{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r03-linear-map-image-0001","batch_id":"batch-20261014-advance-20261102-r03-linear-map-image-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-image"],"title_zh":"線性映射的像","stem_zh":"同一精確2×2線性映射與column-span證書。操作=image-membership；資料={\"matrix\":[[1,1],[2,2]],\"target\":[2,4]}。請依 [target,rank,image_basis,in_image,certificate] 欄序回答。","answer":{"kind":"expression","value":"([2, 4], 1, [[1, 2]], 是, [column_span_cross, [1, 2], 0])","explanation_zh":"Wave5第二層核對=([2, 4], 1, [[1, 2]], 是, [column_span_cross, [1, 2], 0])；依完整明示資料重算；image是full-real column span且不等同codomain；bounded fiber search失敗不證全域nonmembership。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性映射的像"]},"variant":{"family_id":"qf-mod33-image-membership","seed":202611023001,"parameters":{"matrix":[[1,1],[2,2]],"target":[2,4],"domain":"Exact_R2_matrix_image","operation":"image-membership","semantic_template":"qf-mod33-image-membership/{\"matrix\":[[1,1],[2,2]],\"target\":[2,4]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r03-linear-map-image-0002","batch_id":"batch-20261014-advance-20261102-r03-linear-map-image-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-image"],"title_zh":"線性映射的像","stem_zh":"同一精確2×2線性映射與column-span證書。操作=image-membership；資料={\"matrix\":[[1,2],[2,4]],\"target\":[3,6]}。請依 [target,rank,image_basis,in_image,certificate] 欄序回答。","answer":{"kind":"expression","value":"([3, 6], 1, [[1, 2]], 是, [column_span_cross, [1, 2], 0])","explanation_zh":"Wave5第二層核對=([3, 6], 1, [[1, 2]], 是, [column_span_cross, [1, 2], 0])；依完整明示資料重算；image是full-real column span且不等同codomain；bounded fiber search失敗不證全域nonmembership。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性映射的像"]},"variant":{"family_id":"qf-mod33-image-membership","seed":202611023002,"parameters":{"matrix":[[1,2],[2,4]],"target":[3,6],"domain":"Exact_R2_matrix_image","operation":"image-membership","semantic_template":"qf-mod33-image-membership/{\"matrix\":[[1,2],[2,4]],\"target\":[3,6]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r03-linear-map-image-0003","batch_id":"batch-20261014-advance-20261102-r03-linear-map-image-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-image"],"title_zh":"線性映射的像","stem_zh":"同一精確2×2線性映射與column-span證書。操作=image-membership；資料={\"matrix\":[[1,3],[2,6]],\"target\":[4,8]}。請依 [target,rank,image_basis,in_image,certificate] 欄序回答。","answer":{"kind":"expression","value":"([4, 8], 1, [[1, 2]], 是, [column_span_cross, [1, 2], 0])","explanation_zh":"Wave5第二層核對=([4, 8], 1, [[1, 2]], 是, [column_span_cross, [1, 2], 0])；依完整明示資料重算；image是full-real column span且不等同codomain；bounded fiber search失敗不證全域nonmembership。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性映射的像"]},"variant":{"family_id":"qf-mod33-image-membership","seed":202611023003,"parameters":{"matrix":[[1,3],[2,6]],"target":[4,8],"domain":"Exact_R2_matrix_image","operation":"image-membership","semantic_template":"qf-mod33-image-membership/{\"matrix\":[[1,3],[2,6]],\"target\":[4,8]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r03-linear-map-image-0004","batch_id":"batch-20261014-advance-20261102-r03-linear-map-image-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-image"],"title_zh":"線性映射的像","stem_zh":"同一精確2×2線性映射與column-span證書。操作=image-membership；資料={\"matrix\":[[1,4],[2,8]],\"target\":[5,10]}。請依 [target,rank,image_basis,in_image,certificate] 欄序回答。","answer":{"kind":"expression","value":"([5, 10], 1, [[1, 2]], 是, [column_span_cross, [1, 2], 0])","explanation_zh":"Wave5第二層核對=([5, 10], 1, [[1, 2]], 是, [column_span_cross, [1, 2], 0])；依完整明示資料重算；image是full-real column span且不等同codomain；bounded fiber search失敗不證全域nonmembership。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性映射的像"]},"variant":{"family_id":"qf-mod33-image-membership","seed":202611023004,"parameters":{"matrix":[[1,4],[2,8]],"target":[5,10],"domain":"Exact_R2_matrix_image","operation":"image-membership","semantic_template":"qf-mod33-image-membership/{\"matrix\":[[1,4],[2,8]],\"target\":[5,10]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r03-linear-map-image-0005","batch_id":"batch-20261014-advance-20261102-r03-linear-map-image-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-image"],"title_zh":"線性映射的像","stem_zh":"同一精確2×2線性映射與column-span證書。操作=image-membership；資料={\"matrix\":[[1,5],[2,10]],\"target\":[6,12]}。請依 [target,rank,image_basis,in_image,certificate] 欄序回答。","answer":{"kind":"expression","value":"([6, 12], 1, [[1, 2]], 是, [column_span_cross, [1, 2], 0])","explanation_zh":"Wave5第二層核對=([6, 12], 1, [[1, 2]], 是, [column_span_cross, [1, 2], 0])；依完整明示資料重算；image是full-real column span且不等同codomain；bounded fiber search失敗不證全域nonmembership。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性映射的像"]},"variant":{"family_id":"qf-mod33-image-membership","seed":202611023005,"parameters":{"matrix":[[1,5],[2,10]],"target":[6,12],"domain":"Exact_R2_matrix_image","operation":"image-membership","semantic_template":"qf-mod33-image-membership/{\"matrix\":[[1,5],[2,10]],\"target\":[6,12]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r03-linear-map-image-0006","batch_id":"batch-20261014-advance-20261102-r03-linear-map-image-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-image"],"title_zh":"線性映射的像","stem_zh":"同一精確2×2線性映射與column-span證書。操作=image-membership；資料={\"matrix\":[[1,6],[2,12]],\"target\":[7,14]}。請依 [target,rank,image_basis,in_image,certificate] 欄序回答。","answer":{"kind":"expression","value":"([7, 14], 1, [[1, 2]], 是, [column_span_cross, [1, 2], 0])","explanation_zh":"Wave5第二層核對=([7, 14], 1, [[1, 2]], 是, [column_span_cross, [1, 2], 0])；依完整明示資料重算；image是full-real column span且不等同codomain；bounded fiber search失敗不證全域nonmembership。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性映射的像"]},"variant":{"family_id":"qf-mod33-image-membership","seed":202611023006,"parameters":{"matrix":[[1,6],[2,12]],"target":[7,14],"domain":"Exact_R2_matrix_image","operation":"image-membership","semantic_template":"qf-mod33-image-membership/{\"matrix\":[[1,6],[2,12]],\"target\":[7,14]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r03-linear-map-image-0007","batch_id":"batch-20261014-advance-20261102-r03-linear-map-image-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-image"],"title_zh":"線性映射的像","stem_zh":"同一精確2×2線性映射與column-span證書。操作=image-membership；資料={\"matrix\":[[1,7],[2,14]],\"target\":[8,16]}。請依 [target,rank,image_basis,in_image,certificate] 欄序回答。","answer":{"kind":"expression","value":"([8, 16], 1, [[1, 2]], 是, [column_span_cross, [1, 2], 0])","explanation_zh":"Wave5第二層核對=([8, 16], 1, [[1, 2]], 是, [column_span_cross, [1, 2], 0])；依完整明示資料重算；image是full-real column span且不等同codomain；bounded fiber search失敗不證全域nonmembership。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性映射的像"]},"variant":{"family_id":"qf-mod33-image-membership","seed":202611023007,"parameters":{"matrix":[[1,7],[2,14]],"target":[8,16],"domain":"Exact_R2_matrix_image","operation":"image-membership","semantic_template":"qf-mod33-image-membership/{\"matrix\":[[1,7],[2,14]],\"target\":[8,16]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r03-linear-map-image-0008","batch_id":"batch-20261014-advance-20261102-r03-linear-map-image-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-image"],"title_zh":"線性映射的像","stem_zh":"同一精確2×2線性映射與column-span證書。操作=image-membership；資料={\"matrix\":[[1,8],[2,16]],\"target\":[9,18]}。請依 [target,rank,image_basis,in_image,certificate] 欄序回答。","answer":{"kind":"expression","value":"([9, 18], 1, [[1, 2]], 是, [column_span_cross, [1, 2], 0])","explanation_zh":"Wave5第二層核對=([9, 18], 1, [[1, 2]], 是, [column_span_cross, [1, 2], 0])；依完整明示資料重算；image是full-real column span且不等同codomain；bounded fiber search失敗不證全域nonmembership。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性映射的像"]},"variant":{"family_id":"qf-mod33-image-membership","seed":202611023008,"parameters":{"matrix":[[1,8],[2,16]],"target":[9,18],"domain":"Exact_R2_matrix_image","operation":"image-membership","semantic_template":"qf-mod33-image-membership/{\"matrix\":[[1,8],[2,16]],\"target\":[9,18]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r03-linear-map-image-0009","batch_id":"batch-20261014-advance-20261102-r03-linear-map-image-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-image"],"title_zh":"線性映射的像","stem_zh":"同一精確2×2線性映射與column-span證書。操作=image-membership；資料={\"matrix\":[[1,9],[2,18]],\"target\":[10,20]}。請依 [target,rank,image_basis,in_image,certificate] 欄序回答。","answer":{"kind":"expression","value":"([10, 20], 1, [[1, 2]], 是, [column_span_cross, [1, 2], 0])","explanation_zh":"Wave5第二層核對=([10, 20], 1, [[1, 2]], 是, [column_span_cross, [1, 2], 0])；依完整明示資料重算；image是full-real column span且不等同codomain；bounded fiber search失敗不證全域nonmembership。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性映射的像"]},"variant":{"family_id":"qf-mod33-image-membership","seed":202611023009,"parameters":{"matrix":[[1,9],[2,18]],"target":[10,20],"domain":"Exact_R2_matrix_image","operation":"image-membership","semantic_template":"qf-mod33-image-membership/{\"matrix\":[[1,9],[2,18]],\"target\":[10,20]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r03-linear-map-image-0010","batch_id":"batch-20261014-advance-20261102-r03-linear-map-image-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-image"],"title_zh":"線性映射的像","stem_zh":"同一精確2×2線性映射與column-span證書。操作=image-membership；資料={\"matrix\":[[1,10],[2,20]],\"target\":[11,22]}。請依 [target,rank,image_basis,in_image,certificate] 欄序回答。","answer":{"kind":"expression","value":"([11, 22], 1, [[1, 2]], 是, [column_span_cross, [1, 2], 0])","explanation_zh":"Wave5第二層核對=([11, 22], 1, [[1, 2]], 是, [column_span_cross, [1, 2], 0])；依完整明示資料重算；image是full-real column span且不等同codomain；bounded fiber search失敗不證全域nonmembership。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性映射的像"]},"variant":{"family_id":"qf-mod33-image-membership","seed":202611023010,"parameters":{"matrix":[[1,10],[2,20]],"target":[11,22],"domain":"Exact_R2_matrix_image","operation":"image-membership","semantic_template":"qf-mod33-image-membership/{\"matrix\":[[1,10],[2,20]],\"target\":[11,22]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r03-linear-map-image-0011","batch_id":"batch-20261014-advance-20261102-r03-linear-map-image-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-image"],"title_zh":"線性映射的像","stem_zh":"同一精確2×2線性映射與column-span證書。操作=image-membership；資料={\"matrix\":[[1,11],[2,22]],\"target\":[12,24]}。請依 [target,rank,image_basis,in_image,certificate] 欄序回答。","answer":{"kind":"expression","value":"([12, 24], 1, [[1, 2]], 是, [column_span_cross, [1, 2], 0])","explanation_zh":"Wave5第二層核對=([12, 24], 1, [[1, 2]], 是, [column_span_cross, [1, 2], 0])；依完整明示資料重算；image是full-real column span且不等同codomain；bounded fiber search失敗不證全域nonmembership。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性映射的像"]},"variant":{"family_id":"qf-mod33-image-membership","seed":202611023011,"parameters":{"matrix":[[1,11],[2,22]],"target":[12,24],"domain":"Exact_R2_matrix_image","operation":"image-membership","semantic_template":"qf-mod33-image-membership/{\"matrix\":[[1,11],[2,22]],\"target\":[12,24]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r03-linear-map-image-0012","batch_id":"batch-20261014-advance-20261102-r03-linear-map-image-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-image"],"title_zh":"線性映射的像","stem_zh":"同一精確2×2線性映射與column-span證書。操作=image-membership；資料={\"matrix\":[[1,12],[2,24]],\"target\":[13,26]}。請依 [target,rank,image_basis,in_image,certificate] 欄序回答。","answer":{"kind":"expression","value":"([13, 26], 1, [[1, 2]], 是, [column_span_cross, [1, 2], 0])","explanation_zh":"Wave5第二層核對=([13, 26], 1, [[1, 2]], 是, [column_span_cross, [1, 2], 0])；依完整明示資料重算；image是full-real column span且不等同codomain；bounded fiber search失敗不證全域nonmembership。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性映射的像"]},"variant":{"family_id":"qf-mod33-image-membership","seed":202611023012,"parameters":{"matrix":[[1,12],[2,24]],"target":[13,26],"domain":"Exact_R2_matrix_image","operation":"image-membership","semantic_template":"qf-mod33-image-membership/{\"matrix\":[[1,12],[2,24]],\"target\":[13,26]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r03-linear-map-image-0013","batch_id":"batch-20261014-advance-20261102-r03-linear-map-image-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-image"],"title_zh":"線性映射的像","stem_zh":"同一精確2×2線性映射與column-span證書。操作=image-membership；資料={\"matrix\":[[1,13],[2,26]],\"target\":[14,28]}。請依 [target,rank,image_basis,in_image,certificate] 欄序回答。","answer":{"kind":"expression","value":"([14, 28], 1, [[1, 2]], 是, [column_span_cross, [1, 2], 0])","explanation_zh":"Wave5第二層核對=([14, 28], 1, [[1, 2]], 是, [column_span_cross, [1, 2], 0])；依完整明示資料重算；image是full-real column span且不等同codomain；bounded fiber search失敗不證全域nonmembership。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性映射的像"]},"variant":{"family_id":"qf-mod33-image-membership","seed":202611023013,"parameters":{"matrix":[[1,13],[2,26]],"target":[14,28],"domain":"Exact_R2_matrix_image","operation":"image-membership","semantic_template":"qf-mod33-image-membership/{\"matrix\":[[1,13],[2,26]],\"target\":[14,28]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r03-linear-map-image-0014","batch_id":"batch-20261014-advance-20261102-r03-linear-map-image-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-image"],"title_zh":"線性映射的像","stem_zh":"同一精確2×2線性映射與column-span證書。操作=image-membership；資料={\"matrix\":[[1,14],[2,28]],\"target\":[15,30]}。請依 [target,rank,image_basis,in_image,certificate] 欄序回答。","answer":{"kind":"expression","value":"([15, 30], 1, [[1, 2]], 是, [column_span_cross, [1, 2], 0])","explanation_zh":"Wave5第二層核對=([15, 30], 1, [[1, 2]], 是, [column_span_cross, [1, 2], 0])；依完整明示資料重算；image是full-real column span且不等同codomain；bounded fiber search失敗不證全域nonmembership。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性映射的像"]},"variant":{"family_id":"qf-mod33-image-membership","seed":202611023014,"parameters":{"matrix":[[1,14],[2,28]],"target":[15,30],"domain":"Exact_R2_matrix_image","operation":"image-membership","semantic_template":"qf-mod33-image-membership/{\"matrix\":[[1,14],[2,28]],\"target\":[15,30]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r03-linear-map-image-0015","batch_id":"batch-20261014-advance-20261102-r03-linear-map-image-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-image"],"title_zh":"線性映射的像","stem_zh":"同一精確2×2線性映射與column-span證書。操作=image-membership；資料={\"matrix\":[[1,15],[2,30]],\"target\":[16,32]}。請依 [target,rank,image_basis,in_image,certificate] 欄序回答。","answer":{"kind":"expression","value":"([16, 32], 1, [[1, 2]], 是, [column_span_cross, [1, 2], 0])","explanation_zh":"Wave5第二層核對=([16, 32], 1, [[1, 2]], 是, [column_span_cross, [1, 2], 0])；依完整明示資料重算；image是full-real column span且不等同codomain；bounded fiber search失敗不證全域nonmembership。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性映射的像"]},"variant":{"family_id":"qf-mod33-image-membership","seed":202611023015,"parameters":{"matrix":[[1,15],[2,30]],"target":[16,32],"domain":"Exact_R2_matrix_image","operation":"image-membership","semantic_template":"qf-mod33-image-membership/{\"matrix\":[[1,15],[2,30]],\"target\":[16,32]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r03-linear-map-image-0016","batch_id":"batch-20261014-advance-20261102-r03-linear-map-image-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-image"],"title_zh":"線性映射的像","stem_zh":"同一精確2×2線性映射與column-span證書。操作=image-membership；資料={\"matrix\":[[1,16],[2,32]],\"target\":[17,34]}。請依 [target,rank,image_basis,in_image,certificate] 欄序回答。","answer":{"kind":"expression","value":"([17, 34], 1, [[1, 2]], 是, [column_span_cross, [1, 2], 0])","explanation_zh":"Wave5第二層核對=([17, 34], 1, [[1, 2]], 是, [column_span_cross, [1, 2], 0])；依完整明示資料重算；image是full-real column span且不等同codomain；bounded fiber search失敗不證全域nonmembership。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性映射的像"]},"variant":{"family_id":"qf-mod33-image-membership","seed":202611023016,"parameters":{"matrix":[[1,16],[2,32]],"target":[17,34],"domain":"Exact_R2_matrix_image","operation":"image-membership","semantic_template":"qf-mod33-image-membership/{\"matrix\":[[1,16],[2,32]],\"target\":[17,34]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r03-linear-map-image-0017","batch_id":"batch-20261014-advance-20261102-r03-linear-map-image-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-image"],"title_zh":"線性映射的像","stem_zh":"同一精確2×2線性映射與column-span證書。操作=image-membership；資料={\"matrix\":[[1,17],[2,34]],\"target\":[18,36]}。請依 [target,rank,image_basis,in_image,certificate] 欄序回答。","answer":{"kind":"expression","value":"([18, 36], 1, [[1, 2]], 是, [column_span_cross, [1, 2], 0])","explanation_zh":"Wave5第二層核對=([18, 36], 1, [[1, 2]], 是, [column_span_cross, [1, 2], 0])；依完整明示資料重算；image是full-real column span且不等同codomain；bounded fiber search失敗不證全域nonmembership。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性映射的像"]},"variant":{"family_id":"qf-mod33-image-membership","seed":202611023017,"parameters":{"matrix":[[1,17],[2,34]],"target":[18,36],"domain":"Exact_R2_matrix_image","operation":"image-membership","semantic_template":"qf-mod33-image-membership/{\"matrix\":[[1,17],[2,34]],\"target\":[18,36]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r03-linear-map-image-0018","batch_id":"batch-20261014-advance-20261102-r03-linear-map-image-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-image"],"title_zh":"線性映射的像","stem_zh":"同一精確2×2線性映射與column-span證書。操作=image-membership；資料={\"matrix\":[[1,18],[2,36]],\"target\":[19,38]}。請依 [target,rank,image_basis,in_image,certificate] 欄序回答。","answer":{"kind":"expression","value":"([19, 38], 1, [[1, 2]], 是, [column_span_cross, [1, 2], 0])","explanation_zh":"Wave5第二層核對=([19, 38], 1, [[1, 2]], 是, [column_span_cross, [1, 2], 0])；依完整明示資料重算；image是full-real column span且不等同codomain；bounded fiber search失敗不證全域nonmembership。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性映射的像"]},"variant":{"family_id":"qf-mod33-image-membership","seed":202611023018,"parameters":{"matrix":[[1,18],[2,36]],"target":[19,38],"domain":"Exact_R2_matrix_image","operation":"image-membership","semantic_template":"qf-mod33-image-membership/{\"matrix\":[[1,18],[2,36]],\"target\":[19,38]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r03-linear-map-image-0019","batch_id":"batch-20261014-advance-20261102-r03-linear-map-image-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-image"],"title_zh":"線性映射的像","stem_zh":"同一精確2×2線性映射與column-span證書。操作=image-membership；資料={\"matrix\":[[1,19],[2,38]],\"target\":[20,40]}。請依 [target,rank,image_basis,in_image,certificate] 欄序回答。","answer":{"kind":"expression","value":"([20, 40], 1, [[1, 2]], 是, [column_span_cross, [1, 2], 0])","explanation_zh":"Wave5第二層核對=([20, 40], 1, [[1, 2]], 是, [column_span_cross, [1, 2], 0])；依完整明示資料重算；image是full-real column span且不等同codomain；bounded fiber search失敗不證全域nonmembership。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性映射的像"]},"variant":{"family_id":"qf-mod33-image-membership","seed":202611023019,"parameters":{"matrix":[[1,19],[2,38]],"target":[20,40],"domain":"Exact_R2_matrix_image","operation":"image-membership","semantic_template":"qf-mod33-image-membership/{\"matrix\":[[1,19],[2,38]],\"target\":[20,40]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r03-linear-map-image-0020","batch_id":"batch-20261014-advance-20261102-r03-linear-map-image-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-image"],"title_zh":"線性映射的像","stem_zh":"同一精確2×2線性映射與column-span證書。操作=image-membership；資料={\"matrix\":[[1,20],[2,40]],\"target\":[21,42]}。請依 [target,rank,image_basis,in_image,certificate] 欄序回答。","answer":{"kind":"expression","value":"([21, 42], 1, [[1, 2]], 是, [column_span_cross, [1, 2], 0])","explanation_zh":"Wave5第二層核對=([21, 42], 1, [[1, 2]], 是, [column_span_cross, [1, 2], 0])；依完整明示資料重算；image是full-real column span且不等同codomain；bounded fiber search失敗不證全域nonmembership。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性映射的像"]},"variant":{"family_id":"qf-mod33-image-membership","seed":202611023020,"parameters":{"matrix":[[1,20],[2,40]],"target":[21,42],"domain":"Exact_R2_matrix_image","operation":"image-membership","semantic_template":"qf-mod33-image-membership/{\"matrix\":[[1,20],[2,40]],\"target\":[21,42]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r03-linear-map-image-0021","batch_id":"batch-20261014-advance-20261102-r03-linear-map-image-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-image"],"title_zh":"線性映射的像","stem_zh":"同一精確2×2線性映射與column-span證書。操作=image-membership；資料={\"matrix\":[[1,21],[2,42]],\"target\":[22,44]}。請依 [target,rank,image_basis,in_image,certificate] 欄序回答。","answer":{"kind":"expression","value":"([22, 44], 1, [[1, 2]], 是, [column_span_cross, [1, 2], 0])","explanation_zh":"Wave5第二層核對=([22, 44], 1, [[1, 2]], 是, [column_span_cross, [1, 2], 0])；依完整明示資料重算；image是full-real column span且不等同codomain；bounded fiber search失敗不證全域nonmembership。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性映射的像"]},"variant":{"family_id":"qf-mod33-image-membership","seed":202611023021,"parameters":{"matrix":[[1,21],[2,42]],"target":[22,44],"domain":"Exact_R2_matrix_image","operation":"image-membership","semantic_template":"qf-mod33-image-membership/{\"matrix\":[[1,21],[2,42]],\"target\":[22,44]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r03-linear-map-image-0022","batch_id":"batch-20261014-advance-20261102-r03-linear-map-image-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-image"],"title_zh":"線性映射的像","stem_zh":"同一精確2×2線性映射與column-span證書。操作=image-membership；資料={\"matrix\":[[1,22],[2,44]],\"target\":[23,46]}。請依 [target,rank,image_basis,in_image,certificate] 欄序回答。","answer":{"kind":"expression","value":"([23, 46], 1, [[1, 2]], 是, [column_span_cross, [1, 2], 0])","explanation_zh":"Wave5第二層核對=([23, 46], 1, [[1, 2]], 是, [column_span_cross, [1, 2], 0])；依完整明示資料重算；image是full-real column span且不等同codomain；bounded fiber search失敗不證全域nonmembership。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性映射的像"]},"variant":{"family_id":"qf-mod33-image-membership","seed":202611023022,"parameters":{"matrix":[[1,22],[2,44]],"target":[23,46],"domain":"Exact_R2_matrix_image","operation":"image-membership","semantic_template":"qf-mod33-image-membership/{\"matrix\":[[1,22],[2,44]],\"target\":[23,46]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r03-linear-map-image-0023","batch_id":"batch-20261014-advance-20261102-r03-linear-map-image-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-image"],"title_zh":"線性映射的像","stem_zh":"同一精確2×2線性映射與column-span證書。操作=image-membership；資料={\"matrix\":[[1,23],[2,46]],\"target\":[24,48]}。請依 [target,rank,image_basis,in_image,certificate] 欄序回答。","answer":{"kind":"expression","value":"([24, 48], 1, [[1, 2]], 是, [column_span_cross, [1, 2], 0])","explanation_zh":"Wave5第二層核對=([24, 48], 1, [[1, 2]], 是, [column_span_cross, [1, 2], 0])；依完整明示資料重算；image是full-real column span且不等同codomain；bounded fiber search失敗不證全域nonmembership。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性映射的像"]},"variant":{"family_id":"qf-mod33-image-membership","seed":202611023023,"parameters":{"matrix":[[1,23],[2,46]],"target":[24,48],"domain":"Exact_R2_matrix_image","operation":"image-membership","semantic_template":"qf-mod33-image-membership/{\"matrix\":[[1,23],[2,46]],\"target\":[24,48]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r03-linear-map-image-0024","batch_id":"batch-20261014-advance-20261102-r03-linear-map-image-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-image"],"title_zh":"線性映射的像","stem_zh":"同一精確2×2線性映射與column-span證書。操作=image-membership；資料={\"matrix\":[[1,24],[2,48]],\"target\":[25,50]}。請依 [target,rank,image_basis,in_image,certificate] 欄序回答。","answer":{"kind":"expression","value":"([25, 50], 1, [[1, 2]], 是, [column_span_cross, [1, 2], 0])","explanation_zh":"Wave5第二層核對=([25, 50], 1, [[1, 2]], 是, [column_span_cross, [1, 2], 0])；依完整明示資料重算；image是full-real column span且不等同codomain；bounded fiber search失敗不證全域nonmembership。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性映射的像"]},"variant":{"family_id":"qf-mod33-image-membership","seed":202611023024,"parameters":{"matrix":[[1,24],[2,48]],"target":[25,50],"domain":"Exact_R2_matrix_image","operation":"image-membership","semantic_template":"qf-mod33-image-membership/{\"matrix\":[[1,24],[2,48]],\"target\":[25,50]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r03-linear-map-image-0025","batch_id":"batch-20261014-advance-20261102-r03-linear-map-image-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-image"],"title_zh":"線性映射的像","stem_zh":"同一精確2×2線性映射與column-span證書。操作=image-membership；資料={\"matrix\":[[1,25],[2,50]],\"target\":[26,52]}。請依 [target,rank,image_basis,in_image,certificate] 欄序回答。","answer":{"kind":"expression","value":"([26, 52], 1, [[1, 2]], 是, [column_span_cross, [1, 2], 0])","explanation_zh":"Wave5第二層核對=([26, 52], 1, [[1, 2]], 是, [column_span_cross, [1, 2], 0])；依完整明示資料重算；image是full-real column span且不等同codomain；bounded fiber search失敗不證全域nonmembership。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性映射的像"]},"variant":{"family_id":"qf-mod33-image-membership","seed":202611023025,"parameters":{"matrix":[[1,25],[2,50]],"target":[26,52],"domain":"Exact_R2_matrix_image","operation":"image-membership","semantic_template":"qf-mod33-image-membership/{\"matrix\":[[1,25],[2,50]],\"target\":[26,52]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r03-linear-map-image-0026","batch_id":"batch-20261014-advance-20261102-r03-linear-map-image-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-image"],"title_zh":"線性映射的像","stem_zh":"同一精確2×2線性映射與column-span證書。操作=image-basis；資料={\"matrix\":[[1,1],[2,2]]}。請依 [matrix,rank,image_basis,certificate] 欄序回答。","answer":{"kind":"expression","value":"([[1, 1], [2, 2]], 1, [[1, 2]], full_real_column_span)","explanation_zh":"Wave5第二層核對=([[1, 1], [2, 2]], 1, [[1, 2]], full_real_column_span)；依完整明示資料重算；image是full-real column span且不等同codomain；bounded fiber search失敗不證全域nonmembership。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性映射的像"]},"variant":{"family_id":"qf-mod33-image-basis","seed":202611023026,"parameters":{"matrix":[[1,1],[2,2]],"domain":"Exact_R2_matrix_image","operation":"image-basis","semantic_template":"qf-mod33-image-basis/{\"matrix\":[[1,1],[2,2]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r03-linear-map-image-0027","batch_id":"batch-20261014-advance-20261102-r03-linear-map-image-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-image"],"title_zh":"線性映射的像","stem_zh":"同一精確2×2線性映射與column-span證書。操作=image-basis；資料={\"matrix\":[[1,2],[2,4]]}。請依 [matrix,rank,image_basis,certificate] 欄序回答。","answer":{"kind":"expression","value":"([[1, 2], [2, 4]], 1, [[1, 2]], full_real_column_span)","explanation_zh":"Wave5第二層核對=([[1, 2], [2, 4]], 1, [[1, 2]], full_real_column_span)；依完整明示資料重算；image是full-real column span且不等同codomain；bounded fiber search失敗不證全域nonmembership。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性映射的像"]},"variant":{"family_id":"qf-mod33-image-basis","seed":202611023027,"parameters":{"matrix":[[1,2],[2,4]],"domain":"Exact_R2_matrix_image","operation":"image-basis","semantic_template":"qf-mod33-image-basis/{\"matrix\":[[1,2],[2,4]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r03-linear-map-image-0028","batch_id":"batch-20261014-advance-20261102-r03-linear-map-image-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-image"],"title_zh":"線性映射的像","stem_zh":"同一精確2×2線性映射與column-span證書。操作=image-basis；資料={\"matrix\":[[1,3],[2,6]]}。請依 [matrix,rank,image_basis,certificate] 欄序回答。","answer":{"kind":"expression","value":"([[1, 3], [2, 6]], 1, [[1, 2]], full_real_column_span)","explanation_zh":"Wave5第二層核對=([[1, 3], [2, 6]], 1, [[1, 2]], full_real_column_span)；依完整明示資料重算；image是full-real column span且不等同codomain；bounded fiber search失敗不證全域nonmembership。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性映射的像"]},"variant":{"family_id":"qf-mod33-image-basis","seed":202611023028,"parameters":{"matrix":[[1,3],[2,6]],"domain":"Exact_R2_matrix_image","operation":"image-basis","semantic_template":"qf-mod33-image-basis/{\"matrix\":[[1,3],[2,6]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r03-linear-map-image-0029","batch_id":"batch-20261014-advance-20261102-r03-linear-map-image-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-image"],"title_zh":"線性映射的像","stem_zh":"同一精確2×2線性映射與column-span證書。操作=image-basis；資料={\"matrix\":[[1,4],[2,8]]}。請依 [matrix,rank,image_basis,certificate] 欄序回答。","answer":{"kind":"expression","value":"([[1, 4], [2, 8]], 1, [[1, 2]], full_real_column_span)","explanation_zh":"Wave5第二層核對=([[1, 4], [2, 8]], 1, [[1, 2]], full_real_column_span)；依完整明示資料重算；image是full-real column span且不等同codomain；bounded fiber search失敗不證全域nonmembership。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性映射的像"]},"variant":{"family_id":"qf-mod33-image-basis","seed":202611023029,"parameters":{"matrix":[[1,4],[2,8]],"domain":"Exact_R2_matrix_image","operation":"image-basis","semantic_template":"qf-mod33-image-basis/{\"matrix\":[[1,4],[2,8]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r03-linear-map-image-0030","batch_id":"batch-20261014-advance-20261102-r03-linear-map-image-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-image"],"title_zh":"線性映射的像","stem_zh":"同一精確2×2線性映射與column-span證書。操作=image-basis；資料={\"matrix\":[[1,5],[2,10]]}。請依 [matrix,rank,image_basis,certificate] 欄序回答。","answer":{"kind":"expression","value":"([[1, 5], [2, 10]], 1, [[1, 2]], full_real_column_span)","explanation_zh":"Wave5第二層核對=([[1, 5], [2, 10]], 1, [[1, 2]], full_real_column_span)；依完整明示資料重算；image是full-real column span且不等同codomain；bounded fiber search失敗不證全域nonmembership。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性映射的像"]},"variant":{"family_id":"qf-mod33-image-basis","seed":202611023030,"parameters":{"matrix":[[1,5],[2,10]],"domain":"Exact_R2_matrix_image","operation":"image-basis","semantic_template":"qf-mod33-image-basis/{\"matrix\":[[1,5],[2,10]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r03-linear-map-image-0031","batch_id":"batch-20261014-advance-20261102-r03-linear-map-image-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-image"],"title_zh":"線性映射的像","stem_zh":"同一精確2×2線性映射與column-span證書。操作=image-basis；資料={\"matrix\":[[1,6],[2,12]]}。請依 [matrix,rank,image_basis,certificate] 欄序回答。","answer":{"kind":"expression","value":"([[1, 6], [2, 12]], 1, [[1, 2]], full_real_column_span)","explanation_zh":"Wave5第二層核對=([[1, 6], [2, 12]], 1, [[1, 2]], full_real_column_span)；依完整明示資料重算；image是full-real column span且不等同codomain；bounded fiber search失敗不證全域nonmembership。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性映射的像"]},"variant":{"family_id":"qf-mod33-image-basis","seed":202611023031,"parameters":{"matrix":[[1,6],[2,12]],"domain":"Exact_R2_matrix_image","operation":"image-basis","semantic_template":"qf-mod33-image-basis/{\"matrix\":[[1,6],[2,12]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r03-linear-map-image-0032","batch_id":"batch-20261014-advance-20261102-r03-linear-map-image-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-image"],"title_zh":"線性映射的像","stem_zh":"同一精確2×2線性映射與column-span證書。操作=image-basis；資料={\"matrix\":[[1,7],[2,14]]}。請依 [matrix,rank,image_basis,certificate] 欄序回答。","answer":{"kind":"expression","value":"([[1, 7], [2, 14]], 1, [[1, 2]], full_real_column_span)","explanation_zh":"Wave5第二層核對=([[1, 7], [2, 14]], 1, [[1, 2]], full_real_column_span)；依完整明示資料重算；image是full-real column span且不等同codomain；bounded fiber search失敗不證全域nonmembership。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性映射的像"]},"variant":{"family_id":"qf-mod33-image-basis","seed":202611023032,"parameters":{"matrix":[[1,7],[2,14]],"domain":"Exact_R2_matrix_image","operation":"image-basis","semantic_template":"qf-mod33-image-basis/{\"matrix\":[[1,7],[2,14]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r03-linear-map-image-0033","batch_id":"batch-20261014-advance-20261102-r03-linear-map-image-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-image"],"title_zh":"線性映射的像","stem_zh":"同一精確2×2線性映射與column-span證書。操作=image-basis；資料={\"matrix\":[[1,8],[2,16]]}。請依 [matrix,rank,image_basis,certificate] 欄序回答。","answer":{"kind":"expression","value":"([[1, 8], [2, 16]], 1, [[1, 2]], full_real_column_span)","explanation_zh":"Wave5第二層核對=([[1, 8], [2, 16]], 1, [[1, 2]], full_real_column_span)；依完整明示資料重算；image是full-real column span且不等同codomain；bounded fiber search失敗不證全域nonmembership。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性映射的像"]},"variant":{"family_id":"qf-mod33-image-basis","seed":202611023033,"parameters":{"matrix":[[1,8],[2,16]],"domain":"Exact_R2_matrix_image","operation":"image-basis","semantic_template":"qf-mod33-image-basis/{\"matrix\":[[1,8],[2,16]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r03-linear-map-image-0034","batch_id":"batch-20261014-advance-20261102-r03-linear-map-image-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-image"],"title_zh":"線性映射的像","stem_zh":"同一精確2×2線性映射與column-span證書。操作=image-basis；資料={\"matrix\":[[1,9],[2,18]]}。請依 [matrix,rank,image_basis,certificate] 欄序回答。","answer":{"kind":"expression","value":"([[1, 9], [2, 18]], 1, [[1, 2]], full_real_column_span)","explanation_zh":"Wave5第二層核對=([[1, 9], [2, 18]], 1, [[1, 2]], full_real_column_span)；依完整明示資料重算；image是full-real column span且不等同codomain；bounded fiber search失敗不證全域nonmembership。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性映射的像"]},"variant":{"family_id":"qf-mod33-image-basis","seed":202611023034,"parameters":{"matrix":[[1,9],[2,18]],"domain":"Exact_R2_matrix_image","operation":"image-basis","semantic_template":"qf-mod33-image-basis/{\"matrix\":[[1,9],[2,18]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r03-linear-map-image-0035","batch_id":"batch-20261014-advance-20261102-r03-linear-map-image-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-image"],"title_zh":"線性映射的像","stem_zh":"同一精確2×2線性映射與column-span證書。操作=image-basis；資料={\"matrix\":[[1,10],[2,20]]}。請依 [matrix,rank,image_basis,certificate] 欄序回答。","answer":{"kind":"expression","value":"([[1, 10], [2, 20]], 1, [[1, 2]], full_real_column_span)","explanation_zh":"Wave5第二層核對=([[1, 10], [2, 20]], 1, [[1, 2]], full_real_column_span)；依完整明示資料重算；image是full-real column span且不等同codomain；bounded fiber search失敗不證全域nonmembership。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性映射的像"]},"variant":{"family_id":"qf-mod33-image-basis","seed":202611023035,"parameters":{"matrix":[[1,10],[2,20]],"domain":"Exact_R2_matrix_image","operation":"image-basis","semantic_template":"qf-mod33-image-basis/{\"matrix\":[[1,10],[2,20]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r03-linear-map-image-0036","batch_id":"batch-20261014-advance-20261102-r03-linear-map-image-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-image"],"title_zh":"線性映射的像","stem_zh":"同一精確2×2線性映射與column-span證書。操作=image-basis；資料={\"matrix\":[[1,11],[2,22]]}。請依 [matrix,rank,image_basis,certificate] 欄序回答。","answer":{"kind":"expression","value":"([[1, 11], [2, 22]], 1, [[1, 2]], full_real_column_span)","explanation_zh":"Wave5第二層核對=([[1, 11], [2, 22]], 1, [[1, 2]], full_real_column_span)；依完整明示資料重算；image是full-real column span且不等同codomain；bounded fiber search失敗不證全域nonmembership。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性映射的像"]},"variant":{"family_id":"qf-mod33-image-basis","seed":202611023036,"parameters":{"matrix":[[1,11],[2,22]],"domain":"Exact_R2_matrix_image","operation":"image-basis","semantic_template":"qf-mod33-image-basis/{\"matrix\":[[1,11],[2,22]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r03-linear-map-image-0037","batch_id":"batch-20261014-advance-20261102-r03-linear-map-image-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-image"],"title_zh":"線性映射的像","stem_zh":"同一精確2×2線性映射與column-span證書。操作=image-basis；資料={\"matrix\":[[1,12],[2,24]]}。請依 [matrix,rank,image_basis,certificate] 欄序回答。","answer":{"kind":"expression","value":"([[1, 12], [2, 24]], 1, [[1, 2]], full_real_column_span)","explanation_zh":"Wave5第二層核對=([[1, 12], [2, 24]], 1, [[1, 2]], full_real_column_span)；依完整明示資料重算；image是full-real column span且不等同codomain；bounded fiber search失敗不證全域nonmembership。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性映射的像"]},"variant":{"family_id":"qf-mod33-image-basis","seed":202611023037,"parameters":{"matrix":[[1,12],[2,24]],"domain":"Exact_R2_matrix_image","operation":"image-basis","semantic_template":"qf-mod33-image-basis/{\"matrix\":[[1,12],[2,24]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r03-linear-map-image-0038","batch_id":"batch-20261014-advance-20261102-r03-linear-map-image-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-image"],"title_zh":"線性映射的像","stem_zh":"同一精確2×2線性映射與column-span證書。操作=image-basis；資料={\"matrix\":[[1,13],[2,26]]}。請依 [matrix,rank,image_basis,certificate] 欄序回答。","answer":{"kind":"expression","value":"([[1, 13], [2, 26]], 1, [[1, 2]], full_real_column_span)","explanation_zh":"Wave5第二層核對=([[1, 13], [2, 26]], 1, [[1, 2]], full_real_column_span)；依完整明示資料重算；image是full-real column span且不等同codomain；bounded fiber search失敗不證全域nonmembership。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性映射的像"]},"variant":{"family_id":"qf-mod33-image-basis","seed":202611023038,"parameters":{"matrix":[[1,13],[2,26]],"domain":"Exact_R2_matrix_image","operation":"image-basis","semantic_template":"qf-mod33-image-basis/{\"matrix\":[[1,13],[2,26]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r03-linear-map-image-0039","batch_id":"batch-20261014-advance-20261102-r03-linear-map-image-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-image"],"title_zh":"線性映射的像","stem_zh":"同一精確2×2線性映射與column-span證書。操作=image-basis；資料={\"matrix\":[[1,14],[2,28]]}。請依 [matrix,rank,image_basis,certificate] 欄序回答。","answer":{"kind":"expression","value":"([[1, 14], [2, 28]], 1, [[1, 2]], full_real_column_span)","explanation_zh":"Wave5第二層核對=([[1, 14], [2, 28]], 1, [[1, 2]], full_real_column_span)；依完整明示資料重算；image是full-real column span且不等同codomain；bounded fiber search失敗不證全域nonmembership。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性映射的像"]},"variant":{"family_id":"qf-mod33-image-basis","seed":202611023039,"parameters":{"matrix":[[1,14],[2,28]],"domain":"Exact_R2_matrix_image","operation":"image-basis","semantic_template":"qf-mod33-image-basis/{\"matrix\":[[1,14],[2,28]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r03-linear-map-image-0040","batch_id":"batch-20261014-advance-20261102-r03-linear-map-image-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-image"],"title_zh":"線性映射的像","stem_zh":"同一精確2×2線性映射與column-span證書。操作=image-basis；資料={\"matrix\":[[1,15],[2,30]]}。請依 [matrix,rank,image_basis,certificate] 欄序回答。","answer":{"kind":"expression","value":"([[1, 15], [2, 30]], 1, [[1, 2]], full_real_column_span)","explanation_zh":"Wave5第二層核對=([[1, 15], [2, 30]], 1, [[1, 2]], full_real_column_span)；依完整明示資料重算；image是full-real column span且不等同codomain；bounded fiber search失敗不證全域nonmembership。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性映射的像"]},"variant":{"family_id":"qf-mod33-image-basis","seed":202611023040,"parameters":{"matrix":[[1,15],[2,30]],"domain":"Exact_R2_matrix_image","operation":"image-basis","semantic_template":"qf-mod33-image-basis/{\"matrix\":[[1,15],[2,30]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r03-linear-map-image-0041","batch_id":"batch-20261014-advance-20261102-r03-linear-map-image-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-image"],"title_zh":"線性映射的像","stem_zh":"同一精確2×2線性映射與column-span證書。操作=image-basis；資料={\"matrix\":[[1,16],[2,32]]}。請依 [matrix,rank,image_basis,certificate] 欄序回答。","answer":{"kind":"expression","value":"([[1, 16], [2, 32]], 1, [[1, 2]], full_real_column_span)","explanation_zh":"Wave5第二層核對=([[1, 16], [2, 32]], 1, [[1, 2]], full_real_column_span)；依完整明示資料重算；image是full-real column span且不等同codomain；bounded fiber search失敗不證全域nonmembership。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性映射的像"]},"variant":{"family_id":"qf-mod33-image-basis","seed":202611023041,"parameters":{"matrix":[[1,16],[2,32]],"domain":"Exact_R2_matrix_image","operation":"image-basis","semantic_template":"qf-mod33-image-basis/{\"matrix\":[[1,16],[2,32]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r03-linear-map-image-0042","batch_id":"batch-20261014-advance-20261102-r03-linear-map-image-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-image"],"title_zh":"線性映射的像","stem_zh":"同一精確2×2線性映射與column-span證書。操作=image-basis；資料={\"matrix\":[[1,17],[2,34]]}。請依 [matrix,rank,image_basis,certificate] 欄序回答。","answer":{"kind":"expression","value":"([[1, 17], [2, 34]], 1, [[1, 2]], full_real_column_span)","explanation_zh":"Wave5第二層核對=([[1, 17], [2, 34]], 1, [[1, 2]], full_real_column_span)；依完整明示資料重算；image是full-real column span且不等同codomain；bounded fiber search失敗不證全域nonmembership。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性映射的像"]},"variant":{"family_id":"qf-mod33-image-basis","seed":202611023042,"parameters":{"matrix":[[1,17],[2,34]],"domain":"Exact_R2_matrix_image","operation":"image-basis","semantic_template":"qf-mod33-image-basis/{\"matrix\":[[1,17],[2,34]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r03-linear-map-image-0043","batch_id":"batch-20261014-advance-20261102-r03-linear-map-image-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-image"],"title_zh":"線性映射的像","stem_zh":"同一精確2×2線性映射與column-span證書。操作=image-basis；資料={\"matrix\":[[1,18],[2,36]]}。請依 [matrix,rank,image_basis,certificate] 欄序回答。","answer":{"kind":"expression","value":"([[1, 18], [2, 36]], 1, [[1, 2]], full_real_column_span)","explanation_zh":"Wave5第二層核對=([[1, 18], [2, 36]], 1, [[1, 2]], full_real_column_span)；依完整明示資料重算；image是full-real column span且不等同codomain；bounded fiber search失敗不證全域nonmembership。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性映射的像"]},"variant":{"family_id":"qf-mod33-image-basis","seed":202611023043,"parameters":{"matrix":[[1,18],[2,36]],"domain":"Exact_R2_matrix_image","operation":"image-basis","semantic_template":"qf-mod33-image-basis/{\"matrix\":[[1,18],[2,36]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r03-linear-map-image-0044","batch_id":"batch-20261014-advance-20261102-r03-linear-map-image-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-image"],"title_zh":"線性映射的像","stem_zh":"同一精確2×2線性映射與column-span證書。操作=image-basis；資料={\"matrix\":[[1,19],[2,38]]}。請依 [matrix,rank,image_basis,certificate] 欄序回答。","answer":{"kind":"expression","value":"([[1, 19], [2, 38]], 1, [[1, 2]], full_real_column_span)","explanation_zh":"Wave5第二層核對=([[1, 19], [2, 38]], 1, [[1, 2]], full_real_column_span)；依完整明示資料重算；image是full-real column span且不等同codomain；bounded fiber search失敗不證全域nonmembership。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性映射的像"]},"variant":{"family_id":"qf-mod33-image-basis","seed":202611023044,"parameters":{"matrix":[[1,19],[2,38]],"domain":"Exact_R2_matrix_image","operation":"image-basis","semantic_template":"qf-mod33-image-basis/{\"matrix\":[[1,19],[2,38]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r03-linear-map-image-0045","batch_id":"batch-20261014-advance-20261102-r03-linear-map-image-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-image"],"title_zh":"線性映射的像","stem_zh":"同一精確2×2線性映射與column-span證書。操作=image-basis；資料={\"matrix\":[[1,20],[2,40]]}。請依 [matrix,rank,image_basis,certificate] 欄序回答。","answer":{"kind":"expression","value":"([[1, 20], [2, 40]], 1, [[1, 2]], full_real_column_span)","explanation_zh":"Wave5第二層核對=([[1, 20], [2, 40]], 1, [[1, 2]], full_real_column_span)；依完整明示資料重算；image是full-real column span且不等同codomain；bounded fiber search失敗不證全域nonmembership。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性映射的像"]},"variant":{"family_id":"qf-mod33-image-basis","seed":202611023045,"parameters":{"matrix":[[1,20],[2,40]],"domain":"Exact_R2_matrix_image","operation":"image-basis","semantic_template":"qf-mod33-image-basis/{\"matrix\":[[1,20],[2,40]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r03-linear-map-image-0046","batch_id":"batch-20261014-advance-20261102-r03-linear-map-image-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-image"],"title_zh":"線性映射的像","stem_zh":"同一精確2×2線性映射與column-span證書。操作=image-basis；資料={\"matrix\":[[1,21],[2,42]]}。請依 [matrix,rank,image_basis,certificate] 欄序回答。","answer":{"kind":"expression","value":"([[1, 21], [2, 42]], 1, [[1, 2]], full_real_column_span)","explanation_zh":"Wave5第二層核對=([[1, 21], [2, 42]], 1, [[1, 2]], full_real_column_span)；依完整明示資料重算；image是full-real column span且不等同codomain；bounded fiber search失敗不證全域nonmembership。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性映射的像"]},"variant":{"family_id":"qf-mod33-image-basis","seed":202611023046,"parameters":{"matrix":[[1,21],[2,42]],"domain":"Exact_R2_matrix_image","operation":"image-basis","semantic_template":"qf-mod33-image-basis/{\"matrix\":[[1,21],[2,42]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r03-linear-map-image-0047","batch_id":"batch-20261014-advance-20261102-r03-linear-map-image-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-image"],"title_zh":"線性映射的像","stem_zh":"同一精確2×2線性映射與column-span證書。操作=image-basis；資料={\"matrix\":[[1,22],[2,44]]}。請依 [matrix,rank,image_basis,certificate] 欄序回答。","answer":{"kind":"expression","value":"([[1, 22], [2, 44]], 1, [[1, 2]], full_real_column_span)","explanation_zh":"Wave5第二層核對=([[1, 22], [2, 44]], 1, [[1, 2]], full_real_column_span)；依完整明示資料重算；image是full-real column span且不等同codomain；bounded fiber search失敗不證全域nonmembership。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性映射的像"]},"variant":{"family_id":"qf-mod33-image-basis","seed":202611023047,"parameters":{"matrix":[[1,22],[2,44]],"domain":"Exact_R2_matrix_image","operation":"image-basis","semantic_template":"qf-mod33-image-basis/{\"matrix\":[[1,22],[2,44]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r03-linear-map-image-0048","batch_id":"batch-20261014-advance-20261102-r03-linear-map-image-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-image"],"title_zh":"線性映射的像","stem_zh":"同一精確2×2線性映射與column-span證書。操作=image-basis；資料={\"matrix\":[[1,23],[2,46]]}。請依 [matrix,rank,image_basis,certificate] 欄序回答。","answer":{"kind":"expression","value":"([[1, 23], [2, 46]], 1, [[1, 2]], full_real_column_span)","explanation_zh":"Wave5第二層核對=([[1, 23], [2, 46]], 1, [[1, 2]], full_real_column_span)；依完整明示資料重算；image是full-real column span且不等同codomain；bounded fiber search失敗不證全域nonmembership。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性映射的像"]},"variant":{"family_id":"qf-mod33-image-basis","seed":202611023048,"parameters":{"matrix":[[1,23],[2,46]],"domain":"Exact_R2_matrix_image","operation":"image-basis","semantic_template":"qf-mod33-image-basis/{\"matrix\":[[1,23],[2,46]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r03-linear-map-image-0049","batch_id":"batch-20261014-advance-20261102-r03-linear-map-image-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-image"],"title_zh":"線性映射的像","stem_zh":"同一精確2×2線性映射與column-span證書。操作=image-basis；資料={\"matrix\":[[1,24],[2,48]]}。請依 [matrix,rank,image_basis,certificate] 欄序回答。","answer":{"kind":"expression","value":"([[1, 24], [2, 48]], 1, [[1, 2]], full_real_column_span)","explanation_zh":"Wave5第二層核對=([[1, 24], [2, 48]], 1, [[1, 2]], full_real_column_span)；依完整明示資料重算；image是full-real column span且不等同codomain；bounded fiber search失敗不證全域nonmembership。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性映射的像"]},"variant":{"family_id":"qf-mod33-image-basis","seed":202611023049,"parameters":{"matrix":[[1,24],[2,48]],"domain":"Exact_R2_matrix_image","operation":"image-basis","semantic_template":"qf-mod33-image-basis/{\"matrix\":[[1,24],[2,48]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r03-linear-map-image-0050","batch_id":"batch-20261014-advance-20261102-r03-linear-map-image-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-image"],"title_zh":"線性映射的像","stem_zh":"同一精確2×2線性映射與column-span證書。操作=image-basis；資料={\"matrix\":[[1,25],[2,50]]}。請依 [matrix,rank,image_basis,certificate] 欄序回答。","answer":{"kind":"expression","value":"([[1, 25], [2, 50]], 1, [[1, 2]], full_real_column_span)","explanation_zh":"Wave5第二層核對=([[1, 25], [2, 50]], 1, [[1, 2]], full_real_column_span)；依完整明示資料重算；image是full-real column span且不等同codomain；bounded fiber search失敗不證全域nonmembership。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性映射的像"]},"variant":{"family_id":"qf-mod33-image-basis","seed":202611023050,"parameters":{"matrix":[[1,25],[2,50]],"domain":"Exact_R2_matrix_image","operation":"image-basis","semantic_template":"qf-mod33-image-basis/{\"matrix\":[[1,25],[2,50]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r03-linear-map-image-0051","batch_id":"batch-20261014-advance-20261102-r03-linear-map-image-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-image"],"title_zh":"線性映射的像","stem_zh":"同一精確2×2線性映射與column-span證書。操作=image-rank；資料={\"matrix\":[[1,1],[2,2]]}。請依 [rank,basis_count,equal] 欄序回答。","answer":{"kind":"expression","value":"(1, 1, 是)","explanation_zh":"Wave5第二層核對=(1, 1, 是)；依完整明示資料重算；image是full-real column span且不等同codomain；bounded fiber search失敗不證全域nonmembership。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性映射的像"]},"variant":{"family_id":"qf-mod33-image-rank","seed":202611023051,"parameters":{"matrix":[[1,1],[2,2]],"domain":"Exact_R2_matrix_image","operation":"image-rank","semantic_template":"qf-mod33-image-rank/{\"matrix\":[[1,1],[2,2]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r03-linear-map-image-0052","batch_id":"batch-20261014-advance-20261102-r03-linear-map-image-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-image"],"title_zh":"線性映射的像","stem_zh":"同一精確2×2線性映射與column-span證書。操作=image-rank；資料={\"matrix\":[[1,2],[2,4]]}。請依 [rank,basis_count,equal] 欄序回答。","answer":{"kind":"expression","value":"(1, 1, 是)","explanation_zh":"Wave5第二層核對=(1, 1, 是)；依完整明示資料重算；image是full-real column span且不等同codomain；bounded fiber search失敗不證全域nonmembership。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性映射的像"]},"variant":{"family_id":"qf-mod33-image-rank","seed":202611023052,"parameters":{"matrix":[[1,2],[2,4]],"domain":"Exact_R2_matrix_image","operation":"image-rank","semantic_template":"qf-mod33-image-rank/{\"matrix\":[[1,2],[2,4]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r03-linear-map-image-0053","batch_id":"batch-20261014-advance-20261102-r03-linear-map-image-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-image"],"title_zh":"線性映射的像","stem_zh":"同一精確2×2線性映射與column-span證書。操作=image-rank；資料={\"matrix\":[[1,3],[2,6]]}。請依 [rank,basis_count,equal] 欄序回答。","answer":{"kind":"expression","value":"(1, 1, 是)","explanation_zh":"Wave5第二層核對=(1, 1, 是)；依完整明示資料重算；image是full-real column span且不等同codomain；bounded fiber search失敗不證全域nonmembership。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性映射的像"]},"variant":{"family_id":"qf-mod33-image-rank","seed":202611023053,"parameters":{"matrix":[[1,3],[2,6]],"domain":"Exact_R2_matrix_image","operation":"image-rank","semantic_template":"qf-mod33-image-rank/{\"matrix\":[[1,3],[2,6]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r03-linear-map-image-0054","batch_id":"batch-20261014-advance-20261102-r03-linear-map-image-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-image"],"title_zh":"線性映射的像","stem_zh":"同一精確2×2線性映射與column-span證書。操作=image-rank；資料={\"matrix\":[[1,4],[2,8]]}。請依 [rank,basis_count,equal] 欄序回答。","answer":{"kind":"expression","value":"(1, 1, 是)","explanation_zh":"Wave5第二層核對=(1, 1, 是)；依完整明示資料重算；image是full-real column span且不等同codomain；bounded fiber search失敗不證全域nonmembership。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性映射的像"]},"variant":{"family_id":"qf-mod33-image-rank","seed":202611023054,"parameters":{"matrix":[[1,4],[2,8]],"domain":"Exact_R2_matrix_image","operation":"image-rank","semantic_template":"qf-mod33-image-rank/{\"matrix\":[[1,4],[2,8]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r03-linear-map-image-0055","batch_id":"batch-20261014-advance-20261102-r03-linear-map-image-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-image"],"title_zh":"線性映射的像","stem_zh":"同一精確2×2線性映射與column-span證書。操作=image-rank；資料={\"matrix\":[[1,5],[2,10]]}。請依 [rank,basis_count,equal] 欄序回答。","answer":{"kind":"expression","value":"(1, 1, 是)","explanation_zh":"Wave5第二層核對=(1, 1, 是)；依完整明示資料重算；image是full-real column span且不等同codomain；bounded fiber search失敗不證全域nonmembership。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性映射的像"]},"variant":{"family_id":"qf-mod33-image-rank","seed":202611023055,"parameters":{"matrix":[[1,5],[2,10]],"domain":"Exact_R2_matrix_image","operation":"image-rank","semantic_template":"qf-mod33-image-rank/{\"matrix\":[[1,5],[2,10]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r03-linear-map-image-0056","batch_id":"batch-20261014-advance-20261102-r03-linear-map-image-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-image"],"title_zh":"線性映射的像","stem_zh":"同一精確2×2線性映射與column-span證書。操作=image-rank；資料={\"matrix\":[[1,6],[2,12]]}。請依 [rank,basis_count,equal] 欄序回答。","answer":{"kind":"expression","value":"(1, 1, 是)","explanation_zh":"Wave5第二層核對=(1, 1, 是)；依完整明示資料重算；image是full-real column span且不等同codomain；bounded fiber search失敗不證全域nonmembership。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性映射的像"]},"variant":{"family_id":"qf-mod33-image-rank","seed":202611023056,"parameters":{"matrix":[[1,6],[2,12]],"domain":"Exact_R2_matrix_image","operation":"image-rank","semantic_template":"qf-mod33-image-rank/{\"matrix\":[[1,6],[2,12]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r03-linear-map-image-0057","batch_id":"batch-20261014-advance-20261102-r03-linear-map-image-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-image"],"title_zh":"線性映射的像","stem_zh":"同一精確2×2線性映射與column-span證書。操作=image-rank；資料={\"matrix\":[[1,7],[2,14]]}。請依 [rank,basis_count,equal] 欄序回答。","answer":{"kind":"expression","value":"(1, 1, 是)","explanation_zh":"Wave5第二層核對=(1, 1, 是)；依完整明示資料重算；image是full-real column span且不等同codomain；bounded fiber search失敗不證全域nonmembership。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性映射的像"]},"variant":{"family_id":"qf-mod33-image-rank","seed":202611023057,"parameters":{"matrix":[[1,7],[2,14]],"domain":"Exact_R2_matrix_image","operation":"image-rank","semantic_template":"qf-mod33-image-rank/{\"matrix\":[[1,7],[2,14]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r03-linear-map-image-0058","batch_id":"batch-20261014-advance-20261102-r03-linear-map-image-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-image"],"title_zh":"線性映射的像","stem_zh":"同一精確2×2線性映射與column-span證書。操作=image-rank；資料={\"matrix\":[[1,8],[2,16]]}。請依 [rank,basis_count,equal] 欄序回答。","answer":{"kind":"expression","value":"(1, 1, 是)","explanation_zh":"Wave5第二層核對=(1, 1, 是)；依完整明示資料重算；image是full-real column span且不等同codomain；bounded fiber search失敗不證全域nonmembership。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性映射的像"]},"variant":{"family_id":"qf-mod33-image-rank","seed":202611023058,"parameters":{"matrix":[[1,8],[2,16]],"domain":"Exact_R2_matrix_image","operation":"image-rank","semantic_template":"qf-mod33-image-rank/{\"matrix\":[[1,8],[2,16]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r03-linear-map-image-0059","batch_id":"batch-20261014-advance-20261102-r03-linear-map-image-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-image"],"title_zh":"線性映射的像","stem_zh":"同一精確2×2線性映射與column-span證書。操作=image-rank；資料={\"matrix\":[[1,9],[2,18]]}。請依 [rank,basis_count,equal] 欄序回答。","answer":{"kind":"expression","value":"(1, 1, 是)","explanation_zh":"Wave5第二層核對=(1, 1, 是)；依完整明示資料重算；image是full-real column span且不等同codomain；bounded fiber search失敗不證全域nonmembership。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性映射的像"]},"variant":{"family_id":"qf-mod33-image-rank","seed":202611023059,"parameters":{"matrix":[[1,9],[2,18]],"domain":"Exact_R2_matrix_image","operation":"image-rank","semantic_template":"qf-mod33-image-rank/{\"matrix\":[[1,9],[2,18]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r03-linear-map-image-0060","batch_id":"batch-20261014-advance-20261102-r03-linear-map-image-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-image"],"title_zh":"線性映射的像","stem_zh":"同一精確2×2線性映射與column-span證書。操作=image-rank；資料={\"matrix\":[[1,10],[2,20]]}。請依 [rank,basis_count,equal] 欄序回答。","answer":{"kind":"expression","value":"(1, 1, 是)","explanation_zh":"Wave5第二層核對=(1, 1, 是)；依完整明示資料重算；image是full-real column span且不等同codomain；bounded fiber search失敗不證全域nonmembership。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性映射的像"]},"variant":{"family_id":"qf-mod33-image-rank","seed":202611023060,"parameters":{"matrix":[[1,10],[2,20]],"domain":"Exact_R2_matrix_image","operation":"image-rank","semantic_template":"qf-mod33-image-rank/{\"matrix\":[[1,10],[2,20]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r03-linear-map-image-0061","batch_id":"batch-20261014-advance-20261102-r03-linear-map-image-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-image"],"title_zh":"線性映射的像","stem_zh":"同一精確2×2線性映射與column-span證書。操作=image-rank；資料={\"matrix\":[[1,11],[2,22]]}。請依 [rank,basis_count,equal] 欄序回答。","answer":{"kind":"expression","value":"(1, 1, 是)","explanation_zh":"Wave5第二層核對=(1, 1, 是)；依完整明示資料重算；image是full-real column span且不等同codomain；bounded fiber search失敗不證全域nonmembership。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性映射的像"]},"variant":{"family_id":"qf-mod33-image-rank","seed":202611023061,"parameters":{"matrix":[[1,11],[2,22]],"domain":"Exact_R2_matrix_image","operation":"image-rank","semantic_template":"qf-mod33-image-rank/{\"matrix\":[[1,11],[2,22]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r03-linear-map-image-0062","batch_id":"batch-20261014-advance-20261102-r03-linear-map-image-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-image"],"title_zh":"線性映射的像","stem_zh":"同一精確2×2線性映射與column-span證書。操作=image-rank；資料={\"matrix\":[[1,12],[2,24]]}。請依 [rank,basis_count,equal] 欄序回答。","answer":{"kind":"expression","value":"(1, 1, 是)","explanation_zh":"Wave5第二層核對=(1, 1, 是)；依完整明示資料重算；image是full-real column span且不等同codomain；bounded fiber search失敗不證全域nonmembership。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性映射的像"]},"variant":{"family_id":"qf-mod33-image-rank","seed":202611023062,"parameters":{"matrix":[[1,12],[2,24]],"domain":"Exact_R2_matrix_image","operation":"image-rank","semantic_template":"qf-mod33-image-rank/{\"matrix\":[[1,12],[2,24]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r03-linear-map-image-0063","batch_id":"batch-20261014-advance-20261102-r03-linear-map-image-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-image"],"title_zh":"線性映射的像","stem_zh":"同一精確2×2線性映射與column-span證書。操作=image-rank；資料={\"matrix\":[[1,13],[2,26]]}。請依 [rank,basis_count,equal] 欄序回答。","answer":{"kind":"expression","value":"(1, 1, 是)","explanation_zh":"Wave5第二層核對=(1, 1, 是)；依完整明示資料重算；image是full-real column span且不等同codomain；bounded fiber search失敗不證全域nonmembership。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性映射的像"]},"variant":{"family_id":"qf-mod33-image-rank","seed":202611023063,"parameters":{"matrix":[[1,13],[2,26]],"domain":"Exact_R2_matrix_image","operation":"image-rank","semantic_template":"qf-mod33-image-rank/{\"matrix\":[[1,13],[2,26]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r03-linear-map-image-0064","batch_id":"batch-20261014-advance-20261102-r03-linear-map-image-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-image"],"title_zh":"線性映射的像","stem_zh":"同一精確2×2線性映射與column-span證書。操作=image-rank；資料={\"matrix\":[[1,14],[2,28]]}。請依 [rank,basis_count,equal] 欄序回答。","answer":{"kind":"expression","value":"(1, 1, 是)","explanation_zh":"Wave5第二層核對=(1, 1, 是)；依完整明示資料重算；image是full-real column span且不等同codomain；bounded fiber search失敗不證全域nonmembership。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性映射的像"]},"variant":{"family_id":"qf-mod33-image-rank","seed":202611023064,"parameters":{"matrix":[[1,14],[2,28]],"domain":"Exact_R2_matrix_image","operation":"image-rank","semantic_template":"qf-mod33-image-rank/{\"matrix\":[[1,14],[2,28]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r03-linear-map-image-0065","batch_id":"batch-20261014-advance-20261102-r03-linear-map-image-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-image"],"title_zh":"線性映射的像","stem_zh":"同一精確2×2線性映射與column-span證書。操作=image-rank；資料={\"matrix\":[[1,15],[2,30]]}。請依 [rank,basis_count,equal] 欄序回答。","answer":{"kind":"expression","value":"(1, 1, 是)","explanation_zh":"Wave5第二層核對=(1, 1, 是)；依完整明示資料重算；image是full-real column span且不等同codomain；bounded fiber search失敗不證全域nonmembership。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性映射的像"]},"variant":{"family_id":"qf-mod33-image-rank","seed":202611023065,"parameters":{"matrix":[[1,15],[2,30]],"domain":"Exact_R2_matrix_image","operation":"image-rank","semantic_template":"qf-mod33-image-rank/{\"matrix\":[[1,15],[2,30]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r03-linear-map-image-0066","batch_id":"batch-20261014-advance-20261102-r03-linear-map-image-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-image"],"title_zh":"線性映射的像","stem_zh":"同一精確2×2線性映射與column-span證書。操作=image-rank；資料={\"matrix\":[[1,16],[2,32]]}。請依 [rank,basis_count,equal] 欄序回答。","answer":{"kind":"expression","value":"(1, 1, 是)","explanation_zh":"Wave5第二層核對=(1, 1, 是)；依完整明示資料重算；image是full-real column span且不等同codomain；bounded fiber search失敗不證全域nonmembership。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性映射的像"]},"variant":{"family_id":"qf-mod33-image-rank","seed":202611023066,"parameters":{"matrix":[[1,16],[2,32]],"domain":"Exact_R2_matrix_image","operation":"image-rank","semantic_template":"qf-mod33-image-rank/{\"matrix\":[[1,16],[2,32]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r03-linear-map-image-0067","batch_id":"batch-20261014-advance-20261102-r03-linear-map-image-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-image"],"title_zh":"線性映射的像","stem_zh":"同一精確2×2線性映射與column-span證書。操作=image-rank；資料={\"matrix\":[[1,17],[2,34]]}。請依 [rank,basis_count,equal] 欄序回答。","answer":{"kind":"expression","value":"(1, 1, 是)","explanation_zh":"Wave5第二層核對=(1, 1, 是)；依完整明示資料重算；image是full-real column span且不等同codomain；bounded fiber search失敗不證全域nonmembership。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性映射的像"]},"variant":{"family_id":"qf-mod33-image-rank","seed":202611023067,"parameters":{"matrix":[[1,17],[2,34]],"domain":"Exact_R2_matrix_image","operation":"image-rank","semantic_template":"qf-mod33-image-rank/{\"matrix\":[[1,17],[2,34]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r03-linear-map-image-0068","batch_id":"batch-20261014-advance-20261102-r03-linear-map-image-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-image"],"title_zh":"線性映射的像","stem_zh":"同一精確2×2線性映射與column-span證書。操作=image-rank；資料={\"matrix\":[[1,18],[2,36]]}。請依 [rank,basis_count,equal] 欄序回答。","answer":{"kind":"expression","value":"(1, 1, 是)","explanation_zh":"Wave5第二層核對=(1, 1, 是)；依完整明示資料重算；image是full-real column span且不等同codomain；bounded fiber search失敗不證全域nonmembership。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性映射的像"]},"variant":{"family_id":"qf-mod33-image-rank","seed":202611023068,"parameters":{"matrix":[[1,18],[2,36]],"domain":"Exact_R2_matrix_image","operation":"image-rank","semantic_template":"qf-mod33-image-rank/{\"matrix\":[[1,18],[2,36]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r03-linear-map-image-0069","batch_id":"batch-20261014-advance-20261102-r03-linear-map-image-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-image"],"title_zh":"線性映射的像","stem_zh":"同一精確2×2線性映射與column-span證書。操作=image-rank；資料={\"matrix\":[[1,19],[2,38]]}。請依 [rank,basis_count,equal] 欄序回答。","answer":{"kind":"expression","value":"(1, 1, 是)","explanation_zh":"Wave5第二層核對=(1, 1, 是)；依完整明示資料重算；image是full-real column span且不等同codomain；bounded fiber search失敗不證全域nonmembership。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性映射的像"]},"variant":{"family_id":"qf-mod33-image-rank","seed":202611023069,"parameters":{"matrix":[[1,19],[2,38]],"domain":"Exact_R2_matrix_image","operation":"image-rank","semantic_template":"qf-mod33-image-rank/{\"matrix\":[[1,19],[2,38]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r03-linear-map-image-0070","batch_id":"batch-20261014-advance-20261102-r03-linear-map-image-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-image"],"title_zh":"線性映射的像","stem_zh":"同一精確2×2線性映射與column-span證書。操作=image-rank；資料={\"matrix\":[[1,20],[2,40]]}。請依 [rank,basis_count,equal] 欄序回答。","answer":{"kind":"expression","value":"(1, 1, 是)","explanation_zh":"Wave5第二層核對=(1, 1, 是)；依完整明示資料重算；image是full-real column span且不等同codomain；bounded fiber search失敗不證全域nonmembership。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性映射的像"]},"variant":{"family_id":"qf-mod33-image-rank","seed":202611023070,"parameters":{"matrix":[[1,20],[2,40]],"domain":"Exact_R2_matrix_image","operation":"image-rank","semantic_template":"qf-mod33-image-rank/{\"matrix\":[[1,20],[2,40]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r03-linear-map-image-0071","batch_id":"batch-20261014-advance-20261102-r03-linear-map-image-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-image"],"title_zh":"線性映射的像","stem_zh":"同一精確2×2線性映射與column-span證書。操作=image-rank；資料={\"matrix\":[[1,21],[2,42]]}。請依 [rank,basis_count,equal] 欄序回答。","answer":{"kind":"expression","value":"(1, 1, 是)","explanation_zh":"Wave5第二層核對=(1, 1, 是)；依完整明示資料重算；image是full-real column span且不等同codomain；bounded fiber search失敗不證全域nonmembership。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性映射的像"]},"variant":{"family_id":"qf-mod33-image-rank","seed":202611023071,"parameters":{"matrix":[[1,21],[2,42]],"domain":"Exact_R2_matrix_image","operation":"image-rank","semantic_template":"qf-mod33-image-rank/{\"matrix\":[[1,21],[2,42]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r03-linear-map-image-0072","batch_id":"batch-20261014-advance-20261102-r03-linear-map-image-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-image"],"title_zh":"線性映射的像","stem_zh":"同一精確2×2線性映射與column-span證書。操作=image-rank；資料={\"matrix\":[[1,22],[2,44]]}。請依 [rank,basis_count,equal] 欄序回答。","answer":{"kind":"expression","value":"(1, 1, 是)","explanation_zh":"Wave5第二層核對=(1, 1, 是)；依完整明示資料重算；image是full-real column span且不等同codomain；bounded fiber search失敗不證全域nonmembership。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性映射的像"]},"variant":{"family_id":"qf-mod33-image-rank","seed":202611023072,"parameters":{"matrix":[[1,22],[2,44]],"domain":"Exact_R2_matrix_image","operation":"image-rank","semantic_template":"qf-mod33-image-rank/{\"matrix\":[[1,22],[2,44]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r03-linear-map-image-0073","batch_id":"batch-20261014-advance-20261102-r03-linear-map-image-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-image"],"title_zh":"線性映射的像","stem_zh":"同一精確2×2線性映射與column-span證書。操作=image-rank；資料={\"matrix\":[[1,23],[2,46]]}。請依 [rank,basis_count,equal] 欄序回答。","answer":{"kind":"expression","value":"(1, 1, 是)","explanation_zh":"Wave5第二層核對=(1, 1, 是)；依完整明示資料重算；image是full-real column span且不等同codomain；bounded fiber search失敗不證全域nonmembership。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性映射的像"]},"variant":{"family_id":"qf-mod33-image-rank","seed":202611023073,"parameters":{"matrix":[[1,23],[2,46]],"domain":"Exact_R2_matrix_image","operation":"image-rank","semantic_template":"qf-mod33-image-rank/{\"matrix\":[[1,23],[2,46]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r03-linear-map-image-0074","batch_id":"batch-20261014-advance-20261102-r03-linear-map-image-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-image"],"title_zh":"線性映射的像","stem_zh":"同一精確2×2線性映射與column-span證書。操作=image-rank；資料={\"matrix\":[[1,24],[2,48]]}。請依 [rank,basis_count,equal] 欄序回答。","answer":{"kind":"expression","value":"(1, 1, 是)","explanation_zh":"Wave5第二層核對=(1, 1, 是)；依完整明示資料重算；image是full-real column span且不等同codomain；bounded fiber search失敗不證全域nonmembership。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性映射的像"]},"variant":{"family_id":"qf-mod33-image-rank","seed":202611023074,"parameters":{"matrix":[[1,24],[2,48]],"domain":"Exact_R2_matrix_image","operation":"image-rank","semantic_template":"qf-mod33-image-rank/{\"matrix\":[[1,24],[2,48]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r03-linear-map-image-0075","batch_id":"batch-20261014-advance-20261102-r03-linear-map-image-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-image"],"title_zh":"線性映射的像","stem_zh":"同一精確2×2線性映射與column-span證書。操作=image-rank；資料={\"matrix\":[[1,25],[2,50]]}。請依 [rank,basis_count,equal] 欄序回答。","answer":{"kind":"expression","value":"(1, 1, 是)","explanation_zh":"Wave5第二層核對=(1, 1, 是)；依完整明示資料重算；image是full-real column span且不等同codomain；bounded fiber search失敗不證全域nonmembership。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性映射的像"]},"variant":{"family_id":"qf-mod33-image-rank","seed":202611023075,"parameters":{"matrix":[[1,25],[2,50]],"domain":"Exact_R2_matrix_image","operation":"image-rank","semantic_template":"qf-mod33-image-rank/{\"matrix\":[[1,25],[2,50]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r03-linear-map-image-0076","batch_id":"batch-20261014-advance-20261102-r03-linear-map-image-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-image"],"title_zh":"線性映射的像","stem_zh":"同一精確2×2線性映射與column-span證書。操作=image-bounded-search-boundary；資料={\"matrix\":[[2,0],[0,2]],\"target\":[1,1],\"bound\":1}。請依 [target,bound,integer_witnesses,global_membership,boundary] 欄序回答。","answer":{"kind":"expression","value":"([1, 1], 1, [], 是, bounded_failure_not_global_nonmembership)","explanation_zh":"Wave5第二層核對=([1, 1], 1, [], 是, bounded_failure_not_global_nonmembership)；依完整明示資料重算；image是full-real column span且不等同codomain；bounded fiber search失敗不證全域nonmembership。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性映射的像"]},"variant":{"family_id":"qf-mod33-image-bounded-search-boundary","seed":202611023076,"parameters":{"matrix":[[2,0],[0,2]],"target":[1,1],"bound":1,"domain":"Exact_R2_matrix_image","operation":"image-bounded-search-boundary","semantic_template":"qf-mod33-image-bounded-search-boundary/{\"bound\":1,\"matrix\":[[2,0],[0,2]],\"target\":[1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r03-linear-map-image-0077","batch_id":"batch-20261014-advance-20261102-r03-linear-map-image-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-image"],"title_zh":"線性映射的像","stem_zh":"同一精確2×2線性映射與column-span證書。操作=image-bounded-search-boundary；資料={\"matrix\":[[2,0],[0,2]],\"target\":[3,1],\"bound\":2}。請依 [target,bound,integer_witnesses,global_membership,boundary] 欄序回答。","answer":{"kind":"expression","value":"([3, 1], 2, [], 是, bounded_failure_not_global_nonmembership)","explanation_zh":"Wave5第二層核對=([3, 1], 2, [], 是, bounded_failure_not_global_nonmembership)；依完整明示資料重算；image是full-real column span且不等同codomain；bounded fiber search失敗不證全域nonmembership。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性映射的像"]},"variant":{"family_id":"qf-mod33-image-bounded-search-boundary","seed":202611023077,"parameters":{"matrix":[[2,0],[0,2]],"target":[3,1],"bound":2,"domain":"Exact_R2_matrix_image","operation":"image-bounded-search-boundary","semantic_template":"qf-mod33-image-bounded-search-boundary/{\"bound\":2,\"matrix\":[[2,0],[0,2]],\"target\":[1,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r03-linear-map-image-0078","batch_id":"batch-20261014-advance-20261102-r03-linear-map-image-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-image"],"title_zh":"線性映射的像","stem_zh":"同一精確2×2線性映射與column-span證書。操作=image-bounded-search-boundary；資料={\"matrix\":[[2,0],[0,2]],\"target\":[5,1],\"bound\":3}。請依 [target,bound,integer_witnesses,global_membership,boundary] 欄序回答。","answer":{"kind":"expression","value":"([5, 1], 3, [], 是, bounded_failure_not_global_nonmembership)","explanation_zh":"Wave5第二層核對=([5, 1], 3, [], 是, bounded_failure_not_global_nonmembership)；依完整明示資料重算；image是full-real column span且不等同codomain；bounded fiber search失敗不證全域nonmembership。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性映射的像"]},"variant":{"family_id":"qf-mod33-image-bounded-search-boundary","seed":202611023078,"parameters":{"matrix":[[2,0],[0,2]],"target":[5,1],"bound":3,"domain":"Exact_R2_matrix_image","operation":"image-bounded-search-boundary","semantic_template":"qf-mod33-image-bounded-search-boundary/{\"bound\":3,\"matrix\":[[2,0],[0,2]],\"target\":[1,5]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r03-linear-map-image-0079","batch_id":"batch-20261014-advance-20261102-r03-linear-map-image-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-image"],"title_zh":"線性映射的像","stem_zh":"同一精確2×2線性映射與column-span證書。操作=image-bounded-search-boundary；資料={\"matrix\":[[2,0],[0,2]],\"target\":[7,1],\"bound\":4}。請依 [target,bound,integer_witnesses,global_membership,boundary] 欄序回答。","answer":{"kind":"expression","value":"([7, 1], 4, [], 是, bounded_failure_not_global_nonmembership)","explanation_zh":"Wave5第二層核對=([7, 1], 4, [], 是, bounded_failure_not_global_nonmembership)；依完整明示資料重算；image是full-real column span且不等同codomain；bounded fiber search失敗不證全域nonmembership。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性映射的像"]},"variant":{"family_id":"qf-mod33-image-bounded-search-boundary","seed":202611023079,"parameters":{"matrix":[[2,0],[0,2]],"target":[7,1],"bound":4,"domain":"Exact_R2_matrix_image","operation":"image-bounded-search-boundary","semantic_template":"qf-mod33-image-bounded-search-boundary/{\"bound\":4,\"matrix\":[[2,0],[0,2]],\"target\":[1,7]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r03-linear-map-image-0080","batch_id":"batch-20261014-advance-20261102-r03-linear-map-image-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-image"],"title_zh":"線性映射的像","stem_zh":"同一精確2×2線性映射與column-span證書。操作=image-bounded-search-boundary；資料={\"matrix\":[[2,0],[0,2]],\"target\":[9,1],\"bound\":5}。請依 [target,bound,integer_witnesses,global_membership,boundary] 欄序回答。","answer":{"kind":"expression","value":"([9, 1], 5, [], 是, bounded_failure_not_global_nonmembership)","explanation_zh":"Wave5第二層核對=([9, 1], 5, [], 是, bounded_failure_not_global_nonmembership)；依完整明示資料重算；image是full-real column span且不等同codomain；bounded fiber search失敗不證全域nonmembership。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性映射的像"]},"variant":{"family_id":"qf-mod33-image-bounded-search-boundary","seed":202611023080,"parameters":{"matrix":[[2,0],[0,2]],"target":[9,1],"bound":5,"domain":"Exact_R2_matrix_image","operation":"image-bounded-search-boundary","semantic_template":"qf-mod33-image-bounded-search-boundary/{\"bound\":5,\"matrix\":[[2,0],[0,2]],\"target\":[1,9]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r03-linear-map-image-0081","batch_id":"batch-20261014-advance-20261102-r03-linear-map-image-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-image"],"title_zh":"線性映射的像","stem_zh":"同一精確2×2線性映射與column-span證書。操作=image-bounded-search-boundary；資料={\"matrix\":[[2,0],[0,2]],\"target\":[11,1],\"bound\":1}。請依 [target,bound,integer_witnesses,global_membership,boundary] 欄序回答。","answer":{"kind":"expression","value":"([11, 1], 1, [], 是, bounded_failure_not_global_nonmembership)","explanation_zh":"Wave5第二層核對=([11, 1], 1, [], 是, bounded_failure_not_global_nonmembership)；依完整明示資料重算；image是full-real column span且不等同codomain；bounded fiber search失敗不證全域nonmembership。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性映射的像"]},"variant":{"family_id":"qf-mod33-image-bounded-search-boundary","seed":202611023081,"parameters":{"matrix":[[2,0],[0,2]],"target":[11,1],"bound":1,"domain":"Exact_R2_matrix_image","operation":"image-bounded-search-boundary","semantic_template":"qf-mod33-image-bounded-search-boundary/{\"bound\":1,\"matrix\":[[2,0],[0,2]],\"target\":[1,11]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r03-linear-map-image-0082","batch_id":"batch-20261014-advance-20261102-r03-linear-map-image-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-image"],"title_zh":"線性映射的像","stem_zh":"同一精確2×2線性映射與column-span證書。操作=image-bounded-search-boundary；資料={\"matrix\":[[2,0],[0,2]],\"target\":[13,1],\"bound\":2}。請依 [target,bound,integer_witnesses,global_membership,boundary] 欄序回答。","answer":{"kind":"expression","value":"([13, 1], 2, [], 是, bounded_failure_not_global_nonmembership)","explanation_zh":"Wave5第二層核對=([13, 1], 2, [], 是, bounded_failure_not_global_nonmembership)；依完整明示資料重算；image是full-real column span且不等同codomain；bounded fiber search失敗不證全域nonmembership。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性映射的像"]},"variant":{"family_id":"qf-mod33-image-bounded-search-boundary","seed":202611023082,"parameters":{"matrix":[[2,0],[0,2]],"target":[13,1],"bound":2,"domain":"Exact_R2_matrix_image","operation":"image-bounded-search-boundary","semantic_template":"qf-mod33-image-bounded-search-boundary/{\"bound\":2,\"matrix\":[[2,0],[0,2]],\"target\":[1,13]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r03-linear-map-image-0083","batch_id":"batch-20261014-advance-20261102-r03-linear-map-image-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-image"],"title_zh":"線性映射的像","stem_zh":"同一精確2×2線性映射與column-span證書。操作=image-bounded-search-boundary；資料={\"matrix\":[[2,0],[0,2]],\"target\":[15,1],\"bound\":3}。請依 [target,bound,integer_witnesses,global_membership,boundary] 欄序回答。","answer":{"kind":"expression","value":"([15, 1], 3, [], 是, bounded_failure_not_global_nonmembership)","explanation_zh":"Wave5第二層核對=([15, 1], 3, [], 是, bounded_failure_not_global_nonmembership)；依完整明示資料重算；image是full-real column span且不等同codomain；bounded fiber search失敗不證全域nonmembership。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性映射的像"]},"variant":{"family_id":"qf-mod33-image-bounded-search-boundary","seed":202611023083,"parameters":{"matrix":[[2,0],[0,2]],"target":[15,1],"bound":3,"domain":"Exact_R2_matrix_image","operation":"image-bounded-search-boundary","semantic_template":"qf-mod33-image-bounded-search-boundary/{\"bound\":3,\"matrix\":[[2,0],[0,2]],\"target\":[1,15]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r03-linear-map-image-0084","batch_id":"batch-20261014-advance-20261102-r03-linear-map-image-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-image"],"title_zh":"線性映射的像","stem_zh":"同一精確2×2線性映射與column-span證書。操作=image-bounded-search-boundary；資料={\"matrix\":[[2,0],[0,2]],\"target\":[17,1],\"bound\":4}。請依 [target,bound,integer_witnesses,global_membership,boundary] 欄序回答。","answer":{"kind":"expression","value":"([17, 1], 4, [], 是, bounded_failure_not_global_nonmembership)","explanation_zh":"Wave5第二層核對=([17, 1], 4, [], 是, bounded_failure_not_global_nonmembership)；依完整明示資料重算；image是full-real column span且不等同codomain；bounded fiber search失敗不證全域nonmembership。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性映射的像"]},"variant":{"family_id":"qf-mod33-image-bounded-search-boundary","seed":202611023084,"parameters":{"matrix":[[2,0],[0,2]],"target":[17,1],"bound":4,"domain":"Exact_R2_matrix_image","operation":"image-bounded-search-boundary","semantic_template":"qf-mod33-image-bounded-search-boundary/{\"bound\":4,\"matrix\":[[2,0],[0,2]],\"target\":[1,17]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r03-linear-map-image-0085","batch_id":"batch-20261014-advance-20261102-r03-linear-map-image-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-image"],"title_zh":"線性映射的像","stem_zh":"同一精確2×2線性映射與column-span證書。操作=image-bounded-search-boundary；資料={\"matrix\":[[2,0],[0,2]],\"target\":[19,1],\"bound\":5}。請依 [target,bound,integer_witnesses,global_membership,boundary] 欄序回答。","answer":{"kind":"expression","value":"([19, 1], 5, [], 是, bounded_failure_not_global_nonmembership)","explanation_zh":"Wave5第二層核對=([19, 1], 5, [], 是, bounded_failure_not_global_nonmembership)；依完整明示資料重算；image是full-real column span且不等同codomain；bounded fiber search失敗不證全域nonmembership。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性映射的像"]},"variant":{"family_id":"qf-mod33-image-bounded-search-boundary","seed":202611023085,"parameters":{"matrix":[[2,0],[0,2]],"target":[19,1],"bound":5,"domain":"Exact_R2_matrix_image","operation":"image-bounded-search-boundary","semantic_template":"qf-mod33-image-bounded-search-boundary/{\"bound\":5,\"matrix\":[[2,0],[0,2]],\"target\":[1,19]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r03-linear-map-image-0086","batch_id":"batch-20261014-advance-20261102-r03-linear-map-image-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-image"],"title_zh":"線性映射的像","stem_zh":"同一精確2×2線性映射與column-span證書。操作=image-bounded-search-boundary；資料={\"matrix\":[[2,0],[0,2]],\"target\":[21,1],\"bound\":1}。請依 [target,bound,integer_witnesses,global_membership,boundary] 欄序回答。","answer":{"kind":"expression","value":"([21, 1], 1, [], 是, bounded_failure_not_global_nonmembership)","explanation_zh":"Wave5第二層核對=([21, 1], 1, [], 是, bounded_failure_not_global_nonmembership)；依完整明示資料重算；image是full-real column span且不等同codomain；bounded fiber search失敗不證全域nonmembership。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性映射的像"]},"variant":{"family_id":"qf-mod33-image-bounded-search-boundary","seed":202611023086,"parameters":{"matrix":[[2,0],[0,2]],"target":[21,1],"bound":1,"domain":"Exact_R2_matrix_image","operation":"image-bounded-search-boundary","semantic_template":"qf-mod33-image-bounded-search-boundary/{\"bound\":1,\"matrix\":[[2,0],[0,2]],\"target\":[1,21]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r03-linear-map-image-0087","batch_id":"batch-20261014-advance-20261102-r03-linear-map-image-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-image"],"title_zh":"線性映射的像","stem_zh":"同一精確2×2線性映射與column-span證書。操作=image-bounded-search-boundary；資料={\"matrix\":[[2,0],[0,2]],\"target\":[23,1],\"bound\":2}。請依 [target,bound,integer_witnesses,global_membership,boundary] 欄序回答。","answer":{"kind":"expression","value":"([23, 1], 2, [], 是, bounded_failure_not_global_nonmembership)","explanation_zh":"Wave5第二層核對=([23, 1], 2, [], 是, bounded_failure_not_global_nonmembership)；依完整明示資料重算；image是full-real column span且不等同codomain；bounded fiber search失敗不證全域nonmembership。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性映射的像"]},"variant":{"family_id":"qf-mod33-image-bounded-search-boundary","seed":202611023087,"parameters":{"matrix":[[2,0],[0,2]],"target":[23,1],"bound":2,"domain":"Exact_R2_matrix_image","operation":"image-bounded-search-boundary","semantic_template":"qf-mod33-image-bounded-search-boundary/{\"bound\":2,\"matrix\":[[2,0],[0,2]],\"target\":[1,23]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r03-linear-map-image-0088","batch_id":"batch-20261014-advance-20261102-r03-linear-map-image-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-image"],"title_zh":"線性映射的像","stem_zh":"同一精確2×2線性映射與column-span證書。操作=image-bounded-search-boundary；資料={\"matrix\":[[2,0],[0,2]],\"target\":[25,1],\"bound\":3}。請依 [target,bound,integer_witnesses,global_membership,boundary] 欄序回答。","answer":{"kind":"expression","value":"([25, 1], 3, [], 是, bounded_failure_not_global_nonmembership)","explanation_zh":"Wave5第二層核對=([25, 1], 3, [], 是, bounded_failure_not_global_nonmembership)；依完整明示資料重算；image是full-real column span且不等同codomain；bounded fiber search失敗不證全域nonmembership。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性映射的像"]},"variant":{"family_id":"qf-mod33-image-bounded-search-boundary","seed":202611023088,"parameters":{"matrix":[[2,0],[0,2]],"target":[25,1],"bound":3,"domain":"Exact_R2_matrix_image","operation":"image-bounded-search-boundary","semantic_template":"qf-mod33-image-bounded-search-boundary/{\"bound\":3,\"matrix\":[[2,0],[0,2]],\"target\":[1,25]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r03-linear-map-image-0089","batch_id":"batch-20261014-advance-20261102-r03-linear-map-image-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-image"],"title_zh":"線性映射的像","stem_zh":"同一精確2×2線性映射與column-span證書。操作=image-bounded-search-boundary；資料={\"matrix\":[[2,0],[0,2]],\"target\":[27,1],\"bound\":4}。請依 [target,bound,integer_witnesses,global_membership,boundary] 欄序回答。","answer":{"kind":"expression","value":"([27, 1], 4, [], 是, bounded_failure_not_global_nonmembership)","explanation_zh":"Wave5第二層核對=([27, 1], 4, [], 是, bounded_failure_not_global_nonmembership)；依完整明示資料重算；image是full-real column span且不等同codomain；bounded fiber search失敗不證全域nonmembership。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性映射的像"]},"variant":{"family_id":"qf-mod33-image-bounded-search-boundary","seed":202611023089,"parameters":{"matrix":[[2,0],[0,2]],"target":[27,1],"bound":4,"domain":"Exact_R2_matrix_image","operation":"image-bounded-search-boundary","semantic_template":"qf-mod33-image-bounded-search-boundary/{\"bound\":4,\"matrix\":[[2,0],[0,2]],\"target\":[1,27]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r03-linear-map-image-0090","batch_id":"batch-20261014-advance-20261102-r03-linear-map-image-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-image"],"title_zh":"線性映射的像","stem_zh":"同一精確2×2線性映射與column-span證書。操作=image-bounded-search-boundary；資料={\"matrix\":[[2,0],[0,2]],\"target\":[29,1],\"bound\":5}。請依 [target,bound,integer_witnesses,global_membership,boundary] 欄序回答。","answer":{"kind":"expression","value":"([29, 1], 5, [], 是, bounded_failure_not_global_nonmembership)","explanation_zh":"Wave5第二層核對=([29, 1], 5, [], 是, bounded_failure_not_global_nonmembership)；依完整明示資料重算；image是full-real column span且不等同codomain；bounded fiber search失敗不證全域nonmembership。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性映射的像"]},"variant":{"family_id":"qf-mod33-image-bounded-search-boundary","seed":202611023090,"parameters":{"matrix":[[2,0],[0,2]],"target":[29,1],"bound":5,"domain":"Exact_R2_matrix_image","operation":"image-bounded-search-boundary","semantic_template":"qf-mod33-image-bounded-search-boundary/{\"bound\":5,\"matrix\":[[2,0],[0,2]],\"target\":[1,29]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r03-linear-map-image-0091","batch_id":"batch-20261014-advance-20261102-r03-linear-map-image-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-image"],"title_zh":"線性映射的像","stem_zh":"同一精確2×2線性映射與column-span證書。操作=image-bounded-search-boundary；資料={\"matrix\":[[2,0],[0,2]],\"target\":[31,1],\"bound\":1}。請依 [target,bound,integer_witnesses,global_membership,boundary] 欄序回答。","answer":{"kind":"expression","value":"([31, 1], 1, [], 是, bounded_failure_not_global_nonmembership)","explanation_zh":"Wave5第二層核對=([31, 1], 1, [], 是, bounded_failure_not_global_nonmembership)；依完整明示資料重算；image是full-real column span且不等同codomain；bounded fiber search失敗不證全域nonmembership。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性映射的像"]},"variant":{"family_id":"qf-mod33-image-bounded-search-boundary","seed":202611023091,"parameters":{"matrix":[[2,0],[0,2]],"target":[31,1],"bound":1,"domain":"Exact_R2_matrix_image","operation":"image-bounded-search-boundary","semantic_template":"qf-mod33-image-bounded-search-boundary/{\"bound\":1,\"matrix\":[[2,0],[0,2]],\"target\":[1,31]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r03-linear-map-image-0092","batch_id":"batch-20261014-advance-20261102-r03-linear-map-image-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-image"],"title_zh":"線性映射的像","stem_zh":"同一精確2×2線性映射與column-span證書。操作=image-bounded-search-boundary；資料={\"matrix\":[[2,0],[0,2]],\"target\":[33,1],\"bound\":2}。請依 [target,bound,integer_witnesses,global_membership,boundary] 欄序回答。","answer":{"kind":"expression","value":"([33, 1], 2, [], 是, bounded_failure_not_global_nonmembership)","explanation_zh":"Wave5第二層核對=([33, 1], 2, [], 是, bounded_failure_not_global_nonmembership)；依完整明示資料重算；image是full-real column span且不等同codomain；bounded fiber search失敗不證全域nonmembership。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性映射的像"]},"variant":{"family_id":"qf-mod33-image-bounded-search-boundary","seed":202611023092,"parameters":{"matrix":[[2,0],[0,2]],"target":[33,1],"bound":2,"domain":"Exact_R2_matrix_image","operation":"image-bounded-search-boundary","semantic_template":"qf-mod33-image-bounded-search-boundary/{\"bound\":2,\"matrix\":[[2,0],[0,2]],\"target\":[1,33]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r03-linear-map-image-0093","batch_id":"batch-20261014-advance-20261102-r03-linear-map-image-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-image"],"title_zh":"線性映射的像","stem_zh":"同一精確2×2線性映射與column-span證書。操作=image-bounded-search-boundary；資料={\"matrix\":[[2,0],[0,2]],\"target\":[35,1],\"bound\":3}。請依 [target,bound,integer_witnesses,global_membership,boundary] 欄序回答。","answer":{"kind":"expression","value":"([35, 1], 3, [], 是, bounded_failure_not_global_nonmembership)","explanation_zh":"Wave5第二層核對=([35, 1], 3, [], 是, bounded_failure_not_global_nonmembership)；依完整明示資料重算；image是full-real column span且不等同codomain；bounded fiber search失敗不證全域nonmembership。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性映射的像"]},"variant":{"family_id":"qf-mod33-image-bounded-search-boundary","seed":202611023093,"parameters":{"matrix":[[2,0],[0,2]],"target":[35,1],"bound":3,"domain":"Exact_R2_matrix_image","operation":"image-bounded-search-boundary","semantic_template":"qf-mod33-image-bounded-search-boundary/{\"bound\":3,\"matrix\":[[2,0],[0,2]],\"target\":[1,35]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r03-linear-map-image-0094","batch_id":"batch-20261014-advance-20261102-r03-linear-map-image-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-image"],"title_zh":"線性映射的像","stem_zh":"同一精確2×2線性映射與column-span證書。操作=image-bounded-search-boundary；資料={\"matrix\":[[2,0],[0,2]],\"target\":[37,1],\"bound\":4}。請依 [target,bound,integer_witnesses,global_membership,boundary] 欄序回答。","answer":{"kind":"expression","value":"([37, 1], 4, [], 是, bounded_failure_not_global_nonmembership)","explanation_zh":"Wave5第二層核對=([37, 1], 4, [], 是, bounded_failure_not_global_nonmembership)；依完整明示資料重算；image是full-real column span且不等同codomain；bounded fiber search失敗不證全域nonmembership。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性映射的像"]},"variant":{"family_id":"qf-mod33-image-bounded-search-boundary","seed":202611023094,"parameters":{"matrix":[[2,0],[0,2]],"target":[37,1],"bound":4,"domain":"Exact_R2_matrix_image","operation":"image-bounded-search-boundary","semantic_template":"qf-mod33-image-bounded-search-boundary/{\"bound\":4,\"matrix\":[[2,0],[0,2]],\"target\":[1,37]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r03-linear-map-image-0095","batch_id":"batch-20261014-advance-20261102-r03-linear-map-image-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-image"],"title_zh":"線性映射的像","stem_zh":"同一精確2×2線性映射與column-span證書。操作=image-bounded-search-boundary；資料={\"matrix\":[[2,0],[0,2]],\"target\":[39,1],\"bound\":5}。請依 [target,bound,integer_witnesses,global_membership,boundary] 欄序回答。","answer":{"kind":"expression","value":"([39, 1], 5, [], 是, bounded_failure_not_global_nonmembership)","explanation_zh":"Wave5第二層核對=([39, 1], 5, [], 是, bounded_failure_not_global_nonmembership)；依完整明示資料重算；image是full-real column span且不等同codomain；bounded fiber search失敗不證全域nonmembership。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性映射的像"]},"variant":{"family_id":"qf-mod33-image-bounded-search-boundary","seed":202611023095,"parameters":{"matrix":[[2,0],[0,2]],"target":[39,1],"bound":5,"domain":"Exact_R2_matrix_image","operation":"image-bounded-search-boundary","semantic_template":"qf-mod33-image-bounded-search-boundary/{\"bound\":5,\"matrix\":[[2,0],[0,2]],\"target\":[1,39]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r03-linear-map-image-0096","batch_id":"batch-20261014-advance-20261102-r03-linear-map-image-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-image"],"title_zh":"線性映射的像","stem_zh":"同一精確2×2線性映射與column-span證書。操作=image-bounded-search-boundary；資料={\"matrix\":[[2,0],[0,2]],\"target\":[41,1],\"bound\":1}。請依 [target,bound,integer_witnesses,global_membership,boundary] 欄序回答。","answer":{"kind":"expression","value":"([41, 1], 1, [], 是, bounded_failure_not_global_nonmembership)","explanation_zh":"Wave5第二層核對=([41, 1], 1, [], 是, bounded_failure_not_global_nonmembership)；依完整明示資料重算；image是full-real column span且不等同codomain；bounded fiber search失敗不證全域nonmembership。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性映射的像"]},"variant":{"family_id":"qf-mod33-image-bounded-search-boundary","seed":202611023096,"parameters":{"matrix":[[2,0],[0,2]],"target":[41,1],"bound":1,"domain":"Exact_R2_matrix_image","operation":"image-bounded-search-boundary","semantic_template":"qf-mod33-image-bounded-search-boundary/{\"bound\":1,\"matrix\":[[2,0],[0,2]],\"target\":[1,41]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r03-linear-map-image-0097","batch_id":"batch-20261014-advance-20261102-r03-linear-map-image-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-image"],"title_zh":"線性映射的像","stem_zh":"同一精確2×2線性映射與column-span證書。操作=image-bounded-search-boundary；資料={\"matrix\":[[2,0],[0,2]],\"target\":[43,1],\"bound\":2}。請依 [target,bound,integer_witnesses,global_membership,boundary] 欄序回答。","answer":{"kind":"expression","value":"([43, 1], 2, [], 是, bounded_failure_not_global_nonmembership)","explanation_zh":"Wave5第二層核對=([43, 1], 2, [], 是, bounded_failure_not_global_nonmembership)；依完整明示資料重算；image是full-real column span且不等同codomain；bounded fiber search失敗不證全域nonmembership。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性映射的像"]},"variant":{"family_id":"qf-mod33-image-bounded-search-boundary","seed":202611023097,"parameters":{"matrix":[[2,0],[0,2]],"target":[43,1],"bound":2,"domain":"Exact_R2_matrix_image","operation":"image-bounded-search-boundary","semantic_template":"qf-mod33-image-bounded-search-boundary/{\"bound\":2,\"matrix\":[[2,0],[0,2]],\"target\":[1,43]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r03-linear-map-image-0098","batch_id":"batch-20261014-advance-20261102-r03-linear-map-image-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-image"],"title_zh":"線性映射的像","stem_zh":"同一精確2×2線性映射與column-span證書。操作=image-bounded-search-boundary；資料={\"matrix\":[[2,0],[0,2]],\"target\":[45,1],\"bound\":3}。請依 [target,bound,integer_witnesses,global_membership,boundary] 欄序回答。","answer":{"kind":"expression","value":"([45, 1], 3, [], 是, bounded_failure_not_global_nonmembership)","explanation_zh":"Wave5第二層核對=([45, 1], 3, [], 是, bounded_failure_not_global_nonmembership)；依完整明示資料重算；image是full-real column span且不等同codomain；bounded fiber search失敗不證全域nonmembership。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性映射的像"]},"variant":{"family_id":"qf-mod33-image-bounded-search-boundary","seed":202611023098,"parameters":{"matrix":[[2,0],[0,2]],"target":[45,1],"bound":3,"domain":"Exact_R2_matrix_image","operation":"image-bounded-search-boundary","semantic_template":"qf-mod33-image-bounded-search-boundary/{\"bound\":3,\"matrix\":[[2,0],[0,2]],\"target\":[1,45]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r03-linear-map-image-0099","batch_id":"batch-20261014-advance-20261102-r03-linear-map-image-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-image"],"title_zh":"線性映射的像","stem_zh":"同一精確2×2線性映射與column-span證書。操作=image-bounded-search-boundary；資料={\"matrix\":[[2,0],[0,2]],\"target\":[47,1],\"bound\":4}。請依 [target,bound,integer_witnesses,global_membership,boundary] 欄序回答。","answer":{"kind":"expression","value":"([47, 1], 4, [], 是, bounded_failure_not_global_nonmembership)","explanation_zh":"Wave5第二層核對=([47, 1], 4, [], 是, bounded_failure_not_global_nonmembership)；依完整明示資料重算；image是full-real column span且不等同codomain；bounded fiber search失敗不證全域nonmembership。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性映射的像"]},"variant":{"family_id":"qf-mod33-image-bounded-search-boundary","seed":202611023099,"parameters":{"matrix":[[2,0],[0,2]],"target":[47,1],"bound":4,"domain":"Exact_R2_matrix_image","operation":"image-bounded-search-boundary","semantic_template":"qf-mod33-image-bounded-search-boundary/{\"bound\":4,\"matrix\":[[2,0],[0,2]],\"target\":[1,47]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r03-linear-map-image-0100","batch_id":"batch-20261014-advance-20261102-r03-linear-map-image-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-image"],"title_zh":"線性映射的像","stem_zh":"同一精確2×2線性映射與column-span證書。操作=image-bounded-search-boundary；資料={\"matrix\":[[2,0],[0,2]],\"target\":[49,1],\"bound\":5}。請依 [target,bound,integer_witnesses,global_membership,boundary] 欄序回答。","answer":{"kind":"expression","value":"([49, 1], 5, [], 是, bounded_failure_not_global_nonmembership)","explanation_zh":"Wave5第二層核對=([49, 1], 5, [], 是, bounded_failure_not_global_nonmembership)；依完整明示資料重算；image是full-real column span且不等同codomain；bounded fiber search失敗不證全域nonmembership。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性映射的像"]},"variant":{"family_id":"qf-mod33-image-bounded-search-boundary","seed":202611023100,"parameters":{"matrix":[[2,0],[0,2]],"target":[49,1],"bound":5,"domain":"Exact_R2_matrix_image","operation":"image-bounded-search-boundary","semantic_template":"qf-mod33-image-bounded-search-boundary/{\"bound\":5,\"matrix\":[[2,0],[0,2]],\"target\":[1,49]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}