{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r01-matrix-rank-0001","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix-rank"],"title_zh":"矩陣秩","stem_zh":"2×2或3×3完整矩陣與精確rank-preserving消去。操作=rank-exact-rref；資料={\"matrix\":[[1,0],[0,1]]}。請依 [matrix,rref,pivot_columns,rank,certificate] 欄序回答。","answer":{"kind":"expression","value":"([[1, 0], [0, 1]], [[1, 0], [0, 1]], [0, 1], 2, rank_preserving_exact_row_operations)","explanation_zh":"Wave5第三層核對=([[1, 0], [0, 1]], [[1, 0], [0, 1]], [0, 1], 2, rank_preserving_exact_row_operations)；依完整明示資料重算；非零minor只給下界且上界需完整RREF或全部較大minor；det等於零只排除full rank；row-rank等於column-rank依定理。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["矩陣秩"]},"variant":{"family_id":"qf-mod34-rank-exact-rref","seed":202611031001,"parameters":{"matrix":[[1,0],[0,1]],"domain":"Exact_square_matrix_rank","operation":"rank-exact-rref","semantic_template":"qf-mod34-rank-exact-rref/{\"matrix\":[[1,0],[0,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean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-20261103-r01-matrix-rank-0002","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix-rank"],"title_zh":"矩陣秩","stem_zh":"2×2或3×3完整矩陣與精確rank-preserving消去。操作=rank-exact-rref；資料={\"matrix\":[[1,1],[0,2]]}。請依 [matrix,rref,pivot_columns,rank,certificate] 欄序回答。","answer":{"kind":"expression","value":"([[1, 1], [0, 2]], [[1, 0], [0, 1]], [0, 1], 2, rank_preserving_exact_row_operations)","explanation_zh":"Wave5第三層核對=([[1, 1], [0, 2]], [[1, 0], [0, 1]], [0, 1], 2, rank_preserving_exact_row_operations)；依完整明示資料重算；非零minor只給下界且上界需完整RREF或全部較大minor；det等於零只排除full rank；row-rank等於column-rank依定理。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["矩陣秩"]},"variant":{"family_id":"qf-mod34-rank-exact-rref","seed":202611031002,"parameters":{"matrix":[[1,1],[0,2]],"domain":"Exact_square_matrix_rank","operation":"rank-exact-rref","semantic_template":"qf-mod34-rank-exact-rref/{\"matrix\":[[1,1],[0,2]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean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-20261103-r01-matrix-rank-0003","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix-rank"],"title_zh":"矩陣秩","stem_zh":"2×2或3×3完整矩陣與精確rank-preserving消去。操作=rank-exact-rref；資料={\"matrix\":[[1,2],[0,3]]}。請依 [matrix,rref,pivot_columns,rank,certificate] 欄序回答。","answer":{"kind":"expression","value":"([[1, 2], [0, 3]], [[1, 0], [0, 1]], [0, 1], 2, rank_preserving_exact_row_operations)","explanation_zh":"Wave5第三層核對=([[1, 2], [0, 3]], [[1, 0], [0, 1]], [0, 1], 2, rank_preserving_exact_row_operations)；依完整明示資料重算；非零minor只給下界且上界需完整RREF或全部較大minor；det等於零只排除full rank；row-rank等於column-rank依定理。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["矩陣秩"]},"variant":{"family_id":"qf-mod34-rank-exact-rref","seed":202611031003,"parameters":{"matrix":[[1,2],[0,3]],"domain":"Exact_square_matrix_rank","operation":"rank-exact-rref","semantic_template":"qf-mod34-rank-exact-rref/{\"matrix\":[[1,2],[0,3]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean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-20261103-r01-matrix-rank-0004","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix-rank"],"title_zh":"矩陣秩","stem_zh":"2×2或3×3完整矩陣與精確rank-preserving消去。操作=rank-exact-rref；資料={\"matrix\":[[1,3],[0,4]]}。請依 [matrix,rref,pivot_columns,rank,certificate] 欄序回答。","answer":{"kind":"expression","value":"([[1, 3], [0, 4]], [[1, 0], [0, 1]], [0, 1], 2, rank_preserving_exact_row_operations)","explanation_zh":"Wave5第三層核對=([[1, 3], [0, 4]], [[1, 0], [0, 1]], [0, 1], 2, rank_preserving_exact_row_operations)；依完整明示資料重算；非零minor只給下界且上界需完整RREF或全部較大minor；det等於零只排除full rank；row-rank等於column-rank依定理。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["矩陣秩"]},"variant":{"family_id":"qf-mod34-rank-exact-rref","seed":202611031004,"parameters":{"matrix":[[1,3],[0,4]],"domain":"Exact_square_matrix_rank","operation":"rank-exact-rref","semantic_template":"qf-mod34-rank-exact-rref/{\"matrix\":[[1,3],[0,4]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean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-20261103-r01-matrix-rank-0005","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix-rank"],"title_zh":"矩陣秩","stem_zh":"2×2或3×3完整矩陣與精確rank-preserving消去。操作=rank-exact-rref；資料={\"matrix\":[[1,4],[0,5]]}。請依 [matrix,rref,pivot_columns,rank,certificate] 欄序回答。","answer":{"kind":"expression","value":"([[1, 4], [0, 5]], [[1, 0], [0, 1]], [0, 1], 2, rank_preserving_exact_row_operations)","explanation_zh":"Wave5第三層核對=([[1, 4], [0, 5]], [[1, 0], [0, 1]], [0, 1], 2, rank_preserving_exact_row_operations)；依完整明示資料重算；非零minor只給下界且上界需完整RREF或全部較大minor；det等於零只排除full rank；row-rank等於column-rank依定理。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["矩陣秩"]},"variant":{"family_id":"qf-mod34-rank-exact-rref","seed":202611031005,"parameters":{"matrix":[[1,4],[0,5]],"domain":"Exact_square_matrix_rank","operation":"rank-exact-rref","semantic_template":"qf-mod34-rank-exact-rref/{\"matrix\":[[1,4],[0,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean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-20261103-r01-matrix-rank-0006","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix-rank"],"title_zh":"矩陣秩","stem_zh":"2×2或3×3完整矩陣與精確rank-preserving消去。操作=rank-exact-rref；資料={\"matrix\":[[1,5],[0,6]]}。請依 [matrix,rref,pivot_columns,rank,certificate] 欄序回答。","answer":{"kind":"expression","value":"([[1, 5], [0, 6]], [[1, 0], [0, 1]], [0, 1], 2, rank_preserving_exact_row_operations)","explanation_zh":"Wave5第三層核對=([[1, 5], [0, 6]], [[1, 0], [0, 1]], [0, 1], 2, rank_preserving_exact_row_operations)；依完整明示資料重算；非零minor只給下界且上界需完整RREF或全部較大minor；det等於零只排除full rank；row-rank等於column-rank依定理。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["矩陣秩"]},"variant":{"family_id":"qf-mod34-rank-exact-rref","seed":202611031006,"parameters":{"matrix":[[1,5],[0,6]],"domain":"Exact_square_matrix_rank","operation":"rank-exact-rref","semantic_template":"qf-mod34-rank-exact-rref/{\"matrix\":[[1,5],[0,6]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean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-20261103-r01-matrix-rank-0007","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix-rank"],"title_zh":"矩陣秩","stem_zh":"2×2或3×3完整矩陣與精確rank-preserving消去。操作=rank-exact-rref；資料={\"matrix\":[[1,6],[0,7]]}。請依 [matrix,rref,pivot_columns,rank,certificate] 欄序回答。","answer":{"kind":"expression","value":"([[1, 6], [0, 7]], [[1, 0], [0, 1]], [0, 1], 2, rank_preserving_exact_row_operations)","explanation_zh":"Wave5第三層核對=([[1, 6], [0, 7]], [[1, 0], [0, 1]], [0, 1], 2, rank_preserving_exact_row_operations)；依完整明示資料重算；非零minor只給下界且上界需完整RREF或全部較大minor；det等於零只排除full rank；row-rank等於column-rank依定理。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["矩陣秩"]},"variant":{"family_id":"qf-mod34-rank-exact-rref","seed":202611031007,"parameters":{"matrix":[[1,6],[0,7]],"domain":"Exact_square_matrix_rank","operation":"rank-exact-rref","semantic_template":"qf-mod34-rank-exact-rref/{\"matrix\":[[1,6],[0,7]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean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-20261103-r01-matrix-rank-0008","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix-rank"],"title_zh":"矩陣秩","stem_zh":"2×2或3×3完整矩陣與精確rank-preserving消去。操作=rank-exact-rref；資料={\"matrix\":[[1,7],[0,8]]}。請依 [matrix,rref,pivot_columns,rank,certificate] 欄序回答。","answer":{"kind":"expression","value":"([[1, 7], [0, 8]], [[1, 0], [0, 1]], [0, 1], 2, rank_preserving_exact_row_operations)","explanation_zh":"Wave5第三層核對=([[1, 7], [0, 8]], [[1, 0], [0, 1]], [0, 1], 2, rank_preserving_exact_row_operations)；依完整明示資料重算；非零minor只給下界且上界需完整RREF或全部較大minor；det等於零只排除full rank；row-rank等於column-rank依定理。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["矩陣秩"]},"variant":{"family_id":"qf-mod34-rank-exact-rref","seed":202611031008,"parameters":{"matrix":[[1,7],[0,8]],"domain":"Exact_square_matrix_rank","operation":"rank-exact-rref","semantic_template":"qf-mod34-rank-exact-rref/{\"matrix\":[[1,7],[0,8]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean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-20261103-r01-matrix-rank-0009","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix-rank"],"title_zh":"矩陣秩","stem_zh":"2×2或3×3完整矩陣與精確rank-preserving消去。操作=rank-exact-rref；資料={\"matrix\":[[1,8],[0,9]]}。請依 [matrix,rref,pivot_columns,rank,certificate] 欄序回答。","answer":{"kind":"expression","value":"([[1, 8], [0, 9]], [[1, 0], [0, 1]], [0, 1], 2, rank_preserving_exact_row_operations)","explanation_zh":"Wave5第三層核對=([[1, 8], [0, 9]], [[1, 0], [0, 1]], [0, 1], 2, rank_preserving_exact_row_operations)；依完整明示資料重算；非零minor只給下界且上界需完整RREF或全部較大minor；det等於零只排除full rank；row-rank等於column-rank依定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["矩陣秩"]},"variant":{"family_id":"qf-mod34-rank-exact-rref","seed":202611031009,"parameters":{"matrix":[[1,8],[0,9]],"domain":"Exact_square_matrix_rank","operation":"rank-exact-rref","semantic_template":"qf-mod34-rank-exact-rref/{\"matrix\":[[1,8],[0,9]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean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-20261103-r01-matrix-rank-0010","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix-rank"],"title_zh":"矩陣秩","stem_zh":"2×2或3×3完整矩陣與精確rank-preserving消去。操作=rank-exact-rref；資料={\"matrix\":[[1,9],[0,10]]}。請依 [matrix,rref,pivot_columns,rank,certificate] 欄序回答。","answer":{"kind":"expression","value":"([[1, 9], [0, 10]], [[1, 0], [0, 1]], [0, 1], 2, rank_preserving_exact_row_operations)","explanation_zh":"Wave5第三層核對=([[1, 9], [0, 10]], [[1, 0], [0, 1]], [0, 1], 2, rank_preserving_exact_row_operations)；依完整明示資料重算；非零minor只給下界且上界需完整RREF或全部較大minor；det等於零只排除full rank；row-rank等於column-rank依定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["矩陣秩"]},"variant":{"family_id":"qf-mod34-rank-exact-rref","seed":202611031010,"parameters":{"matrix":[[1,9],[0,10]],"domain":"Exact_square_matrix_rank","operation":"rank-exact-rref","semantic_template":"qf-mod34-rank-exact-rref/{\"matrix\":[[1,9],[0,10]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean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-20261103-r01-matrix-rank-0011","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix-rank"],"title_zh":"矩陣秩","stem_zh":"2×2或3×3完整矩陣與精確rank-preserving消去。操作=rank-exact-rref；資料={\"matrix\":[[1,10],[0,11]]}。請依 [matrix,rref,pivot_columns,rank,certificate] 欄序回答。","answer":{"kind":"expression","value":"([[1, 10], [0, 11]], [[1, 0], [0, 1]], [0, 1], 2, rank_preserving_exact_row_operations)","explanation_zh":"Wave5第三層核對=([[1, 10], [0, 11]], [[1, 0], [0, 1]], [0, 1], 2, rank_preserving_exact_row_operations)；依完整明示資料重算；非零minor只給下界且上界需完整RREF或全部較大minor；det等於零只排除full rank；row-rank等於column-rank依定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["矩陣秩"]},"variant":{"family_id":"qf-mod34-rank-exact-rref","seed":202611031011,"parameters":{"matrix":[[1,10],[0,11]],"domain":"Exact_square_matrix_rank","operation":"rank-exact-rref","semantic_template":"qf-mod34-rank-exact-rref/{\"matrix\":[[1,10],[0,11]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean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-20261103-r01-matrix-rank-0012","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix-rank"],"title_zh":"矩陣秩","stem_zh":"2×2或3×3完整矩陣與精確rank-preserving消去。操作=rank-exact-rref；資料={\"matrix\":[[1,11],[0,12]]}。請依 [matrix,rref,pivot_columns,rank,certificate] 欄序回答。","answer":{"kind":"expression","value":"([[1, 11], [0, 12]], [[1, 0], [0, 1]], [0, 1], 2, rank_preserving_exact_row_operations)","explanation_zh":"Wave5第三層核對=([[1, 11], [0, 12]], [[1, 0], [0, 1]], [0, 1], 2, rank_preserving_exact_row_operations)；依完整明示資料重算；非零minor只給下界且上界需完整RREF或全部較大minor；det等於零只排除full rank；row-rank等於column-rank依定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["矩陣秩"]},"variant":{"family_id":"qf-mod34-rank-exact-rref","seed":202611031012,"parameters":{"matrix":[[1,11],[0,12]],"domain":"Exact_square_matrix_rank","operation":"rank-exact-rref","semantic_template":"qf-mod34-rank-exact-rref/{\"matrix\":[[1,11],[0,12]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean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-20261103-r01-matrix-rank-0013","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix-rank"],"title_zh":"矩陣秩","stem_zh":"2×2或3×3完整矩陣與精確rank-preserving消去。操作=rank-exact-rref；資料={\"matrix\":[[1,12],[0,13]]}。請依 [matrix,rref,pivot_columns,rank,certificate] 欄序回答。","answer":{"kind":"expression","value":"([[1, 12], [0, 13]], [[1, 0], [0, 1]], [0, 1], 2, rank_preserving_exact_row_operations)","explanation_zh":"Wave5第三層核對=([[1, 12], [0, 13]], [[1, 0], [0, 1]], [0, 1], 2, rank_preserving_exact_row_operations)；依完整明示資料重算；非零minor只給下界且上界需完整RREF或全部較大minor；det等於零只排除full rank；row-rank等於column-rank依定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["矩陣秩"]},"variant":{"family_id":"qf-mod34-rank-exact-rref","seed":202611031013,"parameters":{"matrix":[[1,12],[0,13]],"domain":"Exact_square_matrix_rank","operation":"rank-exact-rref","semantic_template":"qf-mod34-rank-exact-rref/{\"matrix\":[[1,12],[0,13]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean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-20261103-r01-matrix-rank-0014","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix-rank"],"title_zh":"矩陣秩","stem_zh":"2×2或3×3完整矩陣與精確rank-preserving消去。操作=rank-exact-rref；資料={\"matrix\":[[1,13],[0,14]]}。請依 [matrix,rref,pivot_columns,rank,certificate] 欄序回答。","answer":{"kind":"expression","value":"([[1, 13], [0, 14]], [[1, 0], [0, 1]], [0, 1], 2, rank_preserving_exact_row_operations)","explanation_zh":"Wave5第三層核對=([[1, 13], [0, 14]], [[1, 0], [0, 1]], [0, 1], 2, rank_preserving_exact_row_operations)；依完整明示資料重算；非零minor只給下界且上界需完整RREF或全部較大minor；det等於零只排除full rank；row-rank等於column-rank依定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["矩陣秩"]},"variant":{"family_id":"qf-mod34-rank-exact-rref","seed":202611031014,"parameters":{"matrix":[[1,13],[0,14]],"domain":"Exact_square_matrix_rank","operation":"rank-exact-rref","semantic_template":"qf-mod34-rank-exact-rref/{\"matrix\":[[1,13],[0,14]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean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-20261103-r01-matrix-rank-0015","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix-rank"],"title_zh":"矩陣秩","stem_zh":"2×2或3×3完整矩陣與精確rank-preserving消去。操作=rank-exact-rref；資料={\"matrix\":[[1,14],[0,15]]}。請依 [matrix,rref,pivot_columns,rank,certificate] 欄序回答。","answer":{"kind":"expression","value":"([[1, 14], [0, 15]], [[1, 0], [0, 1]], [0, 1], 2, rank_preserving_exact_row_operations)","explanation_zh":"Wave5第三層核對=([[1, 14], [0, 15]], [[1, 0], [0, 1]], [0, 1], 2, rank_preserving_exact_row_operations)；依完整明示資料重算；非零minor只給下界且上界需完整RREF或全部較大minor；det等於零只排除full rank；row-rank等於column-rank依定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["矩陣秩"]},"variant":{"family_id":"qf-mod34-rank-exact-rref","seed":202611031015,"parameters":{"matrix":[[1,14],[0,15]],"domain":"Exact_square_matrix_rank","operation":"rank-exact-rref","semantic_template":"qf-mod34-rank-exact-rref/{\"matrix\":[[1,14],[0,15]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean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-20261103-r01-matrix-rank-0016","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix-rank"],"title_zh":"矩陣秩","stem_zh":"2×2或3×3完整矩陣與精確rank-preserving消去。操作=rank-exact-rref；資料={\"matrix\":[[1,15],[0,16]]}。請依 [matrix,rref,pivot_columns,rank,certificate] 欄序回答。","answer":{"kind":"expression","value":"([[1, 15], [0, 16]], [[1, 0], [0, 1]], [0, 1], 2, rank_preserving_exact_row_operations)","explanation_zh":"Wave5第三層核對=([[1, 15], [0, 16]], [[1, 0], [0, 1]], [0, 1], 2, rank_preserving_exact_row_operations)；依完整明示資料重算；非零minor只給下界且上界需完整RREF或全部較大minor；det等於零只排除full rank；row-rank等於column-rank依定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["矩陣秩"]},"variant":{"family_id":"qf-mod34-rank-exact-rref","seed":202611031016,"parameters":{"matrix":[[1,15],[0,16]],"domain":"Exact_square_matrix_rank","operation":"rank-exact-rref","semantic_template":"qf-mod34-rank-exact-rref/{\"matrix\":[[1,15],[0,16]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean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-20261103-r01-matrix-rank-0017","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix-rank"],"title_zh":"矩陣秩","stem_zh":"2×2或3×3完整矩陣與精確rank-preserving消去。操作=rank-exact-rref；資料={\"matrix\":[[1,16],[0,17]]}。請依 [matrix,rref,pivot_columns,rank,certificate] 欄序回答。","answer":{"kind":"expression","value":"([[1, 16], [0, 17]], [[1, 0], [0, 1]], [0, 1], 2, rank_preserving_exact_row_operations)","explanation_zh":"Wave5第三層核對=([[1, 16], [0, 17]], [[1, 0], [0, 1]], [0, 1], 2, rank_preserving_exact_row_operations)；依完整明示資料重算；非零minor只給下界且上界需完整RREF或全部較大minor；det等於零只排除full rank；row-rank等於column-rank依定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["矩陣秩"]},"variant":{"family_id":"qf-mod34-rank-exact-rref","seed":202611031017,"parameters":{"matrix":[[1,16],[0,17]],"domain":"Exact_square_matrix_rank","operation":"rank-exact-rref","semantic_template":"qf-mod34-rank-exact-rref/{\"matrix\":[[1,16],[0,17]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean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-20261103-r01-matrix-rank-0018","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix-rank"],"title_zh":"矩陣秩","stem_zh":"2×2或3×3完整矩陣與精確rank-preserving消去。操作=rank-exact-rref；資料={\"matrix\":[[1,17],[0,18]]}。請依 [matrix,rref,pivot_columns,rank,certificate] 欄序回答。","answer":{"kind":"expression","value":"([[1, 17], [0, 18]], [[1, 0], [0, 1]], [0, 1], 2, rank_preserving_exact_row_operations)","explanation_zh":"Wave5第三層核對=([[1, 17], [0, 18]], [[1, 0], [0, 1]], [0, 1], 2, rank_preserving_exact_row_operations)；依完整明示資料重算；非零minor只給下界且上界需完整RREF或全部較大minor；det等於零只排除full rank；row-rank等於column-rank依定理。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["矩陣秩"]},"variant":{"family_id":"qf-mod34-rank-exact-rref","seed":202611031018,"parameters":{"matrix":[[1,17],[0,18]],"domain":"Exact_square_matrix_rank","operation":"rank-exact-rref","semantic_template":"qf-mod34-rank-exact-rref/{\"matrix\":[[1,17],[0,18]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean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-20261103-r01-matrix-rank-0019","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix-rank"],"title_zh":"矩陣秩","stem_zh":"2×2或3×3完整矩陣與精確rank-preserving消去。操作=rank-exact-rref；資料={\"matrix\":[[1,18],[0,19]]}。請依 [matrix,rref,pivot_columns,rank,certificate] 欄序回答。","answer":{"kind":"expression","value":"([[1, 18], [0, 19]], [[1, 0], [0, 1]], [0, 1], 2, rank_preserving_exact_row_operations)","explanation_zh":"Wave5第三層核對=([[1, 18], [0, 19]], [[1, 0], [0, 1]], [0, 1], 2, rank_preserving_exact_row_operations)；依完整明示資料重算；非零minor只給下界且上界需完整RREF或全部較大minor；det等於零只排除full rank；row-rank等於column-rank依定理。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["矩陣秩"]},"variant":{"family_id":"qf-mod34-rank-exact-rref","seed":202611031019,"parameters":{"matrix":[[1,18],[0,19]],"domain":"Exact_square_matrix_rank","operation":"rank-exact-rref","semantic_template":"qf-mod34-rank-exact-rref/{\"matrix\":[[1,18],[0,19]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean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-20261103-r01-matrix-rank-0020","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix-rank"],"title_zh":"矩陣秩","stem_zh":"2×2或3×3完整矩陣與精確rank-preserving消去。操作=rank-exact-rref；資料={\"matrix\":[[1,19],[0,20]]}。請依 [matrix,rref,pivot_columns,rank,certificate] 欄序回答。","answer":{"kind":"expression","value":"([[1, 19], [0, 20]], [[1, 0], [0, 1]], [0, 1], 2, rank_preserving_exact_row_operations)","explanation_zh":"Wave5第三層核對=([[1, 19], [0, 20]], [[1, 0], [0, 1]], [0, 1], 2, rank_preserving_exact_row_operations)；依完整明示資料重算；非零minor只給下界且上界需完整RREF或全部較大minor；det等於零只排除full rank；row-rank等於column-rank依定理。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["矩陣秩"]},"variant":{"family_id":"qf-mod34-rank-exact-rref","seed":202611031020,"parameters":{"matrix":[[1,19],[0,20]],"domain":"Exact_square_matrix_rank","operation":"rank-exact-rref","semantic_template":"qf-mod34-rank-exact-rref/{\"matrix\":[[1,19],[0,20]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean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-20261103-r01-matrix-rank-0021","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix-rank"],"title_zh":"矩陣秩","stem_zh":"2×2或3×3完整矩陣與精確rank-preserving消去。操作=rank-exact-rref；資料={\"matrix\":[[1,20],[0,21]]}。請依 [matrix,rref,pivot_columns,rank,certificate] 欄序回答。","answer":{"kind":"expression","value":"([[1, 20], [0, 21]], [[1, 0], [0, 1]], [0, 1], 2, rank_preserving_exact_row_operations)","explanation_zh":"Wave5第三層核對=([[1, 20], [0, 21]], [[1, 0], [0, 1]], [0, 1], 2, rank_preserving_exact_row_operations)；依完整明示資料重算；非零minor只給下界且上界需完整RREF或全部較大minor；det等於零只排除full rank；row-rank等於column-rank依定理。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["矩陣秩"]},"variant":{"family_id":"qf-mod34-rank-exact-rref","seed":202611031021,"parameters":{"matrix":[[1,20],[0,21]],"domain":"Exact_square_matrix_rank","operation":"rank-exact-rref","semantic_template":"qf-mod34-rank-exact-rref/{\"matrix\":[[1,20],[0,21]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean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-20261103-r01-matrix-rank-0022","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix-rank"],"title_zh":"矩陣秩","stem_zh":"2×2或3×3完整矩陣與精確rank-preserving消去。操作=rank-exact-rref；資料={\"matrix\":[[1,21],[0,22]]}。請依 [matrix,rref,pivot_columns,rank,certificate] 欄序回答。","answer":{"kind":"expression","value":"([[1, 21], [0, 22]], [[1, 0], [0, 1]], [0, 1], 2, rank_preserving_exact_row_operations)","explanation_zh":"Wave5第三層核對=([[1, 21], [0, 22]], [[1, 0], [0, 1]], [0, 1], 2, rank_preserving_exact_row_operations)；依完整明示資料重算；非零minor只給下界且上界需完整RREF或全部較大minor；det等於零只排除full rank；row-rank等於column-rank依定理。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["矩陣秩"]},"variant":{"family_id":"qf-mod34-rank-exact-rref","seed":202611031022,"parameters":{"matrix":[[1,21],[0,22]],"domain":"Exact_square_matrix_rank","operation":"rank-exact-rref","semantic_template":"qf-mod34-rank-exact-rref/{\"matrix\":[[1,21],[0,22]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean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-20261103-r01-matrix-rank-0023","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix-rank"],"title_zh":"矩陣秩","stem_zh":"2×2或3×3完整矩陣與精確rank-preserving消去。操作=rank-exact-rref；資料={\"matrix\":[[1,22],[0,23]]}。請依 [matrix,rref,pivot_columns,rank,certificate] 欄序回答。","answer":{"kind":"expression","value":"([[1, 22], [0, 23]], [[1, 0], [0, 1]], [0, 1], 2, rank_preserving_exact_row_operations)","explanation_zh":"Wave5第三層核對=([[1, 22], [0, 23]], [[1, 0], [0, 1]], [0, 1], 2, rank_preserving_exact_row_operations)；依完整明示資料重算；非零minor只給下界且上界需完整RREF或全部較大minor；det等於零只排除full rank；row-rank等於column-rank依定理。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["矩陣秩"]},"variant":{"family_id":"qf-mod34-rank-exact-rref","seed":202611031023,"parameters":{"matrix":[[1,22],[0,23]],"domain":"Exact_square_matrix_rank","operation":"rank-exact-rref","semantic_template":"qf-mod34-rank-exact-rref/{\"matrix\":[[1,22],[0,23]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean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-20261103-r01-matrix-rank-0024","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix-rank"],"title_zh":"矩陣秩","stem_zh":"2×2或3×3完整矩陣與精確rank-preserving消去。操作=rank-exact-rref；資料={\"matrix\":[[1,23],[0,24]]}。請依 [matrix,rref,pivot_columns,rank,certificate] 欄序回答。","answer":{"kind":"expression","value":"([[1, 23], [0, 24]], [[1, 0], [0, 1]], [0, 1], 2, rank_preserving_exact_row_operations)","explanation_zh":"Wave5第三層核對=([[1, 23], [0, 24]], [[1, 0], [0, 1]], [0, 1], 2, rank_preserving_exact_row_operations)；依完整明示資料重算；非零minor只給下界且上界需完整RREF或全部較大minor；det等於零只排除full rank；row-rank等於column-rank依定理。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["矩陣秩"]},"variant":{"family_id":"qf-mod34-rank-exact-rref","seed":202611031024,"parameters":{"matrix":[[1,23],[0,24]],"domain":"Exact_square_matrix_rank","operation":"rank-exact-rref","semantic_template":"qf-mod34-rank-exact-rref/{\"matrix\":[[1,23],[0,24]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean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-20261103-r01-matrix-rank-0025","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix-rank"],"title_zh":"矩陣秩","stem_zh":"2×2或3×3完整矩陣與精確rank-preserving消去。操作=rank-exact-rref；資料={\"matrix\":[[1,24],[0,25]]}。請依 [matrix,rref,pivot_columns,rank,certificate] 欄序回答。","answer":{"kind":"expression","value":"([[1, 24], [0, 25]], [[1, 0], [0, 1]], [0, 1], 2, rank_preserving_exact_row_operations)","explanation_zh":"Wave5第三層核對=([[1, 24], [0, 25]], [[1, 0], [0, 1]], [0, 1], 2, rank_preserving_exact_row_operations)；依完整明示資料重算；非零minor只給下界且上界需完整RREF或全部較大minor；det等於零只排除full rank；row-rank等於column-rank依定理。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["矩陣秩"]},"variant":{"family_id":"qf-mod34-rank-exact-rref","seed":202611031025,"parameters":{"matrix":[[1,24],[0,25]],"domain":"Exact_square_matrix_rank","operation":"rank-exact-rref","semantic_template":"qf-mod34-rank-exact-rref/{\"matrix\":[[1,24],[0,25]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean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-20261103-r01-matrix-rank-0026","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix-rank"],"title_zh":"矩陣秩","stem_zh":"2×2或3×3完整矩陣與精確rank-preserving消去。操作=rank-minor-lower-bound；資料={\"matrix\":[[1,0],[0,1]],\"rows\":[0],\"cols\":[0]}。請依 [rows,cols,minor,determinant,nonzero,lower_bound] 欄序回答。","answer":{"kind":"expression","value":"([0], [0], [[1]], 1, 是, rank_at_least_1)","explanation_zh":"Wave5第三層核對=([0], [0], [[1]], 1, 是, rank_at_least_1)；依完整明示資料重算；非零minor只給下界且上界需完整RREF或全部較大minor；det等於零只排除full rank；row-rank等於column-rank依定理。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["矩陣秩"]},"variant":{"family_id":"qf-mod34-rank-minor-lower-bound","seed":202611031026,"parameters":{"matrix":[[1,0],[0,1]],"rows":[0],"cols":[0],"domain":"Exact_square_matrix_rank","operation":"rank-minor-lower-bound","semantic_template":"qf-mod34-rank-minor-lower-bound/{\"cols\":[0],\"matrix\":[[1,0],[0,1]],\"rows\":[0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean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-20261103-r01-matrix-rank-0027","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix-rank"],"title_zh":"矩陣秩","stem_zh":"2×2或3×3完整矩陣與精確rank-preserving消去。操作=rank-minor-lower-bound；資料={\"matrix\":[[1,1],[0,2]],\"rows\":[0],\"cols\":[0]}。請依 [rows,cols,minor,determinant,nonzero,lower_bound] 欄序回答。","answer":{"kind":"expression","value":"([0], [0], [[1]], 1, 是, rank_at_least_1)","explanation_zh":"Wave5第三層核對=([0], [0], [[1]], 1, 是, rank_at_least_1)；依完整明示資料重算；非零minor只給下界且上界需完整RREF或全部較大minor；det等於零只排除full rank；row-rank等於column-rank依定理。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["矩陣秩"]},"variant":{"family_id":"qf-mod34-rank-minor-lower-bound","seed":202611031027,"parameters":{"matrix":[[1,1],[0,2]],"rows":[0],"cols":[0],"domain":"Exact_square_matrix_rank","operation":"rank-minor-lower-bound","semantic_template":"qf-mod34-rank-minor-lower-bound/{\"cols\":[0],\"matrix\":[[1,1],[0,2]],\"rows\":[0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean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-20261103-r01-matrix-rank-0028","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix-rank"],"title_zh":"矩陣秩","stem_zh":"2×2或3×3完整矩陣與精確rank-preserving消去。操作=rank-minor-lower-bound；資料={\"matrix\":[[1,2],[0,3]],\"rows\":[0],\"cols\":[0]}。請依 [rows,cols,minor,determinant,nonzero,lower_bound] 欄序回答。","answer":{"kind":"expression","value":"([0], [0], [[1]], 1, 是, rank_at_least_1)","explanation_zh":"Wave5第三層核對=([0], [0], [[1]], 1, 是, rank_at_least_1)；依完整明示資料重算；非零minor只給下界且上界需完整RREF或全部較大minor；det等於零只排除full rank；row-rank等於column-rank依定理。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["矩陣秩"]},"variant":{"family_id":"qf-mod34-rank-minor-lower-bound","seed":202611031028,"parameters":{"matrix":[[1,2],[0,3]],"rows":[0],"cols":[0],"domain":"Exact_square_matrix_rank","operation":"rank-minor-lower-bound","semantic_template":"qf-mod34-rank-minor-lower-bound/{\"cols\":[0],\"matrix\":[[1,2],[0,3]],\"rows\":[0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean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-20261103-r01-matrix-rank-0029","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix-rank"],"title_zh":"矩陣秩","stem_zh":"2×2或3×3完整矩陣與精確rank-preserving消去。操作=rank-minor-lower-bound；資料={\"matrix\":[[1,3],[0,4]],\"rows\":[0],\"cols\":[0]}。請依 [rows,cols,minor,determinant,nonzero,lower_bound] 欄序回答。","answer":{"kind":"expression","value":"([0], [0], [[1]], 1, 是, rank_at_least_1)","explanation_zh":"Wave5第三層核對=([0], [0], [[1]], 1, 是, rank_at_least_1)；依完整明示資料重算；非零minor只給下界且上界需完整RREF或全部較大minor；det等於零只排除full rank；row-rank等於column-rank依定理。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["矩陣秩"]},"variant":{"family_id":"qf-mod34-rank-minor-lower-bound","seed":202611031029,"parameters":{"matrix":[[1,3],[0,4]],"rows":[0],"cols":[0],"domain":"Exact_square_matrix_rank","operation":"rank-minor-lower-bound","semantic_template":"qf-mod34-rank-minor-lower-bound/{\"cols\":[0],\"matrix\":[[1,3],[0,4]],\"rows\":[0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean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-20261103-r01-matrix-rank-0030","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix-rank"],"title_zh":"矩陣秩","stem_zh":"2×2或3×3完整矩陣與精確rank-preserving消去。操作=rank-minor-lower-bound；資料={\"matrix\":[[1,4],[0,5]],\"rows\":[0],\"cols\":[0]}。請依 [rows,cols,minor,determinant,nonzero,lower_bound] 欄序回答。","answer":{"kind":"expression","value":"([0], [0], [[1]], 1, 是, rank_at_least_1)","explanation_zh":"Wave5第三層核對=([0], [0], [[1]], 1, 是, rank_at_least_1)；依完整明示資料重算；非零minor只給下界且上界需完整RREF或全部較大minor；det等於零只排除full rank；row-rank等於column-rank依定理。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["矩陣秩"]},"variant":{"family_id":"qf-mod34-rank-minor-lower-bound","seed":202611031030,"parameters":{"matrix":[[1,4],[0,5]],"rows":[0],"cols":[0],"domain":"Exact_square_matrix_rank","operation":"rank-minor-lower-bound","semantic_template":"qf-mod34-rank-minor-lower-bound/{\"cols\":[0],\"matrix\":[[1,4],[0,5]],\"rows\":[0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean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-20261103-r01-matrix-rank-0031","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix-rank"],"title_zh":"矩陣秩","stem_zh":"2×2或3×3完整矩陣與精確rank-preserving消去。操作=rank-minor-lower-bound；資料={\"matrix\":[[1,5],[0,6]],\"rows\":[0],\"cols\":[0]}。請依 [rows,cols,minor,determinant,nonzero,lower_bound] 欄序回答。","answer":{"kind":"expression","value":"([0], [0], [[1]], 1, 是, rank_at_least_1)","explanation_zh":"Wave5第三層核對=([0], [0], [[1]], 1, 是, rank_at_least_1)；依完整明示資料重算；非零minor只給下界且上界需完整RREF或全部較大minor；det等於零只排除full rank；row-rank等於column-rank依定理。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["矩陣秩"]},"variant":{"family_id":"qf-mod34-rank-minor-lower-bound","seed":202611031031,"parameters":{"matrix":[[1,5],[0,6]],"rows":[0],"cols":[0],"domain":"Exact_square_matrix_rank","operation":"rank-minor-lower-bound","semantic_template":"qf-mod34-rank-minor-lower-bound/{\"cols\":[0],\"matrix\":[[1,5],[0,6]],\"rows\":[0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean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-20261103-r01-matrix-rank-0032","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix-rank"],"title_zh":"矩陣秩","stem_zh":"2×2或3×3完整矩陣與精確rank-preserving消去。操作=rank-minor-lower-bound；資料={\"matrix\":[[1,6],[0,7]],\"rows\":[0],\"cols\":[0]}。請依 [rows,cols,minor,determinant,nonzero,lower_bound] 欄序回答。","answer":{"kind":"expression","value":"([0], [0], [[1]], 1, 是, rank_at_least_1)","explanation_zh":"Wave5第三層核對=([0], [0], [[1]], 1, 是, rank_at_least_1)；依完整明示資料重算；非零minor只給下界且上界需完整RREF或全部較大minor；det等於零只排除full rank；row-rank等於column-rank依定理。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["矩陣秩"]},"variant":{"family_id":"qf-mod34-rank-minor-lower-bound","seed":202611031032,"parameters":{"matrix":[[1,6],[0,7]],"rows":[0],"cols":[0],"domain":"Exact_square_matrix_rank","operation":"rank-minor-lower-bound","semantic_template":"qf-mod34-rank-minor-lower-bound/{\"cols\":[0],\"matrix\":[[1,6],[0,7]],\"rows\":[0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean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-20261103-r01-matrix-rank-0033","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix-rank"],"title_zh":"矩陣秩","stem_zh":"2×2或3×3完整矩陣與精確rank-preserving消去。操作=rank-minor-lower-bound；資料={\"matrix\":[[1,7],[0,8]],\"rows\":[0],\"cols\":[0]}。請依 [rows,cols,minor,determinant,nonzero,lower_bound] 欄序回答。","answer":{"kind":"expression","value":"([0], [0], [[1]], 1, 是, rank_at_least_1)","explanation_zh":"Wave5第三層核對=([0], [0], [[1]], 1, 是, rank_at_least_1)；依完整明示資料重算；非零minor只給下界且上界需完整RREF或全部較大minor；det等於零只排除full rank；row-rank等於column-rank依定理。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["矩陣秩"]},"variant":{"family_id":"qf-mod34-rank-minor-lower-bound","seed":202611031033,"parameters":{"matrix":[[1,7],[0,8]],"rows":[0],"cols":[0],"domain":"Exact_square_matrix_rank","operation":"rank-minor-lower-bound","semantic_template":"qf-mod34-rank-minor-lower-bound/{\"cols\":[0],\"matrix\":[[1,7],[0,8]],\"rows\":[0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean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-20261103-r01-matrix-rank-0034","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix-rank"],"title_zh":"矩陣秩","stem_zh":"2×2或3×3完整矩陣與精確rank-preserving消去。操作=rank-minor-lower-bound；資料={\"matrix\":[[1,8],[0,9]],\"rows\":[0],\"cols\":[0]}。請依 [rows,cols,minor,determinant,nonzero,lower_bound] 欄序回答。","answer":{"kind":"expression","value":"([0], [0], [[1]], 1, 是, rank_at_least_1)","explanation_zh":"Wave5第三層核對=([0], [0], [[1]], 1, 是, rank_at_least_1)；依完整明示資料重算；非零minor只給下界且上界需完整RREF或全部較大minor；det等於零只排除full rank；row-rank等於column-rank依定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["矩陣秩"]},"variant":{"family_id":"qf-mod34-rank-minor-lower-bound","seed":202611031034,"parameters":{"matrix":[[1,8],[0,9]],"rows":[0],"cols":[0],"domain":"Exact_square_matrix_rank","operation":"rank-minor-lower-bound","semantic_template":"qf-mod34-rank-minor-lower-bound/{\"cols\":[0],\"matrix\":[[1,8],[0,9]],\"rows\":[0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean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-20261103-r01-matrix-rank-0035","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix-rank"],"title_zh":"矩陣秩","stem_zh":"2×2或3×3完整矩陣與精確rank-preserving消去。操作=rank-minor-lower-bound；資料={\"matrix\":[[1,9],[0,10]],\"rows\":[0],\"cols\":[0]}。請依 [rows,cols,minor,determinant,nonzero,lower_bound] 欄序回答。","answer":{"kind":"expression","value":"([0], [0], [[1]], 1, 是, rank_at_least_1)","explanation_zh":"Wave5第三層核對=([0], [0], [[1]], 1, 是, rank_at_least_1)；依完整明示資料重算；非零minor只給下界且上界需完整RREF或全部較大minor；det等於零只排除full rank；row-rank等於column-rank依定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["矩陣秩"]},"variant":{"family_id":"qf-mod34-rank-minor-lower-bound","seed":202611031035,"parameters":{"matrix":[[1,9],[0,10]],"rows":[0],"cols":[0],"domain":"Exact_square_matrix_rank","operation":"rank-minor-lower-bound","semantic_template":"qf-mod34-rank-minor-lower-bound/{\"cols\":[0],\"matrix\":[[1,9],[0,10]],\"rows\":[0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean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-20261103-r01-matrix-rank-0036","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix-rank"],"title_zh":"矩陣秩","stem_zh":"2×2或3×3完整矩陣與精確rank-preserving消去。操作=rank-minor-lower-bound；資料={\"matrix\":[[1,10],[0,11]],\"rows\":[0],\"cols\":[0]}。請依 [rows,cols,minor,determinant,nonzero,lower_bound] 欄序回答。","answer":{"kind":"expression","value":"([0], [0], [[1]], 1, 是, rank_at_least_1)","explanation_zh":"Wave5第三層核對=([0], [0], [[1]], 1, 是, rank_at_least_1)；依完整明示資料重算；非零minor只給下界且上界需完整RREF或全部較大minor；det等於零只排除full rank；row-rank等於column-rank依定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["矩陣秩"]},"variant":{"family_id":"qf-mod34-rank-minor-lower-bound","seed":202611031036,"parameters":{"matrix":[[1,10],[0,11]],"rows":[0],"cols":[0],"domain":"Exact_square_matrix_rank","operation":"rank-minor-lower-bound","semantic_template":"qf-mod34-rank-minor-lower-bound/{\"cols\":[0],\"matrix\":[[1,10],[0,11]],\"rows\":[0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean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-20261103-r01-matrix-rank-0037","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix-rank"],"title_zh":"矩陣秩","stem_zh":"2×2或3×3完整矩陣與精確rank-preserving消去。操作=rank-minor-lower-bound；資料={\"matrix\":[[1,11],[0,12]],\"rows\":[0],\"cols\":[0]}。請依 [rows,cols,minor,determinant,nonzero,lower_bound] 欄序回答。","answer":{"kind":"expression","value":"([0], [0], [[1]], 1, 是, rank_at_least_1)","explanation_zh":"Wave5第三層核對=([0], [0], [[1]], 1, 是, rank_at_least_1)；依完整明示資料重算；非零minor只給下界且上界需完整RREF或全部較大minor；det等於零只排除full rank；row-rank等於column-rank依定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["矩陣秩"]},"variant":{"family_id":"qf-mod34-rank-minor-lower-bound","seed":202611031037,"parameters":{"matrix":[[1,11],[0,12]],"rows":[0],"cols":[0],"domain":"Exact_square_matrix_rank","operation":"rank-minor-lower-bound","semantic_template":"qf-mod34-rank-minor-lower-bound/{\"cols\":[0],\"matrix\":[[1,11],[0,12]],\"rows\":[0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean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-20261103-r01-matrix-rank-0038","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix-rank"],"title_zh":"矩陣秩","stem_zh":"2×2或3×3完整矩陣與精確rank-preserving消去。操作=rank-minor-lower-bound；資料={\"matrix\":[[1,12],[0,13]],\"rows\":[0],\"cols\":[0]}。請依 [rows,cols,minor,determinant,nonzero,lower_bound] 欄序回答。","answer":{"kind":"expression","value":"([0], [0], [[1]], 1, 是, rank_at_least_1)","explanation_zh":"Wave5第三層核對=([0], [0], [[1]], 1, 是, rank_at_least_1)；依完整明示資料重算；非零minor只給下界且上界需完整RREF或全部較大minor；det等於零只排除full rank；row-rank等於column-rank依定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["矩陣秩"]},"variant":{"family_id":"qf-mod34-rank-minor-lower-bound","seed":202611031038,"parameters":{"matrix":[[1,12],[0,13]],"rows":[0],"cols":[0],"domain":"Exact_square_matrix_rank","operation":"rank-minor-lower-bound","semantic_template":"qf-mod34-rank-minor-lower-bound/{\"cols\":[0],\"matrix\":[[1,12],[0,13]],\"rows\":[0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean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-20261103-r01-matrix-rank-0039","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix-rank"],"title_zh":"矩陣秩","stem_zh":"2×2或3×3完整矩陣與精確rank-preserving消去。操作=rank-minor-lower-bound；資料={\"matrix\":[[1,13],[0,14]],\"rows\":[0],\"cols\":[0]}。請依 [rows,cols,minor,determinant,nonzero,lower_bound] 欄序回答。","answer":{"kind":"expression","value":"([0], [0], [[1]], 1, 是, rank_at_least_1)","explanation_zh":"Wave5第三層核對=([0], [0], [[1]], 1, 是, rank_at_least_1)；依完整明示資料重算；非零minor只給下界且上界需完整RREF或全部較大minor；det等於零只排除full rank；row-rank等於column-rank依定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["矩陣秩"]},"variant":{"family_id":"qf-mod34-rank-minor-lower-bound","seed":202611031039,"parameters":{"matrix":[[1,13],[0,14]],"rows":[0],"cols":[0],"domain":"Exact_square_matrix_rank","operation":"rank-minor-lower-bound","semantic_template":"qf-mod34-rank-minor-lower-bound/{\"cols\":[0],\"matrix\":[[1,13],[0,14]],\"rows\":[0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean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-20261103-r01-matrix-rank-0040","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix-rank"],"title_zh":"矩陣秩","stem_zh":"2×2或3×3完整矩陣與精確rank-preserving消去。操作=rank-minor-lower-bound；資料={\"matrix\":[[1,14],[0,15]],\"rows\":[0],\"cols\":[0]}。請依 [rows,cols,minor,determinant,nonzero,lower_bound] 欄序回答。","answer":{"kind":"expression","value":"([0], [0], [[1]], 1, 是, rank_at_least_1)","explanation_zh":"Wave5第三層核對=([0], [0], [[1]], 1, 是, rank_at_least_1)；依完整明示資料重算；非零minor只給下界且上界需完整RREF或全部較大minor；det等於零只排除full rank；row-rank等於column-rank依定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["矩陣秩"]},"variant":{"family_id":"qf-mod34-rank-minor-lower-bound","seed":202611031040,"parameters":{"matrix":[[1,14],[0,15]],"rows":[0],"cols":[0],"domain":"Exact_square_matrix_rank","operation":"rank-minor-lower-bound","semantic_template":"qf-mod34-rank-minor-lower-bound/{\"cols\":[0],\"matrix\":[[1,14],[0,15]],\"rows\":[0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean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-20261103-r01-matrix-rank-0041","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix-rank"],"title_zh":"矩陣秩","stem_zh":"2×2或3×3完整矩陣與精確rank-preserving消去。操作=rank-minor-lower-bound；資料={\"matrix\":[[1,15],[0,16]],\"rows\":[0],\"cols\":[0]}。請依 [rows,cols,minor,determinant,nonzero,lower_bound] 欄序回答。","answer":{"kind":"expression","value":"([0], [0], [[1]], 1, 是, rank_at_least_1)","explanation_zh":"Wave5第三層核對=([0], [0], [[1]], 1, 是, rank_at_least_1)；依完整明示資料重算；非零minor只給下界且上界需完整RREF或全部較大minor；det等於零只排除full rank；row-rank等於column-rank依定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["矩陣秩"]},"variant":{"family_id":"qf-mod34-rank-minor-lower-bound","seed":202611031041,"parameters":{"matrix":[[1,15],[0,16]],"rows":[0],"cols":[0],"domain":"Exact_square_matrix_rank","operation":"rank-minor-lower-bound","semantic_template":"qf-mod34-rank-minor-lower-bound/{\"cols\":[0],\"matrix\":[[1,15],[0,16]],\"rows\":[0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean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-20261103-r01-matrix-rank-0042","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix-rank"],"title_zh":"矩陣秩","stem_zh":"2×2或3×3完整矩陣與精確rank-preserving消去。操作=rank-minor-lower-bound；資料={\"matrix\":[[1,16],[0,17]],\"rows\":[0],\"cols\":[0]}。請依 [rows,cols,minor,determinant,nonzero,lower_bound] 欄序回答。","answer":{"kind":"expression","value":"([0], [0], [[1]], 1, 是, rank_at_least_1)","explanation_zh":"Wave5第三層核對=([0], [0], [[1]], 1, 是, rank_at_least_1)；依完整明示資料重算；非零minor只給下界且上界需完整RREF或全部較大minor；det等於零只排除full rank；row-rank等於column-rank依定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["矩陣秩"]},"variant":{"family_id":"qf-mod34-rank-minor-lower-bound","seed":202611031042,"parameters":{"matrix":[[1,16],[0,17]],"rows":[0],"cols":[0],"domain":"Exact_square_matrix_rank","operation":"rank-minor-lower-bound","semantic_template":"qf-mod34-rank-minor-lower-bound/{\"cols\":[0],\"matrix\":[[1,16],[0,17]],\"rows\":[0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean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-20261103-r01-matrix-rank-0043","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix-rank"],"title_zh":"矩陣秩","stem_zh":"2×2或3×3完整矩陣與精確rank-preserving消去。操作=rank-minor-lower-bound；資料={\"matrix\":[[1,17],[0,18]],\"rows\":[0],\"cols\":[0]}。請依 [rows,cols,minor,determinant,nonzero,lower_bound] 欄序回答。","answer":{"kind":"expression","value":"([0], [0], [[1]], 1, 是, rank_at_least_1)","explanation_zh":"Wave5第三層核對=([0], [0], [[1]], 1, 是, rank_at_least_1)；依完整明示資料重算；非零minor只給下界且上界需完整RREF或全部較大minor；det等於零只排除full rank；row-rank等於column-rank依定理。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["矩陣秩"]},"variant":{"family_id":"qf-mod34-rank-minor-lower-bound","seed":202611031043,"parameters":{"matrix":[[1,17],[0,18]],"rows":[0],"cols":[0],"domain":"Exact_square_matrix_rank","operation":"rank-minor-lower-bound","semantic_template":"qf-mod34-rank-minor-lower-bound/{\"cols\":[0],\"matrix\":[[1,17],[0,18]],\"rows\":[0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean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-20261103-r01-matrix-rank-0044","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix-rank"],"title_zh":"矩陣秩","stem_zh":"2×2或3×3完整矩陣與精確rank-preserving消去。操作=rank-minor-lower-bound；資料={\"matrix\":[[1,18],[0,19]],\"rows\":[0],\"cols\":[0]}。請依 [rows,cols,minor,determinant,nonzero,lower_bound] 欄序回答。","answer":{"kind":"expression","value":"([0], [0], [[1]], 1, 是, rank_at_least_1)","explanation_zh":"Wave5第三層核對=([0], [0], [[1]], 1, 是, rank_at_least_1)；依完整明示資料重算；非零minor只給下界且上界需完整RREF或全部較大minor；det等於零只排除full rank；row-rank等於column-rank依定理。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["矩陣秩"]},"variant":{"family_id":"qf-mod34-rank-minor-lower-bound","seed":202611031044,"parameters":{"matrix":[[1,18],[0,19]],"rows":[0],"cols":[0],"domain":"Exact_square_matrix_rank","operation":"rank-minor-lower-bound","semantic_template":"qf-mod34-rank-minor-lower-bound/{\"cols\":[0],\"matrix\":[[1,18],[0,19]],\"rows\":[0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean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-20261103-r01-matrix-rank-0045","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix-rank"],"title_zh":"矩陣秩","stem_zh":"2×2或3×3完整矩陣與精確rank-preserving消去。操作=rank-minor-lower-bound；資料={\"matrix\":[[1,19],[0,20]],\"rows\":[0],\"cols\":[0]}。請依 [rows,cols,minor,determinant,nonzero,lower_bound] 欄序回答。","answer":{"kind":"expression","value":"([0], [0], [[1]], 1, 是, rank_at_least_1)","explanation_zh":"Wave5第三層核對=([0], [0], [[1]], 1, 是, rank_at_least_1)；依完整明示資料重算；非零minor只給下界且上界需完整RREF或全部較大minor；det等於零只排除full rank；row-rank等於column-rank依定理。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["矩陣秩"]},"variant":{"family_id":"qf-mod34-rank-minor-lower-bound","seed":202611031045,"parameters":{"matrix":[[1,19],[0,20]],"rows":[0],"cols":[0],"domain":"Exact_square_matrix_rank","operation":"rank-minor-lower-bound","semantic_template":"qf-mod34-rank-minor-lower-bound/{\"cols\":[0],\"matrix\":[[1,19],[0,20]],\"rows\":[0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean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-20261103-r01-matrix-rank-0046","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix-rank"],"title_zh":"矩陣秩","stem_zh":"2×2或3×3完整矩陣與精確rank-preserving消去。操作=rank-minor-lower-bound；資料={\"matrix\":[[1,20],[0,21]],\"rows\":[0],\"cols\":[0]}。請依 [rows,cols,minor,determinant,nonzero,lower_bound] 欄序回答。","answer":{"kind":"expression","value":"([0], [0], [[1]], 1, 是, rank_at_least_1)","explanation_zh":"Wave5第三層核對=([0], [0], [[1]], 1, 是, rank_at_least_1)；依完整明示資料重算；非零minor只給下界且上界需完整RREF或全部較大minor；det等於零只排除full rank；row-rank等於column-rank依定理。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["矩陣秩"]},"variant":{"family_id":"qf-mod34-rank-minor-lower-bound","seed":202611031046,"parameters":{"matrix":[[1,20],[0,21]],"rows":[0],"cols":[0],"domain":"Exact_square_matrix_rank","operation":"rank-minor-lower-bound","semantic_template":"qf-mod34-rank-minor-lower-bound/{\"cols\":[0],\"matrix\":[[1,20],[0,21]],\"rows\":[0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean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-20261103-r01-matrix-rank-0047","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix-rank"],"title_zh":"矩陣秩","stem_zh":"2×2或3×3完整矩陣與精確rank-preserving消去。操作=rank-minor-lower-bound；資料={\"matrix\":[[1,21],[0,22]],\"rows\":[0],\"cols\":[0]}。請依 [rows,cols,minor,determinant,nonzero,lower_bound] 欄序回答。","answer":{"kind":"expression","value":"([0], [0], [[1]], 1, 是, rank_at_least_1)","explanation_zh":"Wave5第三層核對=([0], [0], [[1]], 1, 是, rank_at_least_1)；依完整明示資料重算；非零minor只給下界且上界需完整RREF或全部較大minor；det等於零只排除full rank；row-rank等於column-rank依定理。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["矩陣秩"]},"variant":{"family_id":"qf-mod34-rank-minor-lower-bound","seed":202611031047,"parameters":{"matrix":[[1,21],[0,22]],"rows":[0],"cols":[0],"domain":"Exact_square_matrix_rank","operation":"rank-minor-lower-bound","semantic_template":"qf-mod34-rank-minor-lower-bound/{\"cols\":[0],\"matrix\":[[1,21],[0,22]],\"rows\":[0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean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-20261103-r01-matrix-rank-0048","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix-rank"],"title_zh":"矩陣秩","stem_zh":"2×2或3×3完整矩陣與精確rank-preserving消去。操作=rank-minor-lower-bound；資料={\"matrix\":[[1,22],[0,23]],\"rows\":[0],\"cols\":[0]}。請依 [rows,cols,minor,determinant,nonzero,lower_bound] 欄序回答。","answer":{"kind":"expression","value":"([0], [0], [[1]], 1, 是, rank_at_least_1)","explanation_zh":"Wave5第三層核對=([0], [0], [[1]], 1, 是, rank_at_least_1)；依完整明示資料重算；非零minor只給下界且上界需完整RREF或全部較大minor；det等於零只排除full rank；row-rank等於column-rank依定理。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["矩陣秩"]},"variant":{"family_id":"qf-mod34-rank-minor-lower-bound","seed":202611031048,"parameters":{"matrix":[[1,22],[0,23]],"rows":[0],"cols":[0],"domain":"Exact_square_matrix_rank","operation":"rank-minor-lower-bound","semantic_template":"qf-mod34-rank-minor-lower-bound/{\"cols\":[0],\"matrix\":[[1,22],[0,23]],\"rows\":[0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean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-20261103-r01-matrix-rank-0049","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix-rank"],"title_zh":"矩陣秩","stem_zh":"2×2或3×3完整矩陣與精確rank-preserving消去。操作=rank-minor-lower-bound；資料={\"matrix\":[[1,23],[0,24]],\"rows\":[0],\"cols\":[0]}。請依 [rows,cols,minor,determinant,nonzero,lower_bound] 欄序回答。","answer":{"kind":"expression","value":"([0], [0], [[1]], 1, 是, rank_at_least_1)","explanation_zh":"Wave5第三層核對=([0], [0], [[1]], 1, 是, rank_at_least_1)；依完整明示資料重算；非零minor只給下界且上界需完整RREF或全部較大minor；det等於零只排除full rank；row-rank等於column-rank依定理。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["矩陣秩"]},"variant":{"family_id":"qf-mod34-rank-minor-lower-bound","seed":202611031049,"parameters":{"matrix":[[1,23],[0,24]],"rows":[0],"cols":[0],"domain":"Exact_square_matrix_rank","operation":"rank-minor-lower-bound","semantic_template":"qf-mod34-rank-minor-lower-bound/{\"cols\":[0],\"matrix\":[[1,23],[0,24]],\"rows\":[0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean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-20261103-r01-matrix-rank-0050","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix-rank"],"title_zh":"矩陣秩","stem_zh":"2×2或3×3完整矩陣與精確rank-preserving消去。操作=rank-minor-lower-bound；資料={\"matrix\":[[1,24],[0,25]],\"rows\":[0],\"cols\":[0]}。請依 [rows,cols,minor,determinant,nonzero,lower_bound] 欄序回答。","answer":{"kind":"expression","value":"([0], [0], [[1]], 1, 是, rank_at_least_1)","explanation_zh":"Wave5第三層核對=([0], [0], [[1]], 1, 是, rank_at_least_1)；依完整明示資料重算；非零minor只給下界且上界需完整RREF或全部較大minor；det等於零只排除full rank；row-rank等於column-rank依定理。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["矩陣秩"]},"variant":{"family_id":"qf-mod34-rank-minor-lower-bound","seed":202611031050,"parameters":{"matrix":[[1,24],[0,25]],"rows":[0],"cols":[0],"domain":"Exact_square_matrix_rank","operation":"rank-minor-lower-bound","semantic_template":"qf-mod34-rank-minor-lower-bound/{\"cols\":[0],\"matrix\":[[1,24],[0,25]],\"rows\":[0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean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-20261103-r01-matrix-rank-0051","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix-rank"],"title_zh":"矩陣秩","stem_zh":"2×2或3×3完整矩陣與精確rank-preserving消去。操作=rank-full-rank-audit；資料={\"matrix\":[[1,0],[0,1]]}。請依 [size,determinant,nonzero,rank,full_rank] 欄序回答。","answer":{"kind":"expression","value":"(2, 1, 是, 2, 是)","explanation_zh":"Wave5第三層核對=(2, 1, 是, 2, 是)；依完整明示資料重算；非零minor只給下界且上界需完整RREF或全部較大minor；det等於零只排除full rank；row-rank等於column-rank依定理。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["矩陣秩"]},"variant":{"family_id":"qf-mod34-rank-full-rank-audit","seed":202611031051,"parameters":{"matrix":[[1,0],[0,1]],"domain":"Exact_square_matrix_rank","operation":"rank-full-rank-audit","semantic_template":"qf-mod34-rank-full-rank-audit/{\"matrix\":[[1,0],[0,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean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-20261103-r01-matrix-rank-0052","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix-rank"],"title_zh":"矩陣秩","stem_zh":"2×2或3×3完整矩陣與精確rank-preserving消去。操作=rank-full-rank-audit；資料={\"matrix\":[[1,1],[0,2]]}。請依 [size,determinant,nonzero,rank,full_rank] 欄序回答。","answer":{"kind":"expression","value":"(2, 2, 是, 2, 是)","explanation_zh":"Wave5第三層核對=(2, 2, 是, 2, 是)；依完整明示資料重算；非零minor只給下界且上界需完整RREF或全部較大minor；det等於零只排除full rank；row-rank等於column-rank依定理。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["矩陣秩"]},"variant":{"family_id":"qf-mod34-rank-full-rank-audit","seed":202611031052,"parameters":{"matrix":[[1,1],[0,2]],"domain":"Exact_square_matrix_rank","operation":"rank-full-rank-audit","semantic_template":"qf-mod34-rank-full-rank-audit/{\"matrix\":[[1,1],[0,2]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean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-20261103-r01-matrix-rank-0053","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix-rank"],"title_zh":"矩陣秩","stem_zh":"2×2或3×3完整矩陣與精確rank-preserving消去。操作=rank-full-rank-audit；資料={\"matrix\":[[1,2],[0,3]]}。請依 [size,determinant,nonzero,rank,full_rank] 欄序回答。","answer":{"kind":"expression","value":"(2, 3, 是, 2, 是)","explanation_zh":"Wave5第三層核對=(2, 3, 是, 2, 是)；依完整明示資料重算；非零minor只給下界且上界需完整RREF或全部較大minor；det等於零只排除full rank；row-rank等於column-rank依定理。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["矩陣秩"]},"variant":{"family_id":"qf-mod34-rank-full-rank-audit","seed":202611031053,"parameters":{"matrix":[[1,2],[0,3]],"domain":"Exact_square_matrix_rank","operation":"rank-full-rank-audit","semantic_template":"qf-mod34-rank-full-rank-audit/{\"matrix\":[[1,2],[0,3]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean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-20261103-r01-matrix-rank-0054","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix-rank"],"title_zh":"矩陣秩","stem_zh":"2×2或3×3完整矩陣與精確rank-preserving消去。操作=rank-full-rank-audit；資料={\"matrix\":[[1,3],[0,4]]}。請依 [size,determinant,nonzero,rank,full_rank] 欄序回答。","answer":{"kind":"expression","value":"(2, 4, 是, 2, 是)","explanation_zh":"Wave5第三層核對=(2, 4, 是, 2, 是)；依完整明示資料重算；非零minor只給下界且上界需完整RREF或全部較大minor；det等於零只排除full rank；row-rank等於column-rank依定理。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["矩陣秩"]},"variant":{"family_id":"qf-mod34-rank-full-rank-audit","seed":202611031054,"parameters":{"matrix":[[1,3],[0,4]],"domain":"Exact_square_matrix_rank","operation":"rank-full-rank-audit","semantic_template":"qf-mod34-rank-full-rank-audit/{\"matrix\":[[1,3],[0,4]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean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-20261103-r01-matrix-rank-0055","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix-rank"],"title_zh":"矩陣秩","stem_zh":"2×2或3×3完整矩陣與精確rank-preserving消去。操作=rank-full-rank-audit；資料={\"matrix\":[[1,4],[0,5]]}。請依 [size,determinant,nonzero,rank,full_rank] 欄序回答。","answer":{"kind":"expression","value":"(2, 5, 是, 2, 是)","explanation_zh":"Wave5第三層核對=(2, 5, 是, 2, 是)；依完整明示資料重算；非零minor只給下界且上界需完整RREF或全部較大minor；det等於零只排除full rank；row-rank等於column-rank依定理。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["矩陣秩"]},"variant":{"family_id":"qf-mod34-rank-full-rank-audit","seed":202611031055,"parameters":{"matrix":[[1,4],[0,5]],"domain":"Exact_square_matrix_rank","operation":"rank-full-rank-audit","semantic_template":"qf-mod34-rank-full-rank-audit/{\"matrix\":[[1,4],[0,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean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-20261103-r01-matrix-rank-0056","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix-rank"],"title_zh":"矩陣秩","stem_zh":"2×2或3×3完整矩陣與精確rank-preserving消去。操作=rank-full-rank-audit；資料={\"matrix\":[[1,5],[0,6]]}。請依 [size,determinant,nonzero,rank,full_rank] 欄序回答。","answer":{"kind":"expression","value":"(2, 6, 是, 2, 是)","explanation_zh":"Wave5第三層核對=(2, 6, 是, 2, 是)；依完整明示資料重算；非零minor只給下界且上界需完整RREF或全部較大minor；det等於零只排除full rank；row-rank等於column-rank依定理。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["矩陣秩"]},"variant":{"family_id":"qf-mod34-rank-full-rank-audit","seed":202611031056,"parameters":{"matrix":[[1,5],[0,6]],"domain":"Exact_square_matrix_rank","operation":"rank-full-rank-audit","semantic_template":"qf-mod34-rank-full-rank-audit/{\"matrix\":[[1,5],[0,6]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean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-20261103-r01-matrix-rank-0057","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix-rank"],"title_zh":"矩陣秩","stem_zh":"2×2或3×3完整矩陣與精確rank-preserving消去。操作=rank-full-rank-audit；資料={\"matrix\":[[1,6],[0,7]]}。請依 [size,determinant,nonzero,rank,full_rank] 欄序回答。","answer":{"kind":"expression","value":"(2, 7, 是, 2, 是)","explanation_zh":"Wave5第三層核對=(2, 7, 是, 2, 是)；依完整明示資料重算；非零minor只給下界且上界需完整RREF或全部較大minor；det等於零只排除full rank；row-rank等於column-rank依定理。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["矩陣秩"]},"variant":{"family_id":"qf-mod34-rank-full-rank-audit","seed":202611031057,"parameters":{"matrix":[[1,6],[0,7]],"domain":"Exact_square_matrix_rank","operation":"rank-full-rank-audit","semantic_template":"qf-mod34-rank-full-rank-audit/{\"matrix\":[[1,6],[0,7]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean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-20261103-r01-matrix-rank-0058","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix-rank"],"title_zh":"矩陣秩","stem_zh":"2×2或3×3完整矩陣與精確rank-preserving消去。操作=rank-full-rank-audit；資料={\"matrix\":[[1,7],[0,8]]}。請依 [size,determinant,nonzero,rank,full_rank] 欄序回答。","answer":{"kind":"expression","value":"(2, 8, 是, 2, 是)","explanation_zh":"Wave5第三層核對=(2, 8, 是, 2, 是)；依完整明示資料重算；非零minor只給下界且上界需完整RREF或全部較大minor；det等於零只排除full rank；row-rank等於column-rank依定理。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["矩陣秩"]},"variant":{"family_id":"qf-mod34-rank-full-rank-audit","seed":202611031058,"parameters":{"matrix":[[1,7],[0,8]],"domain":"Exact_square_matrix_rank","operation":"rank-full-rank-audit","semantic_template":"qf-mod34-rank-full-rank-audit/{\"matrix\":[[1,7],[0,8]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean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-20261103-r01-matrix-rank-0059","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix-rank"],"title_zh":"矩陣秩","stem_zh":"2×2或3×3完整矩陣與精確rank-preserving消去。操作=rank-full-rank-audit；資料={\"matrix\":[[1,8],[0,9]]}。請依 [size,determinant,nonzero,rank,full_rank] 欄序回答。","answer":{"kind":"expression","value":"(2, 9, 是, 2, 是)","explanation_zh":"Wave5第三層核對=(2, 9, 是, 2, 是)；依完整明示資料重算；非零minor只給下界且上界需完整RREF或全部較大minor；det等於零只排除full rank；row-rank等於column-rank依定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["矩陣秩"]},"variant":{"family_id":"qf-mod34-rank-full-rank-audit","seed":202611031059,"parameters":{"matrix":[[1,8],[0,9]],"domain":"Exact_square_matrix_rank","operation":"rank-full-rank-audit","semantic_template":"qf-mod34-rank-full-rank-audit/{\"matrix\":[[1,8],[0,9]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean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-20261103-r01-matrix-rank-0060","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix-rank"],"title_zh":"矩陣秩","stem_zh":"2×2或3×3完整矩陣與精確rank-preserving消去。操作=rank-full-rank-audit；資料={\"matrix\":[[1,9],[0,10]]}。請依 [size,determinant,nonzero,rank,full_rank] 欄序回答。","answer":{"kind":"expression","value":"(2, 10, 是, 2, 是)","explanation_zh":"Wave5第三層核對=(2, 10, 是, 2, 是)；依完整明示資料重算；非零minor只給下界且上界需完整RREF或全部較大minor；det等於零只排除full rank；row-rank等於column-rank依定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["矩陣秩"]},"variant":{"family_id":"qf-mod34-rank-full-rank-audit","seed":202611031060,"parameters":{"matrix":[[1,9],[0,10]],"domain":"Exact_square_matrix_rank","operation":"rank-full-rank-audit","semantic_template":"qf-mod34-rank-full-rank-audit/{\"matrix\":[[1,9],[0,10]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean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-20261103-r01-matrix-rank-0061","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix-rank"],"title_zh":"矩陣秩","stem_zh":"2×2或3×3完整矩陣與精確rank-preserving消去。操作=rank-full-rank-audit；資料={\"matrix\":[[1,10],[0,11]]}。請依 [size,determinant,nonzero,rank,full_rank] 欄序回答。","answer":{"kind":"expression","value":"(2, 11, 是, 2, 是)","explanation_zh":"Wave5第三層核對=(2, 11, 是, 2, 是)；依完整明示資料重算；非零minor只給下界且上界需完整RREF或全部較大minor；det等於零只排除full rank；row-rank等於column-rank依定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["矩陣秩"]},"variant":{"family_id":"qf-mod34-rank-full-rank-audit","seed":202611031061,"parameters":{"matrix":[[1,10],[0,11]],"domain":"Exact_square_matrix_rank","operation":"rank-full-rank-audit","semantic_template":"qf-mod34-rank-full-rank-audit/{\"matrix\":[[1,10],[0,11]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean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-20261103-r01-matrix-rank-0062","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix-rank"],"title_zh":"矩陣秩","stem_zh":"2×2或3×3完整矩陣與精確rank-preserving消去。操作=rank-full-rank-audit；資料={\"matrix\":[[1,11],[0,12]]}。請依 [size,determinant,nonzero,rank,full_rank] 欄序回答。","answer":{"kind":"expression","value":"(2, 12, 是, 2, 是)","explanation_zh":"Wave5第三層核對=(2, 12, 是, 2, 是)；依完整明示資料重算；非零minor只給下界且上界需完整RREF或全部較大minor；det等於零只排除full rank；row-rank等於column-rank依定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["矩陣秩"]},"variant":{"family_id":"qf-mod34-rank-full-rank-audit","seed":202611031062,"parameters":{"matrix":[[1,11],[0,12]],"domain":"Exact_square_matrix_rank","operation":"rank-full-rank-audit","semantic_template":"qf-mod34-rank-full-rank-audit/{\"matrix\":[[1,11],[0,12]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean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-20261103-r01-matrix-rank-0063","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix-rank"],"title_zh":"矩陣秩","stem_zh":"2×2或3×3完整矩陣與精確rank-preserving消去。操作=rank-full-rank-audit；資料={\"matrix\":[[1,12],[0,13]]}。請依 [size,determinant,nonzero,rank,full_rank] 欄序回答。","answer":{"kind":"expression","value":"(2, 13, 是, 2, 是)","explanation_zh":"Wave5第三層核對=(2, 13, 是, 2, 是)；依完整明示資料重算；非零minor只給下界且上界需完整RREF或全部較大minor；det等於零只排除full rank；row-rank等於column-rank依定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["矩陣秩"]},"variant":{"family_id":"qf-mod34-rank-full-rank-audit","seed":202611031063,"parameters":{"matrix":[[1,12],[0,13]],"domain":"Exact_square_matrix_rank","operation":"rank-full-rank-audit","semantic_template":"qf-mod34-rank-full-rank-audit/{\"matrix\":[[1,12],[0,13]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean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-20261103-r01-matrix-rank-0064","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix-rank"],"title_zh":"矩陣秩","stem_zh":"2×2或3×3完整矩陣與精確rank-preserving消去。操作=rank-full-rank-audit；資料={\"matrix\":[[1,13],[0,14]]}。請依 [size,determinant,nonzero,rank,full_rank] 欄序回答。","answer":{"kind":"expression","value":"(2, 14, 是, 2, 是)","explanation_zh":"Wave5第三層核對=(2, 14, 是, 2, 是)；依完整明示資料重算；非零minor只給下界且上界需完整RREF或全部較大minor；det等於零只排除full rank；row-rank等於column-rank依定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["矩陣秩"]},"variant":{"family_id":"qf-mod34-rank-full-rank-audit","seed":202611031064,"parameters":{"matrix":[[1,13],[0,14]],"domain":"Exact_square_matrix_rank","operation":"rank-full-rank-audit","semantic_template":"qf-mod34-rank-full-rank-audit/{\"matrix\":[[1,13],[0,14]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean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-20261103-r01-matrix-rank-0065","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix-rank"],"title_zh":"矩陣秩","stem_zh":"2×2或3×3完整矩陣與精確rank-preserving消去。操作=rank-full-rank-audit；資料={\"matrix\":[[1,14],[0,15]]}。請依 [size,determinant,nonzero,rank,full_rank] 欄序回答。","answer":{"kind":"expression","value":"(2, 15, 是, 2, 是)","explanation_zh":"Wave5第三層核對=(2, 15, 是, 2, 是)；依完整明示資料重算；非零minor只給下界且上界需完整RREF或全部較大minor；det等於零只排除full rank；row-rank等於column-rank依定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["矩陣秩"]},"variant":{"family_id":"qf-mod34-rank-full-rank-audit","seed":202611031065,"parameters":{"matrix":[[1,14],[0,15]],"domain":"Exact_square_matrix_rank","operation":"rank-full-rank-audit","semantic_template":"qf-mod34-rank-full-rank-audit/{\"matrix\":[[1,14],[0,15]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean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-20261103-r01-matrix-rank-0066","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix-rank"],"title_zh":"矩陣秩","stem_zh":"2×2或3×3完整矩陣與精確rank-preserving消去。操作=rank-full-rank-audit；資料={\"matrix\":[[1,15],[0,16]]}。請依 [size,determinant,nonzero,rank,full_rank] 欄序回答。","answer":{"kind":"expression","value":"(2, 16, 是, 2, 是)","explanation_zh":"Wave5第三層核對=(2, 16, 是, 2, 是)；依完整明示資料重算；非零minor只給下界且上界需完整RREF或全部較大minor；det等於零只排除full rank；row-rank等於column-rank依定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["矩陣秩"]},"variant":{"family_id":"qf-mod34-rank-full-rank-audit","seed":202611031066,"parameters":{"matrix":[[1,15],[0,16]],"domain":"Exact_square_matrix_rank","operation":"rank-full-rank-audit","semantic_template":"qf-mod34-rank-full-rank-audit/{\"matrix\":[[1,15],[0,16]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean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-20261103-r01-matrix-rank-0067","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix-rank"],"title_zh":"矩陣秩","stem_zh":"2×2或3×3完整矩陣與精確rank-preserving消去。操作=rank-full-rank-audit；資料={\"matrix\":[[1,16],[0,17]]}。請依 [size,determinant,nonzero,rank,full_rank] 欄序回答。","answer":{"kind":"expression","value":"(2, 17, 是, 2, 是)","explanation_zh":"Wave5第三層核對=(2, 17, 是, 2, 是)；依完整明示資料重算；非零minor只給下界且上界需完整RREF或全部較大minor；det等於零只排除full rank；row-rank等於column-rank依定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["矩陣秩"]},"variant":{"family_id":"qf-mod34-rank-full-rank-audit","seed":202611031067,"parameters":{"matrix":[[1,16],[0,17]],"domain":"Exact_square_matrix_rank","operation":"rank-full-rank-audit","semantic_template":"qf-mod34-rank-full-rank-audit/{\"matrix\":[[1,16],[0,17]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean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-20261103-r01-matrix-rank-0068","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix-rank"],"title_zh":"矩陣秩","stem_zh":"2×2或3×3完整矩陣與精確rank-preserving消去。操作=rank-full-rank-audit；資料={\"matrix\":[[1,17],[0,18]]}。請依 [size,determinant,nonzero,rank,full_rank] 欄序回答。","answer":{"kind":"expression","value":"(2, 18, 是, 2, 是)","explanation_zh":"Wave5第三層核對=(2, 18, 是, 2, 是)；依完整明示資料重算；非零minor只給下界且上界需完整RREF或全部較大minor；det等於零只排除full rank；row-rank等於column-rank依定理。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["矩陣秩"]},"variant":{"family_id":"qf-mod34-rank-full-rank-audit","seed":202611031068,"parameters":{"matrix":[[1,17],[0,18]],"domain":"Exact_square_matrix_rank","operation":"rank-full-rank-audit","semantic_template":"qf-mod34-rank-full-rank-audit/{\"matrix\":[[1,17],[0,18]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean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-20261103-r01-matrix-rank-0069","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix-rank"],"title_zh":"矩陣秩","stem_zh":"2×2或3×3完整矩陣與精確rank-preserving消去。操作=rank-full-rank-audit；資料={\"matrix\":[[1,18],[0,19]]}。請依 [size,determinant,nonzero,rank,full_rank] 欄序回答。","answer":{"kind":"expression","value":"(2, 19, 是, 2, 是)","explanation_zh":"Wave5第三層核對=(2, 19, 是, 2, 是)；依完整明示資料重算；非零minor只給下界且上界需完整RREF或全部較大minor；det等於零只排除full rank；row-rank等於column-rank依定理。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["矩陣秩"]},"variant":{"family_id":"qf-mod34-rank-full-rank-audit","seed":202611031069,"parameters":{"matrix":[[1,18],[0,19]],"domain":"Exact_square_matrix_rank","operation":"rank-full-rank-audit","semantic_template":"qf-mod34-rank-full-rank-audit/{\"matrix\":[[1,18],[0,19]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean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-20261103-r01-matrix-rank-0070","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix-rank"],"title_zh":"矩陣秩","stem_zh":"2×2或3×3完整矩陣與精確rank-preserving消去。操作=rank-full-rank-audit；資料={\"matrix\":[[1,19],[0,20]]}。請依 [size,determinant,nonzero,rank,full_rank] 欄序回答。","answer":{"kind":"expression","value":"(2, 20, 是, 2, 是)","explanation_zh":"Wave5第三層核對=(2, 20, 是, 2, 是)；依完整明示資料重算；非零minor只給下界且上界需完整RREF或全部較大minor；det等於零只排除full rank；row-rank等於column-rank依定理。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["矩陣秩"]},"variant":{"family_id":"qf-mod34-rank-full-rank-audit","seed":202611031070,"parameters":{"matrix":[[1,19],[0,20]],"domain":"Exact_square_matrix_rank","operation":"rank-full-rank-audit","semantic_template":"qf-mod34-rank-full-rank-audit/{\"matrix\":[[1,19],[0,20]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean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-20261103-r01-matrix-rank-0071","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix-rank"],"title_zh":"矩陣秩","stem_zh":"2×2或3×3完整矩陣與精確rank-preserving消去。操作=rank-full-rank-audit；資料={\"matrix\":[[1,20],[0,21]]}。請依 [size,determinant,nonzero,rank,full_rank] 欄序回答。","answer":{"kind":"expression","value":"(2, 21, 是, 2, 是)","explanation_zh":"Wave5第三層核對=(2, 21, 是, 2, 是)；依完整明示資料重算；非零minor只給下界且上界需完整RREF或全部較大minor；det等於零只排除full rank；row-rank等於column-rank依定理。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["矩陣秩"]},"variant":{"family_id":"qf-mod34-rank-full-rank-audit","seed":202611031071,"parameters":{"matrix":[[1,20],[0,21]],"domain":"Exact_square_matrix_rank","operation":"rank-full-rank-audit","semantic_template":"qf-mod34-rank-full-rank-audit/{\"matrix\":[[1,20],[0,21]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean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-20261103-r01-matrix-rank-0072","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix-rank"],"title_zh":"矩陣秩","stem_zh":"2×2或3×3完整矩陣與精確rank-preserving消去。操作=rank-full-rank-audit；資料={\"matrix\":[[1,21],[0,22]]}。請依 [size,determinant,nonzero,rank,full_rank] 欄序回答。","answer":{"kind":"expression","value":"(2, 22, 是, 2, 是)","explanation_zh":"Wave5第三層核對=(2, 22, 是, 2, 是)；依完整明示資料重算；非零minor只給下界且上界需完整RREF或全部較大minor；det等於零只排除full rank；row-rank等於column-rank依定理。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["矩陣秩"]},"variant":{"family_id":"qf-mod34-rank-full-rank-audit","seed":202611031072,"parameters":{"matrix":[[1,21],[0,22]],"domain":"Exact_square_matrix_rank","operation":"rank-full-rank-audit","semantic_template":"qf-mod34-rank-full-rank-audit/{\"matrix\":[[1,21],[0,22]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean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-20261103-r01-matrix-rank-0073","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix-rank"],"title_zh":"矩陣秩","stem_zh":"2×2或3×3完整矩陣與精確rank-preserving消去。操作=rank-full-rank-audit；資料={\"matrix\":[[1,22],[0,23]]}。請依 [size,determinant,nonzero,rank,full_rank] 欄序回答。","answer":{"kind":"expression","value":"(2, 23, 是, 2, 是)","explanation_zh":"Wave5第三層核對=(2, 23, 是, 2, 是)；依完整明示資料重算；非零minor只給下界且上界需完整RREF或全部較大minor；det等於零只排除full rank；row-rank等於column-rank依定理。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["矩陣秩"]},"variant":{"family_id":"qf-mod34-rank-full-rank-audit","seed":202611031073,"parameters":{"matrix":[[1,22],[0,23]],"domain":"Exact_square_matrix_rank","operation":"rank-full-rank-audit","semantic_template":"qf-mod34-rank-full-rank-audit/{\"matrix\":[[1,22],[0,23]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean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-20261103-r01-matrix-rank-0074","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix-rank"],"title_zh":"矩陣秩","stem_zh":"2×2或3×3完整矩陣與精確rank-preserving消去。操作=rank-full-rank-audit；資料={\"matrix\":[[1,23],[0,24]]}。請依 [size,determinant,nonzero,rank,full_rank] 欄序回答。","answer":{"kind":"expression","value":"(2, 24, 是, 2, 是)","explanation_zh":"Wave5第三層核對=(2, 24, 是, 2, 是)；依完整明示資料重算；非零minor只給下界且上界需完整RREF或全部較大minor；det等於零只排除full rank；row-rank等於column-rank依定理。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["矩陣秩"]},"variant":{"family_id":"qf-mod34-rank-full-rank-audit","seed":202611031074,"parameters":{"matrix":[[1,23],[0,24]],"domain":"Exact_square_matrix_rank","operation":"rank-full-rank-audit","semantic_template":"qf-mod34-rank-full-rank-audit/{\"matrix\":[[1,23],[0,24]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean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-20261103-r01-matrix-rank-0075","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix-rank"],"title_zh":"矩陣秩","stem_zh":"2×2或3×3完整矩陣與精確rank-preserving消去。操作=rank-full-rank-audit；資料={\"matrix\":[[1,24],[0,25]]}。請依 [size,determinant,nonzero,rank,full_rank] 欄序回答。","answer":{"kind":"expression","value":"(2, 25, 是, 2, 是)","explanation_zh":"Wave5第三層核對=(2, 25, 是, 2, 是)；依完整明示資料重算；非零minor只給下界且上界需完整RREF或全部較大minor；det等於零只排除full rank；row-rank等於column-rank依定理。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["矩陣秩"]},"variant":{"family_id":"qf-mod34-rank-full-rank-audit","seed":202611031075,"parameters":{"matrix":[[1,24],[0,25]],"domain":"Exact_square_matrix_rank","operation":"rank-full-rank-audit","semantic_template":"qf-mod34-rank-full-rank-audit/{\"matrix\":[[1,24],[0,25]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean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-20261103-r01-matrix-rank-0076","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix-rank"],"title_zh":"矩陣秩","stem_zh":"2×2或3×3完整矩陣與精確rank-preserving消去。操作=rank-determinant-zero-boundary；資料={\"matrix\":[[1,1],[2,2]]}。請依 [determinant,zero,rank,consequence,boundary] 欄序回答。","answer":{"kind":"expression","value":"(0, 是, 1, det_zero_only_excludes_full_rank, zero_determinant_does_not_determine_exact_rank)","explanation_zh":"Wave5第三層核對=(0, 是, 1, det_zero_only_excludes_full_rank, zero_determinant_does_not_determine_exact_rank)；依完整明示資料重算；非零minor只給下界且上界需完整RREF或全部較大minor；det等於零只排除full rank；row-rank等於column-rank依定理。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["矩陣秩"]},"variant":{"family_id":"qf-mod34-rank-determinant-zero-boundary","seed":202611031076,"parameters":{"matrix":[[1,1],[2,2]],"domain":"Exact_square_matrix_rank","operation":"rank-determinant-zero-boundary","semantic_template":"qf-mod34-rank-determinant-zero-boundary/{\"matrix\":[[1,1],[2,2]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean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-20261103-r01-matrix-rank-0077","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix-rank"],"title_zh":"矩陣秩","stem_zh":"2×2或3×3完整矩陣與精確rank-preserving消去。操作=rank-determinant-zero-boundary；資料={\"matrix\":[[1,2],[2,4]]}。請依 [determinant,zero,rank,consequence,boundary] 欄序回答。","answer":{"kind":"expression","value":"(0, 是, 1, det_zero_only_excludes_full_rank, zero_determinant_does_not_determine_exact_rank)","explanation_zh":"Wave5第三層核對=(0, 是, 1, det_zero_only_excludes_full_rank, zero_determinant_does_not_determine_exact_rank)；依完整明示資料重算；非零minor只給下界且上界需完整RREF或全部較大minor；det等於零只排除full rank；row-rank等於column-rank依定理。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["矩陣秩"]},"variant":{"family_id":"qf-mod34-rank-determinant-zero-boundary","seed":202611031077,"parameters":{"matrix":[[1,2],[2,4]],"domain":"Exact_square_matrix_rank","operation":"rank-determinant-zero-boundary","semantic_template":"qf-mod34-rank-determinant-zero-boundary/{\"matrix\":[[1,2],[2,4]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean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-20261103-r01-matrix-rank-0078","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix-rank"],"title_zh":"矩陣秩","stem_zh":"2×2或3×3完整矩陣與精確rank-preserving消去。操作=rank-determinant-zero-boundary；資料={\"matrix\":[[1,3],[2,6]]}。請依 [determinant,zero,rank,consequence,boundary] 欄序回答。","answer":{"kind":"expression","value":"(0, 是, 1, det_zero_only_excludes_full_rank, zero_determinant_does_not_determine_exact_rank)","explanation_zh":"Wave5第三層核對=(0, 是, 1, det_zero_only_excludes_full_rank, zero_determinant_does_not_determine_exact_rank)；依完整明示資料重算；非零minor只給下界且上界需完整RREF或全部較大minor；det等於零只排除full rank；row-rank等於column-rank依定理。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["矩陣秩"]},"variant":{"family_id":"qf-mod34-rank-determinant-zero-boundary","seed":202611031078,"parameters":{"matrix":[[1,3],[2,6]],"domain":"Exact_square_matrix_rank","operation":"rank-determinant-zero-boundary","semantic_template":"qf-mod34-rank-determinant-zero-boundary/{\"matrix\":[[1,3],[2,6]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean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-20261103-r01-matrix-rank-0079","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix-rank"],"title_zh":"矩陣秩","stem_zh":"2×2或3×3完整矩陣與精確rank-preserving消去。操作=rank-determinant-zero-boundary；資料={\"matrix\":[[1,4],[2,8]]}。請依 [determinant,zero,rank,consequence,boundary] 欄序回答。","answer":{"kind":"expression","value":"(0, 是, 1, det_zero_only_excludes_full_rank, zero_determinant_does_not_determine_exact_rank)","explanation_zh":"Wave5第三層核對=(0, 是, 1, det_zero_only_excludes_full_rank, zero_determinant_does_not_determine_exact_rank)；依完整明示資料重算；非零minor只給下界且上界需完整RREF或全部較大minor；det等於零只排除full rank；row-rank等於column-rank依定理。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["矩陣秩"]},"variant":{"family_id":"qf-mod34-rank-determinant-zero-boundary","seed":202611031079,"parameters":{"matrix":[[1,4],[2,8]],"domain":"Exact_square_matrix_rank","operation":"rank-determinant-zero-boundary","semantic_template":"qf-mod34-rank-determinant-zero-boundary/{\"matrix\":[[1,4],[2,8]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean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-20261103-r01-matrix-rank-0080","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix-rank"],"title_zh":"矩陣秩","stem_zh":"2×2或3×3完整矩陣與精確rank-preserving消去。操作=rank-determinant-zero-boundary；資料={\"matrix\":[[1,5],[2,10]]}。請依 [determinant,zero,rank,consequence,boundary] 欄序回答。","answer":{"kind":"expression","value":"(0, 是, 1, det_zero_only_excludes_full_rank, zero_determinant_does_not_determine_exact_rank)","explanation_zh":"Wave5第三層核對=(0, 是, 1, det_zero_only_excludes_full_rank, zero_determinant_does_not_determine_exact_rank)；依完整明示資料重算；非零minor只給下界且上界需完整RREF或全部較大minor；det等於零只排除full rank；row-rank等於column-rank依定理。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["矩陣秩"]},"variant":{"family_id":"qf-mod34-rank-determinant-zero-boundary","seed":202611031080,"parameters":{"matrix":[[1,5],[2,10]],"domain":"Exact_square_matrix_rank","operation":"rank-determinant-zero-boundary","semantic_template":"qf-mod34-rank-determinant-zero-boundary/{\"matrix\":[[1,5],[2,10]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean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-20261103-r01-matrix-rank-0081","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix-rank"],"title_zh":"矩陣秩","stem_zh":"2×2或3×3完整矩陣與精確rank-preserving消去。操作=rank-determinant-zero-boundary；資料={\"matrix\":[[1,6],[2,12]]}。請依 [determinant,zero,rank,consequence,boundary] 欄序回答。","answer":{"kind":"expression","value":"(0, 是, 1, det_zero_only_excludes_full_rank, zero_determinant_does_not_determine_exact_rank)","explanation_zh":"Wave5第三層核對=(0, 是, 1, det_zero_only_excludes_full_rank, zero_determinant_does_not_determine_exact_rank)；依完整明示資料重算；非零minor只給下界且上界需完整RREF或全部較大minor；det等於零只排除full rank；row-rank等於column-rank依定理。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["矩陣秩"]},"variant":{"family_id":"qf-mod34-rank-determinant-zero-boundary","seed":202611031081,"parameters":{"matrix":[[1,6],[2,12]],"domain":"Exact_square_matrix_rank","operation":"rank-determinant-zero-boundary","semantic_template":"qf-mod34-rank-determinant-zero-boundary/{\"matrix\":[[1,6],[2,12]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean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-20261103-r01-matrix-rank-0082","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix-rank"],"title_zh":"矩陣秩","stem_zh":"2×2或3×3完整矩陣與精確rank-preserving消去。操作=rank-determinant-zero-boundary；資料={\"matrix\":[[1,7],[2,14]]}。請依 [determinant,zero,rank,consequence,boundary] 欄序回答。","answer":{"kind":"expression","value":"(0, 是, 1, det_zero_only_excludes_full_rank, zero_determinant_does_not_determine_exact_rank)","explanation_zh":"Wave5第三層核對=(0, 是, 1, det_zero_only_excludes_full_rank, zero_determinant_does_not_determine_exact_rank)；依完整明示資料重算；非零minor只給下界且上界需完整RREF或全部較大minor；det等於零只排除full rank；row-rank等於column-rank依定理。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["矩陣秩"]},"variant":{"family_id":"qf-mod34-rank-determinant-zero-boundary","seed":202611031082,"parameters":{"matrix":[[1,7],[2,14]],"domain":"Exact_square_matrix_rank","operation":"rank-determinant-zero-boundary","semantic_template":"qf-mod34-rank-determinant-zero-boundary/{\"matrix\":[[1,7],[2,14]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean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-20261103-r01-matrix-rank-0083","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix-rank"],"title_zh":"矩陣秩","stem_zh":"2×2或3×3完整矩陣與精確rank-preserving消去。操作=rank-determinant-zero-boundary；資料={\"matrix\":[[1,8],[2,16]]}。請依 [determinant,zero,rank,consequence,boundary] 欄序回答。","answer":{"kind":"expression","value":"(0, 是, 1, det_zero_only_excludes_full_rank, zero_determinant_does_not_determine_exact_rank)","explanation_zh":"Wave5第三層核對=(0, 是, 1, det_zero_only_excludes_full_rank, zero_determinant_does_not_determine_exact_rank)；依完整明示資料重算；非零minor只給下界且上界需完整RREF或全部較大minor；det等於零只排除full rank；row-rank等於column-rank依定理。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["矩陣秩"]},"variant":{"family_id":"qf-mod34-rank-determinant-zero-boundary","seed":202611031083,"parameters":{"matrix":[[1,8],[2,16]],"domain":"Exact_square_matrix_rank","operation":"rank-determinant-zero-boundary","semantic_template":"qf-mod34-rank-determinant-zero-boundary/{\"matrix\":[[1,8],[2,16]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean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-20261103-r01-matrix-rank-0084","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix-rank"],"title_zh":"矩陣秩","stem_zh":"2×2或3×3完整矩陣與精確rank-preserving消去。操作=rank-determinant-zero-boundary；資料={\"matrix\":[[1,9],[2,18]]}。請依 [determinant,zero,rank,consequence,boundary] 欄序回答。","answer":{"kind":"expression","value":"(0, 是, 1, det_zero_only_excludes_full_rank, zero_determinant_does_not_determine_exact_rank)","explanation_zh":"Wave5第三層核對=(0, 是, 1, det_zero_only_excludes_full_rank, zero_determinant_does_not_determine_exact_rank)；依完整明示資料重算；非零minor只給下界且上界需完整RREF或全部較大minor；det等於零只排除full rank；row-rank等於column-rank依定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["矩陣秩"]},"variant":{"family_id":"qf-mod34-rank-determinant-zero-boundary","seed":202611031084,"parameters":{"matrix":[[1,9],[2,18]],"domain":"Exact_square_matrix_rank","operation":"rank-determinant-zero-boundary","semantic_template":"qf-mod34-rank-determinant-zero-boundary/{\"matrix\":[[1,9],[2,18]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean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-20261103-r01-matrix-rank-0085","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix-rank"],"title_zh":"矩陣秩","stem_zh":"2×2或3×3完整矩陣與精確rank-preserving消去。操作=rank-determinant-zero-boundary；資料={\"matrix\":[[1,10],[2,20]]}。請依 [determinant,zero,rank,consequence,boundary] 欄序回答。","answer":{"kind":"expression","value":"(0, 是, 1, det_zero_only_excludes_full_rank, zero_determinant_does_not_determine_exact_rank)","explanation_zh":"Wave5第三層核對=(0, 是, 1, det_zero_only_excludes_full_rank, zero_determinant_does_not_determine_exact_rank)；依完整明示資料重算；非零minor只給下界且上界需完整RREF或全部較大minor；det等於零只排除full rank；row-rank等於column-rank依定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["矩陣秩"]},"variant":{"family_id":"qf-mod34-rank-determinant-zero-boundary","seed":202611031085,"parameters":{"matrix":[[1,10],[2,20]],"domain":"Exact_square_matrix_rank","operation":"rank-determinant-zero-boundary","semantic_template":"qf-mod34-rank-determinant-zero-boundary/{\"matrix\":[[1,10],[2,20]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean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-20261103-r01-matrix-rank-0086","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix-rank"],"title_zh":"矩陣秩","stem_zh":"2×2或3×3完整矩陣與精確rank-preserving消去。操作=rank-determinant-zero-boundary；資料={\"matrix\":[[1,11],[2,22]]}。請依 [determinant,zero,rank,consequence,boundary] 欄序回答。","answer":{"kind":"expression","value":"(0, 是, 1, det_zero_only_excludes_full_rank, zero_determinant_does_not_determine_exact_rank)","explanation_zh":"Wave5第三層核對=(0, 是, 1, det_zero_only_excludes_full_rank, zero_determinant_does_not_determine_exact_rank)；依完整明示資料重算；非零minor只給下界且上界需完整RREF或全部較大minor；det等於零只排除full rank；row-rank等於column-rank依定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["矩陣秩"]},"variant":{"family_id":"qf-mod34-rank-determinant-zero-boundary","seed":202611031086,"parameters":{"matrix":[[1,11],[2,22]],"domain":"Exact_square_matrix_rank","operation":"rank-determinant-zero-boundary","semantic_template":"qf-mod34-rank-determinant-zero-boundary/{\"matrix\":[[1,11],[2,22]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean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-20261103-r01-matrix-rank-0087","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix-rank"],"title_zh":"矩陣秩","stem_zh":"2×2或3×3完整矩陣與精確rank-preserving消去。操作=rank-determinant-zero-boundary；資料={\"matrix\":[[1,12],[2,24]]}。請依 [determinant,zero,rank,consequence,boundary] 欄序回答。","answer":{"kind":"expression","value":"(0, 是, 1, det_zero_only_excludes_full_rank, zero_determinant_does_not_determine_exact_rank)","explanation_zh":"Wave5第三層核對=(0, 是, 1, det_zero_only_excludes_full_rank, zero_determinant_does_not_determine_exact_rank)；依完整明示資料重算；非零minor只給下界且上界需完整RREF或全部較大minor；det等於零只排除full rank；row-rank等於column-rank依定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["矩陣秩"]},"variant":{"family_id":"qf-mod34-rank-determinant-zero-boundary","seed":202611031087,"parameters":{"matrix":[[1,12],[2,24]],"domain":"Exact_square_matrix_rank","operation":"rank-determinant-zero-boundary","semantic_template":"qf-mod34-rank-determinant-zero-boundary/{\"matrix\":[[1,12],[2,24]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean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-20261103-r01-matrix-rank-0088","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix-rank"],"title_zh":"矩陣秩","stem_zh":"2×2或3×3完整矩陣與精確rank-preserving消去。操作=rank-determinant-zero-boundary；資料={\"matrix\":[[1,13],[2,26]]}。請依 [determinant,zero,rank,consequence,boundary] 欄序回答。","answer":{"kind":"expression","value":"(0, 是, 1, det_zero_only_excludes_full_rank, zero_determinant_does_not_determine_exact_rank)","explanation_zh":"Wave5第三層核對=(0, 是, 1, det_zero_only_excludes_full_rank, zero_determinant_does_not_determine_exact_rank)；依完整明示資料重算；非零minor只給下界且上界需完整RREF或全部較大minor；det等於零只排除full rank；row-rank等於column-rank依定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["矩陣秩"]},"variant":{"family_id":"qf-mod34-rank-determinant-zero-boundary","seed":202611031088,"parameters":{"matrix":[[1,13],[2,26]],"domain":"Exact_square_matrix_rank","operation":"rank-determinant-zero-boundary","semantic_template":"qf-mod34-rank-determinant-zero-boundary/{\"matrix\":[[1,13],[2,26]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean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-20261103-r01-matrix-rank-0089","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix-rank"],"title_zh":"矩陣秩","stem_zh":"2×2或3×3完整矩陣與精確rank-preserving消去。操作=rank-determinant-zero-boundary；資料={\"matrix\":[[1,14],[2,28]]}。請依 [determinant,zero,rank,consequence,boundary] 欄序回答。","answer":{"kind":"expression","value":"(0, 是, 1, det_zero_only_excludes_full_rank, zero_determinant_does_not_determine_exact_rank)","explanation_zh":"Wave5第三層核對=(0, 是, 1, det_zero_only_excludes_full_rank, zero_determinant_does_not_determine_exact_rank)；依完整明示資料重算；非零minor只給下界且上界需完整RREF或全部較大minor；det等於零只排除full rank；row-rank等於column-rank依定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["矩陣秩"]},"variant":{"family_id":"qf-mod34-rank-determinant-zero-boundary","seed":202611031089,"parameters":{"matrix":[[1,14],[2,28]],"domain":"Exact_square_matrix_rank","operation":"rank-determinant-zero-boundary","semantic_template":"qf-mod34-rank-determinant-zero-boundary/{\"matrix\":[[1,14],[2,28]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean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-20261103-r01-matrix-rank-0090","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix-rank"],"title_zh":"矩陣秩","stem_zh":"2×2或3×3完整矩陣與精確rank-preserving消去。操作=rank-determinant-zero-boundary；資料={\"matrix\":[[1,15],[2,30]]}。請依 [determinant,zero,rank,consequence,boundary] 欄序回答。","answer":{"kind":"expression","value":"(0, 是, 1, det_zero_only_excludes_full_rank, zero_determinant_does_not_determine_exact_rank)","explanation_zh":"Wave5第三層核對=(0, 是, 1, det_zero_only_excludes_full_rank, zero_determinant_does_not_determine_exact_rank)；依完整明示資料重算；非零minor只給下界且上界需完整RREF或全部較大minor；det等於零只排除full rank；row-rank等於column-rank依定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["矩陣秩"]},"variant":{"family_id":"qf-mod34-rank-determinant-zero-boundary","seed":202611031090,"parameters":{"matrix":[[1,15],[2,30]],"domain":"Exact_square_matrix_rank","operation":"rank-determinant-zero-boundary","semantic_template":"qf-mod34-rank-determinant-zero-boundary/{\"matrix\":[[1,15],[2,30]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean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-20261103-r01-matrix-rank-0091","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix-rank"],"title_zh":"矩陣秩","stem_zh":"2×2或3×3完整矩陣與精確rank-preserving消去。操作=rank-determinant-zero-boundary；資料={\"matrix\":[[1,16],[2,32]]}。請依 [determinant,zero,rank,consequence,boundary] 欄序回答。","answer":{"kind":"expression","value":"(0, 是, 1, det_zero_only_excludes_full_rank, zero_determinant_does_not_determine_exact_rank)","explanation_zh":"Wave5第三層核對=(0, 是, 1, det_zero_only_excludes_full_rank, zero_determinant_does_not_determine_exact_rank)；依完整明示資料重算；非零minor只給下界且上界需完整RREF或全部較大minor；det等於零只排除full rank；row-rank等於column-rank依定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["矩陣秩"]},"variant":{"family_id":"qf-mod34-rank-determinant-zero-boundary","seed":202611031091,"parameters":{"matrix":[[1,16],[2,32]],"domain":"Exact_square_matrix_rank","operation":"rank-determinant-zero-boundary","semantic_template":"qf-mod34-rank-determinant-zero-boundary/{\"matrix\":[[1,16],[2,32]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean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-20261103-r01-matrix-rank-0092","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix-rank"],"title_zh":"矩陣秩","stem_zh":"2×2或3×3完整矩陣與精確rank-preserving消去。操作=rank-determinant-zero-boundary；資料={\"matrix\":[[1,17],[2,34]]}。請依 [determinant,zero,rank,consequence,boundary] 欄序回答。","answer":{"kind":"expression","value":"(0, 是, 1, det_zero_only_excludes_full_rank, zero_determinant_does_not_determine_exact_rank)","explanation_zh":"Wave5第三層核對=(0, 是, 1, det_zero_only_excludes_full_rank, zero_determinant_does_not_determine_exact_rank)；依完整明示資料重算；非零minor只給下界且上界需完整RREF或全部較大minor；det等於零只排除full rank；row-rank等於column-rank依定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["矩陣秩"]},"variant":{"family_id":"qf-mod34-rank-determinant-zero-boundary","seed":202611031092,"parameters":{"matrix":[[1,17],[2,34]],"domain":"Exact_square_matrix_rank","operation":"rank-determinant-zero-boundary","semantic_template":"qf-mod34-rank-determinant-zero-boundary/{\"matrix\":[[1,17],[2,34]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean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-20261103-r01-matrix-rank-0093","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix-rank"],"title_zh":"矩陣秩","stem_zh":"2×2或3×3完整矩陣與精確rank-preserving消去。操作=rank-determinant-zero-boundary；資料={\"matrix\":[[1,18],[2,36]]}。請依 [determinant,zero,rank,consequence,boundary] 欄序回答。","answer":{"kind":"expression","value":"(0, 是, 1, det_zero_only_excludes_full_rank, zero_determinant_does_not_determine_exact_rank)","explanation_zh":"Wave5第三層核對=(0, 是, 1, det_zero_only_excludes_full_rank, zero_determinant_does_not_determine_exact_rank)；依完整明示資料重算；非零minor只給下界且上界需完整RREF或全部較大minor；det等於零只排除full rank；row-rank等於column-rank依定理。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["矩陣秩"]},"variant":{"family_id":"qf-mod34-rank-determinant-zero-boundary","seed":202611031093,"parameters":{"matrix":[[1,18],[2,36]],"domain":"Exact_square_matrix_rank","operation":"rank-determinant-zero-boundary","semantic_template":"qf-mod34-rank-determinant-zero-boundary/{\"matrix\":[[1,18],[2,36]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean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-20261103-r01-matrix-rank-0094","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix-rank"],"title_zh":"矩陣秩","stem_zh":"2×2或3×3完整矩陣與精確rank-preserving消去。操作=rank-determinant-zero-boundary；資料={\"matrix\":[[1,19],[2,38]]}。請依 [determinant,zero,rank,consequence,boundary] 欄序回答。","answer":{"kind":"expression","value":"(0, 是, 1, det_zero_only_excludes_full_rank, zero_determinant_does_not_determine_exact_rank)","explanation_zh":"Wave5第三層核對=(0, 是, 1, det_zero_only_excludes_full_rank, zero_determinant_does_not_determine_exact_rank)；依完整明示資料重算；非零minor只給下界且上界需完整RREF或全部較大minor；det等於零只排除full rank；row-rank等於column-rank依定理。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["矩陣秩"]},"variant":{"family_id":"qf-mod34-rank-determinant-zero-boundary","seed":202611031094,"parameters":{"matrix":[[1,19],[2,38]],"domain":"Exact_square_matrix_rank","operation":"rank-determinant-zero-boundary","semantic_template":"qf-mod34-rank-determinant-zero-boundary/{\"matrix\":[[1,19],[2,38]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean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-20261103-r01-matrix-rank-0095","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix-rank"],"title_zh":"矩陣秩","stem_zh":"2×2或3×3完整矩陣與精確rank-preserving消去。操作=rank-determinant-zero-boundary；資料={\"matrix\":[[1,20],[2,40]]}。請依 [determinant,zero,rank,consequence,boundary] 欄序回答。","answer":{"kind":"expression","value":"(0, 是, 1, det_zero_only_excludes_full_rank, zero_determinant_does_not_determine_exact_rank)","explanation_zh":"Wave5第三層核對=(0, 是, 1, det_zero_only_excludes_full_rank, zero_determinant_does_not_determine_exact_rank)；依完整明示資料重算；非零minor只給下界且上界需完整RREF或全部較大minor；det等於零只排除full rank；row-rank等於column-rank依定理。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["矩陣秩"]},"variant":{"family_id":"qf-mod34-rank-determinant-zero-boundary","seed":202611031095,"parameters":{"matrix":[[1,20],[2,40]],"domain":"Exact_square_matrix_rank","operation":"rank-determinant-zero-boundary","semantic_template":"qf-mod34-rank-determinant-zero-boundary/{\"matrix\":[[1,20],[2,40]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean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-20261103-r01-matrix-rank-0096","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix-rank"],"title_zh":"矩陣秩","stem_zh":"2×2或3×3完整矩陣與精確rank-preserving消去。操作=rank-determinant-zero-boundary；資料={\"matrix\":[[1,21],[2,42]]}。請依 [determinant,zero,rank,consequence,boundary] 欄序回答。","answer":{"kind":"expression","value":"(0, 是, 1, det_zero_only_excludes_full_rank, zero_determinant_does_not_determine_exact_rank)","explanation_zh":"Wave5第三層核對=(0, 是, 1, det_zero_only_excludes_full_rank, zero_determinant_does_not_determine_exact_rank)；依完整明示資料重算；非零minor只給下界且上界需完整RREF或全部較大minor；det等於零只排除full rank；row-rank等於column-rank依定理。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["矩陣秩"]},"variant":{"family_id":"qf-mod34-rank-determinant-zero-boundary","seed":202611031096,"parameters":{"matrix":[[1,21],[2,42]],"domain":"Exact_square_matrix_rank","operation":"rank-determinant-zero-boundary","semantic_template":"qf-mod34-rank-determinant-zero-boundary/{\"matrix\":[[1,21],[2,42]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean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-20261103-r01-matrix-rank-0097","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix-rank"],"title_zh":"矩陣秩","stem_zh":"2×2或3×3完整矩陣與精確rank-preserving消去。操作=rank-determinant-zero-boundary；資料={\"matrix\":[[1,22],[2,44]]}。請依 [determinant,zero,rank,consequence,boundary] 欄序回答。","answer":{"kind":"expression","value":"(0, 是, 1, det_zero_only_excludes_full_rank, zero_determinant_does_not_determine_exact_rank)","explanation_zh":"Wave5第三層核對=(0, 是, 1, det_zero_only_excludes_full_rank, zero_determinant_does_not_determine_exact_rank)；依完整明示資料重算；非零minor只給下界且上界需完整RREF或全部較大minor；det等於零只排除full rank；row-rank等於column-rank依定理。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["矩陣秩"]},"variant":{"family_id":"qf-mod34-rank-determinant-zero-boundary","seed":202611031097,"parameters":{"matrix":[[1,22],[2,44]],"domain":"Exact_square_matrix_rank","operation":"rank-determinant-zero-boundary","semantic_template":"qf-mod34-rank-determinant-zero-boundary/{\"matrix\":[[1,22],[2,44]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean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-20261103-r01-matrix-rank-0098","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix-rank"],"title_zh":"矩陣秩","stem_zh":"2×2或3×3完整矩陣與精確rank-preserving消去。操作=rank-determinant-zero-boundary；資料={\"matrix\":[[1,23],[2,46]]}。請依 [determinant,zero,rank,consequence,boundary] 欄序回答。","answer":{"kind":"expression","value":"(0, 是, 1, det_zero_only_excludes_full_rank, zero_determinant_does_not_determine_exact_rank)","explanation_zh":"Wave5第三層核對=(0, 是, 1, det_zero_only_excludes_full_rank, zero_determinant_does_not_determine_exact_rank)；依完整明示資料重算；非零minor只給下界且上界需完整RREF或全部較大minor；det等於零只排除full rank；row-rank等於column-rank依定理。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["矩陣秩"]},"variant":{"family_id":"qf-mod34-rank-determinant-zero-boundary","seed":202611031098,"parameters":{"matrix":[[1,23],[2,46]],"domain":"Exact_square_matrix_rank","operation":"rank-determinant-zero-boundary","semantic_template":"qf-mod34-rank-determinant-zero-boundary/{\"matrix\":[[1,23],[2,46]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean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-20261103-r01-matrix-rank-0099","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix-rank"],"title_zh":"矩陣秩","stem_zh":"2×2或3×3完整矩陣與精確rank-preserving消去。操作=rank-determinant-zero-boundary；資料={\"matrix\":[[1,24],[2,48]]}。請依 [determinant,zero,rank,consequence,boundary] 欄序回答。","answer":{"kind":"expression","value":"(0, 是, 1, det_zero_only_excludes_full_rank, zero_determinant_does_not_determine_exact_rank)","explanation_zh":"Wave5第三層核對=(0, 是, 1, det_zero_only_excludes_full_rank, zero_determinant_does_not_determine_exact_rank)；依完整明示資料重算；非零minor只給下界且上界需完整RREF或全部較大minor；det等於零只排除full rank；row-rank等於column-rank依定理。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["矩陣秩"]},"variant":{"family_id":"qf-mod34-rank-determinant-zero-boundary","seed":202611031099,"parameters":{"matrix":[[1,24],[2,48]],"domain":"Exact_square_matrix_rank","operation":"rank-determinant-zero-boundary","semantic_template":"qf-mod34-rank-determinant-zero-boundary/{\"matrix\":[[1,24],[2,48]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean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-20261103-r01-matrix-rank-0100","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix-rank"],"title_zh":"矩陣秩","stem_zh":"2×2或3×3完整矩陣與精確rank-preserving消去。操作=rank-determinant-zero-boundary；資料={\"matrix\":[[1,25],[2,50]]}。請依 [determinant,zero,rank,consequence,boundary] 欄序回答。","answer":{"kind":"expression","value":"(0, 是, 1, det_zero_only_excludes_full_rank, zero_determinant_does_not_determine_exact_rank)","explanation_zh":"Wave5第三層核對=(0, 是, 1, det_zero_only_excludes_full_rank, zero_determinant_does_not_determine_exact_rank)；依完整明示資料重算；非零minor只給下界且上界需完整RREF或全部較大minor；det等於零只排除full rank；row-rank等於column-rank依定理。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["矩陣秩"]},"variant":{"family_id":"qf-mod34-rank-determinant-zero-boundary","seed":202611031100,"parameters":{"matrix":[[1,25],[2,50]],"domain":"Exact_square_matrix_rank","operation":"rank-determinant-zero-boundary","semantic_template":"qf-mod34-rank-determinant-zero-boundary/{\"matrix\":[[1,25],[2,50]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean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"}}