":[[0,0],[0,1],[0,3],[1,0],[1,1],[1,3],[2,2],[2,4],[2,5],[3,0],[3,1],[3,3],[4,2],[4,4],[4,5],[5,2],[5,4],[5,5]],\"element\":0}。請依 [element,class] 欄序回答。","answer":{"kind":"expression","value":"(0, [0, 1, 3])","explanation_zh":"Wave4第三層核對=(0, [0, 1, 3])；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；class精確等於全部相關元素且代表元不變；任兩classes相等或不交且不從抽樣relation分群。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["等價類"]},"variant":{"family_id":"qf-mod29-eqclass-of","seed":202610297019,"parameters":{"n":6,"pairs":[[0,0],[0,1],[0,3],[1,0],[1,1],[1,3],[2,2],[2,4],[2,5],[3,0],[3,1],[3,3],[4,2],[4,4],[4,5],[5,2],[5,4],[5,5]],"element":0,"domain":"Complete_finite_equivalence_class","operation":"eqclass-of","semantic_template":"qf-mod29-eqclass-of/{\"element\":0,\"n\":6,\"pairs\":[[0,0],[0,1],[0,3],[1,0],[1,1],[1,3],[2,2],[2,4],[2,5],[3,0],[3,1],[3,3],[4,2],[4,4],[4,5],[5,2],[5,4],[5,5]]}"}},"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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r07-equivalence-class-0020","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","language":"zh-Hant","source_mko_ids":["mko-equivalence-class"],"title_zh":"等價類","stem_zh":"同一完整載體上的已認證等價關係。操作=eqclass-of；資料={\"n\":6,\"pairs\":[[0,0],[1,1],[1,3],[2,2],[2,4],[2,5],[3,1],[3,3],[4,2],[4,4],[4,5],[5,2],[5,4],[5,5]],\"element\":1}。請依 [element,class] 欄序回答。","answer":{"kind":"expression","value":"(1, [1, 3])","explanation_zh":"Wave4第三層核對=(1, [1, 3])；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；class精確等於全部相關元素且代表元不變；任兩classes相等或不交且不從抽樣relation分群。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["等價類"]},"variant":{"family_id":"qf-mod29-eqclass-of","seed":202610297020,"parameters":{"n":6,"pairs":[[0,0],[1,1],[1,3],[2,2],[2,4],[2,5],[3,1],[3,3],[4,2],[4,4],[4,5],[5,2],[5,4],[5,5]],"element":1,"domain":"Complete_finite_equivalence_class","operation":"eqclass-of","semantic_template":"qf-mod29-eqclass-of/{\"element\":1,\"n\":6,\"pairs\":[[0,0],[1,1],[1,3],[2,2],[2,4],[2,5],[3,1],[3,3],[4,2],[4,4],[4,5],[5,2],[5,4],[5,5]]}"}},"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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r07-equivalence-class-0021","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","language":"zh-Hant","source_mko_ids":["mko-equivalence-class"],"title_zh":"等價類","stem_zh":"同一完整載體上的已認證等價關係。操作=eqclass-of；資料={\"n\":6,\"pairs\":[[0,0],[0,2],[0,4],[0,5],[1,1],[2,0],[2,2],[2,4],[2,5],[3,3],[4,0],[4,2],[4,4],[4,5],[5,0],[5,2],[5,4],[5,5]],\"element\":2}。請依 [element,class] 欄序回答。","answer":{"kind":"expression","value":"(2, [0, 2, 4, 5])","explanation_zh":"Wave4第三層核對=(2, [0, 2, 4, 5])；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；class精確等於全部相關元素且代表元不變；任兩classes相等或不交且不從抽樣relation分群。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["等價類"]},"variant":{"family_id":"qf-mod29-eqclass-of","seed":202610297021,"parameters":{"n":6,"pairs":[[0,0],[0,2],[0,4],[0,5],[1,1],[2,0],[2,2],[2,4],[2,5],[3,3],[4,0],[4,2],[4,4],[4,5],[5,0],[5,2],[5,4],[5,5]],"element":2,"domain":"Complete_finite_equivalence_class","operation":"eqclass-of","semantic_template":"qf-mod29-eqclass-of/{\"element\":2,\"n\":6,\"pairs\":[[0,0],[0,2],[0,4],[0,5],[1,1],[2,0],[2,2],[2,4],[2,5],[3,3],[4,0],[4,2],[4,4],[4,5],[5,0],[5,2],[5,4],[5,5]]}"}},"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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r07-equivalence-class-0022","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","language":"zh-Hant","source_mko_ids":["mko-equivalence-class"],"title_zh":"等價類","stem_zh":"同一完整載體上的已認證等價關係。操作=eqclass-of；資料={\"n\":6,\"pairs\":[[0,0],[0,3],[1,1],[2,2],[2,4],[2,5],[3,0],[3,3],[4,2],[4,4],[4,5],[5,2],[5,4],[5,5]],\"element\":3}。請依 [element,class] 欄序回答。","answer":{"kind":"expression","value":"(3, [0, 3])","explanation_zh":"Wave4第三層核對=(3, [0, 3])；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；class精確等於全部相關元素且代表元不變；任兩classes相等或不交且不從抽樣relation分群。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["等價類"]},"variant":{"family_id":"qf-mod29-eqclass-of","seed":202610297022,"parameters":{"n":6,"pairs":[[0,0],[0,3],[1,1],[2,2],[2,4],[2,5],[3,0],[3,3],[4,2],[4,4],[4,5],[5,2],[5,4],[5,5]],"element":3,"domain":"Complete_finite_equivalence_class","operation":"eqclass-of","semantic_template":"qf-mod29-eqclass-of/{\"element\":3,\"n\":6,\"pairs\":[[0,0],[0,3],[1,1],[2,2],[2,4],[2,5],[3,0],[3,3],[4,2],[4,4],[4,5],[5,2],[5,4],[5,5]]}"}},"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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r07-equivalence-class-0023","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","language":"zh-Hant","source_mko_ids":["mko-equivalence-class"],"title_zh":"等價類","stem_zh":"同一完整載體上的已認證等價關係。操作=eqclass-of；資料={\"n\":6,\"pairs\":[[0,0],[0,1],[1,0],[1,1],[2,2],[2,4],[2,5],[3,3],[4,2],[4,4],[4,5],[5,2],[5,4],[5,5]],\"element\":4}。請依 [element,class] 欄序回答。","answer":{"kind":"expression","value":"(4, [2, 4, 5])","explanation_zh":"Wave4第三層核對=(4, [2, 4, 5])；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；class精確等於全部相關元素且代表元不變；任兩classes相等或不交且不從抽樣relation分群。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["等價類"]},"variant":{"family_id":"qf-mod29-eqclass-of","seed":202610297023,"parameters":{"n":6,"pairs":[[0,0],[0,1],[1,0],[1,1],[2,2],[2,4],[2,5],[3,3],[4,2],[4,4],[4,5],[5,2],[5,4],[5,5]],"element":4,"domain":"Complete_finite_equivalence_class","operation":"eqclass-of","semantic_template":"qf-mod29-eqclass-of/{\"element\":4,\"n\":6,\"pairs\":[[0,0],[0,1],[1,0],[1,1],[2,2],[2,4],[2,5],[3,3],[4,2],[4,4],[4,5],[5,2],[5,4],[5,5]]}"}},"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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r07-equivalence-class-0024","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","language":"zh-Hant","source_mko_ids":["mko-equivalence-class"],"title_zh":"等價類","stem_zh":"同一完整載體上的已認證等價關係。操作=eqclass-of；資料={\"n\":6,\"pairs\":[[0,0],[0,1],[0,4],[0,5],[1,0],[1,1],[1,4],[1,5],[2,2],[2,3],[3,2],[3,3],[4,0],[4,1],[4,4],[4,5],[5,0],[5,1],[5,4],[5,5]],\"element\":5}。請依 [element,class] 欄序回答。","answer":{"kind":"expression","value":"(5, [0, 1, 4, 5])","explanation_zh":"Wave4第三層核對=(5, [0, 1, 4, 5])；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；class精確等於全部相關元素且代表元不變；任兩classes相等或不交且不從抽樣relation分群。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["等價類"]},"variant":{"family_id":"qf-mod29-eqclass-of","seed":202610297024,"parameters":{"n":6,"pairs":[[0,0],[0,1],[0,4],[0,5],[1,0],[1,1],[1,4],[1,5],[2,2],[2,3],[3,2],[3,3],[4,0],[4,1],[4,4],[4,5],[5,0],[5,1],[5,4],[5,5]],"element":5,"domain":"Complete_finite_equivalence_class","operation":"eqclass-of","semantic_template":"qf-mod29-eqclass-of/{\"element\":5,\"n\":6,\"pairs\":[[0,0],[0,1],[0,4],[0,5],[1,0],[1,1],[1,4],[1,5],[2,2],[2,3],[3,2],[3,3],[4,0],[4,1],[4,4],[4,5],[5,0],[5,1],[5,4],[5,5]]}"}},"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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r07-equivalence-class-0025","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","language":"zh-Hant","source_mko_ids":["mko-equivalence-class"],"title_zh":"等價類","stem_zh":"同一完整載體上的已認證等價關係。操作=eqclass-of；資料={\"n\":6,\"pairs\":[[0,0],[0,2],[0,3],[1,1],[1,4],[1,5],[2,0],[2,2],[2,3],[3,0],[3,2],[3,3],[4,1],[4,4],[4,5],[5,1],[5,4],[5,5]],\"element\":0}。請依 [element,class] 欄序回答。","answer":{"kind":"expression","value":"(0, [0, 2, 3])","explanation_zh":"Wave4第三層核對=(0, [0, 2, 3])；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；class精確等於全部相關元素且代表元不變；任兩classes相等或不交且不從抽樣relation分群。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["等價類"]},"variant":{"family_id":"qf-mod29-eqclass-of","seed":202610297025,"parameters":{"n":6,"pairs":[[0,0],[0,2],[0,3],[1,1],[1,4],[1,5],[2,0],[2,2],[2,3],[3,0],[3,2],[3,3],[4,1],[4,4],[4,5],[5,1],[5,4],[5,5]],"element":0,"domain":"Complete_finite_equivalence_class","operation":"eqclass-of","semantic_template":"qf-mod29-eqclass-of/{\"element\":0,\"n\":6,\"pairs\":[[0,0],[0,2],[0,3],[1,1],[1,4],[1,5],[2,0],[2,2],[2,3],[3,0],[3,2],[3,3],[4,1],[4,4],[4,5],[5,1],[5,4],[5,5]]}"}},"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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r07-equivalence-class-0026","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","language":"zh-Hant","source_mko_ids":["mko-equivalence-class"],"title_zh":"等價類","stem_zh":"同一完整載體上的已認證等價關係。操作=eqclass-membership；資料={\"n\":6,\"pairs\":[[0,0],[0,1],[0,2],[0,3],[0,4],[0,5],[1,0],[1,1],[1,2],[1,3],[1,4],[1,5],[2,0],[2,1],[2,2],[2,3],[2,4],[2,5],[3,0],[3,1],[3,2],[3,3],[3,4],[3,5],[4,0],[4,1],[4,2],[4,3],[4,4],[4,5],[5,0],[5,1],[5,2],[5,3],[5,4],[5,5]],\"element\":0,\"candidate\":1}。請依 [element,candidate,related,in_class] 欄序回答。","answer":{"kind":"expression","value":"(0, 1, 是, 是)","explanation_zh":"Wave4第三層核對=(0, 1, 是, 是)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；class精確等於全部相關元素且代表元不變；任兩classes相等或不交且不從抽樣relation分群。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["等價類"]},"variant":{"family_id":"qf-mod29-eqclass-membership","seed":202610297026,"parameters":{"n":6,"pairs":[[0,0],[0,1],[0,2],[0,3],[0,4],[0,5],[1,0],[1,1],[1,2],[1,3],[1,4],[1,5],[2,0],[2,1],[2,2],[2,3],[2,4],[2,5],[3,0],[3,1],[3,2],[3,3],[3,4],[3,5],[4,0],[4,1],[4,2],[4,3],[4,4],[4,5],[5,0],[5,1],[5,2],[5,3],[5,4],[5,5]],"element":0,"candidate":1,"domain":"Complete_finite_equivalence_class","operation":"eqclass-membership","semantic_template":"qf-mod29-eqclass-membership/{\"candidate\":1,\"element\":0,\"n\":6,\"pairs\":[[0,0],[0,1],[0,2],[0,3],[0,4],[0,5],[1,0],[1,1],[1,2],[1,3],[1,4],[1,5],[2,0],[2,1],[2,2],[2,3],[2,4],[2,5],[3,0],[3,1],[3,2],[3,3],[3,4],[3,5],[4,0],[4,1],[4,2],[4,3],[4,4],[4,5],[5,0],[5,1],[5,2],[5,3],[5,4],[5,5]]}"}},"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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r07-equivalence-class-0027","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","language":"zh-Hant","source_mko_ids":["mko-equivalence-class"],"title_zh":"等價類","stem_zh":"同一完整載體上的已認證等價關係。操作=eqclass-membership；資料={\"n\":6,\"pairs\":[[0,0],[1,1],[1,2],[1,3],[1,4],[1,5],[2,1],[2,2],[2,3],[2,4],[2,5],[3,1],[3,2],[3,3],[3,4],[3,5],[4,1],[4,2],[4,3],[4,4],[4,5],[5,1],[5,2],[5,3],[5,4],[5,5]],\"element\":1,\"candidate\":4}。請依 [element,candidate,related,in_class] 欄序回答。","answer":{"kind":"expression","value":"(1, 4, 是, 是)","explanation_zh":"Wave4第三層核對=(1, 4, 是, 是)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；class精確等於全部相關元素且代表元不變；任兩classes相等或不交且不從抽樣relation分群。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["等價類"]},"variant":{"family_id":"qf-mod29-eqclass-membership","seed":202610297027,"parameters":{"n":6,"pairs":[[0,0],[1,1],[1,2],[1,3],[1,4],[1,5],[2,1],[2,2],[2,3],[2,4],[2,5],[3,1],[3,2],[3,3],[3,4],[3,5],[4,1],[4,2],[4,3],[4,4],[4,5],[5,1],[5,2],[5,3],[5,4],[5,5]],"element":1,"candidate":4,"domain":"Complete_finite_equivalence_class","operation":"eqclass-membership","semantic_template":"qf-mod29-eqclass-membership/{\"candidate\":4,\"element\":1,\"n\":6,\"pairs\":[[0,0],[1,1],[1,2],[1,3],[1,4],[1,5],[2,1],[2,2],[2,3],[2,4],[2,5],[3,1],[3,2],[3,3],[3,4],[3,5],[4,1],[4,2],[4,3],[4,4],[4,5],[5,1],[5,2],[5,3],[5,4],[5,5]]}"}},"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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r07-equivalence-class-0028","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","language":"zh-Hant","source_mko_ids":["mko-equivalence-class"],"title_zh":"等價類","stem_zh":"同一完整載體上的已認證等價關係。操作=eqclass-membership；資料={\"n\":6,\"pairs\":[[0,0],[0,2],[0,3],[0,4],[0,5],[1,1],[2,0],[2,2],[2,3],[2,4],[2,5],[3,0],[3,2],[3,3],[3,4],[3,5],[4,0],[4,2],[4,3],[4,4],[4,5],[5,0],[5,2],[5,3],[5,4],[5,5]],\"element\":2,\"candidate\":1}。請依 [element,candidate,related,in_class] 欄序回答。","answer":{"kind":"expression","value":"(2, 1, 否, 否)","explanation_zh":"Wave4第三層核對=(2, 1, 否, 否)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；class精確等於全部相關元素且代表元不變；任兩classes相等或不交且不從抽樣relation分群。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["等價類"]},"variant":{"family_id":"qf-mod29-eqclass-membership","seed":202610297028,"parameters":{"n":6,"pairs":[[0,0],[0,2],[0,3],[0,4],[0,5],[1,1],[2,0],[2,2],[2,3],[2,4],[2,5],[3,0],[3,2],[3,3],[3,4],[3,5],[4,0],[4,2],[4,3],[4,4],[4,5],[5,0],[5,2],[5,3],[5,4],[5,5]],"element":2,"candidate":1,"domain":"Complete_finite_equivalence_class","operation":"eqclass-membership","semantic_template":"qf-mod29-eqclass-membership/{\"candidate\":1,\"element\":2,\"n\":6,\"pairs\":[[0,0],[0,2],[0,3],[0,4],[0,5],[1,1],[2,0],[2,2],[2,3],[2,4],[2,5],[3,0],[3,2],[3,3],[3,4],[3,5],[4,0],[4,2],[4,3],[4,4],[4,5],[5,0],[5,2],[5,3],[5,4],[5,5]]}"}},"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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r07-equivalence-class-0029","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","language":"zh-Hant","source_mko_ids":["mko-equivalence-class"],"title_zh":"等價類","stem_zh":"同一完整載體上的已認證等價關係。操作=eqclass-membership；資料={\"n\":6,\"pairs\":[[0,0],[0,1],[1,0],[1,1],[2,2],[2,3],[2,4],[2,5],[3,2],[3,3],[3,4],[3,5],[4,2],[4,3],[4,4],[4,5],[5,2],[5,3],[5,4],[5,5]],\"element\":3,\"candidate\":4}。請依 [element,candidate,related,in_class] 欄序回答。","answer":{"kind":"expression","value":"(3, 4, 是, 是)","explanation_zh":"Wave4第三層核對=(3, 4, 是, 是)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；class精確等於全部相關元素且代表元不變；任兩classes相等或不交且不從抽樣relation分群。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["等價類"]},"variant":{"family_id":"qf-mod29-eqclass-membership","seed":202610297029,"parameters":{"n":6,"pairs":[[0,0],[0,1],[1,0],[1,1],[2,2],[2,3],[2,4],[2,5],[3,2],[3,3],[3,4],[3,5],[4,2],[4,3],[4,4],[4,5],[5,2],[5,3],[5,4],[5,5]],"element":3,"candidate":4,"domain":"Complete_finite_equivalence_class","operation":"eqclass-membership","semantic_template":"qf-mod29-eqclass-membership/{\"candidate\":4,\"element\":3,\"n\":6,\"pairs\":[[0,0],[0,1],[1,0],[1,1],[2,2],[2,3],[2,4],[2,5],[3,2],[3,3],[3,4],[3,5],[4,2],[4,3],[4,4],[4,5],[5,2],[5,3],[5,4],[5,5]]}"}},"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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r07-equivalence-class-0030","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","language":"zh-Hant","source_mko_ids":["mko-equivalence-class"],"title_zh":"等價類","stem_zh":"同一完整載體上的已認證等價關係。操作=eqclass-membership；資料={\"n\":6,\"pairs\":[[0,0],[1,1],[2,2],[2,3],[2,4],[2,5],[3,2],[3,3],[3,4],[3,5],[4,2],[4,3],[4,4],[4,5],[5,2],[5,3],[5,4],[5,5]],\"element\":4,\"candidate\":1}。請依 [element,candidate,related,in_class] 欄序回答。","answer":{"kind":"expression","value":"(4, 1, 否, 否)","explanation_zh":"Wave4第三層核對=(4, 1, 否, 否)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；class精確等於全部相關元素且代表元不變；任兩classes相等或不交且不從抽樣relation分群。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["等價類"]},"variant":{"family_id":"qf-mod29-eqclass-membership","seed":202610297030,"parameters":{"n":6,"pairs":[[0,0],[1,1],[2,2],[2,3],[2,4],[2,5],[3,2],[3,3],[3,4],[3,5],[4,2],[4,3],[4,4],[4,5],[5,2],[5,3],[5,4],[5,5]],"element":4,"candidate":1,"domain":"Complete_finite_equivalence_class","operation":"eqclass-membership","semantic_template":"qf-mod29-eqclass-membership/{\"candidate\":1,\"element\":4,\"n\":6,\"pairs\":[[0,0],[1,1],[2,2],[2,3],[2,4],[2,5],[3,2],[3,3],[3,4],[3,5],[4,2],[4,3],[4,4],[4,5],[5,2],[5,3],[5,4],[5,5]]}"}},"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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r07-equivalence-class-0031","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","language":"zh-Hant","source_mko_ids":["mko-equivalence-class"],"title_zh":"等價類","stem_zh":"同一完整載體上的已認證等價關係。操作=eqclass-membership；資料={\"n\":6,\"pairs\":[[0,0],[0,1],[0,3],[0,4],[0,5],[1,0],[1,1],[1,3],[1,4],[1,5],[2,2],[3,0],[3,1],[3,3],[3,4],[3,5],[4,0],[4,1],[4,3],[4,4],[4,5],[5,0],[5,1],[5,3],[5,4],[5,5]],\"element\":5,\"candidate\":4}。請依 [element,candidate,related,in_class] 欄序回答。","answer":{"kind":"expression","value":"(5, 4, 是, 是)","explanation_zh":"Wave4第三層核對=(5, 4, 是, 是)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；class精確等於全部相關元素且代表元不變；任兩classes相等或不交且不從抽樣relation分群。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["等價類"]},"variant":{"family_id":"qf-mod29-eqclass-membership","seed":202610297031,"parameters":{"n":6,"pairs":[[0,0],[0,1],[0,3],[0,4],[0,5],[1,0],[1,1],[1,3],[1,4],[1,5],[2,2],[3,0],[3,1],[3,3],[3,4],[3,5],[4,0],[4,1],[4,3],[4,4],[4,5],[5,0],[5,1],[5,3],[5,4],[5,5]],"element":5,"candidate":4,"domain":"Complete_finite_equivalence_class","operation":"eqclass-membership","semantic_template":"qf-mod29-eqclass-membership/{\"candidate\":4,\"element\":5,\"n\":6,\"pairs\":[[0,0],[0,1],[0,3],[0,4],[0,5],[1,0],[1,1],[1,3],[1,4],[1,5],[2,2],[3,0],[3,1],[3,3],[3,4],[3,5],[4,0],[4,1],[4,3],[4,4],[4,5],[5,0],[5,1],[5,3],[5,4],[5,5]]}"}},"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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r07-equivalence-class-0032","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","language":"zh-Hant","source_mko_ids":["mko-equivalence-class"],"title_zh":"等價類","stem_zh":"同一完整載體上的已認證等價關係。操作=eqclass-membership；資料={\"n\":6,\"pairs\":[[0,0],[0,2],[1,1],[1,3],[1,4],[1,5],[2,0],[2,2],[3,1],[3,3],[3,4],[3,5],[4,1],[4,3],[4,4],[4,5],[5,1],[5,3],[5,4],[5,5]],\"element\":0,\"candidate\":1}。請依 [element,candidate,related,in_class] 欄序回答。","answer":{"kind":"expression","value":"(0, 1, 否, 否)","explanation_zh":"Wave4第三層核對=(0, 1, 否, 否)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；class精確等於全部相關元素且代表元不變；任兩classes相等或不交且不從抽樣relation分群。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["等價類"]},"variant":{"family_id":"qf-mod29-eqclass-membership","seed":202610297032,"parameters":{"n":6,"pairs":[[0,0],[0,2],[1,1],[1,3],[1,4],[1,5],[2,0],[2,2],[3,1],[3,3],[3,4],[3,5],[4,1],[4,3],[4,4],[4,5],[5,1],[5,3],[5,4],[5,5]],"element":0,"candidate":1,"domain":"Complete_finite_equivalence_class","operation":"eqclass-membership","semantic_template":"qf-mod29-eqclass-membership/{\"candidate\":1,\"element\":0,\"n\":6,\"pairs\":[[0,0],[0,2],[1,1],[1,3],[1,4],[1,5],[2,0],[2,2],[3,1],[3,3],[3,4],[3,5],[4,1],[4,3],[4,4],[4,5],[5,1],[5,3],[5,4],[5,5]]}"}},"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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r07-equivalence-class-0033","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","language":"zh-Hant","source_mko_ids":["mko-equivalence-class"],"title_zh":"等價類","stem_zh":"同一完整載體上的已認證等價關係。操作=eqclass-membership；資料={\"n\":6,\"pairs\":[[0,0],[1,1],[1,3],[1,4],[1,5],[2,2],[3,1],[3,3],[3,4],[3,5],[4,1],[4,3],[4,4],[4,5],[5,1],[5,3],[5,4],[5,5]],\"element\":1,\"candidate\":4}。請依 [element,candidate,related,in_class] 欄序回答。","answer":{"kind":"expression","value":"(1, 4, 是, 是)","explanation_zh":"Wave4第三層核對=(1, 4, 是, 是)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；class精確等於全部相關元素且代表元不變；任兩classes相等或不交且不從抽樣relation分群。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["等價類"]},"variant":{"family_id":"qf-mod29-eqclass-membership","seed":202610297033,"parameters":{"n":6,"pairs":[[0,0],[1,1],[1,3],[1,4],[1,5],[2,2],[3,1],[3,3],[3,4],[3,5],[4,1],[4,3],[4,4],[4,5],[5,1],[5,3],[5,4],[5,5]],"element":1,"candidate":4,"domain":"Complete_finite_equivalence_class","operation":"eqclass-membership","semantic_template":"qf-mod29-eqclass-membership/{\"candidate\":4,\"element\":1,\"n\":6,\"pairs\":[[0,0],[1,1],[1,3],[1,4],[1,5],[2,2],[3,1],[3,3],[3,4],[3,5],[4,1],[4,3],[4,4],[4,5],[5,1],[5,3],[5,4],[5,5]]}"}},"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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r07-equivalence-class-0034","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","language":"zh-Hant","source_mko_ids":["mko-equivalence-class"],"title_zh":"等價類","stem_zh":"同一完整載體上的已認證等價關係。操作=eqclass-membership；資料={\"n\":6,\"pairs\":[[0,0],[0,3],[0,4],[0,5],[1,1],[1,2],[2,1],[2,2],[3,0],[3,3],[3,4],[3,5],[4,0],[4,3],[4,4],[4,5],[5,0],[5,3],[5,4],[5,5]],\"element\":2,\"candidate\":1}。請依 [element,candidate,related,in_class] 欄序回答。","answer":{"kind":"expression","value":"(2, 1, 是, 是)","explanation_zh":"Wave4第三層核對=(2, 1, 是, 是)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；class精確等於全部相關元素且代表元不變；任兩classes相等或不交且不從抽樣relation分群。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["等價類"]},"variant":{"family_id":"qf-mod29-eqclass-membership","seed":202610297034,"parameters":{"n":6,"pairs":[[0,0],[0,3],[0,4],[0,5],[1,1],[1,2],[2,1],[2,2],[3,0],[3,3],[3,4],[3,5],[4,0],[4,3],[4,4],[4,5],[5,0],[5,3],[5,4],[5,5]],"element":2,"candidate":1,"domain":"Complete_finite_equivalence_class","operation":"eqclass-membership","semantic_template":"qf-mod29-eqclass-membership/{\"candidate\":1,\"element\":2,\"n\":6,\"pairs\":[[0,0],[0,3],[0,4],[0,5],[1,1],[1,2],[2,1],[2,2],[3,0],[3,3],[3,4],[3,5],[4,0],[4,3],[4,4],[4,5],[5,0],[5,3],[5,4],[5,5]]}"}},"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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r07-equivalence-class-0035","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","language":"zh-Hant","source_mko_ids":["mko-equivalence-class"],"title_zh":"等價類","stem_zh":"同一完整載體上的已認證等價關係。操作=eqclass-membership；資料={\"n\":6,\"pairs\":[[0,0],[0,1],[0,2],[1,0],[1,1],[1,2],[2,0],[2,1],[2,2],[3,3],[3,4],[3,5],[4,3],[4,4],[4,5],[5,3],[5,4],[5,5]],\"element\":3,\"candidate\":4}。請依 [element,candidate,related,in_class] 欄序回答。","answer":{"kind":"expression","value":"(3, 4, 是, 是)","explanation_zh":"Wave4第三層核對=(3, 4, 是, 是)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；class精確等於全部相關元素且代表元不變；任兩classes相等或不交且不從抽樣relation分群。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["等價類"]},"variant":{"family_id":"qf-mod29-eqclass-membership","seed":202610297035,"parameters":{"n":6,"pairs":[[0,0],[0,1],[0,2],[1,0],[1,1],[1,2],[2,0],[2,1],[2,2],[3,3],[3,4],[3,5],[4,3],[4,4],[4,5],[5,3],[5,4],[5,5]],"element":3,"candidate":4,"domain":"Complete_finite_equivalence_class","operation":"eqclass-membership","semantic_template":"qf-mod29-eqclass-membership/{\"candidate\":4,\"element\":3,\"n\":6,\"pairs\":[[0,0],[0,1],[0,2],[1,0],[1,1],[1,2],[2,0],[2,1],[2,2],[3,3],[3,4],[3,5],[4,3],[4,4],[4,5],[5,3],[5,4],[5,5]]}"}},"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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r07-equivalence-class-0036","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","language":"zh-Hant","source_mko_ids":["mko-equivalence-class"],"title_zh":"等價類","stem_zh":"同一完整載體上的已認證等價關係。操作=eqclass-membership；資料={\"n\":6,\"pairs\":[[0,0],[1,1],[1,2],[2,1],[2,2],[3,3],[3,4],[3,5],[4,3],[4,4],[4,5],[5,3],[5,4],[5,5]],\"element\":4,\"candidate\":1}。請依 [element,candidate,related,in_class] 欄序回答。","answer":{"kind":"expression","value":"(4, 1, 否, 否)","explanation_zh":"Wave4第三層核對=(4, 1, 否, 否)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；class精確等於全部相關元素且代表元不變；任兩classes相等或不交且不從抽樣relation分群。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["等價類"]},"variant":{"family_id":"qf-mod29-eqclass-membership","seed":202610297036,"parameters":{"n":6,"pairs":[[0,0],[1,1],[1,2],[2,1],[2,2],[3,3],[3,4],[3,5],[4,3],[4,4],[4,5],[5,3],[5,4],[5,5]],"element":4,"candidate":1,"domain":"Complete_finite_equivalence_class","operation":"eqclass-membership","semantic_template":"qf-mod29-eqclass-membership/{\"candidate\":1,\"element\":4,\"n\":6,\"pairs\":[[0,0],[1,1],[1,2],[2,1],[2,2],[3,3],[3,4],[3,5],[4,3],[4,4],[4,5],[5,3],[5,4],[5,5]]}"}},"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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r07-equivalence-class-0037","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","language":"zh-Hant","source_mko_ids":["mko-equivalence-class"],"title_zh":"等價類","stem_zh":"同一完整載體上的已認證等價關係。操作=eqclass-membership；資料={\"n\":6,\"pairs\":[[0,0],[0,3],[0,4],[0,5],[1,1],[2,2],[3,0],[3,3],[3,4],[3,5],[4,0],[4,3],[4,4],[4,5],[5,0],[5,3],[5,4],[5,5]],\"element\":5,\"candidate\":4}。請依 [element,candidate,related,in_class] 欄序回答。","answer":{"kind":"expression","value":"(5, 4, 是, 是)","explanation_zh":"Wave4第三層核對=(5, 4, 是, 是)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；class精確等於全部相關元素且代表元不變；任兩classes相等或不交且不從抽樣relation分群。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["等價類"]},"variant":{"family_id":"qf-mod29-eqclass-membership","seed":202610297037,"parameters":{"n":6,"pairs":[[0,0],[0,3],[0,4],[0,5],[1,1],[2,2],[3,0],[3,3],[3,4],[3,5],[4,0],[4,3],[4,4],[4,5],[5,0],[5,3],[5,4],[5,5]],"element":5,"candidate":4,"domain":"Complete_finite_equivalence_class","operation":"eqclass-membership","semantic_template":"qf-mod29-eqclass-membership/{\"candidate\":4,\"element\":5,\"n\":6,\"pairs\":[[0,0],[0,3],[0,4],[0,5],[1,1],[2,2],[3,0],[3,3],[3,4],[3,5],[4,0],[4,3],[4,4],[4,5],[5,0],[5,3],[5,4],[5,5]]}"}},"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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r07-equivalence-class-0038","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","language":"zh-Hant","source_mko_ids":["mko-equivalence-class"],"title_zh":"等價類","stem_zh":"同一完整載體上的已認證等價關係。操作=eqclass-membership；資料={\"n\":6,\"pairs\":[[0,0],[0,2],[1,1],[2,0],[2,2],[3,3],[3,4],[3,5],[4,3],[4,4],[4,5],[5,3],[5,4],[5,5]],\"element\":0,\"candidate\":1}。請依 [element,candidate,related,in_class] 欄序回答。","answer":{"kind":"expression","value":"(0, 1, 否, 否)","explanation_zh":"Wave4第三層核對=(0, 1, 否, 否)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；class精確等於全部相關元素且代表元不變；任兩classes相等或不交且不從抽樣relation分群。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["等價類"]},"variant":{"family_id":"qf-mod29-eqclass-membership","seed":202610297038,"parameters":{"n":6,"pairs":[[0,0],[0,2],[1,1],[2,0],[2,2],[3,3],[3,4],[3,5],[4,3],[4,4],[4,5],[5,3],[5,4],[5,5]],"element":0,"candidate":1,"domain":"Complete_finite_equivalence_class","operation":"eqclass-membership","semantic_template":"qf-mod29-eqclass-membership/{\"candidate\":1,\"element\":0,\"n\":6,\"pairs\":[[0,0],[0,2],[1,1],[2,0],[2,2],[3,3],[3,4],[3,5],[4,3],[4,4],[4,5],[5,3],[5,4],[5,5]]}"}},"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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r07-equivalence-class-0039","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","language":"zh-Hant","source_mko_ids":["mko-equivalence-class"],"title_zh":"等價類","stem_zh":"同一完整載體上的已認證等價關係。操作=eqclass-membership；資料={\"n\":6,\"pairs\":[[0,0],[0,1],[1,0],[1,1],[2,2],[3,3],[3,4],[3,5],[4,3],[4,4],[4,5],[5,3],[5,4],[5,5]],\"element\":1,\"candidate\":4}。請依 [element,candidate,related,in_class] 欄序回答。","answer":{"kind":"expression","value":"(1, 4, 否, 否)","explanation_zh":"Wave4第三層核對=(1, 4, 否, 否)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；class精確等於全部相關元素且代表元不變；任兩classes相等或不交且不從抽樣relation分群。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["等價類"]},"variant":{"family_id":"qf-mod29-eqclass-membership","seed":202610297039,"parameters":{"n":6,"pairs":[[0,0],[0,1],[1,0],[1,1],[2,2],[3,3],[3,4],[3,5],[4,3],[4,4],[4,5],[5,3],[5,4],[5,5]],"element":1,"candidate":4,"domain":"Complete_finite_equivalence_class","operation":"eqclass-membership","semantic_template":"qf-mod29-eqclass-membership/{\"candidate\":4,\"element\":1,\"n\":6,\"pairs\":[[0,0],[0,1],[1,0],[1,1],[2,2],[3,3],[3,4],[3,5],[4,3],[4,4],[4,5],[5,3],[5,4],[5,5]]}"}},"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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r07-equivalence-class-0040","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","language":"zh-Hant","source_mko_ids":["mko-equivalence-class"],"title_zh":"等價類","stem_zh":"同一完整載體上的已認證等價關係。操作=eqclass-membership；資料={\"n\":6,\"pairs\":[[0,0],[0,1],[0,2],[0,4],[0,5],[1,0],[1,1],[1,2],[1,4],[1,5],[2,0],[2,1],[2,2],[2,4],[2,5],[3,3],[4,0],[4,1],[4,2],[4,4],[4,5],[5,0],[5,1],[5,2],[5,4],[5,5]],\"element\":2,\"candidate\":1}。請依 [element,candidate,related,in_class] 欄序回答。","answer":{"kind":"expression","value":"(2, 1, 是, 是)","explanation_zh":"Wave4第三層核對=(2, 1, 是, 是)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；class精確等於全部相關元素且代表元不變；任兩classes相等或不交且不從抽樣relation分群。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["等價類"]},"variant":{"family_id":"qf-mod29-eqclass-membership","seed":202610297040,"parameters":{"n":6,"pairs":[[0,0],[0,1],[0,2],[0,4],[0,5],[1,0],[1,1],[1,2],[1,4],[1,5],[2,0],[2,1],[2,2],[2,4],[2,5],[3,3],[4,0],[4,1],[4,2],[4,4],[4,5],[5,0],[5,1],[5,2],[5,4],[5,5]],"element":2,"candidate":1,"domain":"Complete_finite_equivalence_class","operation":"eqclass-membership","semantic_template":"qf-mod29-eqclass-membership/{\"candidate\":1,\"element\":2,\"n\":6,\"pairs\":[[0,0],[0,1],[0,2],[0,4],[0,5],[1,0],[1,1],[1,2],[1,4],[1,5],[2,0],[2,1],[2,2],[2,4],[2,5],[3,3],[4,0],[4,1],[4,2],[4,4],[4,5],[5,0],[5,1],[5,2],[5,4],[5,5]]}"}},"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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r07-equivalence-class-0041","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","language":"zh-Hant","source_mko_ids":["mko-equivalence-class"],"title_zh":"等價類","stem_zh":"同一完整載體上的已認證等價關係。操作=eqclass-membership；資料={\"n\":6,\"pairs\":[[0,0],[0,3],[1,1],[1,2],[1,4],[1,5],[2,1],[2,2],[2,4],[2,5],[3,0],[3,3],[4,1],[4,2],[4,4],[4,5],[5,1],[5,2],[5,4],[5,5]],\"element\":3,\"candidate\":4}。請依 [element,candidate,related,in_class] 欄序回答。","answer":{"kind":"expression","value":"(3, 4, 否, 否)","explanation_zh":"Wave4第三層核對=(3, 4, 否, 否)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；class精確等於全部相關元素且代表元不變；任兩classes相等或不交且不從抽樣relation分群。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["等價類"]},"variant":{"family_id":"qf-mod29-eqclass-membership","seed":202610297041,"parameters":{"n":6,"pairs":[[0,0],[0,3],[1,1],[1,2],[1,4],[1,5],[2,1],[2,2],[2,4],[2,5],[3,0],[3,3],[4,1],[4,2],[4,4],[4,5],[5,1],[5,2],[5,4],[5,5]],"element":3,"candidate":4,"domain":"Complete_finite_equivalence_class","operation":"eqclass-membership","semantic_template":"qf-mod29-eqclass-membership/{\"candidate\":4,\"element\":3,\"n\":6,\"pairs\":[[0,0],[0,3],[1,1],[1,2],[1,4],[1,5],[2,1],[2,2],[2,4],[2,5],[3,0],[3,3],[4,1],[4,2],[4,4],[4,5],[5,1],[5,2],[5,4],[5,5]]}"}},"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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r07-equivalence-class-0042","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","language":"zh-Hant","source_mko_ids":["mko-equivalence-class"],"title_zh":"等價類","stem_zh":"同一完整載體上的已認證等價關係。操作=eqclass-membership；資料={\"n\":6,\"pairs\":[[0,0],[1,1],[1,2],[1,4],[1,5],[2,1],[2,2],[2,4],[2,5],[3,3],[4,1],[4,2],[4,4],[4,5],[5,1],[5,2],[5,4],[5,5]],\"element\":4,\"candidate\":1}。請依 [element,candidate,related,in_class] 欄序回答。","answer":{"kind":"expression","value":"(4, 1, 是, 是)","explanation_zh":"Wave4第三層核對=(4, 1, 是, 是)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；class精確等於全部相關元素且代表元不變；任兩classes相等或不交且不從抽樣relation分群。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["等價類"]},"variant":{"family_id":"qf-mod29-eqclass-membership","seed":202610297042,"parameters":{"n":6,"pairs":[[0,0],[1,1],[1,2],[1,4],[1,5],[2,1],[2,2],[2,4],[2,5],[3,3],[4,1],[4,2],[4,4],[4,5],[5,1],[5,2],[5,4],[5,5]],"element":4,"candidate":1,"domain":"Complete_finite_equivalence_class","operation":"eqclass-membership","semantic_template":"qf-mod29-eqclass-membership/{\"candidate\":1,\"element\":4,\"n\":6,\"pairs\":[[0,0],[1,1],[1,2],[1,4],[1,5],[2,1],[2,2],[2,4],[2,5],[3,3],[4,1],[4,2],[4,4],[4,5],[5,1],[5,2],[5,4],[5,5]]}"}},"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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r07-equivalence-class-0043","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","language":"zh-Hant","source_mko_ids":["mko-equivalence-class"],"title_zh":"等價類","stem_zh":"同一完整載體上的已認證等價關係。操作=eqclass-membership；資料={\"n\":6,\"pairs\":[[0,0],[0,2],[0,4],[0,5],[1,1],[1,3],[2,0],[2,2],[2,4],[2,5],[3,1],[3,3],[4,0],[4,2],[4,4],[4,5],[5,0],[5,2],[5,4],[5,5]],\"element\":5,\"candidate\":4}。請依 [element,candidate,related,in_class] 欄序回答。","answer":{"kind":"expression","value":"(5, 4, 是, 是)","explanation_zh":"Wave4第三層核對=(5, 4, 是, 是)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；class精確等於全部相關元素且代表元不變；任兩classes相等或不交且不從抽樣relation分群。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["等價類"]},"variant":{"family_id":"qf-mod29-eqclass-membership","seed":202610297043,"parameters":{"n":6,"pairs":[[0,0],[0,2],[0,4],[0,5],[1,1],[1,3],[2,0],[2,2],[2,4],[2,5],[3,1],[3,3],[4,0],[4,2],[4,4],[4,5],[5,0],[5,2],[5,4],[5,5]],"element":5,"candidate":4,"domain":"Complete_finite_equivalence_class","operation":"eqclass-membership","semantic_template":"qf-mod29-eqclass-membership/{\"candidate\":4,\"element\":5,\"n\":6,\"pairs\":[[0,0],[0,2],[0,4],[0,5],[1,1],[1,3],[2,0],[2,2],[2,4],[2,5],[3,1],[3,3],[4,0],[4,2],[4,4],[4,5],[5,0],[5,2],[5,4],[5,5]]}"}},"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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r07-equivalence-class-0044","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","language":"zh-Hant","source_mko_ids":["mko-equivalence-class"],"title_zh":"等價類","stem_zh":"同一完整載體上的已認證等價關係。操作=eqclass-membership；資料={\"n\":6,\"pairs\":[[0,0],[0,1],[0,3],[1,0],[1,1],[1,3],[2,2],[2,4],[2,5],[3,0],[3,1],[3,3],[4,2],[4,4],[4,5],[5,2],[5,4],[5,5]],\"element\":0,\"candidate\":1}。請依 [element,candidate,related,in_class] 欄序回答。","answer":{"kind":"expression","value":"(0, 1, 是, 是)","explanation_zh":"Wave4第三層核對=(0, 1, 是, 是)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；class精確等於全部相關元素且代表元不變；任兩classes相等或不交且不從抽樣relation分群。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["等價類"]},"variant":{"family_id":"qf-mod29-eqclass-membership","seed":202610297044,"parameters":{"n":6,"pairs":[[0,0],[0,1],[0,3],[1,0],[1,1],[1,3],[2,2],[2,4],[2,5],[3,0],[3,1],[3,3],[4,2],[4,4],[4,5],[5,2],[5,4],[5,5]],"element":0,"candidate":1,"domain":"Complete_finite_equivalence_class","operation":"eqclass-membership","semantic_template":"qf-mod29-eqclass-membership/{\"candidate\":1,\"element\":0,\"n\":6,\"pairs\":[[0,0],[0,1],[0,3],[1,0],[1,1],[1,3],[2,2],[2,4],[2,5],[3,0],[3,1],[3,3],[4,2],[4,4],[4,5],[5,2],[5,4],[5,5]]}"}},"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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r07-equivalence-class-0045","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","language":"zh-Hant","source_mko_ids":["mko-equivalence-class"],"title_zh":"等價類","stem_zh":"同一完整載體上的已認證等價關係。操作=eqclass-membership；資料={\"n\":6,\"pairs\":[[0,0],[1,1],[1,3],[2,2],[2,4],[2,5],[3,1],[3,3],[4,2],[4,4],[4,5],[5,2],[5,4],[5,5]],\"element\":1,\"candidate\":4}。請依 [element,candidate,related,in_class] 欄序回答。","answer":{"kind":"expression","value":"(1, 4, 否, 否)","explanation_zh":"Wave4第三層核對=(1, 4, 否, 否)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；class精確等於全部相關元素且代表元不變；任兩classes相等或不交且不從抽樣relation分群。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["等價類"]},"variant":{"family_id":"qf-mod29-eqclass-membership","seed":202610297045,"parameters":{"n":6,"pairs":[[0,0],[1,1],[1,3],[2,2],[2,4],[2,5],[3,1],[3,3],[4,2],[4,4],[4,5],[5,2],[5,4],[5,5]],"element":1,"candidate":4,"domain":"Complete_finite_equivalence_class","operation":"eqclass-membership","semantic_template":"qf-mod29-eqclass-membership/{\"candidate\":4,\"element\":1,\"n\":6,\"pairs\":[[0,0],[1,1],[1,3],[2,2],[2,4],[2,5],[3,1],[3,3],[4,2],[4,4],[4,5],[5,2],[5,4],[5,5]]}"}},"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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r07-equivalence-class-0046","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","language":"zh-Hant","source_mko_ids":["mko-equivalence-class"],"title_zh":"等價類","stem_zh":"同一完整載體上的已認證等價關係。操作=eqclass-membership；資料={\"n\":6,\"pairs\":[[0,0],[0,2],[0,4],[0,5],[1,1],[2,0],[2,2],[2,4],[2,5],[3,3],[4,0],[4,2],[4,4],[4,5],[5,0],[5,2],[5,4],[5,5]],\"element\":2,\"candidate\":1}。請依 [element,candidate,related,in_class] 欄序回答。","answer":{"kind":"expression","value":"(2, 1, 否, 否)","explanation_zh":"Wave4第三層核對=(2, 1, 否, 否)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；class精確等於全部相關元素且代表元不變；任兩classes相等或不交且不從抽樣relation分群。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["等價類"]},"variant":{"family_id":"qf-mod29-eqclass-membership","seed":202610297046,"parameters":{"n":6,"pairs":[[0,0],[0,2],[0,4],[0,5],[1,1],[2,0],[2,2],[2,4],[2,5],[3,3],[4,0],[4,2],[4,4],[4,5],[5,0],[5,2],[5,4],[5,5]],"element":2,"candidate":1,"domain":"Complete_finite_equivalence_class","operation":"eqclass-membership","semantic_template":"qf-mod29-eqclass-membership/{\"candidate\":1,\"element\":2,\"n\":6,\"pairs\":[[0,0],[0,2],[0,4],[0,5],[1,1],[2,0],[2,2],[2,4],[2,5],[3,3],[4,0],[4,2],[4,4],[4,5],[5,0],[5,2],[5,4],[5,5]]}"}},"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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r07-equivalence-class-0047","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","language":"zh-Hant","source_mko_ids":["mko-equivalence-class"],"title_zh":"等價類","stem_zh":"同一完整載體上的已認證等價關係。操作=eqclass-membership；資料={\"n\":6,\"pairs\":[[0,0],[0,3],[1,1],[2,2],[2,4],[2,5],[3,0],[3,3],[4,2],[4,4],[4,5],[5,2],[5,4],[5,5]],\"element\":3,\"candidate\":4}。請依 [element,candidate,related,in_class] 欄序回答。","answer":{"kind":"expression","value":"(3, 4, 否, 否)","explanation_zh":"Wave4第三層核對=(3, 4, 否, 否)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；class精確等於全部相關元素且代表元不變；任兩classes相等或不交且不從抽樣relation分群。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["等價類"]},"variant":{"family_id":"qf-mod29-eqclass-membership","seed":202610297047,"parameters":{"n":6,"pairs":[[0,0],[0,3],[1,1],[2,2],[2,4],[2,5],[3,0],[3,3],[4,2],[4,4],[4,5],[5,2],[5,4],[5,5]],"element":3,"candidate":4,"domain":"Complete_finite_equivalence_class","operation":"eqclass-membership","semantic_template":"qf-mod29-eqclass-membership/{\"candidate\":4,\"element\":3,\"n\":6,\"pairs\":[[0,0],[0,3],[1,1],[2,2],[2,4],[2,5],[3,0],[3,3],[4,2],[4,4],[4,5],[5,2],[5,4],[5,5]]}"}},"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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r07-equivalence-class-0048","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","language":"zh-Hant","source_mko_ids":["mko-equivalence-class"],"title_zh":"等價類","stem_zh":"同一完整載體上的已認證等價關係。操作=eqclass-membership；資料={\"n\":6,\"pairs\":[[0,0],[0,1],[1,0],[1,1],[2,2],[2,4],[2,5],[3,3],[4,2],[4,4],[4,5],[5,2],[5,4],[5,5]],\"element\":4,\"candidate\":1}。請依 [element,candidate,related,in_class] 欄序回答。","answer":{"kind":"expression","value":"(4, 1, 否, 否)","explanation_zh":"Wave4第三層核對=(4, 1, 否, 否)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；class精確等於全部相關元素且代表元不變；任兩classes相等或不交且不從抽樣relation分群。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["等價類"]},"variant":{"family_id":"qf-mod29-eqclass-membership","seed":202610297048,"parameters":{"n":6,"pairs":[[0,0],[0,1],[1,0],[1,1],[2,2],[2,4],[2,5],[3,3],[4,2],[4,4],[4,5],[5,2],[5,4],[5,5]],"element":4,"candidate":1,"domain":"Complete_finite_equivalence_class","operation":"eqclass-membership","semantic_template":"qf-mod29-eqclass-membership/{\"candidate\":1,\"element\":4,\"n\":6,\"pairs\":[[0,0],[0,1],[1,0],[1,1],[2,2],[2,4],[2,5],[3,3],[4,2],[4,4],[4,5],[5,2],[5,4],[5,5]]}"}},"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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r07-equivalence-class-0049","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","language":"zh-Hant","source_mko_ids":["mko-equivalence-class"],"title_zh":"等價類","stem_zh":"同一完整載體上的已認證等價關係。操作=eqclass-membership；資料={\"n\":6,\"pairs\":[[0,0],[0,1],[0,4],[0,5],[1,0],[1,1],[1,4],[1,5],[2,2],[2,3],[3,2],[3,3],[4,0],[4,1],[4,4],[4,5],[5,0],[5,1],[5,4],[5,5]],\"element\":5,\"candidate\":4}。請依 [element,candidate,related,in_class] 欄序回答。","answer":{"kind":"expression","value":"(5, 4, 是, 是)","explanation_zh":"Wave4第三層核對=(5, 4, 是, 是)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；class精確等於全部相關元素且代表元不變；任兩classes相等或不交且不從抽樣relation分群。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["等價類"]},"variant":{"family_id":"qf-mod29-eqclass-membership","seed":202610297049,"parameters":{"n":6,"pairs":[[0,0],[0,1],[0,4],[0,5],[1,0],[1,1],[1,4],[1,5],[2,2],[2,3],[3,2],[3,3],[4,0],[4,1],[4,4],[4,5],[5,0],[5,1],[5,4],[5,5]],"element":5,"candidate":4,"domain":"Complete_finite_equivalence_class","operation":"eqclass-membership","semantic_template":"qf-mod29-eqclass-membership/{\"candidate\":4,\"element\":5,\"n\":6,\"pairs\":[[0,0],[0,1],[0,4],[0,5],[1,0],[1,1],[1,4],[1,5],[2,2],[2,3],[3,2],[3,3],[4,0],[4,1],[4,4],[4,5],[5,0],[5,1],[5,4],[5,5]]}"}},"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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r07-equivalence-class-0050","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","language":"zh-Hant","source_mko_ids":["mko-equivalence-class"],"title_zh":"等價類","stem_zh":"同一完整載體上的已認證等價關係。操作=eqclass-membership；資料={\"n\":6,\"pairs\":[[0,0],[0,2],[0,3],[1,1],[1,4],[1,5],[2,0],[2,2],[2,3],[3,0],[3,2],[3,3],[4,1],[4,4],[4,5],[5,1],[5,4],[5,5]],\"element\":0,\"candidate\":1}。請依 [element,candidate,related,in_class] 欄序回答。","answer":{"kind":"expression","value":"(0, 1, 否, 否)","explanation_zh":"Wave4第三層核對=(0, 1, 否, 否)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；class精確等於全部相關元素且代表元不變；任兩classes相等或不交且不從抽樣relation分群。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["等價類"]},"variant":{"family_id":"qf-mod29-eqclass-membership","seed":202610297050,"parameters":{"n":6,"pairs":[[0,0],[0,2],[0,3],[1,1],[1,4],[1,5],[2,0],[2,2],[2,3],[3,0],[3,2],[3,3],[4,1],[4,4],[4,5],[5,1],[5,4],[5,5]],"element":0,"candidate":1,"domain":"Complete_finite_equivalence_class","operation":"eqclass-membership","semantic_template":"qf-mod29-eqclass-membership/{\"candidate\":1,\"element\":0,\"n\":6,\"pairs\":[[0,0],[0,2],[0,3],[1,1],[1,4],[1,5],[2,0],[2,2],[2,3],[3,0],[3,2],[3,3],[4,1],[4,4],[4,5],[5,1],[5,4],[5,5]]}"}},"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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r07-equivalence-class-0051","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","language":"zh-Hant","source_mko_ids":["mko-equivalence-class"],"title_zh":"等價類","stem_zh":"同一完整載體上的已認證等價關係。操作=eqclass-representative-invariance；資料={\"n\":6,\"pairs\":[[0,0],[0,1],[0,2],[0,3],[0,4],[0,5],[1,0],[1,1],[1,2],[1,3],[1,4],[1,5],[2,0],[2,1],[2,2],[2,3],[2,4],[2,5],[3,0],[3,1],[3,2],[3,3],[3,4],[3,5],[4,0],[4,1],[4,2],[4,3],[4,4],[4,5],[5,0],[5,1],[5,2],[5,3],[5,4],[5,5]],\"a\":0,\"b\":1}。請依 [a,b,related,class_a,class_b,invariance] 欄序回答。","answer":{"kind":"expression","value":"(0, 1, 是, [0, 1, 2, 3, 4, 5], [0, 1, 2, 3, 4, 5], 是)","explanation_zh":"Wave4第三層核對=(0, 1, 是, [0, 1, 2, 3, 4, 5], [0, 1, 2, 3, 4, 5], 是)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；class精確等於全部相關元素且代表元不變；任兩classes相等或不交且不從抽樣relation分群。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["等價類"]},"variant":{"family_id":"qf-mod29-eqclass-representative-invariance","seed":202610297051,"parameters":{"n":6,"pairs":[[0,0],[0,1],[0,2],[0,3],[0,4],[0,5],[1,0],[1,1],[1,2],[1,3],[1,4],[1,5],[2,0],[2,1],[2,2],[2,3],[2,4],[2,5],[3,0],[3,1],[3,2],[3,3],[3,4],[3,5],[4,0],[4,1],[4,2],[4,3],[4,4],[4,5],[5,0],[5,1],[5,2],[5,3],[5,4],[5,5]],"a":0,"b":1,"domain":"Complete_finite_equivalence_class","operation":"eqclass-representative-invariance","semantic_template":"qf-mod29-eqclass-representative-invariance/{\"a\":0,\"b\":1,\"n\":6,\"pairs\":[[0,0],[0,1],[0,2],[0,3],[0,4],[0,5],[1,0],[1,1],[1,2],[1,3],[1,4],[1,5],[2,0],[2,1],[2,2],[2,3],[2,4],[2,5],[3,0],[3,1],[3,2],[3,3],[3,4],[3,5],[4,0],[4,1],[4,2],[4,3],[4,4],[4,5],[5,0],[5,1],[5,2],[5,3],[5,4],[5,5]]}"}},"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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r07-equivalence-class-0052","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","language":"zh-Hant","source_mko_ids":["mko-equivalence-class"],"title_zh":"等價類","stem_zh":"同一完整載體上的已認證等價關係。操作=eqclass-representative-invariance；資料={\"n\":6,\"pairs\":[[0,0],[1,1],[1,2],[1,3],[1,4],[1,5],[2,1],[2,2],[2,3],[2,4],[2,5],[3,1],[3,2],[3,3],[3,4],[3,5],[4,1],[4,2],[4,3],[4,4],[4,5],[5,1],[5,2],[5,3],[5,4],[5,5]],\"a\":1,\"b\":4}。請依 [a,b,related,class_a,class_b,invariance] 欄序回答。","answer":{"kind":"expression","value":"(1, 4, 是, [1, 2, 3, 4, 5], [1, 2, 3, 4, 5], 是)","explanation_zh":"Wave4第三層核對=(1, 4, 是, [1, 2, 3, 4, 5], [1, 2, 3, 4, 5], 是)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；class精確等於全部相關元素且代表元不變；任兩classes相等或不交且不從抽樣relation分群。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["等價類"]},"variant":{"family_id":"qf-mod29-eqclass-representative-invariance","seed":202610297052,"parameters":{"n":6,"pairs":[[0,0],[1,1],[1,2],[1,3],[1,4],[1,5],[2,1],[2,2],[2,3],[2,4],[2,5],[3,1],[3,2],[3,3],[3,4],[3,5],[4,1],[4,2],[4,3],[4,4],[4,5],[5,1],[5,2],[5,3],[5,4],[5,5]],"a":1,"b":4,"domain":"Complete_finite_equivalence_class","operation":"eqclass-representative-invariance","semantic_template":"qf-mod29-eqclass-representative-invariance/{\"a\":1,\"b\":4,\"n\":6,\"pairs\":[[0,0],[1,1],[1,2],[1,3],[1,4],[1,5],[2,1],[2,2],[2,3],[2,4],[2,5],[3,1],[3,2],[3,3],[3,4],[3,5],[4,1],[4,2],[4,3],[4,4],[4,5],[5,1],[5,2],[5,3],[5,4],[5,5]]}"}},"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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r07-equivalence-class-0053","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","language":"zh-Hant","source_mko_ids":["mko-equivalence-class"],"title_zh":"等價類","stem_zh":"同一完整載體上的已認證等價關係。操作=eqclass-representative-invariance；資料={\"n\":6,\"pairs\":[[0,0],[0,2],[0,3],[0,4],[0,5],[1,1],[2,0],[2,2],[2,3],[2,4],[2,5],[3,0],[3,2],[3,3],[3,4],[3,5],[4,0],[4,2],[4,3],[4,4],[4,5],[5,0],[5,2],[5,3],[5,4],[5,5]],\"a\":2,\"b\":1}。請依 [a,b,related,class_a,class_b,invariance] 欄序回答。","answer":{"kind":"expression","value":"(2, 1, 否, [0, 2, 3, 4, 5], [1], 是)","explanation_zh":"Wave4第三層核對=(2, 1, 否, [0, 2, 3, 4, 5], [1], 是)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；class精確等於全部相關元素且代表元不變；任兩classes相等或不交且不從抽樣relation分群。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["等價類"]},"variant":{"family_id":"qf-mod29-eqclass-representative-invariance","seed":202610297053,"parameters":{"n":6,"pairs":[[0,0],[0,2],[0,3],[0,4],[0,5],[1,1],[2,0],[2,2],[2,3],[2,4],[2,5],[3,0],[3,2],[3,3],[3,4],[3,5],[4,0],[4,2],[4,3],[4,4],[4,5],[5,0],[5,2],[5,3],[5,4],[5,5]],"a":2,"b":1,"domain":"Complete_finite_equivalence_class","operation":"eqclass-representative-invariance","semantic_template":"qf-mod29-eqclass-representative-invariance/{\"a\":2,\"b\":1,\"n\":6,\"pairs\":[[0,0],[0,2],[0,3],[0,4],[0,5],[1,1],[2,0],[2,2],[2,3],[2,4],[2,5],[3,0],[3,2],[3,3],[3,4],[3,5],[4,0],[4,2],[4,3],[4,4],[4,5],[5,0],[5,2],[5,3],[5,4],[5,5]]}"}},"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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r07-equivalence-class-0054","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","language":"zh-Hant","source_mko_ids":["mko-equivalence-class"],"title_zh":"等價類","stem_zh":"同一完整載體上的已認證等價關係。操作=eqclass-representative-invariance；資料={\"n\":6,\"pairs\":[[0,0],[0,1],[1,0],[1,1],[2,2],[2,3],[2,4],[2,5],[3,2],[3,3],[3,4],[3,5],[4,2],[4,3],[4,4],[4,5],[5,2],[5,3],[5,4],[5,5]],\"a\":3,\"b\":4}。請依 [a,b,related,class_a,class_b,invariance] 欄序回答。","answer":{"kind":"expression","value":"(3, 4, 是, [2, 3, 4, 5], [2, 3, 4, 5], 是)","explanation_zh":"Wave4第三層核對=(3, 4, 是, [2, 3, 4, 5], [2, 3, 4, 5], 是)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；class精確等於全部相關元素且代表元不變；任兩classes相等或不交且不從抽樣relation分群。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["等價類"]},"variant":{"family_id":"qf-mod29-eqclass-representative-invariance","seed":202610297054,"parameters":{"n":6,"pairs":[[0,0],[0,1],[1,0],[1,1],[2,2],[2,3],[2,4],[2,5],[3,2],[3,3],[3,4],[3,5],[4,2],[4,3],[4,4],[4,5],[5,2],[5,3],[5,4],[5,5]],"a":3,"b":4,"domain":"Complete_finite_equivalence_class","operation":"eqclass-representative-invariance","semantic_template":"qf-mod29-eqclass-representative-invariance/{\"a\":3,\"b\":4,\"n\":6,\"pairs\":[[0,0],[0,1],[1,0],[1,1],[2,2],[2,3],[2,4],[2,5],[3,2],[3,3],[3,4],[3,5],[4,2],[4,3],[4,4],[4,5],[5,2],[5,3],[5,4],[5,5]]}"}},"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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r07-equivalence-class-0055","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","language":"zh-Hant","source_mko_ids":["mko-equivalence-class"],"title_zh":"等價類","stem_zh":"同一完整載體上的已認證等價關係。操作=eqclass-representative-invariance；資料={\"n\":6,\"pairs\":[[0,0],[1,1],[2,2],[2,3],[2,4],[2,5],[3,2],[3,3],[3,4],[3,5],[4,2],[4,3],[4,4],[4,5],[5,2],[5,3],[5,4],[5,5]],\"a\":4,\"b\":1}。請依 [a,b,related,class_a,class_b,invariance] 欄序回答。","answer":{"kind":"expression","value":"(4, 1, 否, [2, 3, 4, 5], [1], 是)","explanation_zh":"Wave4第三層核對=(4, 1, 否, [2, 3, 4, 5], [1], 是)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；class精確等於全部相關元素且代表元不變；任兩classes相等或不交且不從抽樣relation分群。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["等價類"]},"variant":{"family_id":"qf-mod29-eqclass-representative-invariance","seed":202610297055,"parameters":{"n":6,"pairs":[[0,0],[1,1],[2,2],[2,3],[2,4],[2,5],[3,2],[3,3],[3,4],[3,5],[4,2],[4,3],[4,4],[4,5],[5,2],[5,3],[5,4],[5,5]],"a":4,"b":1,"domain":"Complete_finite_equivalence_class","operation":"eqclass-representative-invariance","semantic_template":"qf-mod29-eqclass-representative-invariance/{\"a\":4,\"b\":1,\"n\":6,\"pairs\":[[0,0],[1,1],[2,2],[2,3],[2,4],[2,5],[3,2],[3,3],[3,4],[3,5],[4,2],[4,3],[4,4],[4,5],[5,2],[5,3],[5,4],[5,5]]}"}},"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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r07-equivalence-class-0056","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","language":"zh-Hant","source_mko_ids":["mko-equivalence-class"],"title_zh":"等價類","stem_zh":"同一完整載體上的已認證等價關係。操作=eqclass-representative-invariance；資料={\"n\":6,\"pairs\":[[0,0],[0,1],[0,3],[0,4],[0,5],[1,0],[1,1],[1,3],[1,4],[1,5],[2,2],[3,0],[3,1],[3,3],[3,4],[3,5],[4,0],[4,1],[4,3],[4,4],[4,5],[5,0],[5,1],[5,3],[5,4],[5,5]],\"a\":5,\"b\":4}。請依 [a,b,related,class_a,class_b,invariance] 欄序回答。","answer":{"kind":"expression","value":"(5, 4, 是, [0, 1, 3, 4, 5], [0, 1, 3, 4, 5], 是)","explanation_zh":"Wave4第三層核對=(5, 4, 是, [0, 1, 3, 4, 5], [0, 1, 3, 4, 5], 是)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；class精確等於全部相關元素且代表元不變；任兩classes相等或不交且不從抽樣relation分群。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["等價類"]},"variant":{"family_id":"qf-mod29-eqclass-representative-invariance","seed":202610297056,"parameters":{"n":6,"pairs":[[0,0],[0,1],[0,3],[0,4],[0,5],[1,0],[1,1],[1,3],[1,4],[1,5],[2,2],[3,0],[3,1],[3,3],[3,4],[3,5],[4,0],[4,1],[4,3],[4,4],[4,5],[5,0],[5,1],[5,3],[5,4],[5,5]],"a":5,"b":4,"domain":"Complete_finite_equivalence_class","operation":"eqclass-representative-invariance","semantic_template":"qf-mod29-eqclass-representative-invariance/{\"a\":5,\"b\":4,\"n\":6,\"pairs\":[[0,0],[0,1],[0,3],[0,4],[0,5],[1,0],[1,1],[1,3],[1,4],[1,5],[2,2],[3,0],[3,1],[3,3],[3,4],[3,5],[4,0],[4,1],[4,3],[4,4],[4,5],[5,0],[5,1],[5,3],[5,4],[5,5]]}"}},"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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r07-equivalence-class-0057","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","language":"zh-Hant","source_mko_ids":["mko-equivalence-class"],"title_zh":"等價類","stem_zh":"同一完整載體上的已認證等價關係。操作=eqclass-representative-invariance；資料={\"n\":6,\"pairs\":[[0,0],[0,2],[1,1],[1,3],[1,4],[1,5],[2,0],[2,2],[3,1],[3,3],[3,4],[3,5],[4,1],[4,3],[4,4],[4,5],[5,1],[5,3],[5,4],[5,5]],\"a\":0,\"b\":1}。請依 [a,b,related,class_a,class_b,invariance] 欄序回答。","answer":{"kind":"expression","value":"(0, 1, 否, [0, 2], [1, 3, 4, 5], 是)","explanation_zh":"Wave4第三層核對=(0, 1, 否, [0, 2], [1, 3, 4, 5], 是)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；class精確等於全部相關元素且代表元不變；任兩classes相等或不交且不從抽樣relation分群。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["等價類"]},"variant":{"family_id":"qf-mod29-eqclass-representative-invariance","seed":202610297057,"parameters":{"n":6,"pairs":[[0,0],[0,2],[1,1],[1,3],[1,4],[1,5],[2,0],[2,2],[3,1],[3,3],[3,4],[3,5],[4,1],[4,3],[4,4],[4,5],[5,1],[5,3],[5,4],[5,5]],"a":0,"b":1,"domain":"Complete_finite_equivalence_class","operation":"eqclass-representative-invariance","semantic_template":"qf-mod29-eqclass-representative-invariance/{\"a\":0,\"b\":1,\"n\":6,\"pairs\":[[0,0],[0,2],[1,1],[1,3],[1,4],[1,5],[2,0],[2,2],[3,1],[3,3],[3,4],[3,5],[4,1],[4,3],[4,4],[4,5],[5,1],[5,3],[5,4],[5,5]]}"}},"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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r07-equivalence-class-0058","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","language":"zh-Hant","source_mko_ids":["mko-equivalence-class"],"title_zh":"等價類","stem_zh":"同一完整載體上的已認證等價關係。操作=eqclass-representative-invariance；資料={\"n\":6,\"pairs\":[[0,0],[1,1],[1,3],[1,4],[1,5],[2,2],[3,1],[3,3],[3,4],[3,5],[4,1],[4,3],[4,4],[4,5],[5,1],[5,3],[5,4],[5,5]],\"a\":1,\"b\":4}。請依 [a,b,related,class_a,class_b,invariance] 欄序回答。","answer":{"kind":"expression","value":"(1, 4, 是, [1, 3, 4, 5], [1, 3, 4, 5], 是)","explanation_zh":"Wave4第三層核對=(1, 4, 是, [1, 3, 4, 5], [1, 3, 4, 5], 是)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；class精確等於全部相關元素且代表元不變；任兩classes相等或不交且不從抽樣relation分群。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["等價類"]},"variant":{"family_id":"qf-mod29-eqclass-representative-invariance","seed":202610297058,"parameters":{"n":6,"pairs":[[0,0],[1,1],[1,3],[1,4],[1,5],[2,2],[3,1],[3,3],[3,4],[3,5],[4,1],[4,3],[4,4],[4,5],[5,1],[5,3],[5,4],[5,5]],"a":1,"b":4,"domain":"Complete_finite_equivalence_class","operation":"eqclass-representative-invariance","semantic_template":"qf-mod29-eqclass-representative-invariance/{\"a\":1,\"b\":4,\"n\":6,\"pairs\":[[0,0],[1,1],[1,3],[1,4],[1,5],[2,2],[3,1],[3,3],[3,4],[3,5],[4,1],[4,3],[4,4],[4,5],[5,1],[5,3],[5,4],[5,5]]}"}},"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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r07-equivalence-class-0059","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","language":"zh-Hant","source_mko_ids":["mko-equivalence-class"],"title_zh":"等價類","stem_zh":"同一完整載體上的已認證等價關係。操作=eqclass-representative-invariance；資料={\"n\":6,\"pairs\":[[0,0],[0,3],[0,4],[0,5],[1,1],[1,2],[2,1],[2,2],[3,0],[3,3],[3,4],[3,5],[4,0],[4,3],[4,4],[4,5],[5,0],[5,3],[5,4],[5,5]],\"a\":2,\"b\":1}。請依 [a,b,related,class_a,class_b,invariance] 欄序回答。","answer":{"kind":"expression","value":"(2, 1, 是, [1, 2], [1, 2], 是)","explanation_zh":"Wave4第三層核對=(2, 1, 是, [1, 2], [1, 2], 是)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；class精確等於全部相關元素且代表元不變；任兩classes相等或不交且不從抽樣relation分群。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["等價類"]},"variant":{"family_id":"qf-mod29-eqclass-representative-invariance","seed":202610297059,"parameters":{"n":6,"pairs":[[0,0],[0,3],[0,4],[0,5],[1,1],[1,2],[2,1],[2,2],[3,0],[3,3],[3,4],[3,5],[4,0],[4,3],[4,4],[4,5],[5,0],[5,3],[5,4],[5,5]],"a":2,"b":1,"domain":"Complete_finite_equivalence_class","operation":"eqclass-representative-invariance","semantic_template":"qf-mod29-eqclass-representative-invariance/{\"a\":2,\"b\":1,\"n\":6,\"pairs\":[[0,0],[0,3],[0,4],[0,5],[1,1],[1,2],[2,1],[2,2],[3,0],[3,3],[3,4],[3,5],[4,0],[4,3],[4,4],[4,5],[5,0],[5,3],[5,4],[5,5]]}"}},"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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r07-equivalence-class-0060","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","language":"zh-Hant","source_mko_ids":["mko-equivalence-class"],"title_zh":"等價類","stem_zh":"同一完整載體上的已認證等價關係。操作=eqclass-representative-invariance；資料={\"n\":6,\"pairs\":[[0,0],[0,1],[0,2],[1,0],[1,1],[1,2],[2,0],[2,1],[2,2],[3,3],[3,4],[3,5],[4,3],[4,4],[4,5],[5,3],[5,4],[5,5]],\"a\":3,\"b\":4}。請依 [a,b,related,class_a,class_b,invariance] 欄序回答。","answer":{"kind":"expression","value":"(3, 4, 是, [3, 4, 5], [3, 4, 5], 是)","explanation_zh":"Wave4第三層核對=(3, 4, 是, [3, 4, 5], [3, 4, 5], 是)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；class精確等於全部相關元素且代表元不變；任兩classes相等或不交且不從抽樣relation分群。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["等價類"]},"variant":{"family_id":"qf-mod29-eqclass-representative-invariance","seed":202610297060,"parameters":{"n":6,"pairs":[[0,0],[0,1],[0,2],[1,0],[1,1],[1,2],[2,0],[2,1],[2,2],[3,3],[3,4],[3,5],[4,3],[4,4],[4,5],[5,3],[5,4],[5,5]],"a":3,"b":4,"domain":"Complete_finite_equivalence_class","operation":"eqclass-representative-invariance","semantic_template":"qf-mod29-eqclass-representative-invariance/{\"a\":3,\"b\":4,\"n\":6,\"pairs\":[[0,0],[0,1],[0,2],[1,0],[1,1],[1,2],[2,0],[2,1],[2,2],[3,3],[3,4],[3,5],[4,3],[4,4],[4,5],[5,3],[5,4],[5,5]]}"}},"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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r07-equivalence-class-0061","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","language":"zh-Hant","source_mko_ids":["mko-equivalence-class"],"title_zh":"等價類","stem_zh":"同一完整載體上的已認證等價關係。操作=eqclass-representative-invariance；資料={\"n\":6,\"pairs\":[[0,0],[1,1],[1,2],[2,1],[2,2],[3,3],[3,4],[3,5],[4,3],[4,4],[4,5],[5,3],[5,4],[5,5]],\"a\":4,\"b\":1}。請依 [a,b,related,class_a,class_b,invariance] 欄序回答。","answer":{"kind":"expression","value":"(4, 1, 否, [3, 4, 5], [1, 2], 是)","explanation_zh":"Wave4第三層核對=(4, 1, 否, [3, 4, 5], [1, 2], 是)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；class精確等於全部相關元素且代表元不變；任兩classes相等或不交且不從抽樣relation分群。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["等價類"]},"variant":{"family_id":"qf-mod29-eqclass-representative-invariance","seed":202610297061,"parameters":{"n":6,"pairs":[[0,0],[1,1],[1,2],[2,1],[2,2],[3,3],[3,4],[3,5],[4,3],[4,4],[4,5],[5,3],[5,4],[5,5]],"a":4,"b":1,"domain":"Complete_finite_equivalence_class","operation":"eqclass-representative-invariance","semantic_template":"qf-mod29-eqclass-representative-invariance/{\"a\":4,\"b\":1,\"n\":6,\"pairs\":[[0,0],[1,1],[1,2],[2,1],[2,2],[3,3],[3,4],[3,5],[4,3],[4,4],[4,5],[5,3],[5,4],[5,5]]}"}},"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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r07-equivalence-class-0062","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","language":"zh-Hant","source_mko_ids":["mko-equivalence-class"],"title_zh":"等價類","stem_zh":"同一完整載體上的已認證等價關係。操作=eqclass-representative-invariance；資料={\"n\":6,\"pairs\":[[0,0],[0,3],[0,4],[0,5],[1,1],[2,2],[3,0],[3,3],[3,4],[3,5],[4,0],[4,3],[4,4],[4,5],[5,0],[5,3],[5,4],[5,5]],\"a\":5,\"b\":4}。請依 [a,b,related,class_a,class_b,invariance] 欄序回答。","answer":{"kind":"expression","value":"(5, 4, 是, [0, 3, 4, 5], [0, 3, 4, 5], 是)","explanation_zh":"Wave4第三層核對=(5, 4, 是, [0, 3, 4, 5], [0, 3, 4, 5], 是)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；class精確等於全部相關元素且代表元不變；任兩classes相等或不交且不從抽樣relation分群。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["等價類"]},"variant":{"family_id":"qf-mod29-eqclass-representative-invariance","seed":202610297062,"parameters":{"n":6,"pairs":[[0,0],[0,3],[0,4],[0,5],[1,1],[2,2],[3,0],[3,3],[3,4],[3,5],[4,0],[4,3],[4,4],[4,5],[5,0],[5,3],[5,4],[5,5]],"a":5,"b":4,"domain":"Complete_finite_equivalence_class","operation":"eqclass-representative-invariance","semantic_template":"qf-mod29-eqclass-representative-invariance/{\"a\":5,\"b\":4,\"n\":6,\"pairs\":[[0,0],[0,3],[0,4],[0,5],[1,1],[2,2],[3,0],[3,3],[3,4],[3,5],[4,0],[4,3],[4,4],[4,5],[5,0],[5,3],[5,4],[5,5]]}"}},"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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r07-equivalence-class-0063","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","language":"zh-Hant","source_mko_ids":["mko-equivalence-class"],"title_zh":"等價類","stem_zh":"同一完整載體上的已認證等價關係。操作=eqclass-representative-invariance；資料={\"n\":6,\"pairs\":[[0,0],[0,2],[1,1],[2,0],[2,2],[3,3],[3,4],[3,5],[4,3],[4,4],[4,5],[5,3],[5,4],[5,5]],\"a\":0,\"b\":1}。請依 [a,b,related,class_a,class_b,invariance] 欄序回答。","answer":{"kind":"expression","value":"(0, 1, 否, [0, 2], [1], 是)","explanation_zh":"Wave4第三層核對=(0, 1, 否, [0, 2], [1], 是)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；class精確等於全部相關元素且代表元不變；任兩classes相等或不交且不從抽樣relation分群。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["等價類"]},"variant":{"family_id":"qf-mod29-eqclass-representative-invariance","seed":202610297063,"parameters":{"n":6,"pairs":[[0,0],[0,2],[1,1],[2,0],[2,2],[3,3],[3,4],[3,5],[4,3],[4,4],[4,5],[5,3],[5,4],[5,5]],"a":0,"b":1,"domain":"Complete_finite_equivalence_class","operation":"eqclass-representative-invariance","semantic_template":"qf-mod29-eqclass-representative-invariance/{\"a\":0,\"b\":1,\"n\":6,\"pairs\":[[0,0],[0,2],[1,1],[2,0],[2,2],[3,3],[3,4],[3,5],[4,3],[4,4],[4,5],[5,3],[5,4],[5,5]]}"}},"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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r07-equivalence-class-0064","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","language":"zh-Hant","source_mko_ids":["mko-equivalence-class"],"title_zh":"等價類","stem_zh":"同一完整載體上的已認證等價關係。操作=eqclass-representative-invariance；資料={\"n\":6,\"pairs\":[[0,0],[0,1],[1,0],[1,1],[2,2],[3,3],[3,4],[3,5],[4,3],[4,4],[4,5],[5,3],[5,4],[5,5]],\"a\":1,\"b\":4}。請依 [a,b,related,class_a,class_b,invariance] 欄序回答。","answer":{"kind":"expression","value":"(1, 4, 否, [0, 1], [3, 4, 5], 是)","explanation_zh":"Wave4第三層核對=(1, 4, 否, [0, 1], [3, 4, 5], 是)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；class精確等於全部相關元素且代表元不變；任兩classes相等或不交且不從抽樣relation分群。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["等價類"]},"variant":{"family_id":"qf-mod29-eqclass-representative-invariance","seed":202610297064,"parameters":{"n":6,"pairs":[[0,0],[0,1],[1,0],[1,1],[2,2],[3,3],[3,4],[3,5],[4,3],[4,4],[4,5],[5,3],[5,4],[5,5]],"a":1,"b":4,"domain":"Complete_finite_equivalence_class","operation":"eqclass-representative-invariance","semantic_template":"qf-mod29-eqclass-representative-invariance/{\"a\":1,\"b\":4,\"n\":6,\"pairs\":[[0,0],[0,1],[1,0],[1,1],[2,2],[3,3],[3,4],[3,5],[4,3],[4,4],[4,5],[5,3],[5,4],[5,5]]}"}},"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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r07-equivalence-class-0065","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","language":"zh-Hant","source_mko_ids":["mko-equivalence-class"],"title_zh":"等價類","stem_zh":"同一完整載體上的已認證等價關係。操作=eqclass-representative-invariance；資料={\"n\":6,\"pairs\":[[0,0],[0,1],[0,2],[0,4],[0,5],[1,0],[1,1],[1,2],[1,4],[1,5],[2,0],[2,1],[2,2],[2,4],[2,5],[3,3],[4,0],[4,1],[4,2],[4,4],[4,5],[5,0],[5,1],[5,2],[5,4],[5,5]],\"a\":2,\"b\":1}。請依 [a,b,related,class_a,class_b,invariance] 欄序回答。","answer":{"kind":"expression","value":"(2, 1, 是, [0, 1, 2, 4, 5], [0, 1, 2, 4, 5], 是)","explanation_zh":"Wave4第三層核對=(2, 1, 是, [0, 1, 2, 4, 5], [0, 1, 2, 4, 5], 是)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；class精確等於全部相關元素且代表元不變；任兩classes相等或不交且不從抽樣relation分群。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["等價類"]},"variant":{"family_id":"qf-mod29-eqclass-representative-invariance","seed":202610297065,"parameters":{"n":6,"pairs":[[0,0],[0,1],[0,2],[0,4],[0,5],[1,0],[1,1],[1,2],[1,4],[1,5],[2,0],[2,1],[2,2],[2,4],[2,5],[3,3],[4,0],[4,1],[4,2],[4,4],[4,5],[5,0],[5,1],[5,2],[5,4],[5,5]],"a":2,"b":1,"domain":"Complete_finite_equivalence_class","operation":"eqclass-representative-invariance","semantic_template":"qf-mod29-eqclass-representative-invariance/{\"a\":2,\"b\":1,\"n\":6,\"pairs\":[[0,0],[0,1],[0,2],[0,4],[0,5],[1,0],[1,1],[1,2],[1,4],[1,5],[2,0],[2,1],[2,2],[2,4],[2,5],[3,3],[4,0],[4,1],[4,2],[4,4],[4,5],[5,0],[5,1],[5,2],[5,4],[5,5]]}"}},"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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r07-equivalence-class-0066","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","language":"zh-Hant","source_mko_ids":["mko-equivalence-class"],"title_zh":"等價類","stem_zh":"同一完整載體上的已認證等價關係。操作=eqclass-representative-invariance；資料={\"n\":6,\"pairs\":[[0,0],[0,3],[1,1],[1,2],[1,4],[1,5],[2,1],[2,2],[2,4],[2,5],[3,0],[3,3],[4,1],[4,2],[4,4],[4,5],[5,1],[5,2],[5,4],[5,5]],\"a\":3,\"b\":4}。請依 [a,b,related,class_a,class_b,invariance] 欄序回答。","answer":{"kind":"expression","value":"(3, 4, 否, [0, 3], [1, 2, 4, 5], 是)","explanation_zh":"Wave4第三層核對=(3, 4, 否, [0, 3], [1, 2, 4, 5], 是)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；class精確等於全部相關元素且代表元不變；任兩classes相等或不交且不從抽樣relation分群。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["等價類"]},"variant":{"family_id":"qf-mod29-eqclass-representative-invariance","seed":202610297066,"parameters":{"n":6,"pairs":[[0,0],[0,3],[1,1],[1,2],[1,4],[1,5],[2,1],[2,2],[2,4],[2,5],[3,0],[3,3],[4,1],[4,2],[4,4],[4,5],[5,1],[5,2],[5,4],[5,5]],"a":3,"b":4,"domain":"Complete_finite_equivalence_class","operation":"eqclass-representative-invariance","semantic_template":"qf-mod29-eqclass-representative-invariance/{\"a\":3,\"b\":4,\"n\":6,\"pairs\":[[0,0],[0,3],[1,1],[1,2],[1,4],[1,5],[2,1],[2,2],[2,4],[2,5],[3,0],[3,3],[4,1],[4,2],[4,4],[4,5],[5,1],[5,2],[5,4],[5,5]]}"}},"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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r07-equivalence-class-0067","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","language":"zh-Hant","source_mko_ids":["mko-equivalence-class"],"title_zh":"等價類","stem_zh":"同一完整載體上的已認證等價關係。操作=eqclass-representative-invariance；資料={\"n\":6,\"pairs\":[[0,0],[1,1],[1,2],[1,4],[1,5],[2,1],[2,2],[2,4],[2,5],[3,3],[4,1],[4,2],[4,4],[4,5],[5,1],[5,2],[5,4],[5,5]],\"a\":4,\"b\":1}。請依 [a,b,related,class_a,class_b,invariance] 欄序回答。","answer":{"kind":"expression","value":"(4, 1, 是, [1, 2, 4, 5], [1, 2, 4, 5], 是)","explanation_zh":"Wave4第三層核對=(4, 1, 是, [1, 2, 4, 5], [1, 2, 4, 5], 是)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；class精確等於全部相關元素且代表元不變；任兩classes相等或不交且不從抽樣relation分群。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["等價類"]},"variant":{"family_id":"qf-mod29-eqclass-representative-invariance","seed":202610297067,"parameters":{"n":6,"pairs":[[0,0],[1,1],[1,2],[1,4],[1,5],[2,1],[2,2],[2,4],[2,5],[3,3],[4,1],[4,2],[4,4],[4,5],[5,1],[5,2],[5,4],[5,5]],"a":4,"b":1,"domain":"Complete_finite_equivalence_class","operation":"eqclass-representative-invariance","semantic_template":"qf-mod29-eqclass-representative-invariance/{\"a\":4,\"b\":1,\"n\":6,\"pairs\":[[0,0],[1,1],[1,2],[1,4],[1,5],[2,1],[2,2],[2,4],[2,5],[3,3],[4,1],[4,2],[4,4],[4,5],[5,1],[5,2],[5,4],[5,5]]}"}},"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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r07-equivalence-class-0068","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","language":"zh-Hant","source_mko_ids":["mko-equivalence-class"],"title_zh":"等價類","stem_zh":"同一完整載體上的已認證等價關係。操作=eqclass-representative-invariance；資料={\"n\":6,\"pairs\":[[0,0],[0,2],[0,4],[0,5],[1,1],[1,3],[2,0],[2,2],[2,4],[2,5],[3,1],[3,3],[4,0],[4,2],[4,4],[4,5],[5,0],[5,2],[5,4],[5,5]],\"a\":5,\"b\":4}。請依 [a,b,related,class_a,class_b,invariance] 欄序回答。","answer":{"kind":"expression","value":"(5, 4, 是, [0, 2, 4, 5], [0, 2, 4, 5], 是)","explanation_zh":"Wave4第三層核對=(5, 4, 是, [0, 2, 4, 5], [0, 2, 4, 5], 是)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；class精確等於全部相關元素且代表元不變；任兩classes相等或不交且不從抽樣relation分群。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["等價類"]},"variant":{"family_id":"qf-mod29-eqclass-representative-invariance","seed":202610297068,"parameters":{"n":6,"pairs":[[0,0],[0,2],[0,4],[0,5],[1,1],[1,3],[2,0],[2,2],[2,4],[2,5],[3,1],[3,3],[4,0],[4,2],[4,4],[4,5],[5,0],[5,2],[5,4],[5,5]],"a":5,"b":4,"domain":"Complete_finite_equivalence_class","operation":"eqclass-representative-invariance","semantic_template":"qf-mod29-eqclass-representative-invariance/{\"a\":5,\"b\":4,\"n\":6,\"pairs\":[[0,0],[0,2],[0,4],[0,5],[1,1],[1,3],[2,0],[2,2],[2,4],[2,5],[3,1],[3,3],[4,0],[4,2],[4,4],[4,5],[5,0],[5,2],[5,4],[5,5]]}"}},"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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r07-equivalence-class-0069","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","language":"zh-Hant","source_mko_ids":["mko-equivalence-class"],"title_zh":"等價類","stem_zh":"同一完整載體上的已認證等價關係。操作=eqclass-representative-invariance；資料={\"n\":6,\"pairs\":[[0,0],[0,1],[0,3],[1,0],[1,1],[1,3],[2,2],[2,4],[2,5],[3,0],[3,1],[3,3],[4,2],[4,4],[4,5],[5,2],[5,4],[5,5]],\"a\":0,\"b\":1}。請依 [a,b,related,class_a,class_b,invariance] 欄序回答。","answer":{"kind":"expression","value":"(0, 1, 是, [0, 1, 3], [0, 1, 3], 是)","explanation_zh":"Wave4第三層核對=(0, 1, 是, [0, 1, 3], [0, 1, 3], 是)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；class精確等於全部相關元素且代表元不變；任兩classes相等或不交且不從抽樣relation分群。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["等價類"]},"variant":{"family_id":"qf-mod29-eqclass-representative-invariance","seed":202610297069,"parameters":{"n":6,"pairs":[[0,0],[0,1],[0,3],[1,0],[1,1],[1,3],[2,2],[2,4],[2,5],[3,0],[3,1],[3,3],[4,2],[4,4],[4,5],[5,2],[5,4],[5,5]],"a":0,"b":1,"domain":"Complete_finite_equivalence_class","operation":"eqclass-representative-invariance","semantic_template":"qf-mod29-eqclass-representative-invariance/{\"a\":0,\"b\":1,\"n\":6,\"pairs\":[[0,0],[0,1],[0,3],[1,0],[1,1],[1,3],[2,2],[2,4],[2,5],[3,0],[3,1],[3,3],[4,2],[4,4],[4,5],[5,2],[5,4],[5,5]]}"}},"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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r07-equivalence-class-0070","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","language":"zh-Hant","source_mko_ids":["mko-equivalence-class"],"title_zh":"等價類","stem_zh":"同一完整載體上的已認證等價關係。操作=eqclass-representative-invariance；資料={\"n\":6,\"pairs\":[[0,0],[1,1],[1,3],[2,2],[2,4],[2,5],[3,1],[3,3],[4,2],[4,4],[4,5],[5,2],[5,4],[5,5]],\"a\":1,\"b\":4}。請依 [a,b,related,class_a,class_b,invariance] 欄序回答。","answer":{"kind":"expression","value":"(1, 4, 否, [1, 3], [2, 4, 5], 是)","explanation_zh":"Wave4第三層核對=(1, 4, 否, [1, 3], [2, 4, 5], 是)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；class精確等於全部相關元素且代表元不變；任兩classes相等或不交且不從抽樣relation分群。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["等價類"]},"variant":{"family_id":"qf-mod29-eqclass-representative-invariance","seed":202610297070,"parameters":{"n":6,"pairs":[[0,0],[1,1],[1,3],[2,2],[2,4],[2,5],[3,1],[3,3],[4,2],[4,4],[4,5],[5,2],[5,4],[5,5]],"a":1,"b":4,"domain":"Complete_finite_equivalence_class","operation":"eqclass-representative-invariance","semantic_template":"qf-mod29-eqclass-representative-invariance/{\"a\":1,\"b\":4,\"n\":6,\"pairs\":[[0,0],[1,1],[1,3],[2,2],[2,4],[2,5],[3,1],[3,3],[4,2],[4,4],[4,5],[5,2],[5,4],[5,5]]}"}},"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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r07-equivalence-class-0071","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","language":"zh-Hant","source_mko_ids":["mko-equivalence-class"],"title_zh":"等價類","stem_zh":"同一完整載體上的已認證等價關係。操作=eqclass-representative-invariance；資料={\"n\":6,\"pairs\":[[0,0],[0,2],[0,4],[0,5],[1,1],[2,0],[2,2],[2,4],[2,5],[3,3],[4,0],[4,2],[4,4],[4,5],[5,0],[5,2],[5,4],[5,5]],\"a\":2,\"b\":1}。請依 [a,b,related,class_a,class_b,invariance] 欄序回答。","answer":{"kind":"expression","value":"(2, 1, 否, [0, 2, 4, 5], [1], 是)","explanation_zh":"Wave4第三層核對=(2, 1, 否, [0, 2, 4, 5], [1], 是)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；class精確等於全部相關元素且代表元不變；任兩classes相等或不交且不從抽樣relation分群。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["等價類"]},"variant":{"family_id":"qf-mod29-eqclass-representative-invariance","seed":202610297071,"parameters":{"n":6,"pairs":[[0,0],[0,2],[0,4],[0,5],[1,1],[2,0],[2,2],[2,4],[2,5],[3,3],[4,0],[4,2],[4,4],[4,5],[5,0],[5,2],[5,4],[5,5]],"a":2,"b":1,"domain":"Complete_finite_equivalence_class","operation":"eqclass-representative-invariance","semantic_template":"qf-mod29-eqclass-representative-invariance/{\"a\":2,\"b\":1,\"n\":6,\"pairs\":[[0,0],[0,2],[0,4],[0,5],[1,1],[2,0],[2,2],[2,4],[2,5],[3,3],[4,0],[4,2],[4,4],[4,5],[5,0],[5,2],[5,4],[5,5]]}"}},"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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r07-equivalence-class-0072","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","language":"zh-Hant","source_mko_ids":["mko-equivalence-class"],"title_zh":"等價類","stem_zh":"同一完整載體上的已認證等價關係。操作=eqclass-representative-invariance；資料={\"n\":6,\"pairs\":[[0,0],[0,3],[1,1],[2,2],[2,4],[2,5],[3,0],[3,3],[4,2],[4,4],[4,5],[5,2],[5,4],[5,5]],\"a\":3,\"b\":4}。請依 [a,b,related,class_a,class_b,invariance] 欄序回答。","answer":{"kind":"expression","value":"(3, 4, 否, [0, 3], [2, 4, 5], 是)","explanation_zh":"Wave4第三層核對=(3, 4, 否, [0, 3], [2, 4, 5], 是)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；class精確等於全部相關元素且代表元不變；任兩classes相等或不交且不從抽樣relation分群。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["等價類"]},"variant":{"family_id":"qf-mod29-eqclass-representative-invariance","seed":202610297072,"parameters":{"n":6,"pairs":[[0,0],[0,3],[1,1],[2,2],[2,4],[2,5],[3,0],[3,3],[4,2],[4,4],[4,5],[5,2],[5,4],[5,5]],"a":3,"b":4,"domain":"Complete_finite_equivalence_class","operation":"eqclass-representative-invariance","semantic_template":"qf-mod29-eqclass-representative-invariance/{\"a\":3,\"b\":4,\"n\":6,\"pairs\":[[0,0],[0,3],[1,1],[2,2],[2,4],[2,5],[3,0],[3,3],[4,2],[4,4],[4,5],[5,2],[5,4],[5,5]]}"}},"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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r07-equivalence-class-0073","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","language":"zh-Hant","source_mko_ids":["mko-equivalence-class"],"title_zh":"等價類","stem_zh":"同一完整載體上的已認證等價關係。操作=eqclass-representative-invariance；資料={\"n\":6,\"pairs\":[[0,0],[0,1],[1,0],[1,1],[2,2],[2,4],[2,5],[3,3],[4,2],[4,4],[4,5],[5,2],[5,4],[5,5]],\"a\":4,\"b\":1}。請依 [a,b,related,class_a,class_b,invariance] 欄序回答。","answer":{"kind":"expression","value":"(4, 1, 否, [2, 4, 5], [0, 1], 是)","explanation_zh":"Wave4第三層核對=(4, 1, 否, [2, 4, 5], [0, 1], 是)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；class精確等於全部相關元素且代表元不變；任兩classes相等或不交且不從抽樣relation分群。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["等價類"]},"variant":{"family_id":"qf-mod29-eqclass-representative-invariance","seed":202610297073,"parameters":{"n":6,"pairs":[[0,0],[0,1],[1,0],[1,1],[2,2],[2,4],[2,5],[3,3],[4,2],[4,4],[4,5],[5,2],[5,4],[5,5]],"a":4,"b":1,"domain":"Complete_finite_equivalence_class","operation":"eqclass-representative-invariance","semantic_template":"qf-mod29-eqclass-representative-invariance/{\"a\":4,\"b\":1,\"n\":6,\"pairs\":[[0,0],[0,1],[1,0],[1,1],[2,2],[2,4],[2,5],[3,3],[4,2],[4,4],[4,5],[5,2],[5,4],[5,5]]}"}},"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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r07-equivalence-class-0074","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","language":"zh-Hant","source_mko_ids":["mko-equivalence-class"],"title_zh":"等價類","stem_zh":"同一完整載體上的已認證等價關係。操作=eqclass-representative-invariance；資料={\"n\":6,\"pairs\":[[0,0],[0,1],[0,4],[0,5],[1,0],[1,1],[1,4],[1,5],[2,2],[2,3],[3,2],[3,3],[4,0],[4,1],[4,4],[4,5],[5,0],[5,1],[5,4],[5,5]],\"a\":5,\"b\":4}。請依 [a,b,related,class_a,class_b,invariance] 欄序回答。","answer":{"kind":"expression","value":"(5, 4, 是, [0, 1, 4, 5], [0, 1, 4, 5], 是)","explanation_zh":"Wave4第三層核對=(5, 4, 是, [0, 1, 4, 5], [0, 1, 4, 5], 是)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；class精確等於全部相關元素且代表元不變；任兩classes相等或不交且不從抽樣relation分群。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["等價類"]},"variant":{"family_id":"qf-mod29-eqclass-representative-invariance","seed":202610297074,"parameters":{"n":6,"pairs":[[0,0],[0,1],[0,4],[0,5],[1,0],[1,1],[1,4],[1,5],[2,2],[2,3],[3,2],[3,3],[4,0],[4,1],[4,4],[4,5],[5,0],[5,1],[5,4],[5,5]],"a":5,"b":4,"domain":"Complete_finite_equivalence_class","operation":"eqclass-representative-invariance","semantic_template":"qf-mod29-eqclass-representative-invariance/{\"a\":5,\"b\":4,\"n\":6,\"pairs\":[[0,0],[0,1],[0,4],[0,5],[1,0],[1,1],[1,4],[1,5],[2,2],[2,3],[3,2],[3,3],[4,0],[4,1],[4,4],[4,5],[5,0],[5,1],[5,4],[5,5]]}"}},"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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r07-equivalence-class-0075","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","language":"zh-Hant","source_mko_ids":["mko-equivalence-class"],"title_zh":"等價類","stem_zh":"同一完整載體上的已認證等價關係。操作=eqclass-representative-invariance；資料={\"n\":6,\"pairs\":[[0,0],[0,2],[0,3],[1,1],[1,4],[1,5],[2,0],[2,2],[2,3],[3,0],[3,2],[3,3],[4,1],[4,4],[4,5],[5,1],[5,4],[5,5]],\"a\":0,\"b\":1}。請依 [a,b,related,class_a,class_b,invariance] 欄序回答。","answer":{"kind":"expression","value":"(0, 1, 否, [0, 2, 3], [1, 4, 5], 是)","explanation_zh":"Wave4第三層核對=(0, 1, 否, [0, 2, 3], [1, 4, 5], 是)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；class精確等於全部相關元素且代表元不變；任兩classes相等或不交且不從抽樣relation分群。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["等價類"]},"variant":{"family_id":"qf-mod29-eqclass-representative-invariance","seed":202610297075,"parameters":{"n":6,"pairs":[[0,0],[0,2],[0,3],[1,1],[1,4],[1,5],[2,0],[2,2],[2,3],[3,0],[3,2],[3,3],[4,1],[4,4],[4,5],[5,1],[5,4],[5,5]],"a":0,"b":1,"domain":"Complete_finite_equivalence_class","operation":"eqclass-representative-invariance","semantic_template":"qf-mod29-eqclass-representative-invariance/{\"a\":0,\"b\":1,\"n\":6,\"pairs\":[[0,0],[0,2],[0,3],[1,1],[1,4],[1,5],[2,0],[2,2],[2,3],[3,0],[3,2],[3,3],[4,1],[4,4],[4,5],[5,1],[5,4],[5,5]]}"}},"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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r07-equivalence-class-0076","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","language":"zh-Hant","source_mko_ids":["mko-equivalence-class"],"title_zh":"等價類","stem_zh":"同一完整載體上的已認證等價關係。操作=eqclass-equal-or-disjoint；資料={\"n\":6,\"pairs\":[[0,0],[0,1],[0,2],[0,3],[0,4],[0,5],[1,0],[1,1],[1,2],[1,3],[1,4],[1,5],[2,0],[2,1],[2,2],[2,3],[2,4],[2,5],[3,0],[3,1],[3,2],[3,3],[3,4],[3,5],[4,0],[4,1],[4,2],[4,3],[4,4],[4,5],[5,0],[5,1],[5,2],[5,3],[5,4],[5,5]],\"a\":0,\"b\":1}。請依 [a,b,class_a,class_b,intersection,equal_or_disjoint,classification] 欄序回答。","answer":{"kind":"expression","value":"(0, 1, [0, 1, 2, 3, 4, 5], [0, 1, 2, 3, 4, 5], [0, 1, 2, 3, 4, 5], 是, equal)","explanation_zh":"Wave4第三層核對=(0, 1, [0, 1, 2, 3, 4, 5], [0, 1, 2, 3, 4, 5], [0, 1, 2, 3, 4, 5], 是, equal)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；class精確等於全部相關元素且代表元不變；任兩classes相等或不交且不從抽樣relation分群。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["等價類"]},"variant":{"family_id":"qf-mod29-eqclass-equal-or-disjoint","seed":202610297076,"parameters":{"n":6,"pairs":[[0,0],[0,1],[0,2],[0,3],[0,4],[0,5],[1,0],[1,1],[1,2],[1,3],[1,4],[1,5],[2,0],[2,1],[2,2],[2,3],[2,4],[2,5],[3,0],[3,1],[3,2],[3,3],[3,4],[3,5],[4,0],[4,1],[4,2],[4,3],[4,4],[4,5],[5,0],[5,1],[5,2],[5,3],[5,4],[5,5]],"a":0,"b":1,"domain":"Complete_finite_equivalence_class","operation":"eqclass-equal-or-disjoint","semantic_template":"qf-mod29-eqclass-equal-or-disjoint/{\"a\":0,\"b\":1,\"n\":6,\"pairs\":[[0,0],[0,1],[0,2],[0,3],[0,4],[0,5],[1,0],[1,1],[1,2],[1,3],[1,4],[1,5],[2,0],[2,1],[2,2],[2,3],[2,4],[2,5],[3,0],[3,1],[3,2],[3,3],[3,4],[3,5],[4,0],[4,1],[4,2],[4,3],[4,4],[4,5],[5,0],[5,1],[5,2],[5,3],[5,4],[5,5]]}"}},"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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r07-equivalence-class-0077","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","language":"zh-Hant","source_mko_ids":["mko-equivalence-class"],"title_zh":"等價類","stem_zh":"同一完整載體上的已認證等價關係。操作=eqclass-equal-or-disjoint；資料={\"n\":6,\"pairs\":[[0,0],[1,1],[1,2],[1,3],[1,4],[1,5],[2,1],[2,2],[2,3],[2,4],[2,5],[3,1],[3,2],[3,3],[3,4],[3,5],[4,1],[4,2],[4,3],[4,4],[4,5],[5,1],[5,2],[5,3],[5,4],[5,5]],\"a\":1,\"b\":4}。請依 [a,b,class_a,class_b,intersection,equal_or_disjoint,classification] 欄序回答。","answer":{"kind":"expression","value":"(1, 4, [1, 2, 3, 4, 5], [1, 2, 3, 4, 5], [1, 2, 3, 4, 5], 是, equal)","explanation_zh":"Wave4第三層核對=(1, 4, [1, 2, 3, 4, 5], [1, 2, 3, 4, 5], [1, 2, 3, 4, 5], 是, equal)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；class精確等於全部相關元素且代表元不變；任兩classes相等或不交且不從抽樣relation分群。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["等價類"]},"variant":{"family_id":"qf-mod29-eqclass-equal-or-disjoint","seed":202610297077,"parameters":{"n":6,"pairs":[[0,0],[1,1],[1,2],[1,3],[1,4],[1,5],[2,1],[2,2],[2,3],[2,4],[2,5],[3,1],[3,2],[3,3],[3,4],[3,5],[4,1],[4,2],[4,3],[4,4],[4,5],[5,1],[5,2],[5,3],[5,4],[5,5]],"a":1,"b":4,"domain":"Complete_finite_equivalence_class","operation":"eqclass-equal-or-disjoint","semantic_template":"qf-mod29-eqclass-equal-or-disjoint/{\"a\":1,\"b\":4,\"n\":6,\"pairs\":[[0,0],[1,1],[1,2],[1,3],[1,4],[1,5],[2,1],[2,2],[2,3],[2,4],[2,5],[3,1],[3,2],[3,3],[3,4],[3,5],[4,1],[4,2],[4,3],[4,4],[4,5],[5,1],[5,2],[5,3],[5,4],[5,5]]}"}},"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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r07-equivalence-class-0078","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","language":"zh-Hant","source_mko_ids":["mko-equivalence-class"],"title_zh":"等價類","stem_zh":"同一完整載體上的已認證等價關係。操作=eqclass-equal-or-disjoint；資料={\"n\":6,\"pairs\":[[0,0],[0,2],[0,3],[0,4],[0,5],[1,1],[2,0],[2,2],[2,3],[2,4],[2,5],[3,0],[3,2],[3,3],[3,4],[3,5],[4,0],[4,2],[4,3],[4,4],[4,5],[5,0],[5,2],[5,3],[5,4],[5,5]],\"a\":2,\"b\":1}。請依 [a,b,class_a,class_b,intersection,equal_or_disjoint,classification] 欄序回答。","answer":{"kind":"expression","value":"(2, 1, [0, 2, 3, 4, 5], [1], [], 是, disjoint)","explanation_zh":"Wave4第三層核對=(2, 1, [0, 2, 3, 4, 5], [1], [], 是, disjoint)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；class精確等於全部相關元素且代表元不變；任兩classes相等或不交且不從抽樣relation分群。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["等價類"]},"variant":{"family_id":"qf-mod29-eqclass-equal-or-disjoint","seed":202610297078,"parameters":{"n":6,"pairs":[[0,0],[0,2],[0,3],[0,4],[0,5],[1,1],[2,0],[2,2],[2,3],[2,4],[2,5],[3,0],[3,2],[3,3],[3,4],[3,5],[4,0],[4,2],[4,3],[4,4],[4,5],[5,0],[5,2],[5,3],[5,4],[5,5]],"a":2,"b":1,"domain":"Complete_finite_equivalence_class","operation":"eqclass-equal-or-disjoint","semantic_template":"qf-mod29-eqclass-equal-or-disjoint/{\"a\":2,\"b\":1,\"n\":6,\"pairs\":[[0,0],[0,2],[0,3],[0,4],[0,5],[1,1],[2,0],[2,2],[2,3],[2,4],[2,5],[3,0],[3,2],[3,3],[3,4],[3,5],[4,0],[4,2],[4,3],[4,4],[4,5],[5,0],[5,2],[5,3],[5,4],[5,5]]}"}},"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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r07-equivalence-class-0079","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","language":"zh-Hant","source_mko_ids":["mko-equivalence-class"],"title_zh":"等價類","stem_zh":"同一完整載體上的已認證等價關係。操作=eqclass-equal-or-disjoint；資料={\"n\":6,\"pairs\":[[0,0],[0,1],[1,0],[1,1],[2,2],[2,3],[2,4],[2,5],[3,2],[3,3],[3,4],[3,5],[4,2],[4,3],[4,4],[4,5],[5,2],[5,3],[5,4],[5,5]],\"a\":3,\"b\":4}。請依 [a,b,class_a,class_b,intersection,equal_or_disjoint,classification] 欄序回答。","answer":{"kind":"expression","value":"(3, 4, [2, 3, 4, 5], [2, 3, 4, 5], [2, 3, 4, 5], 是, equal)","explanation_zh":"Wave4第三層核對=(3, 4, [2, 3, 4, 5], [2, 3, 4, 5], [2, 3, 4, 5], 是, equal)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；class精確等於全部相關元素且代表元不變；任兩classes相等或不交且不從抽樣relation分群。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["等價類"]},"variant":{"family_id":"qf-mod29-eqclass-equal-or-disjoint","seed":202610297079,"parameters":{"n":6,"pairs":[[0,0],[0,1],[1,0],[1,1],[2,2],[2,3],[2,4],[2,5],[3,2],[3,3],[3,4],[3,5],[4,2],[4,3],[4,4],[4,5],[5,2],[5,3],[5,4],[5,5]],"a":3,"b":4,"domain":"Complete_finite_equivalence_class","operation":"eqclass-equal-or-disjoint","semantic_template":"qf-mod29-eqclass-equal-or-disjoint/{\"a\":3,\"b\":4,\"n\":6,\"pairs\":[[0,0],[0,1],[1,0],[1,1],[2,2],[2,3],[2,4],[2,5],[3,2],[3,3],[3,4],[3,5],[4,2],[4,3],[4,4],[4,5],[5,2],[5,3],[5,4],[5,5]]}"}},"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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r07-equivalence-class-0080","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","language":"zh-Hant","source_mko_ids":["mko-equivalence-class"],"title_zh":"等價類","stem_zh":"同一完整載體上的已認證等價關係。操作=eqclass-equal-or-disjoint；資料={\"n\":6,\"pairs\":[[0,0],[1,1],[2,2],[2,3],[2,4],[2,5],[3,2],[3,3],[3,4],[3,5],[4,2],[4,3],[4,4],[4,5],[5,2],[5,3],[5,4],[5,5]],\"a\":4,\"b\":1}。請依 [a,b,class_a,class_b,intersection,equal_or_disjoint,classification] 欄序回答。","answer":{"kind":"expression","value":"(4, 1, [2, 3, 4, 5], [1], [], 是, disjoint)","explanation_zh":"Wave4第三層核對=(4, 1, [2, 3, 4, 5], [1], [], 是, disjoint)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；class精確等於全部相關元素且代表元不變；任兩classes相等或不交且不從抽樣relation分群。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["等價類"]},"variant":{"family_id":"qf-mod29-eqclass-equal-or-disjoint","seed":202610297080,"parameters":{"n":6,"pairs":[[0,0],[1,1],[2,2],[2,3],[2,4],[2,5],[3,2],[3,3],[3,4],[3,5],[4,2],[4,3],[4,4],[4,5],[5,2],[5,3],[5,4],[5,5]],"a":4,"b":1,"domain":"Complete_finite_equivalence_class","operation":"eqclass-equal-or-disjoint","semantic_template":"qf-mod29-eqclass-equal-or-disjoint/{\"a\":4,\"b\":1,\"n\":6,\"pairs\":[[0,0],[1,1],[2,2],[2,3],[2,4],[2,5],[3,2],[3,3],[3,4],[3,5],[4,2],[4,3],[4,4],[4,5],[5,2],[5,3],[5,4],[5,5]]}"}},"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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r07-equivalence-class-0081","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","language":"zh-Hant","source_mko_ids":["mko-equivalence-class"],"title_zh":"等價類","stem_zh":"同一完整載體上的已認證等價關係。操作=eqclass-equal-or-disjoint；資料={\"n\":6,\"pairs\":[[0,0],[0,1],[0,3],[0,4],[0,5],[1,0],[1,1],[1,3],[1,4],[1,5],[2,2],[3,0],[3,1],[3,3],[3,4],[3,5],[4,0],[4,1],[4,3],[4,4],[4,5],[5,0],[5,1],[5,3],[5,4],[5,5]],\"a\":5,\"b\":4}。請依 [a,b,class_a,class_b,intersection,equal_or_disjoint,classification] 欄序回答。","answer":{"kind":"expression","value":"(5, 4, [0, 1, 3, 4, 5], [0, 1, 3, 4, 5], [0, 1, 3, 4, 5], 是, equal)","explanation_zh":"Wave4第三層核對=(5, 4, [0, 1, 3, 4, 5], [0, 1, 3, 4, 5], [0, 1, 3, 4, 5], 是, equal)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；class精確等於全部相關元素且代表元不變；任兩classes相等或不交且不從抽樣relation分群。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["等價類"]},"variant":{"family_id":"qf-mod29-eqclass-equal-or-disjoint","seed":202610297081,"parameters":{"n":6,"pairs":[[0,0],[0,1],[0,3],[0,4],[0,5],[1,0],[1,1],[1,3],[1,4],[1,5],[2,2],[3,0],[3,1],[3,3],[3,4],[3,5],[4,0],[4,1],[4,3],[4,4],[4,5],[5,0],[5,1],[5,3],[5,4],[5,5]],"a":5,"b":4,"domain":"Complete_finite_equivalence_class","operation":"eqclass-equal-or-disjoint","semantic_template":"qf-mod29-eqclass-equal-or-disjoint/{\"a\":5,\"b\":4,\"n\":6,\"pairs\":[[0,0],[0,1],[0,3],[0,4],[0,5],[1,0],[1,1],[1,3],[1,4],[1,5],[2,2],[3,0],[3,1],[3,3],[3,4],[3,5],[4,0],[4,1],[4,3],[4,4],[4,5],[5,0],[5,1],[5,3],[5,4],[5,5]]}"}},"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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r07-equivalence-class-0082","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","language":"zh-Hant","source_mko_ids":["mko-equivalence-class"],"title_zh":"等價類","stem_zh":"同一完整載體上的已認證等價關係。操作=eqclass-equal-or-disjoint；資料={\"n\":6,\"pairs\":[[0,0],[0,2],[1,1],[1,3],[1,4],[1,5],[2,0],[2,2],[3,1],[3,3],[3,4],[3,5],[4,1],[4,3],[4,4],[4,5],[5,1],[5,3],[5,4],[5,5]],\"a\":0,\"b\":1}。請依 [a,b,class_a,class_b,intersection,equal_or_disjoint,classification] 欄序回答。","answer":{"kind":"expression","value":"(0, 1, [0, 2], [1, 3, 4, 5], [], 是, disjoint)","explanation_zh":"Wave4第三層核對=(0, 1, [0, 2], [1, 3, 4, 5], [], 是, disjoint)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；class精確等於全部相關元素且代表元不變；任兩classes相等或不交且不從抽樣relation分群。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["等價類"]},"variant":{"family_id":"qf-mod29-eqclass-equal-or-disjoint","seed":202610297082,"parameters":{"n":6,"pairs":[[0,0],[0,2],[1,1],[1,3],[1,4],[1,5],[2,0],[2,2],[3,1],[3,3],[3,4],[3,5],[4,1],[4,3],[4,4],[4,5],[5,1],[5,3],[5,4],[5,5]],"a":0,"b":1,"domain":"Complete_finite_equivalence_class","operation":"eqclass-equal-or-disjoint","semantic_template":"qf-mod29-eqclass-equal-or-disjoint/{\"a\":0,\"b\":1,\"n\":6,\"pairs\":[[0,0],[0,2],[1,1],[1,3],[1,4],[1,5],[2,0],[2,2],[3,1],[3,3],[3,4],[3,5],[4,1],[4,3],[4,4],[4,5],[5,1],[5,3],[5,4],[5,5]]}"}},"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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r07-equivalence-class-0083","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","language":"zh-Hant","source_mko_ids":["mko-equivalence-class"],"title_zh":"等價類","stem_zh":"同一完整載體上的已認證等價關係。操作=eqclass-equal-or-disjoint；資料={\"n\":6,\"pairs\":[[0,0],[1,1],[1,3],[1,4],[1,5],[2,2],[3,1],[3,3],[3,4],[3,5],[4,1],[4,3],[4,4],[4,5],[5,1],[5,3],[5,4],[5,5]],\"a\":1,\"b\":4}。請依 [a,b,class_a,class_b,intersection,equal_or_disjoint,classification] 欄序回答。","answer":{"kind":"expression","value":"(1, 4, [1, 3, 4, 5], [1, 3, 4, 5], [1, 3, 4, 5], 是, equal)","explanation_zh":"Wave4第三層核對=(1, 4, [1, 3, 4, 5], [1, 3, 4, 5], [1, 3, 4, 5], 是, equal)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；class精確等於全部相關元素且代表元不變；任兩classes相等或不交且不從抽樣relation分群。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["等價類"]},"variant":{"family_id":"qf-mod29-eqclass-equal-or-disjoint","seed":202610297083,"parameters":{"n":6,"pairs":[[0,0],[1,1],[1,3],[1,4],[1,5],[2,2],[3,1],[3,3],[3,4],[3,5],[4,1],[4,3],[4,4],[4,5],[5,1],[5,3],[5,4],[5,5]],"a":1,"b":4,"domain":"Complete_finite_equivalence_class","operation":"eqclass-equal-or-disjoint","semantic_template":"qf-mod29-eqclass-equal-or-disjoint/{\"a\":1,\"b\":4,\"n\":6,\"pairs\":[[0,0],[1,1],[1,3],[1,4],[1,5],[2,2],[3,1],[3,3],[3,4],[3,5],[4,1],[4,3],[4,4],[4,5],[5,1],[5,3],[5,4],[5,5]]}"}},"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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r07-equivalence-class-0084","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","language":"zh-Hant","source_mko_ids":["mko-equivalence-class"],"title_zh":"等價類","stem_zh":"同一完整載體上的已認證等價關係。操作=eqclass-equal-or-disjoint；資料={\"n\":6,\"pairs\":[[0,0],[0,3],[0,4],[0,5],[1,1],[1,2],[2,1],[2,2],[3,0],[3,3],[3,4],[3,5],[4,0],[4,3],[4,4],[4,5],[5,0],[5,3],[5,4],[5,5]],\"a\":2,\"b\":1}。請依 [a,b,class_a,class_b,intersection,equal_or_disjoint,classification] 欄序回答。","answer":{"kind":"expression","value":"(2, 1, [1, 2], [1, 2], [1, 2], 是, equal)","explanation_zh":"Wave4第三層核對=(2, 1, [1, 2], [1, 2], [1, 2], 是, equal)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；class精確等於全部相關元素且代表元不變；任兩classes相等或不交且不從抽樣relation分群。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["等價類"]},"variant":{"family_id":"qf-mod29-eqclass-equal-or-disjoint","seed":202610297084,"parameters":{"n":6,"pairs":[[0,0],[0,3],[0,4],[0,5],[1,1],[1,2],[2,1],[2,2],[3,0],[3,3],[3,4],[3,5],[4,0],[4,3],[4,4],[4,5],[5,0],[5,3],[5,4],[5,5]],"a":2,"b":1,"domain":"Complete_finite_equivalence_class","operation":"eqclass-equal-or-disjoint","semantic_template":"qf-mod29-eqclass-equal-or-disjoint/{\"a\":2,\"b\":1,\"n\":6,\"pairs\":[[0,0],[0,3],[0,4],[0,5],[1,1],[1,2],[2,1],[2,2],[3,0],[3,3],[3,4],[3,5],[4,0],[4,3],[4,4],[4,5],[5,0],[5,3],[5,4],[5,5]]}"}},"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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r07-equivalence-class-0085","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","language":"zh-Hant","source_mko_ids":["mko-equivalence-class"],"title_zh":"等價類","stem_zh":"同一完整載體上的已認證等價關係。操作=eqclass-equal-or-disjoint；資料={\"n\":6,\"pairs\":[[0,0],[0,1],[0,2],[1,0],[1,1],[1,2],[2,0],[2,1],[2,2],[3,3],[3,4],[3,5],[4,3],[4,4],[4,5],[5,3],[5,4],[5,5]],\"a\":3,\"b\":4}。請依 [a,b,class_a,class_b,intersection,equal_or_disjoint,classification] 欄序回答。","answer":{"kind":"expression","value":"(3, 4, [3, 4, 5], [3, 4, 5], [3, 4, 5], 是, equal)","explanation_zh":"Wave4第三層核對=(3, 4, [3, 4, 5], [3, 4, 5], [3, 4, 5], 是, equal)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；class精確等於全部相關元素且代表元不變；任兩classes相等或不交且不從抽樣relation分群。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["等價類"]},"variant":{"family_id":"qf-mod29-eqclass-equal-or-disjoint","seed":202610297085,"parameters":{"n":6,"pairs":[[0,0],[0,1],[0,2],[1,0],[1,1],[1,2],[2,0],[2,1],[2,2],[3,3],[3,4],[3,5],[4,3],[4,4],[4,5],[5,3],[5,4],[5,5]],"a":3,"b":4,"domain":"Complete_finite_equivalence_class","operation":"eqclass-equal-or-disjoint","semantic_template":"qf-mod29-eqclass-equal-or-disjoint/{\"a\":3,\"b\":4,\"n\":6,\"pairs\":[[0,0],[0,1],[0,2],[1,0],[1,1],[1,2],[2,0],[2,1],[2,2],[3,3],[3,4],[3,5],[4,3],[4,4],[4,5],[5,3],[5,4],[5,5]]}"}},"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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r07-equivalence-class-0086","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","language":"zh-Hant","source_mko_ids":["mko-equivalence-class"],"title_zh":"等價類","stem_zh":"同一完整載體上的已認證等價關係。操作=eqclass-equal-or-disjoint；資料={\"n\":6,\"pairs\":[[0,0],[1,1],[1,2],[2,1],[2,2],[3,3],[3,4],[3,5],[4,3],[4,4],[4,5],[5,3],[5,4],[5,5]],\"a\":4,\"b\":1}。請依 [a,b,class_a,class_b,intersection,equal_or_disjoint,classification] 欄序回答。","answer":{"kind":"expression","value":"(4, 1, [3, 4, 5], [1, 2], [], 是, disjoint)","explanation_zh":"Wave4第三層核對=(4, 1, [3, 4, 5], [1, 2], [], 是, disjoint)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；class精確等於全部相關元素且代表元不變；任兩classes相等或不交且不從抽樣relation分群。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["等價類"]},"variant":{"family_id":"qf-mod29-eqclass-equal-or-disjoint","seed":202610297086,"parameters":{"n":6,"pairs":[[0,0],[1,1],[1,2],[2,1],[2,2],[3,3],[3,4],[3,5],[4,3],[4,4],[4,5],[5,3],[5,4],[5,5]],"a":4,"b":1,"domain":"Complete_finite_equivalence_class","operation":"eqclass-equal-or-disjoint","semantic_template":"qf-mod29-eqclass-equal-or-disjoint/{\"a\":4,\"b\":1,\"n\":6,\"pairs\":[[0,0],[1,1],[1,2],[2,1],[2,2],[3,3],[3,4],[3,5],[4,3],[4,4],[4,5],[5,3],[5,4],[5,5]]}"}},"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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r07-equivalence-class-0087","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","language":"zh-Hant","source_mko_ids":["mko-equivalence-class"],"title_zh":"等價類","stem_zh":"同一完整載體上的已認證等價關係。操作=eqclass-equal-or-disjoint；資料={\"n\":6,\"pairs\":[[0,0],[0,3],[0,4],[0,5],[1,1],[2,2],[3,0],[3,3],[3,4],[3,5],[4,0],[4,3],[4,4],[4,5],[5,0],[5,3],[5,4],[5,5]],\"a\":5,\"b\":4}。請依 [a,b,class_a,class_b,intersection,equal_or_disjoint,classification] 欄序回答。","answer":{"kind":"expression","value":"(5, 4, [0, 3, 4, 5], [0, 3, 4, 5], [0, 3, 4, 5], 是, equal)","explanation_zh":"Wave4第三層核對=(5, 4, [0, 3, 4, 5], [0, 3, 4, 5], [0, 3, 4, 5], 是, equal)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；class精確等於全部相關元素且代表元不變；任兩classes相等或不交且不從抽樣relation分群。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["等價類"]},"variant":{"family_id":"qf-mod29-eqclass-equal-or-disjoint","seed":202610297087,"parameters":{"n":6,"pairs":[[0,0],[0,3],[0,4],[0,5],[1,1],[2,2],[3,0],[3,3],[3,4],[3,5],[4,0],[4,3],[4,4],[4,5],[5,0],[5,3],[5,4],[5,5]],"a":5,"b":4,"domain":"Complete_finite_equivalence_class","operation":"eqclass-equal-or-disjoint","semantic_template":"qf-mod29-eqclass-equal-or-disjoint/{\"a\":5,\"b\":4,\"n\":6,\"pairs\":[[0,0],[0,3],[0,4],[0,5],[1,1],[2,2],[3,0],[3,3],[3,4],[3,5],[4,0],[4,3],[4,4],[4,5],[5,0],[5,3],[5,4],[5,5]]}"}},"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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r07-equivalence-class-0088","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","language":"zh-Hant","source_mko_ids":["mko-equivalence-class"],"title_zh":"等價類","stem_zh":"同一完整載體上的已認證等價關係。操作=eqclass-equal-or-disjoint；資料={\"n\":6,\"pairs\":[[0,0],[0,2],[1,1],[2,0],[2,2],[3,3],[3,4],[3,5],[4,3],[4,4],[4,5],[5,3],[5,4],[5,5]],\"a\":0,\"b\":1}。請依 [a,b,class_a,class_b,intersection,equal_or_disjoint,classification] 欄序回答。","answer":{"kind":"expression","value":"(0, 1, [0, 2], [1], [], 是, disjoint)","explanation_zh":"Wave4第三層核對=(0, 1, [0, 2], [1], [], 是, disjoint)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；class精確等於全部相關元素且代表元不變；任兩classes相等或不交且不從抽樣relation分群。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["等價類"]},"variant":{"family_id":"qf-mod29-eqclass-equal-or-disjoint","seed":202610297088,"parameters":{"n":6,"pairs":[[0,0],[0,2],[1,1],[2,0],[2,2],[3,3],[3,4],[3,5],[4,3],[4,4],[4,5],[5,3],[5,4],[5,5]],"a":0,"b":1,"domain":"Complete_finite_equivalence_class","operation":"eqclass-equal-or-disjoint","semantic_template":"qf-mod29-eqclass-equal-or-disjoint/{\"a\":0,\"b\":1,\"n\":6,\"pairs\":[[0,0],[0,2],[1,1],[2,0],[2,2],[3,3],[3,4],[3,5],[4,3],[4,4],[4,5],[5,3],[5,4],[5,5]]}"}},"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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r07-equivalence-class-0089","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","language":"zh-Hant","source_mko_ids":["mko-equivalence-class"],"title_zh":"等價類","stem_zh":"同一完整載體上的已認證等價關係。操作=eqclass-equal-or-disjoint；資料={\"n\":6,\"pairs\":[[0,0],[0,1],[1,0],[1,1],[2,2],[3,3],[3,4],[3,5],[4,3],[4,4],[4,5],[5,3],[5,4],[5,5]],\"a\":1,\"b\":4}。請依 [a,b,class_a,class_b,intersection,equal_or_disjoint,classification] 欄序回答。","answer":{"kind":"expression","value":"(1, 4, [0, 1], [3, 4, 5], [], 是, disjoint)","explanation_zh":"Wave4第三層核對=(1, 4, [0, 1], [3, 4, 5], [], 是, disjoint)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；class精確等於全部相關元素且代表元不變；任兩classes相等或不交且不從抽樣relation分群。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["等價類"]},"variant":{"family_id":"qf-mod29-eqclass-equal-or-disjoint","seed":202610297089,"parameters":{"n":6,"pairs":[[0,0],[0,1],[1,0],[1,1],[2,2],[3,3],[3,4],[3,5],[4,3],[4,4],[4,5],[5,3],[5,4],[5,5]],"a":1,"b":4,"domain":"Complete_finite_equivalence_class","operation":"eqclass-equal-or-disjoint","semantic_template":"qf-mod29-eqclass-equal-or-disjoint/{\"a\":1,\"b\":4,\"n\":6,\"pairs\":[[0,0],[0,1],[1,0],[1,1],[2,2],[3,3],[3,4],[3,5],[4,3],[4,4],[4,5],[5,3],[5,4],[5,5]]}"}},"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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r07-equivalence-class-0090","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","language":"zh-Hant","source_mko_ids":["mko-equivalence-class"],"title_zh":"等價類","stem_zh":"同一完整載體上的已認證等價關係。操作=eqclass-equal-or-disjoint；資料={\"n\":6,\"pairs\":[[0,0],[0,1],[0,2],[0,4],[0,5],[1,0],[1,1],[1,2],[1,4],[1,5],[2,0],[2,1],[2,2],[2,4],[2,5],[3,3],[4,0],[4,1],[4,2],[4,4],[4,5],[5,0],[5,1],[5,2],[5,4],[5,5]],\"a\":2,\"b\":1}。請依 [a,b,class_a,class_b,intersection,equal_or_disjoint,classification] 欄序回答。","answer":{"kind":"expression","value":"(2, 1, [0, 1, 2, 4, 5], [0, 1, 2, 4, 5], [0, 1, 2, 4, 5], 是, equal)","explanation_zh":"Wave4第三層核對=(2, 1, [0, 1, 2, 4, 5], [0, 1, 2, 4, 5], [0, 1, 2, 4, 5], 是, equal)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；class精確等於全部相關元素且代表元不變；任兩classes相等或不交且不從抽樣relation分群。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["等價類"]},"variant":{"family_id":"qf-mod29-eqclass-equal-or-disjoint","seed":202610297090,"parameters":{"n":6,"pairs":[[0,0],[0,1],[0,2],[0,4],[0,5],[1,0],[1,1],[1,2],[1,4],[1,5],[2,0],[2,1],[2,2],[2,4],[2,5],[3,3],[4,0],[4,1],[4,2],[4,4],[4,5],[5,0],[5,1],[5,2],[5,4],[5,5]],"a":2,"b":1,"domain":"Complete_finite_equivalence_class","operation":"eqclass-equal-or-disjoint","semantic_template":"qf-mod29-eqclass-equal-or-disjoint/{\"a\":2,\"b\":1,\"n\":6,\"pairs\":[[0,0],[0,1],[0,2],[0,4],[0,5],[1,0],[1,1],[1,2],[1,4],[1,5],[2,0],[2,1],[2,2],[2,4],[2,5],[3,3],[4,0],[4,1],[4,2],[4,4],[4,5],[5,0],[5,1],[5,2],[5,4],[5,5]]}"}},"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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r07-equivalence-class-0091","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","language":"zh-Hant","source_mko_ids":["mko-equivalence-class"],"title_zh":"等價類","stem_zh":"同一完整載體上的已認證等價關係。操作=eqclass-equal-or-disjoint；資料={\"n\":6,\"pairs\":[[0,0],[0,3],[1,1],[1,2],[1,4],[1,5],[2,1],[2,2],[2,4],[2,5],[3,0],[3,3],[4,1],[4,2],[4,4],[4,5],[5,1],[5,2],[5,4],[5,5]],\"a\":3,\"b\":4}。請依 [a,b,class_a,class_b,intersection,equal_or_disjoint,classification] 欄序回答。","answer":{"kind":"expression","value":"(3, 4, [0, 3], [1, 2, 4, 5], [], 是, disjoint)","explanation_zh":"Wave4第三層核對=(3, 4, [0, 3], [1, 2, 4, 5], [], 是, disjoint)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；class精確等於全部相關元素且代表元不變；任兩classes相等或不交且不從抽樣relation分群。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["等價類"]},"variant":{"family_id":"qf-mod29-eqclass-equal-or-disjoint","seed":202610297091,"parameters":{"n":6,"pairs":[[0,0],[0,3],[1,1],[1,2],[1,4],[1,5],[2,1],[2,2],[2,4],[2,5],[3,0],[3,3],[4,1],[4,2],[4,4],[4,5],[5,1],[5,2],[5,4],[5,5]],"a":3,"b":4,"domain":"Complete_finite_equivalence_class","operation":"eqclass-equal-or-disjoint","semantic_template":"qf-mod29-eqclass-equal-or-disjoint/{\"a\":3,\"b\":4,\"n\":6,\"pairs\":[[0,0],[0,3],[1,1],[1,2],[1,4],[1,5],[2,1],[2,2],[2,4],[2,5],[3,0],[3,3],[4,1],[4,2],[4,4],[4,5],[5,1],[5,2],[5,4],[5,5]]}"}},"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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r07-equivalence-class-0092","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","language":"zh-Hant","source_mko_ids":["mko-equivalence-class"],"title_zh":"等價類","stem_zh":"同一完整載體上的已認證等價關係。操作=eqclass-equal-or-disjoint；資料={\"n\":6,\"pairs\":[[0,0],[1,1],[1,2],[1,4],[1,5],[2,1],[2,2],[2,4],[2,5],[3,3],[4,1],[4,2],[4,4],[4,5],[5,1],[5,2],[5,4],[5,5]],\"a\":4,\"b\":1}。請依 [a,b,class_a,class_b,intersection,equal_or_disjoint,classification] 欄序回答。","answer":{"kind":"expression","value":"(4, 1, [1, 2, 4, 5], [1, 2, 4, 5], [1, 2, 4, 5], 是, equal)","explanation_zh":"Wave4第三層核對=(4, 1, [1, 2, 4, 5], [1, 2, 4, 5], [1, 2, 4, 5], 是, equal)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；class精確等於全部相關元素且代表元不變；任兩classes相等或不交且不從抽樣relation分群。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["等價類"]},"variant":{"family_id":"qf-mod29-eqclass-equal-or-disjoint","seed":202610297092,"parameters":{"n":6,"pairs":[[0,0],[1,1],[1,2],[1,4],[1,5],[2,1],[2,2],[2,4],[2,5],[3,3],[4,1],[4,2],[4,4],[4,5],[5,1],[5,2],[5,4],[5,5]],"a":4,"b":1,"domain":"Complete_finite_equivalence_class","operation":"eqclass-equal-or-disjoint","semantic_template":"qf-mod29-eqclass-equal-or-disjoint/{\"a\":4,\"b\":1,\"n\":6,\"pairs\":[[0,0],[1,1],[1,2],[1,4],[1,5],[2,1],[2,2],[2,4],[2,5],[3,3],[4,1],[4,2],[4,4],[4,5],[5,1],[5,2],[5,4],[5,5]]}"}},"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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r07-equivalence-class-0093","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","language":"zh-Hant","source_mko_ids":["mko-equivalence-class"],"title_zh":"等價類","stem_zh":"同一完整載體上的已認證等價關係。操作=eqclass-equal-or-disjoint；資料={\"n\":6,\"pairs\":[[0,0],[0,2],[0,4],[0,5],[1,1],[1,3],[2,0],[2,2],[2,4],[2,5],[3,1],[3,3],[4,0],[4,2],[4,4],[4,5],[5,0],[5,2],[5,4],[5,5]],\"a\":5,\"b\":4}。請依 [a,b,class_a,class_b,intersection,equal_or_disjoint,classification] 欄序回答。","answer":{"kind":"expression","value":"(5, 4, [0, 2, 4, 5], [0, 2, 4, 5], [0, 2, 4, 5], 是, equal)","explanation_zh":"Wave4第三層核對=(5, 4, [0, 2, 4, 5], [0, 2, 4, 5], [0, 2, 4, 5], 是, equal)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；class精確等於全部相關元素且代表元不變；任兩classes相等或不交且不從抽樣relation分群。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["等價類"]},"variant":{"family_id":"qf-mod29-eqclass-equal-or-disjoint","seed":202610297093,"parameters":{"n":6,"pairs":[[0,0],[0,2],[0,4],[0,5],[1,1],[1,3],[2,0],[2,2],[2,4],[2,5],[3,1],[3,3],[4,0],[4,2],[4,4],[4,5],[5,0],[5,2],[5,4],[5,5]],"a":5,"b":4,"domain":"Complete_finite_equivalence_class","operation":"eqclass-equal-or-disjoint","semantic_template":"qf-mod29-eqclass-equal-or-disjoint/{\"a\":5,\"b\":4,\"n\":6,\"pairs\":[[0,0],[0,2],[0,4],[0,5],[1,1],[1,3],[2,0],[2,2],[2,4],[2,5],[3,1],[3,3],[4,0],[4,2],[4,4],[4,5],[5,0],[5,2],[5,4],[5,5]]}"}},"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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r07-equivalence-class-0094","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","language":"zh-Hant","source_mko_ids":["mko-equivalence-class"],"title_zh":"等價類","stem_zh":"同一完整載體上的已認證等價關係。操作=eqclass-equal-or-disjoint；資料={\"n\":6,\"pairs\":[[0,0],[0,1],[0,3],[1,0],[1,1],[1,3],[2,2],[2,4],[2,5],[3,0],[3,1],[3,3],[4,2],[4,4],[4,5],[5,2],[5,4],[5,5]],\"a\":0,\"b\":1}。請依 [a,b,class_a,class_b,intersection,equal_or_disjoint,classification] 欄序回答。","answer":{"kind":"expression","value":"(0, 1, [0, 1, 3], [0, 1, 3], [0, 1, 3], 是, equal)","explanation_zh":"Wave4第三層核對=(0, 1, [0, 1, 3], [0, 1, 3], [0, 1, 3], 是, equal)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；class精確等於全部相關元素且代表元不變；任兩classes相等或不交且不從抽樣relation分群。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["等價類"]},"variant":{"family_id":"qf-mod29-eqclass-equal-or-disjoint","seed":202610297094,"parameters":{"n":6,"pairs":[[0,0],[0,1],[0,3],[1,0],[1,1],[1,3],[2,2],[2,4],[2,5],[3,0],[3,1],[3,3],[4,2],[4,4],[4,5],[5,2],[5,4],[5,5]],"a":0,"b":1,"domain":"Complete_finite_equivalence_class","operation":"eqclass-equal-or-disjoint","semantic_template":"qf-mod29-eqclass-equal-or-disjoint/{\"a\":0,\"b\":1,\"n\":6,\"pairs\":[[0,0],[0,1],[0,3],[1,0],[1,1],[1,3],[2,2],[2,4],[2,5],[3,0],[3,1],[3,3],[4,2],[4,4],[4,5],[5,2],[5,4],[5,5]]}"}},"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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r07-equivalence-class-0095","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","language":"zh-Hant","source_mko_ids":["mko-equivalence-class"],"title_zh":"等價類","stem_zh":"同一完整載體上的已認證等價關係。操作=eqclass-equal-or-disjoint；資料={\"n\":6,\"pairs\":[[0,0],[1,1],[1,3],[2,2],[2,4],[2,5],[3,1],[3,3],[4,2],[4,4],[4,5],[5,2],[5,4],[5,5]],\"a\":1,\"b\":4}。請依 [a,b,class_a,class_b,intersection,equal_or_disjoint,classification] 欄序回答。","answer":{"kind":"expression","value":"(1, 4, [1, 3], [2, 4, 5], [], 是, disjoint)","explanation_zh":"Wave4第三層核對=(1, 4, [1, 3], [2, 4, 5], [], 是, disjoint)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；class精確等於全部相關元素且代表元不變；任兩classes相等或不交且不從抽樣relation分群。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["等價類"]},"variant":{"family_id":"qf-mod29-eqclass-equal-or-disjoint","seed":202610297095,"parameters":{"n":6,"pairs":[[0,0],[1,1],[1,3],[2,2],[2,4],[2,5],[3,1],[3,3],[4,2],[4,4],[4,5],[5,2],[5,4],[5,5]],"a":1,"b":4,"domain":"Complete_finite_equivalence_class","operation":"eqclass-equal-or-disjoint","semantic_template":"qf-mod29-eqclass-equal-or-disjoint/{\"a\":1,\"b\":4,\"n\":6,\"pairs\":[[0,0],[1,1],[1,3],[2,2],[2,4],[2,5],[3,1],[3,3],[4,2],[4,4],[4,5],[5,2],[5,4],[5,5]]}"}},"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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r07-equivalence-class-0096","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","language":"zh-Hant","source_mko_ids":["mko-equivalence-class"],"title_zh":"等價類","stem_zh":"同一完整載體上的已認證等價關係。操作=eqclass-equal-or-disjoint；資料={\"n\":6,\"pairs\":[[0,0],[0,2],[0,4],[0,5],[1,1],[2,0],[2,2],[2,4],[2,5],[3,3],[4,0],[4,2],[4,4],[4,5],[5,0],[5,2],[5,4],[5,5]],\"a\":2,\"b\":1}。請依 [a,b,class_a,class_b,intersection,equal_or_disjoint,classification] 欄序回答。","answer":{"kind":"expression","value":"(2, 1, [0, 2, 4, 5], [1], [], 是, disjoint)","explanation_zh":"Wave4第三層核對=(2, 1, [0, 2, 4, 5], [1], [], 是, disjoint)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；class精確等於全部相關元素且代表元不變；任兩classes相等或不交且不從抽樣relation分群。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["等價類"]},"variant":{"family_id":"qf-mod29-eqclass-equal-or-disjoint","seed":202610297096,"parameters":{"n":6,"pairs":[[0,0],[0,2],[0,4],[0,5],[1,1],[2,0],[2,2],[2,4],[2,5],[3,3],[4,0],[4,2],[4,4],[4,5],[5,0],[5,2],[5,4],[5,5]],"a":2,"b":1,"domain":"Complete_finite_equivalence_class","operation":"eqclass-equal-or-disjoint","semantic_template":"qf-mod29-eqclass-equal-or-disjoint/{\"a\":2,\"b\":1,\"n\":6,\"pairs\":[[0,0],[0,2],[0,4],[0,5],[1,1],[2,0],[2,2],[2,4],[2,5],[3,3],[4,0],[4,2],[4,4],[4,5],[5,0],[5,2],[5,4],[5,5]]}"}},"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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r07-equivalence-class-0097","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","language":"zh-Hant","source_mko_ids":["mko-equivalence-class"],"title_zh":"等價類","stem_zh":"同一完整載體上的已認證等價關係。操作=eqclass-equal-or-disjoint；資料={\"n\":6,\"pairs\":[[0,0],[0,3],[1,1],[2,2],[2,4],[2,5],[3,0],[3,3],[4,2],[4,4],[4,5],[5,2],[5,4],[5,5]],\"a\":3,\"b\":4}。請依 [a,b,class_a,class_b,intersection,equal_or_disjoint,classification] 欄序回答。","answer":{"kind":"expression","value":"(3, 4, [0, 3], [2, 4, 5], [], 是, disjoint)","explanation_zh":"Wave4第三層核對=(3, 4, [0, 3], [2, 4, 5], [], 是, disjoint)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；class精確等於全部相關元素且代表元不變；任兩classes相等或不交且不從抽樣relation分群。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["等價類"]},"variant":{"family_id":"qf-mod29-eqclass-equal-or-disjoint","seed":202610297097,"parameters":{"n":6,"pairs":[[0,0],[0,3],[1,1],[2,2],[2,4],[2,5],[3,0],[3,3],[4,2],[4,4],[4,5],[5,2],[5,4],[5,5]],"a":3,"b":4,"domain":"Complete_finite_equivalence_class","operation":"eqclass-equal-or-disjoint","semantic_template":"qf-mod29-eqclass-equal-or-disjoint/{\"a\":3,\"b\":4,\"n\":6,\"pairs\":[[0,0],[0,3],[1,1],[2,2],[2,4],[2,5],[3,0],[3,3],[4,2],[4,4],[4,5],[5,2],[5,4],[5,5]]}"}},"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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r07-equivalence-class-0098","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","language":"zh-Hant","source_mko_ids":["mko-equivalence-class"],"title_zh":"等價類","stem_zh":"同一完整載體上的已認證等價關係。操作=eqclass-equal-or-disjoint；資料={\"n\":6,\"pairs\":[[0,0],[0,1],[1,0],[1,1],[2,2],[2,4],[2,5],[3,3],[4,2],[4,4],[4,5],[5,2],[5,4],[5,5]],\"a\":4,\"b\":1}。請依 [a,b,class_a,class_b,intersection,equal_or_disjoint,classification] 欄序回答。","answer":{"kind":"expression","value":"(4, 1, [2, 4, 5], [0, 1], [], 是, disjoint)","explanation_zh":"Wave4第三層核對=(4, 1, [2, 4, 5], [0, 1], [], 是, disjoint)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；class精確等於全部相關元素且代表元不變；任兩classes相等或不交且不從抽樣relation分群。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["等價類"]},"variant":{"family_id":"qf-mod29-eqclass-equal-or-disjoint","seed":202610297098,"parameters":{"n":6,"pairs":[[0,0],[0,1],[1,0],[1,1],[2,2],[2,4],[2,5],[3,3],[4,2],[4,4],[4,5],[5,2],[5,4],[5,5]],"a":4,"b":1,"domain":"Complete_finite_equivalence_class","operation":"eqclass-equal-or-disjoint","semantic_template":"qf-mod29-eqclass-equal-or-disjoint/{\"a\":4,\"b\":1,\"n\":6,\"pairs\":[[0,0],[0,1],[1,0],[1,1],[2,2],[2,4],[2,5],[3,3],[4,2],[4,4],[4,5],[5,2],[5,4],[5,5]]}"}},"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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r07-equivalence-class-0099","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","language":"zh-Hant","source_mko_ids":["mko-equivalence-class"],"title_zh":"等價類","stem_zh":"同一完整載體上的已認證等價關係。操作=eqclass-equal-or-disjoint；資料={\"n\":6,\"pairs\":[[0,0],[0,1],[0,4],[0,5],[1,0],[1,1],[1,4],[1,5],[2,2],[2,3],[3,2],[3,3],[4,0],[4,1],[4,4],[4,5],[5,0],[5,1],[5,4],[5,5]],\"a\":5,\"b\":4}。請依 [a,b,class_a,class_b,intersection,equal_or_disjoint,classification] 欄序回答。","answer":{"kind":"expression","value":"(5, 4, [0, 1, 4, 5], [0, 1, 4, 5], [0, 1, 4, 5], 是, equal)","explanation_zh":"Wave4第三層核對=(5, 4, [0, 1, 4, 5], [0, 1, 4, 5], [0, 1, 4, 5], 是, equal)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；class精確等於全部相關元素且代表元不變；任兩classes相等或不交且不從抽樣relation分群。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["等價類"]},"variant":{"family_id":"qf-mod29-eqclass-equal-or-disjoint","seed":202610297099,"parameters":{"n":6,"pairs":[[0,0],[0,1],[0,4],[0,5],[1,0],[1,1],[1,4],[1,5],[2,2],[2,3],[3,2],[3,3],[4,0],[4,1],[4,4],[4,5],[5,0],[5,1],[5,4],[5,5]],"a":5,"b":4,"domain":"Complete_finite_equivalence_class","operation":"eqclass-equal-or-disjoint","semantic_template":"qf-mod29-eqclass-equal-or-disjoint/{\"a\":5,\"b\":4,\"n\":6,\"pairs\":[[0,0],[0,1],[0,4],[0,5],[1,0],[1,1],[1,4],[1,5],[2,2],[2,3],[3,2],[3,3],[4,0],[4,1],[4,4],[4,5],[5,0],[5,1],[5,4],[5,5]]}"}},"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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r07-equivalence-class-0100","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","language":"zh-Hant","source_mko_ids":["mko-equivalence-class"],"title_zh":"等價類","stem_zh":"同一完整載體上的已認證等價關係。操作=eqclass-equal-or-disjoint；資料={\"n\":6,\"pairs\":[[0,0],[0,2],[0,3],[1,1],[1,4],[1,5],[2,0],[2,2],[2,3],[3,0],[3,2],[3,3],[4,1],[4,4],[4,5],[5,1],[5,4],[5,5]],\"a\":0,\"b\":1}。請依 [a,b,class_a,class_b,intersection,equal_or_disjoint,classification] 欄序回答。","answer":{"kind":"expression","value":"(0, 1, [0, 2, 3], [1, 4, 5], [], 是, disjoint)","explanation_zh":"Wave4第三層核對=(0, 1, [0, 2, 3], [1, 4, 5], [], 是, disjoint)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；class精確等於全部相關元素且代表元不變；任兩classes相等或不交且不從抽樣relation分群。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["等價類"]},"variant":{"family_id":"qf-mod29-eqclass-equal-or-disjoint","seed":202610297100,"parameters":{"n":6,"pairs":[[0,0],[0,2],[0,3],[1,1],[1,4],[1,5],[2,0],[2,2],[2,3],[3,0],[3,2],[3,3],[4,1],[4,4],[4,5],[5,1],[5,4],[5,5]],"a":0,"b":1,"domain":"Complete_finite_equivalence_class","operation":"eqclass-equal-or-disjoint","semantic_template":"qf-mod29-eqclass-equal-or-disjoint/{\"a\":0,\"b\":1,\"n\":6,\"pairs\":[[0,0],[0,2],[0,3],[1,1],[1,4],[1,5],[2,0],[2,2],[2,3],[3,0],[3,2],[3,3],[4,1],[4,4],[4,5],[5,1],[5,4],[5,5]]}"}},"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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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"}}