{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0001","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-rank-nullity-theorem"],"title_zh":"秩零度定理","stem_zh":"明示m×n矩陣代表T:R^n到R^m並使用exact RREF。操作=ranknull-qualification；資料={\"matrix\":[[1,0,0],[0,1,1]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, 3, [[1, 0, 0], [0, 1, 1]], [0, 1], [2], 2, 1)","explanation_zh":"Wave5第四層核對=(2, 3, [[1, 0, 0], [0, 1, 1]], [0, 1], [2], 2, 1)；依完整明示資料重算；n等於kernel dimension加rank且nullity不是解的個數；kernel image rank皆須symbolic certificate；sample不足。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["秩零度定理"]},"variant":{"family_id":"qf-mod35-ranknull-qualification","seed":202611041001,"parameters":{"matrix":[[1,0,0],[0,1,1]],"domain":"Exact_finite_matrix_rank_nullity","operation":"ranknull-qualification","semantic_template":"qf-mod35-ranknull-qualification/{\"matrix\":[[1,0,0],[0,1,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0002","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-rank-nullity-theorem"],"title_zh":"秩零度定理","stem_zh":"明示m×n矩陣代表T:R^n到R^m並使用exact RREF。操作=ranknull-qualification；資料={\"matrix\":[[1,0,1],[0,1,1]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, 3, [[1, 0, 1], [0, 1, 1]], [0, 1], [2], 2, 1)","explanation_zh":"Wave5第四層核對=(2, 3, [[1, 0, 1], [0, 1, 1]], [0, 1], [2], 2, 1)；依完整明示資料重算；n等於kernel dimension加rank且nullity不是解的個數；kernel image rank皆須symbolic certificate；sample不足。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["秩零度定理"]},"variant":{"family_id":"qf-mod35-ranknull-qualification","seed":202611041002,"parameters":{"matrix":[[1,0,1],[0,1,1]],"domain":"Exact_finite_matrix_rank_nullity","operation":"ranknull-qualification","semantic_template":"qf-mod35-ranknull-qualification/{\"matrix\":[[1,0,1],[0,1,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0003","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-rank-nullity-theorem"],"title_zh":"秩零度定理","stem_zh":"明示m×n矩陣代表T:R^n到R^m並使用exact RREF。操作=ranknull-qualification；資料={\"matrix\":[[1,0,2],[0,1,1]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, 3, [[1, 0, 2], [0, 1, 1]], [0, 1], [2], 2, 1)","explanation_zh":"Wave5第四層核對=(2, 3, [[1, 0, 2], [0, 1, 1]], [0, 1], [2], 2, 1)；依完整明示資料重算；n等於kernel dimension加rank且nullity不是解的個數；kernel image rank皆須symbolic certificate；sample不足。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["秩零度定理"]},"variant":{"family_id":"qf-mod35-ranknull-qualification","seed":202611041003,"parameters":{"matrix":[[1,0,2],[0,1,1]],"domain":"Exact_finite_matrix_rank_nullity","operation":"ranknull-qualification","semantic_template":"qf-mod35-ranknull-qualification/{\"matrix\":[[1,0,2],[0,1,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0004","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-rank-nullity-theorem"],"title_zh":"秩零度定理","stem_zh":"明示m×n矩陣代表T:R^n到R^m並使用exact RREF。操作=ranknull-qualification；資料={\"matrix\":[[1,0,3],[0,1,1]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, 3, [[1, 0, 3], [0, 1, 1]], [0, 1], [2], 2, 1)","explanation_zh":"Wave5第四層核對=(2, 3, [[1, 0, 3], [0, 1, 1]], [0, 1], [2], 2, 1)；依完整明示資料重算；n等於kernel dimension加rank且nullity不是解的個數；kernel image rank皆須symbolic certificate；sample不足。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["秩零度定理"]},"variant":{"family_id":"qf-mod35-ranknull-qualification","seed":202611041004,"parameters":{"matrix":[[1,0,3],[0,1,1]],"domain":"Exact_finite_matrix_rank_nullity","operation":"ranknull-qualification","semantic_template":"qf-mod35-ranknull-qualification/{\"matrix\":[[1,0,3],[0,1,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0005","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-rank-nullity-theorem"],"title_zh":"秩零度定理","stem_zh":"明示m×n矩陣代表T:R^n到R^m並使用exact RREF。操作=ranknull-qualification；資料={\"matrix\":[[1,0,4],[0,1,1]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, 3, [[1, 0, 4], [0, 1, 1]], [0, 1], [2], 2, 1)","explanation_zh":"Wave5第四層核對=(2, 3, [[1, 0, 4], [0, 1, 1]], [0, 1], [2], 2, 1)；依完整明示資料重算；n等於kernel dimension加rank且nullity不是解的個數；kernel image rank皆須symbolic certificate；sample不足。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["秩零度定理"]},"variant":{"family_id":"qf-mod35-ranknull-qualification","seed":202611041005,"parameters":{"matrix":[[1,0,4],[0,1,1]],"domain":"Exact_finite_matrix_rank_nullity","operation":"ranknull-qualification","semantic_template":"qf-mod35-ranknull-qualification/{\"matrix\":[[1,0,4],[0,1,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0006","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-rank-nullity-theorem"],"title_zh":"秩零度定理","stem_zh":"明示m×n矩陣代表T:R^n到R^m並使用exact RREF。操作=ranknull-qualification；資料={\"matrix\":[[1,0,5],[0,1,1]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, 3, [[1, 0, 5], [0, 1, 1]], [0, 1], [2], 2, 1)","explanation_zh":"Wave5第四層核對=(2, 3, [[1, 0, 5], [0, 1, 1]], [0, 1], [2], 2, 1)；依完整明示資料重算；n等於kernel dimension加rank且nullity不是解的個數；kernel image rank皆須symbolic certificate；sample不足。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["秩零度定理"]},"variant":{"family_id":"qf-mod35-ranknull-qualification","seed":202611041006,"parameters":{"matrix":[[1,0,5],[0,1,1]],"domain":"Exact_finite_matrix_rank_nullity","operation":"ranknull-qualification","semantic_template":"qf-mod35-ranknull-qualification/{\"matrix\":[[1,0,5],[0,1,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0007","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-rank-nullity-theorem"],"title_zh":"秩零度定理","stem_zh":"明示m×n矩陣代表T:R^n到R^m並使用exact RREF。操作=ranknull-qualification；資料={\"matrix\":[[1,0,6],[0,1,1]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, 3, [[1, 0, 6], [0, 1, 1]], [0, 1], [2], 2, 1)","explanation_zh":"Wave5第四層核對=(2, 3, [[1, 0, 6], [0, 1, 1]], [0, 1], [2], 2, 1)；依完整明示資料重算；n等於kernel dimension加rank且nullity不是解的個數；kernel image rank皆須symbolic certificate；sample不足。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["秩零度定理"]},"variant":{"family_id":"qf-mod35-ranknull-qualification","seed":202611041007,"parameters":{"matrix":[[1,0,6],[0,1,1]],"domain":"Exact_finite_matrix_rank_nullity","operation":"ranknull-qualification","semantic_template":"qf-mod35-ranknull-qualification/{\"matrix\":[[1,0,6],[0,1,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0008","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-rank-nullity-theorem"],"title_zh":"秩零度定理","stem_zh":"明示m×n矩陣代表T:R^n到R^m並使用exact RREF。操作=ranknull-qualification；資料={\"matrix\":[[1,0,7],[0,1,1]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, 3, [[1, 0, 7], [0, 1, 1]], [0, 1], [2], 2, 1)","explanation_zh":"Wave5第四層核對=(2, 3, [[1, 0, 7], [0, 1, 1]], [0, 1], [2], 2, 1)；依完整明示資料重算；n等於kernel dimension加rank且nullity不是解的個數；kernel image rank皆須symbolic certificate；sample不足。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["秩零度定理"]},"variant":{"family_id":"qf-mod35-ranknull-qualification","seed":202611041008,"parameters":{"matrix":[[1,0,7],[0,1,1]],"domain":"Exact_finite_matrix_rank_nullity","operation":"ranknull-qualification","semantic_template":"qf-mod35-ranknull-qualification/{\"matrix\":[[1,0,7],[0,1,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0009","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-rank-nullity-theorem"],"title_zh":"秩零度定理","stem_zh":"明示m×n矩陣代表T:R^n到R^m並使用exact RREF。操作=ranknull-qualification；資料={\"matrix\":[[1,0,8],[0,1,1]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, 3, [[1, 0, 8], [0, 1, 1]], [0, 1], [2], 2, 1)","explanation_zh":"Wave5第四層核對=(2, 3, [[1, 0, 8], [0, 1, 1]], [0, 1], [2], 2, 1)；依完整明示資料重算；n等於kernel dimension加rank且nullity不是解的個數；kernel image rank皆須symbolic certificate；sample不足。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["秩零度定理"]},"variant":{"family_id":"qf-mod35-ranknull-qualification","seed":202611041009,"parameters":{"matrix":[[1,0,8],[0,1,1]],"domain":"Exact_finite_matrix_rank_nullity","operation":"ranknull-qualification","semantic_template":"qf-mod35-ranknull-qualification/{\"matrix\":[[1,0,8],[0,1,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0010","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-rank-nullity-theorem"],"title_zh":"秩零度定理","stem_zh":"明示m×n矩陣代表T:R^n到R^m並使用exact RREF。操作=ranknull-qualification；資料={\"matrix\":[[1,0,9],[0,1,1]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, 3, [[1, 0, 9], [0, 1, 1]], [0, 1], [2], 2, 1)","explanation_zh":"Wave5第四層核對=(2, 3, [[1, 0, 9], [0, 1, 1]], [0, 1], [2], 2, 1)；依完整明示資料重算；n等於kernel dimension加rank且nullity不是解的個數；kernel image rank皆須symbolic certificate；sample不足。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["秩零度定理"]},"variant":{"family_id":"qf-mod35-ranknull-qualification","seed":202611041010,"parameters":{"matrix":[[1,0,9],[0,1,1]],"domain":"Exact_finite_matrix_rank_nullity","operation":"ranknull-qualification","semantic_template":"qf-mod35-ranknull-qualification/{\"matrix\":[[1,0,9],[0,1,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0011","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-rank-nullity-theorem"],"title_zh":"秩零度定理","stem_zh":"明示m×n矩陣代表T:R^n到R^m並使用exact RREF。操作=ranknull-qualification；資料={\"matrix\":[[1,0,10],[0,1,1]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, 3, [[1, 0, 10], [0, 1, 1]], [0, 1], [2], 2, 1)","explanation_zh":"Wave5第四層核對=(2, 3, [[1, 0, 10], [0, 1, 1]], [0, 1], [2], 2, 1)；依完整明示資料重算；n等於kernel dimension加rank且nullity不是解的個數；kernel image rank皆須symbolic certificate；sample不足。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["秩零度定理"]},"variant":{"family_id":"qf-mod35-ranknull-qualification","seed":202611041011,"parameters":{"matrix":[[1,0,10],[0,1,1]],"domain":"Exact_finite_matrix_rank_nullity","operation":"ranknull-qualification","semantic_template":"qf-mod35-ranknull-qualification/{\"matrix\":[[1,0,10],[0,1,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0012","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-rank-nullity-theorem"],"title_zh":"秩零度定理","stem_zh":"明示m×n矩陣代表T:R^n到R^m並使用exact RREF。操作=ranknull-qualification；資料={\"matrix\":[[1,0,11],[0,1,1]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, 3, [[1, 0, 11], [0, 1, 1]], [0, 1], [2], 2, 1)","explanation_zh":"Wave5第四層核對=(2, 3, [[1, 0, 11], [0, 1, 1]], [0, 1], [2], 2, 1)；依完整明示資料重算；n等於kernel dimension加rank且nullity不是解的個數；kernel image rank皆須symbolic certificate；sample不足。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["秩零度定理"]},"variant":{"family_id":"qf-mod35-ranknull-qualification","seed":202611041012,"parameters":{"matrix":[[1,0,11],[0,1,1]],"domain":"Exact_finite_matrix_rank_nullity","operation":"ranknull-qualification","semantic_template":"qf-mod35-ranknull-qualification/{\"matrix\":[[1,0,11],[0,1,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0013","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-rank-nullity-theorem"],"title_zh":"秩零度定理","stem_zh":"明示m×n矩陣代表T:R^n到R^m並使用exact RREF。操作=ranknull-qualification；資料={\"matrix\":[[1,0,12],[0,1,1]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, 3, [[1, 0, 12], [0, 1, 1]], [0, 1], [2], 2, 1)","explanation_zh":"Wave5第四層核對=(2, 3, [[1, 0, 12], [0, 1, 1]], [0, 1], [2], 2, 1)；依完整明示資料重算；n等於kernel dimension加rank且nullity不是解的個數；kernel image rank皆須symbolic certificate；sample不足。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["秩零度定理"]},"variant":{"family_id":"qf-mod35-ranknull-qualification","seed":202611041013,"parameters":{"matrix":[[1,0,12],[0,1,1]],"domain":"Exact_finite_matrix_rank_nullity","operation":"ranknull-qualification","semantic_template":"qf-mod35-ranknull-qualification/{\"matrix\":[[1,0,12],[0,1,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0014","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-rank-nullity-theorem"],"title_zh":"秩零度定理","stem_zh":"明示m×n矩陣代表T:R^n到R^m並使用exact RREF。操作=ranknull-qualification；資料={\"matrix\":[[1,0,13],[0,1,1]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, 3, [[1, 0, 13], [0, 1, 1]], [0, 1], [2], 2, 1)","explanation_zh":"Wave5第四層核對=(2, 3, [[1, 0, 13], [0, 1, 1]], [0, 1], [2], 2, 1)；依完整明示資料重算；n等於kernel dimension加rank且nullity不是解的個數；kernel image rank皆須symbolic certificate；sample不足。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["秩零度定理"]},"variant":{"family_id":"qf-mod35-ranknull-qualification","seed":202611041014,"parameters":{"matrix":[[1,0,13],[0,1,1]],"domain":"Exact_finite_matrix_rank_nullity","operation":"ranknull-qualification","semantic_template":"qf-mod35-ranknull-qualification/{\"matrix\":[[1,0,13],[0,1,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0015","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-rank-nullity-theorem"],"title_zh":"秩零度定理","stem_zh":"明示m×n矩陣代表T:R^n到R^m並使用exact RREF。操作=ranknull-qualification；資料={\"matrix\":[[1,0,14],[0,1,1]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, 3, [[1, 0, 14], [0, 1, 1]], [0, 1], [2], 2, 1)","explanation_zh":"Wave5第四層核對=(2, 3, [[1, 0, 14], [0, 1, 1]], [0, 1], [2], 2, 1)；依完整明示資料重算；n等於kernel dimension加rank且nullity不是解的個數；kernel image rank皆須symbolic certificate；sample不足。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["秩零度定理"]},"variant":{"family_id":"qf-mod35-ranknull-qualification","seed":202611041015,"parameters":{"matrix":[[1,0,14],[0,1,1]],"domain":"Exact_finite_matrix_rank_nullity","operation":"ranknull-qualification","semantic_template":"qf-mod35-ranknull-qualification/{\"matrix\":[[1,0,14],[0,1,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0016","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-rank-nullity-theorem"],"title_zh":"秩零度定理","stem_zh":"明示m×n矩陣代表T:R^n到R^m並使用exact RREF。操作=ranknull-qualification；資料={\"matrix\":[[1,0,15],[0,1,1]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, 3, [[1, 0, 15], [0, 1, 1]], [0, 1], [2], 2, 1)","explanation_zh":"Wave5第四層核對=(2, 3, [[1, 0, 15], [0, 1, 1]], [0, 1], [2], 2, 1)；依完整明示資料重算；n等於kernel dimension加rank且nullity不是解的個數；kernel image rank皆須symbolic certificate；sample不足。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["秩零度定理"]},"variant":{"family_id":"qf-mod35-ranknull-qualification","seed":202611041016,"parameters":{"matrix":[[1,0,15],[0,1,1]],"domain":"Exact_finite_matrix_rank_nullity","operation":"ranknull-qualification","semantic_template":"qf-mod35-ranknull-qualification/{\"matrix\":[[1,0,15],[0,1,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0017","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-rank-nullity-theorem"],"title_zh":"秩零度定理","stem_zh":"明示m×n矩陣代表T:R^n到R^m並使用exact RREF。操作=ranknull-qualification；資料={\"matrix\":[[1,0,16],[0,1,1]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, 3, [[1, 0, 16], [0, 1, 1]], [0, 1], [2], 2, 1)","explanation_zh":"Wave5第四層核對=(2, 3, [[1, 0, 16], [0, 1, 1]], [0, 1], [2], 2, 1)；依完整明示資料重算；n等於kernel dimension加rank且nullity不是解的個數；kernel image rank皆須symbolic certificate；sample不足。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["秩零度定理"]},"variant":{"family_id":"qf-mod35-ranknull-qualification","seed":202611041017,"parameters":{"matrix":[[1,0,16],[0,1,1]],"domain":"Exact_finite_matrix_rank_nullity","operation":"ranknull-qualification","semantic_template":"qf-mod35-ranknull-qualification/{\"matrix\":[[1,0,16],[0,1,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0018","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-rank-nullity-theorem"],"title_zh":"秩零度定理","stem_zh":"明示m×n矩陣代表T:R^n到R^m並使用exact RREF。操作=ranknull-qualification；資料={\"matrix\":[[1,0,17],[0,1,1]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, 3, [[1, 0, 17], [0, 1, 1]], [0, 1], [2], 2, 1)","explanation_zh":"Wave5第四層核對=(2, 3, [[1, 0, 17], [0, 1, 1]], [0, 1], [2], 2, 1)；依完整明示資料重算；n等於kernel dimension加rank且nullity不是解的個數；kernel image rank皆須symbolic certificate；sample不足。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["秩零度定理"]},"variant":{"family_id":"qf-mod35-ranknull-qualification","seed":202611041018,"parameters":{"matrix":[[1,0,17],[0,1,1]],"domain":"Exact_finite_matrix_rank_nullity","operation":"ranknull-qualification","semantic_template":"qf-mod35-ranknull-qualification/{\"matrix\":[[1,0,17],[0,1,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0019","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-rank-nullity-theorem"],"title_zh":"秩零度定理","stem_zh":"明示m×n矩陣代表T:R^n到R^m並使用exact RREF。操作=ranknull-qualification；資料={\"matrix\":[[1,0,18],[0,1,1]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, 3, [[1, 0, 18], [0, 1, 1]], [0, 1], [2], 2, 1)","explanation_zh":"Wave5第四層核對=(2, 3, [[1, 0, 18], [0, 1, 1]], [0, 1], [2], 2, 1)；依完整明示資料重算；n等於kernel dimension加rank且nullity不是解的個數；kernel image rank皆須symbolic certificate；sample不足。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["秩零度定理"]},"variant":{"family_id":"qf-mod35-ranknull-qualification","seed":202611041019,"parameters":{"matrix":[[1,0,18],[0,1,1]],"domain":"Exact_finite_matrix_rank_nullity","operation":"ranknull-qualification","semantic_template":"qf-mod35-ranknull-qualification/{\"matrix\":[[1,0,18],[0,1,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0020","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-rank-nullity-theorem"],"title_zh":"秩零度定理","stem_zh":"明示m×n矩陣代表T:R^n到R^m並使用exact RREF。操作=ranknull-qualification；資料={\"matrix\":[[1,0,19],[0,1,1]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, 3, [[1, 0, 19], [0, 1, 1]], [0, 1], [2], 2, 1)","explanation_zh":"Wave5第四層核對=(2, 3, [[1, 0, 19], [0, 1, 1]], [0, 1], [2], 2, 1)；依完整明示資料重算；n等於kernel dimension加rank且nullity不是解的個數；kernel image rank皆須symbolic certificate；sample不足。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["秩零度定理"]},"variant":{"family_id":"qf-mod35-ranknull-qualification","seed":202611041020,"parameters":{"matrix":[[1,0,19],[0,1,1]],"domain":"Exact_finite_matrix_rank_nullity","operation":"ranknull-qualification","semantic_template":"qf-mod35-ranknull-qualification/{\"matrix\":[[1,0,19],[0,1,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0021","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-rank-nullity-theorem"],"title_zh":"秩零度定理","stem_zh":"明示m×n矩陣代表T:R^n到R^m並使用exact RREF。操作=ranknull-qualification；資料={\"matrix\":[[1,0,20],[0,1,1]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, 3, [[1, 0, 20], [0, 1, 1]], [0, 1], [2], 2, 1)","explanation_zh":"Wave5第四層核對=(2, 3, [[1, 0, 20], [0, 1, 1]], [0, 1], [2], 2, 1)；依完整明示資料重算；n等於kernel dimension加rank且nullity不是解的個數；kernel image rank皆須symbolic certificate；sample不足。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["秩零度定理"]},"variant":{"family_id":"qf-mod35-ranknull-qualification","seed":202611041021,"parameters":{"matrix":[[1,0,20],[0,1,1]],"domain":"Exact_finite_matrix_rank_nullity","operation":"ranknull-qualification","semantic_template":"qf-mod35-ranknull-qualification/{\"matrix\":[[1,0,20],[0,1,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0022","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-rank-nullity-theorem"],"title_zh":"秩零度定理","stem_zh":"明示m×n矩陣代表T:R^n到R^m並使用exact RREF。操作=ranknull-qualification；資料={\"matrix\":[[1,0,21],[0,1,1]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, 3, [[1, 0, 21], [0, 1, 1]], [0, 1], [2], 2, 1)","explanation_zh":"Wave5第四層核對=(2, 3, [[1, 0, 21], [0, 1, 1]], [0, 1], [2], 2, 1)；依完整明示資料重算；n等於kernel dimension加rank且nullity不是解的個數；kernel image rank皆須symbolic certificate；sample不足。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["秩零度定理"]},"variant":{"family_id":"qf-mod35-ranknull-qualification","seed":202611041022,"parameters":{"matrix":[[1,0,21],[0,1,1]],"domain":"Exact_finite_matrix_rank_nullity","operation":"ranknull-qualification","semantic_template":"qf-mod35-ranknull-qualification/{\"matrix\":[[1,0,21],[0,1,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0023","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-rank-nullity-theorem"],"title_zh":"秩零度定理","stem_zh":"明示m×n矩陣代表T:R^n到R^m並使用exact RREF。操作=ranknull-qualification；資料={\"matrix\":[[1,0,22],[0,1,1]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, 3, [[1, 0, 22], [0, 1, 1]], [0, 1], [2], 2, 1)","explanation_zh":"Wave5第四層核對=(2, 3, [[1, 0, 22], [0, 1, 1]], [0, 1], [2], 2, 1)；依完整明示資料重算；n等於kernel dimension加rank且nullity不是解的個數；kernel image rank皆須symbolic certificate；sample不足。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["秩零度定理"]},"variant":{"family_id":"qf-mod35-ranknull-qualification","seed":202611041023,"parameters":{"matrix":[[1,0,22],[0,1,1]],"domain":"Exact_finite_matrix_rank_nullity","operation":"ranknull-qualification","semantic_template":"qf-mod35-ranknull-qualification/{\"matrix\":[[1,0,22],[0,1,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0024","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-rank-nullity-theorem"],"title_zh":"秩零度定理","stem_zh":"明示m×n矩陣代表T:R^n到R^m並使用exact RREF。操作=ranknull-qualification；資料={\"matrix\":[[1,0,23],[0,1,1]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, 3, [[1, 0, 23], [0, 1, 1]], [0, 1], [2], 2, 1)","explanation_zh":"Wave5第四層核對=(2, 3, [[1, 0, 23], [0, 1, 1]], [0, 1], [2], 2, 1)；依完整明示資料重算；n等於kernel dimension加rank且nullity不是解的個數；kernel image rank皆須symbolic certificate；sample不足。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["秩零度定理"]},"variant":{"family_id":"qf-mod35-ranknull-qualification","seed":202611041024,"parameters":{"matrix":[[1,0,23],[0,1,1]],"domain":"Exact_finite_matrix_rank_nullity","operation":"ranknull-qualification","semantic_template":"qf-mod35-ranknull-qualification/{\"matrix\":[[1,0,23],[0,1,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0025","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-rank-nullity-theorem"],"title_zh":"秩零度定理","stem_zh":"明示m×n矩陣代表T:R^n到R^m並使用exact RREF。操作=ranknull-qualification；資料={\"matrix\":[[1,0,24],[0,1,1]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, 3, [[1, 0, 24], [0, 1, 1]], [0, 1], [2], 2, 1)","explanation_zh":"Wave5第四層核對=(2, 3, [[1, 0, 24], [0, 1, 1]], [0, 1], [2], 2, 1)；依完整明示資料重算；n等於kernel dimension加rank且nullity不是解的個數；kernel image rank皆須symbolic certificate；sample不足。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["秩零度定理"]},"variant":{"family_id":"qf-mod35-ranknull-qualification","seed":202611041025,"parameters":{"matrix":[[1,0,24],[0,1,1]],"domain":"Exact_finite_matrix_rank_nullity","operation":"ranknull-qualification","semantic_template":"qf-mod35-ranknull-qualification/{\"matrix\":[[1,0,24],[0,1,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0026","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-rank-nullity-theorem"],"title_zh":"秩零度定理","stem_zh":"明示m×n矩陣代表T:R^n到R^m並使用exact RREF。操作=ranknull-dimension-identity；資料={\"matrix\":[[1,0,0],[0,1,1]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 1, 2, 3, 是)","explanation_zh":"Wave5第四層核對=(3, 1, 2, 3, 是)；依完整明示資料重算；n等於kernel dimension加rank且nullity不是解的個數；kernel image rank皆須symbolic certificate；sample不足。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["秩零度定理"]},"variant":{"family_id":"qf-mod35-ranknull-dimension-identity","seed":202611041026,"parameters":{"matrix":[[1,0,0],[0,1,1]],"domain":"Exact_finite_matrix_rank_nullity","operation":"ranknull-dimension-identity","semantic_template":"qf-mod35-ranknull-dimension-identity/{\"matrix\":[[1,0,0],[0,1,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0027","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-rank-nullity-theorem"],"title_zh":"秩零度定理","stem_zh":"明示m×n矩陣代表T:R^n到R^m並使用exact RREF。操作=ranknull-dimension-identity；資料={\"matrix\":[[1,0,1],[0,1,1]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 1, 2, 3, 是)","explanation_zh":"Wave5第四層核對=(3, 1, 2, 3, 是)；依完整明示資料重算；n等於kernel dimension加rank且nullity不是解的個數；kernel image rank皆須symbolic certificate；sample不足。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["秩零度定理"]},"variant":{"family_id":"qf-mod35-ranknull-dimension-identity","seed":202611041027,"parameters":{"matrix":[[1,0,1],[0,1,1]],"domain":"Exact_finite_matrix_rank_nullity","operation":"ranknull-dimension-identity","semantic_template":"qf-mod35-ranknull-dimension-identity/{\"matrix\":[[1,0,1],[0,1,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0028","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-rank-nullity-theorem"],"title_zh":"秩零度定理","stem_zh":"明示m×n矩陣代表T:R^n到R^m並使用exact RREF。操作=ranknull-dimension-identity；資料={\"matrix\":[[1,0,2],[0,1,1]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 1, 2, 3, 是)","explanation_zh":"Wave5第四層核對=(3, 1, 2, 3, 是)；依完整明示資料重算；n等於kernel dimension加rank且nullity不是解的個數；kernel image rank皆須symbolic certificate；sample不足。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["秩零度定理"]},"variant":{"family_id":"qf-mod35-ranknull-dimension-identity","seed":202611041028,"parameters":{"matrix":[[1,0,2],[0,1,1]],"domain":"Exact_finite_matrix_rank_nullity","operation":"ranknull-dimension-identity","semantic_template":"qf-mod35-ranknull-dimension-identity/{\"matrix\":[[1,0,2],[0,1,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0029","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-rank-nullity-theorem"],"title_zh":"秩零度定理","stem_zh":"明示m×n矩陣代表T:R^n到R^m並使用exact RREF。操作=ranknull-dimension-identity；資料={\"matrix\":[[1,0,3],[0,1,1]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 1, 2, 3, 是)","explanation_zh":"Wave5第四層核對=(3, 1, 2, 3, 是)；依完整明示資料重算；n等於kernel dimension加rank且nullity不是解的個數；kernel image rank皆須symbolic certificate；sample不足。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["秩零度定理"]},"variant":{"family_id":"qf-mod35-ranknull-dimension-identity","seed":202611041029,"parameters":{"matrix":[[1,0,3],[0,1,1]],"domain":"Exact_finite_matrix_rank_nullity","operation":"ranknull-dimension-identity","semantic_template":"qf-mod35-ranknull-dimension-identity/{\"matrix\":[[1,0,3],[0,1,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0030","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-rank-nullity-theorem"],"title_zh":"秩零度定理","stem_zh":"明示m×n矩陣代表T:R^n到R^m並使用exact RREF。操作=ranknull-dimension-identity；資料={\"matrix\":[[1,0,4],[0,1,1]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 1, 2, 3, 是)","explanation_zh":"Wave5第四層核對=(3, 1, 2, 3, 是)；依完整明示資料重算；n等於kernel dimension加rank且nullity不是解的個數；kernel image rank皆須symbolic certificate；sample不足。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["秩零度定理"]},"variant":{"family_id":"qf-mod35-ranknull-dimension-identity","seed":202611041030,"parameters":{"matrix":[[1,0,4],[0,1,1]],"domain":"Exact_finite_matrix_rank_nullity","operation":"ranknull-dimension-identity","semantic_template":"qf-mod35-ranknull-dimension-identity/{\"matrix\":[[1,0,4],[0,1,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0031","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-rank-nullity-theorem"],"title_zh":"秩零度定理","stem_zh":"明示m×n矩陣代表T:R^n到R^m並使用exact RREF。操作=ranknull-dimension-identity；資料={\"matrix\":[[1,0,5],[0,1,1]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 1, 2, 3, 是)","explanation_zh":"Wave5第四層核對=(3, 1, 2, 3, 是)；依完整明示資料重算；n等於kernel dimension加rank且nullity不是解的個數；kernel image rank皆須symbolic certificate；sample不足。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["秩零度定理"]},"variant":{"family_id":"qf-mod35-ranknull-dimension-identity","seed":202611041031,"parameters":{"matrix":[[1,0,5],[0,1,1]],"domain":"Exact_finite_matrix_rank_nullity","operation":"ranknull-dimension-identity","semantic_template":"qf-mod35-ranknull-dimension-identity/{\"matrix\":[[1,0,5],[0,1,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0032","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-rank-nullity-theorem"],"title_zh":"秩零度定理","stem_zh":"明示m×n矩陣代表T:R^n到R^m並使用exact RREF。操作=ranknull-dimension-identity；資料={\"matrix\":[[1,0,6],[0,1,1]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 1, 2, 3, 是)","explanation_zh":"Wave5第四層核對=(3, 1, 2, 3, 是)；依完整明示資料重算；n等於kernel dimension加rank且nullity不是解的個數；kernel image rank皆須symbolic certificate；sample不足。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["秩零度定理"]},"variant":{"family_id":"qf-mod35-ranknull-dimension-identity","seed":202611041032,"parameters":{"matrix":[[1,0,6],[0,1,1]],"domain":"Exact_finite_matrix_rank_nullity","operation":"ranknull-dimension-identity","semantic_template":"qf-mod35-ranknull-dimension-identity/{\"matrix\":[[1,0,6],[0,1,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0033","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-rank-nullity-theorem"],"title_zh":"秩零度定理","stem_zh":"明示m×n矩陣代表T:R^n到R^m並使用exact RREF。操作=ranknull-dimension-identity；資料={\"matrix\":[[1,0,7],[0,1,1]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 1, 2, 3, 是)","explanation_zh":"Wave5第四層核對=(3, 1, 2, 3, 是)；依完整明示資料重算；n等於kernel dimension加rank且nullity不是解的個數；kernel image rank皆須symbolic certificate；sample不足。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["秩零度定理"]},"variant":{"family_id":"qf-mod35-ranknull-dimension-identity","seed":202611041033,"parameters":{"matrix":[[1,0,7],[0,1,1]],"domain":"Exact_finite_matrix_rank_nullity","operation":"ranknull-dimension-identity","semantic_template":"qf-mod35-ranknull-dimension-identity/{\"matrix\":[[1,0,7],[0,1,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0034","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-rank-nullity-theorem"],"title_zh":"秩零度定理","stem_zh":"明示m×n矩陣代表T:R^n到R^m並使用exact RREF。操作=ranknull-dimension-identity；資料={\"matrix\":[[1,0,8],[0,1,1]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 1, 2, 3, 是)","explanation_zh":"Wave5第四層核對=(3, 1, 2, 3, 是)；依完整明示資料重算；n等於kernel dimension加rank且nullity不是解的個數；kernel image rank皆須symbolic certificate；sample不足。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["秩零度定理"]},"variant":{"family_id":"qf-mod35-ranknull-dimension-identity","seed":202611041034,"parameters":{"matrix":[[1,0,8],[0,1,1]],"domain":"Exact_finite_matrix_rank_nullity","operation":"ranknull-dimension-identity","semantic_template":"qf-mod35-ranknull-dimension-identity/{\"matrix\":[[1,0,8],[0,1,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0035","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-rank-nullity-theorem"],"title_zh":"秩零度定理","stem_zh":"明示m×n矩陣代表T:R^n到R^m並使用exact RREF。操作=ranknull-dimension-identity；資料={\"matrix\":[[1,0,9],[0,1,1]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 1, 2, 3, 是)","explanation_zh":"Wave5第四層核對=(3, 1, 2, 3, 是)；依完整明示資料重算；n等於kernel dimension加rank且nullity不是解的個數；kernel image rank皆須symbolic certificate；sample不足。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["秩零度定理"]},"variant":{"family_id":"qf-mod35-ranknull-dimension-identity","seed":202611041035,"parameters":{"matrix":[[1,0,9],[0,1,1]],"domain":"Exact_finite_matrix_rank_nullity","operation":"ranknull-dimension-identity","semantic_template":"qf-mod35-ranknull-dimension-identity/{\"matrix\":[[1,0,9],[0,1,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0036","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-rank-nullity-theorem"],"title_zh":"秩零度定理","stem_zh":"明示m×n矩陣代表T:R^n到R^m並使用exact RREF。操作=ranknull-dimension-identity；資料={\"matrix\":[[1,0,10],[0,1,1]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 1, 2, 3, 是)","explanation_zh":"Wave5第四層核對=(3, 1, 2, 3, 是)；依完整明示資料重算；n等於kernel dimension加rank且nullity不是解的個數；kernel image rank皆須symbolic certificate；sample不足。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["秩零度定理"]},"variant":{"family_id":"qf-mod35-ranknull-dimension-identity","seed":202611041036,"parameters":{"matrix":[[1,0,10],[0,1,1]],"domain":"Exact_finite_matrix_rank_nullity","operation":"ranknull-dimension-identity","semantic_template":"qf-mod35-ranknull-dimension-identity/{\"matrix\":[[1,0,10],[0,1,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0037","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-rank-nullity-theorem"],"title_zh":"秩零度定理","stem_zh":"明示m×n矩陣代表T:R^n到R^m並使用exact RREF。操作=ranknull-dimension-identity；資料={\"matrix\":[[1,0,11],[0,1,1]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 1, 2, 3, 是)","explanation_zh":"Wave5第四層核對=(3, 1, 2, 3, 是)；依完整明示資料重算；n等於kernel dimension加rank且nullity不是解的個數；kernel image rank皆須symbolic certificate；sample不足。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["秩零度定理"]},"variant":{"family_id":"qf-mod35-ranknull-dimension-identity","seed":202611041037,"parameters":{"matrix":[[1,0,11],[0,1,1]],"domain":"Exact_finite_matrix_rank_nullity","operation":"ranknull-dimension-identity","semantic_template":"qf-mod35-ranknull-dimension-identity/{\"matrix\":[[1,0,11],[0,1,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0038","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-rank-nullity-theorem"],"title_zh":"秩零度定理","stem_zh":"明示m×n矩陣代表T:R^n到R^m並使用exact RREF。操作=ranknull-dimension-identity；資料={\"matrix\":[[1,0,12],[0,1,1]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 1, 2, 3, 是)","explanation_zh":"Wave5第四層核對=(3, 1, 2, 3, 是)；依完整明示資料重算；n等於kernel dimension加rank且nullity不是解的個數；kernel image rank皆須symbolic certificate；sample不足。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["秩零度定理"]},"variant":{"family_id":"qf-mod35-ranknull-dimension-identity","seed":202611041038,"parameters":{"matrix":[[1,0,12],[0,1,1]],"domain":"Exact_finite_matrix_rank_nullity","operation":"ranknull-dimension-identity","semantic_template":"qf-mod35-ranknull-dimension-identity/{\"matrix\":[[1,0,12],[0,1,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0039","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-rank-nullity-theorem"],"title_zh":"秩零度定理","stem_zh":"明示m×n矩陣代表T:R^n到R^m並使用exact RREF。操作=ranknull-dimension-identity；資料={\"matrix\":[[1,0,13],[0,1,1]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 1, 2, 3, 是)","explanation_zh":"Wave5第四層核對=(3, 1, 2, 3, 是)；依完整明示資料重算；n等於kernel dimension加rank且nullity不是解的個數；kernel image rank皆須symbolic certificate；sample不足。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["秩零度定理"]},"variant":{"family_id":"qf-mod35-ranknull-dimension-identity","seed":202611041039,"parameters":{"matrix":[[1,0,13],[0,1,1]],"domain":"Exact_finite_matrix_rank_nullity","operation":"ranknull-dimension-identity","semantic_template":"qf-mod35-ranknull-dimension-identity/{\"matrix\":[[1,0,13],[0,1,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0040","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-rank-nullity-theorem"],"title_zh":"秩零度定理","stem_zh":"明示m×n矩陣代表T:R^n到R^m並使用exact RREF。操作=ranknull-dimension-identity；資料={\"matrix\":[[1,0,14],[0,1,1]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 1, 2, 3, 是)","explanation_zh":"Wave5第四層核對=(3, 1, 2, 3, 是)；依完整明示資料重算；n等於kernel dimension加rank且nullity不是解的個數；kernel image rank皆須symbolic certificate；sample不足。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["秩零度定理"]},"variant":{"family_id":"qf-mod35-ranknull-dimension-identity","seed":202611041040,"parameters":{"matrix":[[1,0,14],[0,1,1]],"domain":"Exact_finite_matrix_rank_nullity","operation":"ranknull-dimension-identity","semantic_template":"qf-mod35-ranknull-dimension-identity/{\"matrix\":[[1,0,14],[0,1,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0041","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-rank-nullity-theorem"],"title_zh":"秩零度定理","stem_zh":"明示m×n矩陣代表T:R^n到R^m並使用exact RREF。操作=ranknull-dimension-identity；資料={\"matrix\":[[1,0,15],[0,1,1]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 1, 2, 3, 是)","explanation_zh":"Wave5第四層核對=(3, 1, 2, 3, 是)；依完整明示資料重算；n等於kernel dimension加rank且nullity不是解的個數；kernel image rank皆須symbolic certificate；sample不足。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["秩零度定理"]},"variant":{"family_id":"qf-mod35-ranknull-dimension-identity","seed":202611041041,"parameters":{"matrix":[[1,0,15],[0,1,1]],"domain":"Exact_finite_matrix_rank_nullity","operation":"ranknull-dimension-identity","semantic_template":"qf-mod35-ranknull-dimension-identity/{\"matrix\":[[1,0,15],[0,1,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0042","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-rank-nullity-theorem"],"title_zh":"秩零度定理","stem_zh":"明示m×n矩陣代表T:R^n到R^m並使用exact RREF。操作=ranknull-dimension-identity；資料={\"matrix\":[[1,0,16],[0,1,1]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 1, 2, 3, 是)","explanation_zh":"Wave5第四層核對=(3, 1, 2, 3, 是)；依完整明示資料重算；n等於kernel dimension加rank且nullity不是解的個數；kernel image rank皆須symbolic certificate；sample不足。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["秩零度定理"]},"variant":{"family_id":"qf-mod35-ranknull-dimension-identity","seed":202611041042,"parameters":{"matrix":[[1,0,16],[0,1,1]],"domain":"Exact_finite_matrix_rank_nullity","operation":"ranknull-dimension-identity","semantic_template":"qf-mod35-ranknull-dimension-identity/{\"matrix\":[[1,0,16],[0,1,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0043","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-rank-nullity-theorem"],"title_zh":"秩零度定理","stem_zh":"明示m×n矩陣代表T:R^n到R^m並使用exact RREF。操作=ranknull-dimension-identity；資料={\"matrix\":[[1,0,17],[0,1,1]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 1, 2, 3, 是)","explanation_zh":"Wave5第四層核對=(3, 1, 2, 3, 是)；依完整明示資料重算；n等於kernel dimension加rank且nullity不是解的個數；kernel image rank皆須symbolic certificate；sample不足。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["秩零度定理"]},"variant":{"family_id":"qf-mod35-ranknull-dimension-identity","seed":202611041043,"parameters":{"matrix":[[1,0,17],[0,1,1]],"domain":"Exact_finite_matrix_rank_nullity","operation":"ranknull-dimension-identity","semantic_template":"qf-mod35-ranknull-dimension-identity/{\"matrix\":[[1,0,17],[0,1,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0044","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-rank-nullity-theorem"],"title_zh":"秩零度定理","stem_zh":"明示m×n矩陣代表T:R^n到R^m並使用exact RREF。操作=ranknull-dimension-identity；資料={\"matrix\":[[1,0,18],[0,1,1]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 1, 2, 3, 是)","explanation_zh":"Wave5第四層核對=(3, 1, 2, 3, 是)；依完整明示資料重算；n等於kernel dimension加rank且nullity不是解的個數；kernel image rank皆須symbolic certificate；sample不足。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["秩零度定理"]},"variant":{"family_id":"qf-mod35-ranknull-dimension-identity","seed":202611041044,"parameters":{"matrix":[[1,0,18],[0,1,1]],"domain":"Exact_finite_matrix_rank_nullity","operation":"ranknull-dimension-identity","semantic_template":"qf-mod35-ranknull-dimension-identity/{\"matrix\":[[1,0,18],[0,1,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0045","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-rank-nullity-theorem"],"title_zh":"秩零度定理","stem_zh":"明示m×n矩陣代表T:R^n到R^m並使用exact RREF。操作=ranknull-dimension-identity；資料={\"matrix\":[[1,0,19],[0,1,1]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 1, 2, 3, 是)","explanation_zh":"Wave5第四層核對=(3, 1, 2, 3, 是)；依完整明示資料重算；n等於kernel dimension加rank且nullity不是解的個數；kernel image rank皆須symbolic certificate；sample不足。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["秩零度定理"]},"variant":{"family_id":"qf-mod35-ranknull-dimension-identity","seed":202611041045,"parameters":{"matrix":[[1,0,19],[0,1,1]],"domain":"Exact_finite_matrix_rank_nullity","operation":"ranknull-dimension-identity","semantic_template":"qf-mod35-ranknull-dimension-identity/{\"matrix\":[[1,0,19],[0,1,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0046","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-rank-nullity-theorem"],"title_zh":"秩零度定理","stem_zh":"明示m×n矩陣代表T:R^n到R^m並使用exact RREF。操作=ranknull-dimension-identity；資料={\"matrix\":[[1,0,20],[0,1,1]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 1, 2, 3, 是)","explanation_zh":"Wave5第四層核對=(3, 1, 2, 3, 是)；依完整明示資料重算；n等於kernel dimension加rank且nullity不是解的個數；kernel image rank皆須symbolic certificate；sample不足。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["秩零度定理"]},"variant":{"family_id":"qf-mod35-ranknull-dimension-identity","seed":202611041046,"parameters":{"matrix":[[1,0,20],[0,1,1]],"domain":"Exact_finite_matrix_rank_nullity","operation":"ranknull-dimension-identity","semantic_template":"qf-mod35-ranknull-dimension-identity/{\"matrix\":[[1,0,20],[0,1,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0047","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-rank-nullity-theorem"],"title_zh":"秩零度定理","stem_zh":"明示m×n矩陣代表T:R^n到R^m並使用exact RREF。操作=ranknull-dimension-identity；資料={\"matrix\":[[1,0,21],[0,1,1]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 1, 2, 3, 是)","explanation_zh":"Wave5第四層核對=(3, 1, 2, 3, 是)；依完整明示資料重算；n等於kernel dimension加rank且nullity不是解的個數；kernel image rank皆須symbolic certificate；sample不足。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["秩零度定理"]},"variant":{"family_id":"qf-mod35-ranknull-dimension-identity","seed":202611041047,"parameters":{"matrix":[[1,0,21],[0,1,1]],"domain":"Exact_finite_matrix_rank_nullity","operation":"ranknull-dimension-identity","semantic_template":"qf-mod35-ranknull-dimension-identity/{\"matrix\":[[1,0,21],[0,1,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0048","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-rank-nullity-theorem"],"title_zh":"秩零度定理","stem_zh":"明示m×n矩陣代表T:R^n到R^m並使用exact RREF。操作=ranknull-dimension-identity；資料={\"matrix\":[[1,0,22],[0,1,1]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 1, 2, 3, 是)","explanation_zh":"Wave5第四層核對=(3, 1, 2, 3, 是)；依完整明示資料重算；n等於kernel dimension加rank且nullity不是解的個數；kernel image rank皆須symbolic certificate；sample不足。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["秩零度定理"]},"variant":{"family_id":"qf-mod35-ranknull-dimension-identity","seed":202611041048,"parameters":{"matrix":[[1,0,22],[0,1,1]],"domain":"Exact_finite_matrix_rank_nullity","operation":"ranknull-dimension-identity","semantic_template":"qf-mod35-ranknull-dimension-identity/{\"matrix\":[[1,0,22],[0,1,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0049","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-rank-nullity-theorem"],"title_zh":"秩零度定理","stem_zh":"明示m×n矩陣代表T:R^n到R^m並使用exact RREF。操作=ranknull-dimension-identity；資料={\"matrix\":[[1,0,23],[0,1,1]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 1, 2, 3, 是)","explanation_zh":"Wave5第四層核對=(3, 1, 2, 3, 是)；依完整明示資料重算；n等於kernel dimension加rank且nullity不是解的個數；kernel image rank皆須symbolic certificate；sample不足。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["秩零度定理"]},"variant":{"family_id":"qf-mod35-ranknull-dimension-identity","seed":202611041049,"parameters":{"matrix":[[1,0,23],[0,1,1]],"domain":"Exact_finite_matrix_rank_nullity","operation":"ranknull-dimension-identity","semantic_template":"qf-mod35-ranknull-dimension-identity/{\"matrix\":[[1,0,23],[0,1,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0050","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-rank-nullity-theorem"],"title_zh":"秩零度定理","stem_zh":"明示m×n矩陣代表T:R^n到R^m並使用exact RREF。操作=ranknull-dimension-identity；資料={\"matrix\":[[1,0,24],[0,1,1]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 1, 2, 3, 是)","explanation_zh":"Wave5第四層核對=(3, 1, 2, 3, 是)；依完整明示資料重算；n等於kernel dimension加rank且nullity不是解的個數；kernel image rank皆須symbolic certificate；sample不足。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["秩零度定理"]},"variant":{"family_id":"qf-mod35-ranknull-dimension-identity","seed":202611041050,"parameters":{"matrix":[[1,0,24],[0,1,1]],"domain":"Exact_finite_matrix_rank_nullity","operation":"ranknull-dimension-identity","semantic_template":"qf-mod35-ranknull-dimension-identity/{\"matrix\":[[1,0,24],[0,1,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0051","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-rank-nullity-theorem"],"title_zh":"秩零度定理","stem_zh":"明示m×n矩陣代表T:R^n到R^m並使用exact RREF。操作=ranknull-kernel-image-certificate；資料={\"matrix\":[[1,0,0],[0,1,1]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, -1, 1]], [[1, 0], [0, 1]], 1, 2, 是, exact_symbolic_certificates)","explanation_zh":"Wave5第四層核對=([[0, -1, 1]], [[1, 0], [0, 1]], 1, 2, 是, exact_symbolic_certificates)；依完整明示資料重算；n等於kernel dimension加rank且nullity不是解的個數；kernel image rank皆須symbolic certificate；sample不足。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["秩零度定理"]},"variant":{"family_id":"qf-mod35-ranknull-kernel-image-certificate","seed":202611041051,"parameters":{"matrix":[[1,0,0],[0,1,1]],"domain":"Exact_finite_matrix_rank_nullity","operation":"ranknull-kernel-image-certificate","semantic_template":"qf-mod35-ranknull-kernel-image-certificate/{\"matrix\":[[1,0,0],[0,1,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0052","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-rank-nullity-theorem"],"title_zh":"秩零度定理","stem_zh":"明示m×n矩陣代表T:R^n到R^m並使用exact RREF。操作=ranknull-kernel-image-certificate；資料={\"matrix\":[[1,0,1],[0,1,1]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[-1, -1, 1]], [[1, 0], [0, 1]], 1, 2, 是, exact_symbolic_certificates)","explanation_zh":"Wave5第四層核對=([[-1, -1, 1]], [[1, 0], [0, 1]], 1, 2, 是, exact_symbolic_certificates)；依完整明示資料重算；n等於kernel dimension加rank且nullity不是解的個數；kernel image rank皆須symbolic certificate；sample不足。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["秩零度定理"]},"variant":{"family_id":"qf-mod35-ranknull-kernel-image-certificate","seed":202611041052,"parameters":{"matrix":[[1,0,1],[0,1,1]],"domain":"Exact_finite_matrix_rank_nullity","operation":"ranknull-kernel-image-certificate","semantic_template":"qf-mod35-ranknull-kernel-image-certificate/{\"matrix\":[[1,0,1],[0,1,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0053","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-rank-nullity-theorem"],"title_zh":"秩零度定理","stem_zh":"明示m×n矩陣代表T:R^n到R^m並使用exact RREF。操作=ranknull-kernel-image-certificate；資料={\"matrix\":[[1,0,2],[0,1,1]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[-2, -1, 1]], [[1, 0], [0, 1]], 1, 2, 是, exact_symbolic_certificates)","explanation_zh":"Wave5第四層核對=([[-2, -1, 1]], [[1, 0], [0, 1]], 1, 2, 是, exact_symbolic_certificates)；依完整明示資料重算；n等於kernel dimension加rank且nullity不是解的個數；kernel image rank皆須symbolic certificate；sample不足。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["秩零度定理"]},"variant":{"family_id":"qf-mod35-ranknull-kernel-image-certificate","seed":202611041053,"parameters":{"matrix":[[1,0,2],[0,1,1]],"domain":"Exact_finite_matrix_rank_nullity","operation":"ranknull-kernel-image-certificate","semantic_template":"qf-mod35-ranknull-kernel-image-certificate/{\"matrix\":[[1,0,2],[0,1,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0054","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-rank-nullity-theorem"],"title_zh":"秩零度定理","stem_zh":"明示m×n矩陣代表T:R^n到R^m並使用exact RREF。操作=ranknull-kernel-image-certificate；資料={\"matrix\":[[1,0,3],[0,1,1]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[-3, -1, 1]], [[1, 0], [0, 1]], 1, 2, 是, exact_symbolic_certificates)","explanation_zh":"Wave5第四層核對=([[-3, -1, 1]], [[1, 0], [0, 1]], 1, 2, 是, exact_symbolic_certificates)；依完整明示資料重算；n等於kernel dimension加rank且nullity不是解的個數；kernel image rank皆須symbolic certificate；sample不足。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["秩零度定理"]},"variant":{"family_id":"qf-mod35-ranknull-kernel-image-certificate","seed":202611041054,"parameters":{"matrix":[[1,0,3],[0,1,1]],"domain":"Exact_finite_matrix_rank_nullity","operation":"ranknull-kernel-image-certificate","semantic_template":"qf-mod35-ranknull-kernel-image-certificate/{\"matrix\":[[1,0,3],[0,1,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0055","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-rank-nullity-theorem"],"title_zh":"秩零度定理","stem_zh":"明示m×n矩陣代表T:R^n到R^m並使用exact RREF。操作=ranknull-kernel-image-certificate；資料={\"matrix\":[[1,0,4],[0,1,1]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[-4, -1, 1]], [[1, 0], [0, 1]], 1, 2, 是, exact_symbolic_certificates)","explanation_zh":"Wave5第四層核對=([[-4, -1, 1]], [[1, 0], [0, 1]], 1, 2, 是, exact_symbolic_certificates)；依完整明示資料重算；n等於kernel dimension加rank且nullity不是解的個數；kernel image rank皆須symbolic certificate；sample不足。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["秩零度定理"]},"variant":{"family_id":"qf-mod35-ranknull-kernel-image-certificate","seed":202611041055,"parameters":{"matrix":[[1,0,4],[0,1,1]],"domain":"Exact_finite_matrix_rank_nullity","operation":"ranknull-kernel-image-certificate","semantic_template":"qf-mod35-ranknull-kernel-image-certificate/{\"matrix\":[[1,0,4],[0,1,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0056","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-rank-nullity-theorem"],"title_zh":"秩零度定理","stem_zh":"明示m×n矩陣代表T:R^n到R^m並使用exact RREF。操作=ranknull-kernel-image-certificate；資料={\"matrix\":[[1,0,5],[0,1,1]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[-5, -1, 1]], [[1, 0], [0, 1]], 1, 2, 是, exact_symbolic_certificates)","explanation_zh":"Wave5第四層核對=([[-5, -1, 1]], [[1, 0], [0, 1]], 1, 2, 是, exact_symbolic_certificates)；依完整明示資料重算；n等於kernel dimension加rank且nullity不是解的個數；kernel image rank皆須symbolic certificate；sample不足。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["秩零度定理"]},"variant":{"family_id":"qf-mod35-ranknull-kernel-image-certificate","seed":202611041056,"parameters":{"matrix":[[1,0,5],[0,1,1]],"domain":"Exact_finite_matrix_rank_nullity","operation":"ranknull-kernel-image-certificate","semantic_template":"qf-mod35-ranknull-kernel-image-certificate/{\"matrix\":[[1,0,5],[0,1,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0057","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-rank-nullity-theorem"],"title_zh":"秩零度定理","stem_zh":"明示m×n矩陣代表T:R^n到R^m並使用exact RREF。操作=ranknull-kernel-image-certificate；資料={\"matrix\":[[1,0,6],[0,1,1]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[-6, -1, 1]], [[1, 0], [0, 1]], 1, 2, 是, exact_symbolic_certificates)","explanation_zh":"Wave5第四層核對=([[-6, -1, 1]], [[1, 0], [0, 1]], 1, 2, 是, exact_symbolic_certificates)；依完整明示資料重算；n等於kernel dimension加rank且nullity不是解的個數；kernel image rank皆須symbolic certificate；sample不足。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["秩零度定理"]},"variant":{"family_id":"qf-mod35-ranknull-kernel-image-certificate","seed":202611041057,"parameters":{"matrix":[[1,0,6],[0,1,1]],"domain":"Exact_finite_matrix_rank_nullity","operation":"ranknull-kernel-image-certificate","semantic_template":"qf-mod35-ranknull-kernel-image-certificate/{\"matrix\":[[1,0,6],[0,1,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0058","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-rank-nullity-theorem"],"title_zh":"秩零度定理","stem_zh":"明示m×n矩陣代表T:R^n到R^m並使用exact RREF。操作=ranknull-kernel-image-certificate；資料={\"matrix\":[[1,0,7],[0,1,1]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[-7, -1, 1]], [[1, 0], [0, 1]], 1, 2, 是, exact_symbolic_certificates)","explanation_zh":"Wave5第四層核對=([[-7, -1, 1]], [[1, 0], [0, 1]], 1, 2, 是, exact_symbolic_certificates)；依完整明示資料重算；n等於kernel dimension加rank且nullity不是解的個數；kernel image rank皆須symbolic certificate；sample不足。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["秩零度定理"]},"variant":{"family_id":"qf-mod35-ranknull-kernel-image-certificate","seed":202611041058,"parameters":{"matrix":[[1,0,7],[0,1,1]],"domain":"Exact_finite_matrix_rank_nullity","operation":"ranknull-kernel-image-certificate","semantic_template":"qf-mod35-ranknull-kernel-image-certificate/{\"matrix\":[[1,0,7],[0,1,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0059","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-rank-nullity-theorem"],"title_zh":"秩零度定理","stem_zh":"明示m×n矩陣代表T:R^n到R^m並使用exact RREF。操作=ranknull-kernel-image-certificate；資料={\"matrix\":[[1,0,8],[0,1,1]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[-8, -1, 1]], [[1, 0], [0, 1]], 1, 2, 是, exact_symbolic_certificates)","explanation_zh":"Wave5第四層核對=([[-8, -1, 1]], [[1, 0], [0, 1]], 1, 2, 是, exact_symbolic_certificates)；依完整明示資料重算；n等於kernel dimension加rank且nullity不是解的個數；kernel image rank皆須symbolic certificate；sample不足。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["秩零度定理"]},"variant":{"family_id":"qf-mod35-ranknull-kernel-image-certificate","seed":202611041059,"parameters":{"matrix":[[1,0,8],[0,1,1]],"domain":"Exact_finite_matrix_rank_nullity","operation":"ranknull-kernel-image-certificate","semantic_template":"qf-mod35-ranknull-kernel-image-certificate/{\"matrix\":[[1,0,8],[0,1,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0060","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-rank-nullity-theorem"],"title_zh":"秩零度定理","stem_zh":"明示m×n矩陣代表T:R^n到R^m並使用exact RREF。操作=ranknull-kernel-image-certificate；資料={\"matrix\":[[1,0,9],[0,1,1]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[-9, -1, 1]], [[1, 0], [0, 1]], 1, 2, 是, exact_symbolic_certificates)","explanation_zh":"Wave5第四層核對=([[-9, -1, 1]], [[1, 0], [0, 1]], 1, 2, 是, exact_symbolic_certificates)；依完整明示資料重算；n等於kernel dimension加rank且nullity不是解的個數；kernel image rank皆須symbolic certificate；sample不足。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["秩零度定理"]},"variant":{"family_id":"qf-mod35-ranknull-kernel-image-certificate","seed":202611041060,"parameters":{"matrix":[[1,0,9],[0,1,1]],"domain":"Exact_finite_matrix_rank_nullity","operation":"ranknull-kernel-image-certificate","semantic_template":"qf-mod35-ranknull-kernel-image-certificate/{\"matrix\":[[1,0,9],[0,1,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0061","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-rank-nullity-theorem"],"title_zh":"秩零度定理","stem_zh":"明示m×n矩陣代表T:R^n到R^m並使用exact RREF。操作=ranknull-kernel-image-certificate；資料={\"matrix\":[[1,0,10],[0,1,1]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[-10, -1, 1]], [[1, 0], [0, 1]], 1, 2, 是, exact_symbolic_certificates)","explanation_zh":"Wave5第四層核對=([[-10, -1, 1]], [[1, 0], [0, 1]], 1, 2, 是, exact_symbolic_certificates)；依完整明示資料重算；n等於kernel dimension加rank且nullity不是解的個數；kernel image rank皆須symbolic certificate；sample不足。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["秩零度定理"]},"variant":{"family_id":"qf-mod35-ranknull-kernel-image-certificate","seed":202611041061,"parameters":{"matrix":[[1,0,10],[0,1,1]],"domain":"Exact_finite_matrix_rank_nullity","operation":"ranknull-kernel-image-certificate","semantic_template":"qf-mod35-ranknull-kernel-image-certificate/{\"matrix\":[[1,0,10],[0,1,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0062","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-rank-nullity-theorem"],"title_zh":"秩零度定理","stem_zh":"明示m×n矩陣代表T:R^n到R^m並使用exact RREF。操作=ranknull-kernel-image-certificate；資料={\"matrix\":[[1,0,11],[0,1,1]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[-11, -1, 1]], [[1, 0], [0, 1]], 1, 2, 是, exact_symbolic_certificates)","explanation_zh":"Wave5第四層核對=([[-11, -1, 1]], [[1, 0], [0, 1]], 1, 2, 是, exact_symbolic_certificates)；依完整明示資料重算；n等於kernel dimension加rank且nullity不是解的個數；kernel image rank皆須symbolic certificate；sample不足。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["秩零度定理"]},"variant":{"family_id":"qf-mod35-ranknull-kernel-image-certificate","seed":202611041062,"parameters":{"matrix":[[1,0,11],[0,1,1]],"domain":"Exact_finite_matrix_rank_nullity","operation":"ranknull-kernel-image-certificate","semantic_template":"qf-mod35-ranknull-kernel-image-certificate/{\"matrix\":[[1,0,11],[0,1,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0063","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-rank-nullity-theorem"],"title_zh":"秩零度定理","stem_zh":"明示m×n矩陣代表T:R^n到R^m並使用exact RREF。操作=ranknull-kernel-image-certificate；資料={\"matrix\":[[1,0,12],[0,1,1]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[-12, -1, 1]], [[1, 0], [0, 1]], 1, 2, 是, exact_symbolic_certificates)","explanation_zh":"Wave5第四層核對=([[-12, -1, 1]], [[1, 0], [0, 1]], 1, 2, 是, exact_symbolic_certificates)；依完整明示資料重算；n等於kernel dimension加rank且nullity不是解的個數；kernel image rank皆須symbolic certificate；sample不足。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["秩零度定理"]},"variant":{"family_id":"qf-mod35-ranknull-kernel-image-certificate","seed":202611041063,"parameters":{"matrix":[[1,0,12],[0,1,1]],"domain":"Exact_finite_matrix_rank_nullity","operation":"ranknull-kernel-image-certificate","semantic_template":"qf-mod35-ranknull-kernel-image-certificate/{\"matrix\":[[1,0,12],[0,1,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0064","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-rank-nullity-theorem"],"title_zh":"秩零度定理","stem_zh":"明示m×n矩陣代表T:R^n到R^m並使用exact RREF。操作=ranknull-kernel-image-certificate；資料={\"matrix\":[[1,0,13],[0,1,1]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[-13, -1, 1]], [[1, 0], [0, 1]], 1, 2, 是, exact_symbolic_certificates)","explanation_zh":"Wave5第四層核對=([[-13, -1, 1]], [[1, 0], [0, 1]], 1, 2, 是, exact_symbolic_certificates)；依完整明示資料重算；n等於kernel dimension加rank且nullity不是解的個數；kernel image rank皆須symbolic certificate；sample不足。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["秩零度定理"]},"variant":{"family_id":"qf-mod35-ranknull-kernel-image-certificate","seed":202611041064,"parameters":{"matrix":[[1,0,13],[0,1,1]],"domain":"Exact_finite_matrix_rank_nullity","operation":"ranknull-kernel-image-certificate","semantic_template":"qf-mod35-ranknull-kernel-image-certificate/{\"matrix\":[[1,0,13],[0,1,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0065","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-rank-nullity-theorem"],"title_zh":"秩零度定理","stem_zh":"明示m×n矩陣代表T:R^n到R^m並使用exact RREF。操作=ranknull-kernel-image-certificate；資料={\"matrix\":[[1,0,14],[0,1,1]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[-14, -1, 1]], [[1, 0], [0, 1]], 1, 2, 是, exact_symbolic_certificates)","explanation_zh":"Wave5第四層核對=([[-14, -1, 1]], [[1, 0], [0, 1]], 1, 2, 是, exact_symbolic_certificates)；依完整明示資料重算；n等於kernel dimension加rank且nullity不是解的個數；kernel image rank皆須symbolic certificate；sample不足。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["秩零度定理"]},"variant":{"family_id":"qf-mod35-ranknull-kernel-image-certificate","seed":202611041065,"parameters":{"matrix":[[1,0,14],[0,1,1]],"domain":"Exact_finite_matrix_rank_nullity","operation":"ranknull-kernel-image-certificate","semantic_template":"qf-mod35-ranknull-kernel-image-certificate/{\"matrix\":[[1,0,14],[0,1,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0066","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-rank-nullity-theorem"],"title_zh":"秩零度定理","stem_zh":"明示m×n矩陣代表T:R^n到R^m並使用exact RREF。操作=ranknull-kernel-image-certificate；資料={\"matrix\":[[1,0,15],[0,1,1]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[-15, -1, 1]], [[1, 0], [0, 1]], 1, 2, 是, exact_symbolic_certificates)","explanation_zh":"Wave5第四層核對=([[-15, -1, 1]], [[1, 0], [0, 1]], 1, 2, 是, exact_symbolic_certificates)；依完整明示資料重算；n等於kernel dimension加rank且nullity不是解的個數；kernel image rank皆須symbolic certificate；sample不足。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["秩零度定理"]},"variant":{"family_id":"qf-mod35-ranknull-kernel-image-certificate","seed":202611041066,"parameters":{"matrix":[[1,0,15],[0,1,1]],"domain":"Exact_finite_matrix_rank_nullity","operation":"ranknull-kernel-image-certificate","semantic_template":"qf-mod35-ranknull-kernel-image-certificate/{\"matrix\":[[1,0,15],[0,1,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0067","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-rank-nullity-theorem"],"title_zh":"秩零度定理","stem_zh":"明示m×n矩陣代表T:R^n到R^m並使用exact RREF。操作=ranknull-kernel-image-certificate；資料={\"matrix\":[[1,0,16],[0,1,1]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[-16, -1, 1]], [[1, 0], [0, 1]], 1, 2, 是, exact_symbolic_certificates)","explanation_zh":"Wave5第四層核對=([[-16, -1, 1]], [[1, 0], [0, 1]], 1, 2, 是, exact_symbolic_certificates)；依完整明示資料重算；n等於kernel dimension加rank且nullity不是解的個數；kernel image rank皆須symbolic certificate；sample不足。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["秩零度定理"]},"variant":{"family_id":"qf-mod35-ranknull-kernel-image-certificate","seed":202611041067,"parameters":{"matrix":[[1,0,16],[0,1,1]],"domain":"Exact_finite_matrix_rank_nullity","operation":"ranknull-kernel-image-certificate","semantic_template":"qf-mod35-ranknull-kernel-image-certificate/{\"matrix\":[[1,0,16],[0,1,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0068","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-rank-nullity-theorem"],"title_zh":"秩零度定理","stem_zh":"明示m×n矩陣代表T:R^n到R^m並使用exact RREF。操作=ranknull-kernel-image-certificate；資料={\"matrix\":[[1,0,17],[0,1,1]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[-17, -1, 1]], [[1, 0], [0, 1]], 1, 2, 是, exact_symbolic_certificates)","explanation_zh":"Wave5第四層核對=([[-17, -1, 1]], [[1, 0], [0, 1]], 1, 2, 是, exact_symbolic_certificates)；依完整明示資料重算；n等於kernel dimension加rank且nullity不是解的個數；kernel image rank皆須symbolic certificate；sample不足。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["秩零度定理"]},"variant":{"family_id":"qf-mod35-ranknull-kernel-image-certificate","seed":202611041068,"parameters":{"matrix":[[1,0,17],[0,1,1]],"domain":"Exact_finite_matrix_rank_nullity","operation":"ranknull-kernel-image-certificate","semantic_template":"qf-mod35-ranknull-kernel-image-certificate/{\"matrix\":[[1,0,17],[0,1,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0069","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-rank-nullity-theorem"],"title_zh":"秩零度定理","stem_zh":"明示m×n矩陣代表T:R^n到R^m並使用exact RREF。操作=ranknull-kernel-image-certificate；資料={\"matrix\":[[1,0,18],[0,1,1]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[-18, -1, 1]], [[1, 0], [0, 1]], 1, 2, 是, exact_symbolic_certificates)","explanation_zh":"Wave5第四層核對=([[-18, -1, 1]], [[1, 0], [0, 1]], 1, 2, 是, exact_symbolic_certificates)；依完整明示資料重算；n等於kernel dimension加rank且nullity不是解的個數；kernel image rank皆須symbolic certificate；sample不足。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["秩零度定理"]},"variant":{"family_id":"qf-mod35-ranknull-kernel-image-certificate","seed":202611041069,"parameters":{"matrix":[[1,0,18],[0,1,1]],"domain":"Exact_finite_matrix_rank_nullity","operation":"ranknull-kernel-image-certificate","semantic_template":"qf-mod35-ranknull-kernel-image-certificate/{\"matrix\":[[1,0,18],[0,1,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0070","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-rank-nullity-theorem"],"title_zh":"秩零度定理","stem_zh":"明示m×n矩陣代表T:R^n到R^m並使用exact RREF。操作=ranknull-kernel-image-certificate；資料={\"matrix\":[[1,0,19],[0,1,1]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[-19, -1, 1]], [[1, 0], [0, 1]], 1, 2, 是, exact_symbolic_certificates)","explanation_zh":"Wave5第四層核對=([[-19, -1, 1]], [[1, 0], [0, 1]], 1, 2, 是, exact_symbolic_certificates)；依完整明示資料重算；n等於kernel dimension加rank且nullity不是解的個數；kernel image rank皆須symbolic certificate；sample不足。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["秩零度定理"]},"variant":{"family_id":"qf-mod35-ranknull-kernel-image-certificate","seed":202611041070,"parameters":{"matrix":[[1,0,19],[0,1,1]],"domain":"Exact_finite_matrix_rank_nullity","operation":"ranknull-kernel-image-certificate","semantic_template":"qf-mod35-ranknull-kernel-image-certificate/{\"matrix\":[[1,0,19],[0,1,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0071","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-rank-nullity-theorem"],"title_zh":"秩零度定理","stem_zh":"明示m×n矩陣代表T:R^n到R^m並使用exact RREF。操作=ranknull-kernel-image-certificate；資料={\"matrix\":[[1,0,20],[0,1,1]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[-20, -1, 1]], [[1, 0], [0, 1]], 1, 2, 是, exact_symbolic_certificates)","explanation_zh":"Wave5第四層核對=([[-20, -1, 1]], [[1, 0], [0, 1]], 1, 2, 是, exact_symbolic_certificates)；依完整明示資料重算；n等於kernel dimension加rank且nullity不是解的個數；kernel image rank皆須symbolic certificate；sample不足。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["秩零度定理"]},"variant":{"family_id":"qf-mod35-ranknull-kernel-image-certificate","seed":202611041071,"parameters":{"matrix":[[1,0,20],[0,1,1]],"domain":"Exact_finite_matrix_rank_nullity","operation":"ranknull-kernel-image-certificate","semantic_template":"qf-mod35-ranknull-kernel-image-certificate/{\"matrix\":[[1,0,20],[0,1,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0072","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-rank-nullity-theorem"],"title_zh":"秩零度定理","stem_zh":"明示m×n矩陣代表T:R^n到R^m並使用exact RREF。操作=ranknull-kernel-image-certificate；資料={\"matrix\":[[1,0,21],[0,1,1]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[-21, -1, 1]], [[1, 0], [0, 1]], 1, 2, 是, exact_symbolic_certificates)","explanation_zh":"Wave5第四層核對=([[-21, -1, 1]], [[1, 0], [0, 1]], 1, 2, 是, exact_symbolic_certificates)；依完整明示資料重算；n等於kernel dimension加rank且nullity不是解的個數；kernel image rank皆須symbolic certificate；sample不足。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["秩零度定理"]},"variant":{"family_id":"qf-mod35-ranknull-kernel-image-certificate","seed":202611041072,"parameters":{"matrix":[[1,0,21],[0,1,1]],"domain":"Exact_finite_matrix_rank_nullity","operation":"ranknull-kernel-image-certificate","semantic_template":"qf-mod35-ranknull-kernel-image-certificate/{\"matrix\":[[1,0,21],[0,1,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0073","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-rank-nullity-theorem"],"title_zh":"秩零度定理","stem_zh":"明示m×n矩陣代表T:R^n到R^m並使用exact RREF。操作=ranknull-kernel-image-certificate；資料={\"matrix\":[[1,0,22],[0,1,1]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[-22, -1, 1]], [[1, 0], [0, 1]], 1, 2, 是, exact_symbolic_certificates)","explanation_zh":"Wave5第四層核對=([[-22, -1, 1]], [[1, 0], [0, 1]], 1, 2, 是, exact_symbolic_certificates)；依完整明示資料重算；n等於kernel dimension加rank且nullity不是解的個數；kernel image rank皆須symbolic certificate；sample不足。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["秩零度定理"]},"variant":{"family_id":"qf-mod35-ranknull-kernel-image-certificate","seed":202611041073,"parameters":{"matrix":[[1,0,22],[0,1,1]],"domain":"Exact_finite_matrix_rank_nullity","operation":"ranknull-kernel-image-certificate","semantic_template":"qf-mod35-ranknull-kernel-image-certificate/{\"matrix\":[[1,0,22],[0,1,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0074","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-rank-nullity-theorem"],"title_zh":"秩零度定理","stem_zh":"明示m×n矩陣代表T:R^n到R^m並使用exact RREF。操作=ranknull-kernel-image-certificate；資料={\"matrix\":[[1,0,23],[0,1,1]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[-23, -1, 1]], [[1, 0], [0, 1]], 1, 2, 是, exact_symbolic_certificates)","explanation_zh":"Wave5第四層核對=([[-23, -1, 1]], [[1, 0], [0, 1]], 1, 2, 是, exact_symbolic_certificates)；依完整明示資料重算；n等於kernel dimension加rank且nullity不是解的個數；kernel image rank皆須symbolic certificate；sample不足。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["秩零度定理"]},"variant":{"family_id":"qf-mod35-ranknull-kernel-image-certificate","seed":202611041074,"parameters":{"matrix":[[1,0,23],[0,1,1]],"domain":"Exact_finite_matrix_rank_nullity","operation":"ranknull-kernel-image-certificate","semantic_template":"qf-mod35-ranknull-kernel-image-certificate/{\"matrix\":[[1,0,23],[0,1,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0075","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-rank-nullity-theorem"],"title_zh":"秩零度定理","stem_zh":"明示m×n矩陣代表T:R^n到R^m並使用exact RREF。操作=ranknull-kernel-image-certificate；資料={\"matrix\":[[1,0,24],[0,1,1]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[-24, -1, 1]], [[1, 0], [0, 1]], 1, 2, 是, exact_symbolic_certificates)","explanation_zh":"Wave5第四層核對=([[-24, -1, 1]], [[1, 0], [0, 1]], 1, 2, 是, exact_symbolic_certificates)；依完整明示資料重算；n等於kernel dimension加rank且nullity不是解的個數；kernel image rank皆須symbolic certificate；sample不足。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["秩零度定理"]},"variant":{"family_id":"qf-mod35-ranknull-kernel-image-certificate","seed":202611041075,"parameters":{"matrix":[[1,0,24],[0,1,1]],"domain":"Exact_finite_matrix_rank_nullity","operation":"ranknull-kernel-image-certificate","semantic_template":"qf-mod35-ranknull-kernel-image-certificate/{\"matrix\":[[1,0,24],[0,1,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0076","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-rank-nullity-theorem"],"title_zh":"秩零度定理","stem_zh":"明示m×n矩陣代表T:R^n到R^m並使用exact RREF。操作=ranknull-sampled-boundary；資料={\"matrix\":[[1,0,0],[0,1,1]],\"samples\":[[0,0,0],[1,0,0]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 0, 0], [1, 0, 0]], 否, sampled_vectors_do_not_prove_kernel_image_or_rank_nullity)","explanation_zh":"Wave5第四層核對=([[0, 0, 0], [1, 0, 0]], 否, sampled_vectors_do_not_prove_kernel_image_or_rank_nullity)；依完整明示資料重算；n等於kernel dimension加rank且nullity不是解的個數；kernel image rank皆須symbolic certificate；sample不足。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["秩零度定理"]},"variant":{"family_id":"qf-mod35-ranknull-sampled-boundary","seed":202611041076,"parameters":{"matrix":[[1,0,0],[0,1,1]],"samples":[[0,0,0],[1,0,0]],"domain":"Exact_finite_matrix_rank_nullity","operation":"ranknull-sampled-boundary","semantic_template":"qf-mod35-ranknull-sampled-boundary/{\"matrix\":[[1,0,0],[0,1,1]],\"samples\":[[0,0,0],[1,0,0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0077","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-rank-nullity-theorem"],"title_zh":"秩零度定理","stem_zh":"明示m×n矩陣代表T:R^n到R^m並使用exact RREF。操作=ranknull-sampled-boundary；資料={\"matrix\":[[1,0,1],[0,1,1]],\"samples\":[[0,0,0],[1,0,0]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 0, 0], [1, 0, 0]], 否, sampled_vectors_do_not_prove_kernel_image_or_rank_nullity)","explanation_zh":"Wave5第四層核對=([[0, 0, 0], [1, 0, 0]], 否, sampled_vectors_do_not_prove_kernel_image_or_rank_nullity)；依完整明示資料重算；n等於kernel dimension加rank且nullity不是解的個數；kernel image rank皆須symbolic certificate；sample不足。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["秩零度定理"]},"variant":{"family_id":"qf-mod35-ranknull-sampled-boundary","seed":202611041077,"parameters":{"matrix":[[1,0,1],[0,1,1]],"samples":[[0,0,0],[1,0,0]],"domain":"Exact_finite_matrix_rank_nullity","operation":"ranknull-sampled-boundary","semantic_template":"qf-mod35-ranknull-sampled-boundary/{\"matrix\":[[1,0,1],[0,1,1]],\"samples\":[[0,0,0],[1,0,0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0078","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-rank-nullity-theorem"],"title_zh":"秩零度定理","stem_zh":"明示m×n矩陣代表T:R^n到R^m並使用exact RREF。操作=ranknull-sampled-boundary；資料={\"matrix\":[[1,0,2],[0,1,1]],\"samples\":[[0,0,0],[1,0,0]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 0, 0], [1, 0, 0]], 否, sampled_vectors_do_not_prove_kernel_image_or_rank_nullity)","explanation_zh":"Wave5第四層核對=([[0, 0, 0], [1, 0, 0]], 否, sampled_vectors_do_not_prove_kernel_image_or_rank_nullity)；依完整明示資料重算；n等於kernel dimension加rank且nullity不是解的個數；kernel image rank皆須symbolic certificate；sample不足。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["秩零度定理"]},"variant":{"family_id":"qf-mod35-ranknull-sampled-boundary","seed":202611041078,"parameters":{"matrix":[[1,0,2],[0,1,1]],"samples":[[0,0,0],[1,0,0]],"domain":"Exact_finite_matrix_rank_nullity","operation":"ranknull-sampled-boundary","semantic_template":"qf-mod35-ranknull-sampled-boundary/{\"matrix\":[[1,0,2],[0,1,1]],\"samples\":[[0,0,0],[1,0,0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0079","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-rank-nullity-theorem"],"title_zh":"秩零度定理","stem_zh":"明示m×n矩陣代表T:R^n到R^m並使用exact RREF。操作=ranknull-sampled-boundary；資料={\"matrix\":[[1,0,3],[0,1,1]],\"samples\":[[0,0,0],[1,0,0]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 0, 0], [1, 0, 0]], 否, sampled_vectors_do_not_prove_kernel_image_or_rank_nullity)","explanation_zh":"Wave5第四層核對=([[0, 0, 0], [1, 0, 0]], 否, sampled_vectors_do_not_prove_kernel_image_or_rank_nullity)；依完整明示資料重算；n等於kernel dimension加rank且nullity不是解的個數；kernel image rank皆須symbolic certificate；sample不足。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["秩零度定理"]},"variant":{"family_id":"qf-mod35-ranknull-sampled-boundary","seed":202611041079,"parameters":{"matrix":[[1,0,3],[0,1,1]],"samples":[[0,0,0],[1,0,0]],"domain":"Exact_finite_matrix_rank_nullity","operation":"ranknull-sampled-boundary","semantic_template":"qf-mod35-ranknull-sampled-boundary/{\"matrix\":[[1,0,3],[0,1,1]],\"samples\":[[0,0,0],[1,0,0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0080","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-rank-nullity-theorem"],"title_zh":"秩零度定理","stem_zh":"明示m×n矩陣代表T:R^n到R^m並使用exact RREF。操作=ranknull-sampled-boundary；資料={\"matrix\":[[1,0,4],[0,1,1]],\"samples\":[[0,0,0],[1,0,0]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 0, 0], [1, 0, 0]], 否, sampled_vectors_do_not_prove_kernel_image_or_rank_nullity)","explanation_zh":"Wave5第四層核對=([[0, 0, 0], [1, 0, 0]], 否, sampled_vectors_do_not_prove_kernel_image_or_rank_nullity)；依完整明示資料重算；n等於kernel dimension加rank且nullity不是解的個數；kernel image rank皆須symbolic certificate；sample不足。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["秩零度定理"]},"variant":{"family_id":"qf-mod35-ranknull-sampled-boundary","seed":202611041080,"parameters":{"matrix":[[1,0,4],[0,1,1]],"samples":[[0,0,0],[1,0,0]],"domain":"Exact_finite_matrix_rank_nullity","operation":"ranknull-sampled-boundary","semantic_template":"qf-mod35-ranknull-sampled-boundary/{\"matrix\":[[1,0,4],[0,1,1]],\"samples\":[[0,0,0],[1,0,0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0081","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-rank-nullity-theorem"],"title_zh":"秩零度定理","stem_zh":"明示m×n矩陣代表T:R^n到R^m並使用exact RREF。操作=ranknull-sampled-boundary；資料={\"matrix\":[[1,0,5],[0,1,1]],\"samples\":[[0,0,0],[1,0,0]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 0, 0], [1, 0, 0]], 否, sampled_vectors_do_not_prove_kernel_image_or_rank_nullity)","explanation_zh":"Wave5第四層核對=([[0, 0, 0], [1, 0, 0]], 否, sampled_vectors_do_not_prove_kernel_image_or_rank_nullity)；依完整明示資料重算；n等於kernel dimension加rank且nullity不是解的個數；kernel image rank皆須symbolic certificate；sample不足。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["秩零度定理"]},"variant":{"family_id":"qf-mod35-ranknull-sampled-boundary","seed":202611041081,"parameters":{"matrix":[[1,0,5],[0,1,1]],"samples":[[0,0,0],[1,0,0]],"domain":"Exact_finite_matrix_rank_nullity","operation":"ranknull-sampled-boundary","semantic_template":"qf-mod35-ranknull-sampled-boundary/{\"matrix\":[[1,0,5],[0,1,1]],\"samples\":[[0,0,0],[1,0,0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0082","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-rank-nullity-theorem"],"title_zh":"秩零度定理","stem_zh":"明示m×n矩陣代表T:R^n到R^m並使用exact RREF。操作=ranknull-sampled-boundary；資料={\"matrix\":[[1,0,6],[0,1,1]],\"samples\":[[0,0,0],[1,0,0]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 0, 0], [1, 0, 0]], 否, sampled_vectors_do_not_prove_kernel_image_or_rank_nullity)","explanation_zh":"Wave5第四層核對=([[0, 0, 0], [1, 0, 0]], 否, sampled_vectors_do_not_prove_kernel_image_or_rank_nullity)；依完整明示資料重算；n等於kernel dimension加rank且nullity不是解的個數；kernel image rank皆須symbolic certificate；sample不足。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["秩零度定理"]},"variant":{"family_id":"qf-mod35-ranknull-sampled-boundary","seed":202611041082,"parameters":{"matrix":[[1,0,6],[0,1,1]],"samples":[[0,0,0],[1,0,0]],"domain":"Exact_finite_matrix_rank_nullity","operation":"ranknull-sampled-boundary","semantic_template":"qf-mod35-ranknull-sampled-boundary/{\"matrix\":[[1,0,6],[0,1,1]],\"samples\":[[0,0,0],[1,0,0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0083","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-rank-nullity-theorem"],"title_zh":"秩零度定理","stem_zh":"明示m×n矩陣代表T:R^n到R^m並使用exact RREF。操作=ranknull-sampled-boundary；資料={\"matrix\":[[1,0,7],[0,1,1]],\"samples\":[[0,0,0],[1,0,0]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 0, 0], [1, 0, 0]], 否, sampled_vectors_do_not_prove_kernel_image_or_rank_nullity)","explanation_zh":"Wave5第四層核對=([[0, 0, 0], [1, 0, 0]], 否, sampled_vectors_do_not_prove_kernel_image_or_rank_nullity)；依完整明示資料重算；n等於kernel dimension加rank且nullity不是解的個數；kernel image rank皆須symbolic certificate；sample不足。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["秩零度定理"]},"variant":{"family_id":"qf-mod35-ranknull-sampled-boundary","seed":202611041083,"parameters":{"matrix":[[1,0,7],[0,1,1]],"samples":[[0,0,0],[1,0,0]],"domain":"Exact_finite_matrix_rank_nullity","operation":"ranknull-sampled-boundary","semantic_template":"qf-mod35-ranknull-sampled-boundary/{\"matrix\":[[1,0,7],[0,1,1]],\"samples\":[[0,0,0],[1,0,0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0084","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-rank-nullity-theorem"],"title_zh":"秩零度定理","stem_zh":"明示m×n矩陣代表T:R^n到R^m並使用exact RREF。操作=ranknull-sampled-boundary；資料={\"matrix\":[[1,0,8],[0,1,1]],\"samples\":[[0,0,0],[1,0,0]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 0, 0], [1, 0, 0]], 否, sampled_vectors_do_not_prove_kernel_image_or_rank_nullity)","explanation_zh":"Wave5第四層核對=([[0, 0, 0], [1, 0, 0]], 否, sampled_vectors_do_not_prove_kernel_image_or_rank_nullity)；依完整明示資料重算；n等於kernel dimension加rank且nullity不是解的個數；kernel image rank皆須symbolic certificate；sample不足。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["秩零度定理"]},"variant":{"family_id":"qf-mod35-ranknull-sampled-boundary","seed":202611041084,"parameters":{"matrix":[[1,0,8],[0,1,1]],"samples":[[0,0,0],[1,0,0]],"domain":"Exact_finite_matrix_rank_nullity","operation":"ranknull-sampled-boundary","semantic_template":"qf-mod35-ranknull-sampled-boundary/{\"matrix\":[[1,0,8],[0,1,1]],\"samples\":[[0,0,0],[1,0,0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0085","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-rank-nullity-theorem"],"title_zh":"秩零度定理","stem_zh":"明示m×n矩陣代表T:R^n到R^m並使用exact RREF。操作=ranknull-sampled-boundary；資料={\"matrix\":[[1,0,9],[0,1,1]],\"samples\":[[0,0,0],[1,0,0]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 0, 0], [1, 0, 0]], 否, sampled_vectors_do_not_prove_kernel_image_or_rank_nullity)","explanation_zh":"Wave5第四層核對=([[0, 0, 0], [1, 0, 0]], 否, sampled_vectors_do_not_prove_kernel_image_or_rank_nullity)；依完整明示資料重算；n等於kernel dimension加rank且nullity不是解的個數；kernel image rank皆須symbolic certificate；sample不足。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["秩零度定理"]},"variant":{"family_id":"qf-mod35-ranknull-sampled-boundary","seed":202611041085,"parameters":{"matrix":[[1,0,9],[0,1,1]],"samples":[[0,0,0],[1,0,0]],"domain":"Exact_finite_matrix_rank_nullity","operation":"ranknull-sampled-boundary","semantic_template":"qf-mod35-ranknull-sampled-boundary/{\"matrix\":[[1,0,9],[0,1,1]],\"samples\":[[0,0,0],[1,0,0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0086","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-rank-nullity-theorem"],"title_zh":"秩零度定理","stem_zh":"明示m×n矩陣代表T:R^n到R^m並使用exact RREF。操作=ranknull-sampled-boundary；資料={\"matrix\":[[1,0,10],[0,1,1]],\"samples\":[[0,0,0],[1,0,0]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 0, 0], [1, 0, 0]], 否, sampled_vectors_do_not_prove_kernel_image_or_rank_nullity)","explanation_zh":"Wave5第四層核對=([[0, 0, 0], [1, 0, 0]], 否, sampled_vectors_do_not_prove_kernel_image_or_rank_nullity)；依完整明示資料重算；n等於kernel dimension加rank且nullity不是解的個數；kernel image rank皆須symbolic certificate；sample不足。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["秩零度定理"]},"variant":{"family_id":"qf-mod35-ranknull-sampled-boundary","seed":202611041086,"parameters":{"matrix":[[1,0,10],[0,1,1]],"samples":[[0,0,0],[1,0,0]],"domain":"Exact_finite_matrix_rank_nullity","operation":"ranknull-sampled-boundary","semantic_template":"qf-mod35-ranknull-sampled-boundary/{\"matrix\":[[1,0,10],[0,1,1]],\"samples\":[[0,0,0],[1,0,0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0087","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-rank-nullity-theorem"],"title_zh":"秩零度定理","stem_zh":"明示m×n矩陣代表T:R^n到R^m並使用exact RREF。操作=ranknull-sampled-boundary；資料={\"matrix\":[[1,0,11],[0,1,1]],\"samples\":[[0,0,0],[1,0,0]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 0, 0], [1, 0, 0]], 否, sampled_vectors_do_not_prove_kernel_image_or_rank_nullity)","explanation_zh":"Wave5第四層核對=([[0, 0, 0], [1, 0, 0]], 否, sampled_vectors_do_not_prove_kernel_image_or_rank_nullity)；依完整明示資料重算；n等於kernel dimension加rank且nullity不是解的個數；kernel image rank皆須symbolic certificate；sample不足。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["秩零度定理"]},"variant":{"family_id":"qf-mod35-ranknull-sampled-boundary","seed":202611041087,"parameters":{"matrix":[[1,0,11],[0,1,1]],"samples":[[0,0,0],[1,0,0]],"domain":"Exact_finite_matrix_rank_nullity","operation":"ranknull-sampled-boundary","semantic_template":"qf-mod35-ranknull-sampled-boundary/{\"matrix\":[[1,0,11],[0,1,1]],\"samples\":[[0,0,0],[1,0,0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0088","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-rank-nullity-theorem"],"title_zh":"秩零度定理","stem_zh":"明示m×n矩陣代表T:R^n到R^m並使用exact RREF。操作=ranknull-sampled-boundary；資料={\"matrix\":[[1,0,12],[0,1,1]],\"samples\":[[0,0,0],[1,0,0]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 0, 0], [1, 0, 0]], 否, sampled_vectors_do_not_prove_kernel_image_or_rank_nullity)","explanation_zh":"Wave5第四層核對=([[0, 0, 0], [1, 0, 0]], 否, sampled_vectors_do_not_prove_kernel_image_or_rank_nullity)；依完整明示資料重算；n等於kernel dimension加rank且nullity不是解的個數；kernel image rank皆須symbolic certificate；sample不足。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["秩零度定理"]},"variant":{"family_id":"qf-mod35-ranknull-sampled-boundary","seed":202611041088,"parameters":{"matrix":[[1,0,12],[0,1,1]],"samples":[[0,0,0],[1,0,0]],"domain":"Exact_finite_matrix_rank_nullity","operation":"ranknull-sampled-boundary","semantic_template":"qf-mod35-ranknull-sampled-boundary/{\"matrix\":[[1,0,12],[0,1,1]],\"samples\":[[0,0,0],[1,0,0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0089","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-rank-nullity-theorem"],"title_zh":"秩零度定理","stem_zh":"明示m×n矩陣代表T:R^n到R^m並使用exact RREF。操作=ranknull-sampled-boundary；資料={\"matrix\":[[1,0,13],[0,1,1]],\"samples\":[[0,0,0],[1,0,0]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 0, 0], [1, 0, 0]], 否, sampled_vectors_do_not_prove_kernel_image_or_rank_nullity)","explanation_zh":"Wave5第四層核對=([[0, 0, 0], [1, 0, 0]], 否, sampled_vectors_do_not_prove_kernel_image_or_rank_nullity)；依完整明示資料重算；n等於kernel dimension加rank且nullity不是解的個數；kernel image rank皆須symbolic certificate；sample不足。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["秩零度定理"]},"variant":{"family_id":"qf-mod35-ranknull-sampled-boundary","seed":202611041089,"parameters":{"matrix":[[1,0,13],[0,1,1]],"samples":[[0,0,0],[1,0,0]],"domain":"Exact_finite_matrix_rank_nullity","operation":"ranknull-sampled-boundary","semantic_template":"qf-mod35-ranknull-sampled-boundary/{\"matrix\":[[1,0,13],[0,1,1]],\"samples\":[[0,0,0],[1,0,0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0090","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-rank-nullity-theorem"],"title_zh":"秩零度定理","stem_zh":"明示m×n矩陣代表T:R^n到R^m並使用exact RREF。操作=ranknull-sampled-boundary；資料={\"matrix\":[[1,0,14],[0,1,1]],\"samples\":[[0,0,0],[1,0,0]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 0, 0], [1, 0, 0]], 否, sampled_vectors_do_not_prove_kernel_image_or_rank_nullity)","explanation_zh":"Wave5第四層核對=([[0, 0, 0], [1, 0, 0]], 否, sampled_vectors_do_not_prove_kernel_image_or_rank_nullity)；依完整明示資料重算；n等於kernel dimension加rank且nullity不是解的個數；kernel image rank皆須symbolic certificate；sample不足。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["秩零度定理"]},"variant":{"family_id":"qf-mod35-ranknull-sampled-boundary","seed":202611041090,"parameters":{"matrix":[[1,0,14],[0,1,1]],"samples":[[0,0,0],[1,0,0]],"domain":"Exact_finite_matrix_rank_nullity","operation":"ranknull-sampled-boundary","semantic_template":"qf-mod35-ranknull-sampled-boundary/{\"matrix\":[[1,0,14],[0,1,1]],\"samples\":[[0,0,0],[1,0,0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0091","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-rank-nullity-theorem"],"title_zh":"秩零度定理","stem_zh":"明示m×n矩陣代表T:R^n到R^m並使用exact RREF。操作=ranknull-sampled-boundary；資料={\"matrix\":[[1,0,15],[0,1,1]],\"samples\":[[0,0,0],[1,0,0]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 0, 0], [1, 0, 0]], 否, sampled_vectors_do_not_prove_kernel_image_or_rank_nullity)","explanation_zh":"Wave5第四層核對=([[0, 0, 0], [1, 0, 0]], 否, sampled_vectors_do_not_prove_kernel_image_or_rank_nullity)；依完整明示資料重算；n等於kernel dimension加rank且nullity不是解的個數；kernel image rank皆須symbolic certificate；sample不足。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["秩零度定理"]},"variant":{"family_id":"qf-mod35-ranknull-sampled-boundary","seed":202611041091,"parameters":{"matrix":[[1,0,15],[0,1,1]],"samples":[[0,0,0],[1,0,0]],"domain":"Exact_finite_matrix_rank_nullity","operation":"ranknull-sampled-boundary","semantic_template":"qf-mod35-ranknull-sampled-boundary/{\"matrix\":[[1,0,15],[0,1,1]],\"samples\":[[0,0,0],[1,0,0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0092","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-rank-nullity-theorem"],"title_zh":"秩零度定理","stem_zh":"明示m×n矩陣代表T:R^n到R^m並使用exact RREF。操作=ranknull-sampled-boundary；資料={\"matrix\":[[1,0,16],[0,1,1]],\"samples\":[[0,0,0],[1,0,0]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 0, 0], [1, 0, 0]], 否, sampled_vectors_do_not_prove_kernel_image_or_rank_nullity)","explanation_zh":"Wave5第四層核對=([[0, 0, 0], [1, 0, 0]], 否, sampled_vectors_do_not_prove_kernel_image_or_rank_nullity)；依完整明示資料重算；n等於kernel dimension加rank且nullity不是解的個數；kernel image rank皆須symbolic certificate；sample不足。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["秩零度定理"]},"variant":{"family_id":"qf-mod35-ranknull-sampled-boundary","seed":202611041092,"parameters":{"matrix":[[1,0,16],[0,1,1]],"samples":[[0,0,0],[1,0,0]],"domain":"Exact_finite_matrix_rank_nullity","operation":"ranknull-sampled-boundary","semantic_template":"qf-mod35-ranknull-sampled-boundary/{\"matrix\":[[1,0,16],[0,1,1]],\"samples\":[[0,0,0],[1,0,0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0093","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-rank-nullity-theorem"],"title_zh":"秩零度定理","stem_zh":"明示m×n矩陣代表T:R^n到R^m並使用exact RREF。操作=ranknull-sampled-boundary；資料={\"matrix\":[[1,0,17],[0,1,1]],\"samples\":[[0,0,0],[1,0,0]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 0, 0], [1, 0, 0]], 否, sampled_vectors_do_not_prove_kernel_image_or_rank_nullity)","explanation_zh":"Wave5第四層核對=([[0, 0, 0], [1, 0, 0]], 否, sampled_vectors_do_not_prove_kernel_image_or_rank_nullity)；依完整明示資料重算；n等於kernel dimension加rank且nullity不是解的個數；kernel image rank皆須symbolic certificate；sample不足。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["秩零度定理"]},"variant":{"family_id":"qf-mod35-ranknull-sampled-boundary","seed":202611041093,"parameters":{"matrix":[[1,0,17],[0,1,1]],"samples":[[0,0,0],[1,0,0]],"domain":"Exact_finite_matrix_rank_nullity","operation":"ranknull-sampled-boundary","semantic_template":"qf-mod35-ranknull-sampled-boundary/{\"matrix\":[[1,0,17],[0,1,1]],\"samples\":[[0,0,0],[1,0,0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0094","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-rank-nullity-theorem"],"title_zh":"秩零度定理","stem_zh":"明示m×n矩陣代表T:R^n到R^m並使用exact RREF。操作=ranknull-sampled-boundary；資料={\"matrix\":[[1,0,18],[0,1,1]],\"samples\":[[0,0,0],[1,0,0]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 0, 0], [1, 0, 0]], 否, sampled_vectors_do_not_prove_kernel_image_or_rank_nullity)","explanation_zh":"Wave5第四層核對=([[0, 0, 0], [1, 0, 0]], 否, sampled_vectors_do_not_prove_kernel_image_or_rank_nullity)；依完整明示資料重算；n等於kernel dimension加rank且nullity不是解的個數；kernel image rank皆須symbolic certificate；sample不足。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["秩零度定理"]},"variant":{"family_id":"qf-mod35-ranknull-sampled-boundary","seed":202611041094,"parameters":{"matrix":[[1,0,18],[0,1,1]],"samples":[[0,0,0],[1,0,0]],"domain":"Exact_finite_matrix_rank_nullity","operation":"ranknull-sampled-boundary","semantic_template":"qf-mod35-ranknull-sampled-boundary/{\"matrix\":[[1,0,18],[0,1,1]],\"samples\":[[0,0,0],[1,0,0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0095","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-rank-nullity-theorem"],"title_zh":"秩零度定理","stem_zh":"明示m×n矩陣代表T:R^n到R^m並使用exact RREF。操作=ranknull-sampled-boundary；資料={\"matrix\":[[1,0,19],[0,1,1]],\"samples\":[[0,0,0],[1,0,0]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 0, 0], [1, 0, 0]], 否, sampled_vectors_do_not_prove_kernel_image_or_rank_nullity)","explanation_zh":"Wave5第四層核對=([[0, 0, 0], [1, 0, 0]], 否, sampled_vectors_do_not_prove_kernel_image_or_rank_nullity)；依完整明示資料重算；n等於kernel dimension加rank且nullity不是解的個數；kernel image rank皆須symbolic certificate；sample不足。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["秩零度定理"]},"variant":{"family_id":"qf-mod35-ranknull-sampled-boundary","seed":202611041095,"parameters":{"matrix":[[1,0,19],[0,1,1]],"samples":[[0,0,0],[1,0,0]],"domain":"Exact_finite_matrix_rank_nullity","operation":"ranknull-sampled-boundary","semantic_template":"qf-mod35-ranknull-sampled-boundary/{\"matrix\":[[1,0,19],[0,1,1]],\"samples\":[[0,0,0],[1,0,0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0096","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-rank-nullity-theorem"],"title_zh":"秩零度定理","stem_zh":"明示m×n矩陣代表T:R^n到R^m並使用exact RREF。操作=ranknull-sampled-boundary；資料={\"matrix\":[[1,0,20],[0,1,1]],\"samples\":[[0,0,0],[1,0,0]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 0, 0], [1, 0, 0]], 否, sampled_vectors_do_not_prove_kernel_image_or_rank_nullity)","explanation_zh":"Wave5第四層核對=([[0, 0, 0], [1, 0, 0]], 否, sampled_vectors_do_not_prove_kernel_image_or_rank_nullity)；依完整明示資料重算；n等於kernel dimension加rank且nullity不是解的個數；kernel image rank皆須symbolic certificate；sample不足。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["秩零度定理"]},"variant":{"family_id":"qf-mod35-ranknull-sampled-boundary","seed":202611041096,"parameters":{"matrix":[[1,0,20],[0,1,1]],"samples":[[0,0,0],[1,0,0]],"domain":"Exact_finite_matrix_rank_nullity","operation":"ranknull-sampled-boundary","semantic_template":"qf-mod35-ranknull-sampled-boundary/{\"matrix\":[[1,0,20],[0,1,1]],\"samples\":[[0,0,0],[1,0,0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0097","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-rank-nullity-theorem"],"title_zh":"秩零度定理","stem_zh":"明示m×n矩陣代表T:R^n到R^m並使用exact RREF。操作=ranknull-sampled-boundary；資料={\"matrix\":[[1,0,21],[0,1,1]],\"samples\":[[0,0,0],[1,0,0]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 0, 0], [1, 0, 0]], 否, sampled_vectors_do_not_prove_kernel_image_or_rank_nullity)","explanation_zh":"Wave5第四層核對=([[0, 0, 0], [1, 0, 0]], 否, sampled_vectors_do_not_prove_kernel_image_or_rank_nullity)；依完整明示資料重算；n等於kernel dimension加rank且nullity不是解的個數；kernel image rank皆須symbolic certificate；sample不足。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["秩零度定理"]},"variant":{"family_id":"qf-mod35-ranknull-sampled-boundary","seed":202611041097,"parameters":{"matrix":[[1,0,21],[0,1,1]],"samples":[[0,0,0],[1,0,0]],"domain":"Exact_finite_matrix_rank_nullity","operation":"ranknull-sampled-boundary","semantic_template":"qf-mod35-ranknull-sampled-boundary/{\"matrix\":[[1,0,21],[0,1,1]],\"samples\":[[0,0,0],[1,0,0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0098","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-rank-nullity-theorem"],"title_zh":"秩零度定理","stem_zh":"明示m×n矩陣代表T:R^n到R^m並使用exact RREF。操作=ranknull-sampled-boundary；資料={\"matrix\":[[1,0,22],[0,1,1]],\"samples\":[[0,0,0],[1,0,0]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 0, 0], [1, 0, 0]], 否, sampled_vectors_do_not_prove_kernel_image_or_rank_nullity)","explanation_zh":"Wave5第四層核對=([[0, 0, 0], [1, 0, 0]], 否, sampled_vectors_do_not_prove_kernel_image_or_rank_nullity)；依完整明示資料重算；n等於kernel dimension加rank且nullity不是解的個數；kernel image rank皆須symbolic certificate；sample不足。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["秩零度定理"]},"variant":{"family_id":"qf-mod35-ranknull-sampled-boundary","seed":202611041098,"parameters":{"matrix":[[1,0,22],[0,1,1]],"samples":[[0,0,0],[1,0,0]],"domain":"Exact_finite_matrix_rank_nullity","operation":"ranknull-sampled-boundary","semantic_template":"qf-mod35-ranknull-sampled-boundary/{\"matrix\":[[1,0,22],[0,1,1]],\"samples\":[[0,0,0],[1,0,0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0099","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-rank-nullity-theorem"],"title_zh":"秩零度定理","stem_zh":"明示m×n矩陣代表T:R^n到R^m並使用exact RREF。操作=ranknull-sampled-boundary；資料={\"matrix\":[[1,0,23],[0,1,1]],\"samples\":[[0,0,0],[1,0,0]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 0, 0], [1, 0, 0]], 否, sampled_vectors_do_not_prove_kernel_image_or_rank_nullity)","explanation_zh":"Wave5第四層核對=([[0, 0, 0], [1, 0, 0]], 否, sampled_vectors_do_not_prove_kernel_image_or_rank_nullity)；依完整明示資料重算；n等於kernel dimension加rank且nullity不是解的個數；kernel image rank皆須symbolic certificate；sample不足。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["秩零度定理"]},"variant":{"family_id":"qf-mod35-ranknull-sampled-boundary","seed":202611041099,"parameters":{"matrix":[[1,0,23],[0,1,1]],"samples":[[0,0,0],[1,0,0]],"domain":"Exact_finite_matrix_rank_nullity","operation":"ranknull-sampled-boundary","semantic_template":"qf-mod35-ranknull-sampled-boundary/{\"matrix\":[[1,0,23],[0,1,1]],\"samples\":[[0,0,0],[1,0,0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0100","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-rank-nullity-theorem"],"title_zh":"秩零度定理","stem_zh":"明示m×n矩陣代表T:R^n到R^m並使用exact RREF。操作=ranknull-sampled-boundary；資料={\"matrix\":[[1,0,24],[0,1,1]],\"samples\":[[0,0,0],[1,0,0]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 0, 0], [1, 0, 0]], 否, sampled_vectors_do_not_prove_kernel_image_or_rank_nullity)","explanation_zh":"Wave5第四層核對=([[0, 0, 0], [1, 0, 0]], 否, sampled_vectors_do_not_prove_kernel_image_or_rank_nullity)；依完整明示資料重算；n等於kernel dimension加rank且nullity不是解的個數；kernel image rank皆須symbolic certificate；sample不足。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["秩零度定理"]},"variant":{"family_id":"qf-mod35-ranknull-sampled-boundary","seed":202611041100,"parameters":{"matrix":[[1,0,24],[0,1,1]],"samples":[[0,0,0],[1,0,0]],"domain":"Exact_finite_matrix_rank_nullity","operation":"ranknull-sampled-boundary","semantic_template":"qf-mod35-ranknull-sampled-boundary/{\"matrix\":[[1,0,24],[0,1,1]],\"samples\":[[0,0,0],[1,0,0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}