{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r02-linear-map-kernel-0001","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-kernel"],"title_zh":"線性映射的核","stem_zh":"同一精確2×2線性映射T:R²到R²。操作=kernel-membership；資料={\"matrix\":[[1,1],[2,2]],\"vector\":[-1,1]}。請依 [vector,image,in_kernel,boundary] 欄序回答。","answer":{"kind":"expression","value":"([-1, 1], [0, 0], 是, kernel_target_is_zero_vector)","explanation_zh":"Wave5第二層核對=([-1, 1], [0, 0], 是, kernel_target_is_zero_vector)；依完整明示資料重算；kernel只解T(x)=0且使用symbolic full-real certificate；bounded integer search只回答bounded問題；非零fiber不是kernel。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性映射的核"]},"variant":{"family_id":"qf-mod33-kernel-membership","seed":202611022001,"parameters":{"matrix":[[1,1],[2,2]],"vector":[-1,1],"domain":"Exact_R2_matrix_kernel","operation":"kernel-membership","semantic_template":"qf-mod33-kernel-membership/{\"matrix\":[[1,1],[2,2]],\"vector\":[-1,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；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-r02-linear-map-kernel-0002","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-kernel"],"title_zh":"線性映射的核","stem_zh":"同一精確2×2線性映射T:R²到R²。操作=kernel-membership；資料={\"matrix\":[[1,2],[2,4]],\"vector\":[-2,1]}。請依 [vector,image,in_kernel,boundary] 欄序回答。","answer":{"kind":"expression","value":"([-2, 1], [0, 0], 是, kernel_target_is_zero_vector)","explanation_zh":"Wave5第二層核對=([-2, 1], [0, 0], 是, kernel_target_is_zero_vector)；依完整明示資料重算；kernel只解T(x)=0且使用symbolic full-real certificate；bounded integer search只回答bounded問題；非零fiber不是kernel。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性映射的核"]},"variant":{"family_id":"qf-mod33-kernel-membership","seed":202611022002,"parameters":{"matrix":[[1,2],[2,4]],"vector":[-2,1],"domain":"Exact_R2_matrix_kernel","operation":"kernel-membership","semantic_template":"qf-mod33-kernel-membership/{\"matrix\":[[1,2],[2,4]],\"vector\":[-2,1]}"}},"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-r02-linear-map-kernel-0003","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-kernel"],"title_zh":"線性映射的核","stem_zh":"同一精確2×2線性映射T:R²到R²。操作=kernel-membership；資料={\"matrix\":[[1,3],[2,6]],\"vector\":[-3,1]}。請依 [vector,image,in_kernel,boundary] 欄序回答。","answer":{"kind":"expression","value":"([-3, 1], [0, 0], 是, kernel_target_is_zero_vector)","explanation_zh":"Wave5第二層核對=([-3, 1], [0, 0], 是, kernel_target_is_zero_vector)；依完整明示資料重算；kernel只解T(x)=0且使用symbolic full-real certificate；bounded integer search只回答bounded問題；非零fiber不是kernel。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性映射的核"]},"variant":{"family_id":"qf-mod33-kernel-membership","seed":202611022003,"parameters":{"matrix":[[1,3],[2,6]],"vector":[-3,1],"domain":"Exact_R2_matrix_kernel","operation":"kernel-membership","semantic_template":"qf-mod33-kernel-membership/{\"matrix\":[[1,3],[2,6]],\"vector\":[-3,1]}"}},"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-r02-linear-map-kernel-0004","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-kernel"],"title_zh":"線性映射的核","stem_zh":"同一精確2×2線性映射T:R²到R²。操作=kernel-membership；資料={\"matrix\":[[1,4],[2,8]],\"vector\":[-4,1]}。請依 [vector,image,in_kernel,boundary] 欄序回答。","answer":{"kind":"expression","value":"([-4, 1], [0, 0], 是, kernel_target_is_zero_vector)","explanation_zh":"Wave5第二層核對=([-4, 1], [0, 0], 是, kernel_target_is_zero_vector)；依完整明示資料重算；kernel只解T(x)=0且使用symbolic full-real certificate；bounded integer search只回答bounded問題；非零fiber不是kernel。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性映射的核"]},"variant":{"family_id":"qf-mod33-kernel-membership","seed":202611022004,"parameters":{"matrix":[[1,4],[2,8]],"vector":[-4,1],"domain":"Exact_R2_matrix_kernel","operation":"kernel-membership","semantic_template":"qf-mod33-kernel-membership/{\"matrix\":[[1,4],[2,8]],\"vector\":[-4,1]}"}},"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-r02-linear-map-kernel-0005","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-kernel"],"title_zh":"線性映射的核","stem_zh":"同一精確2×2線性映射T:R²到R²。操作=kernel-membership；資料={\"matrix\":[[1,5],[2,10]],\"vector\":[-5,1]}。請依 [vector,image,in_kernel,boundary] 欄序回答。","answer":{"kind":"expression","value":"([-5, 1], [0, 0], 是, kernel_target_is_zero_vector)","explanation_zh":"Wave5第二層核對=([-5, 1], [0, 0], 是, kernel_target_is_zero_vector)；依完整明示資料重算；kernel只解T(x)=0且使用symbolic full-real certificate；bounded integer search只回答bounded問題；非零fiber不是kernel。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性映射的核"]},"variant":{"family_id":"qf-mod33-kernel-membership","seed":202611022005,"parameters":{"matrix":[[1,5],[2,10]],"vector":[-5,1],"domain":"Exact_R2_matrix_kernel","operation":"kernel-membership","semantic_template":"qf-mod33-kernel-membership/{\"matrix\":[[1,5],[2,10]],\"vector\":[-5,1]}"}},"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-r02-linear-map-kernel-0006","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-kernel"],"title_zh":"線性映射的核","stem_zh":"同一精確2×2線性映射T:R²到R²。操作=kernel-membership；資料={\"matrix\":[[1,6],[2,12]],\"vector\":[-6,1]}。請依 [vector,image,in_kernel,boundary] 欄序回答。","answer":{"kind":"expression","value":"([-6, 1], [0, 0], 是, kernel_target_is_zero_vector)","explanation_zh":"Wave5第二層核對=([-6, 1], [0, 0], 是, kernel_target_is_zero_vector)；依完整明示資料重算；kernel只解T(x)=0且使用symbolic full-real certificate；bounded integer search只回答bounded問題；非零fiber不是kernel。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性映射的核"]},"variant":{"family_id":"qf-mod33-kernel-membership","seed":202611022006,"parameters":{"matrix":[[1,6],[2,12]],"vector":[-6,1],"domain":"Exact_R2_matrix_kernel","operation":"kernel-membership","semantic_template":"qf-mod33-kernel-membership/{\"matrix\":[[1,6],[2,12]],\"vector\":[-6,1]}"}},"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-r02-linear-map-kernel-0007","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-kernel"],"title_zh":"線性映射的核","stem_zh":"同一精確2×2線性映射T:R²到R²。操作=kernel-membership；資料={\"matrix\":[[1,7],[2,14]],\"vector\":[-7,1]}。請依 [vector,image,in_kernel,boundary] 欄序回答。","answer":{"kind":"expression","value":"([-7, 1], [0, 0], 是, kernel_target_is_zero_vector)","explanation_zh":"Wave5第二層核對=([-7, 1], [0, 0], 是, kernel_target_is_zero_vector)；依完整明示資料重算；kernel只解T(x)=0且使用symbolic full-real certificate；bounded integer search只回答bounded問題；非零fiber不是kernel。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性映射的核"]},"variant":{"family_id":"qf-mod33-kernel-membership","seed":202611022007,"parameters":{"matrix":[[1,7],[2,14]],"vector":[-7,1],"domain":"Exact_R2_matrix_kernel","operation":"kernel-membership","semantic_template":"qf-mod33-kernel-membership/{\"matrix\":[[1,7],[2,14]],\"vector\":[-7,1]}"}},"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-r02-linear-map-kernel-0008","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-kernel"],"title_zh":"線性映射的核","stem_zh":"同一精確2×2線性映射T:R²到R²。操作=kernel-membership；資料={\"matrix\":[[1,8],[2,16]],\"vector\":[-8,1]}。請依 [vector,image,in_kernel,boundary] 欄序回答。","answer":{"kind":"expression","value":"([-8, 1], [0, 0], 是, kernel_target_is_zero_vector)","explanation_zh":"Wave5第二層核對=([-8, 1], [0, 0], 是, kernel_target_is_zero_vector)；依完整明示資料重算；kernel只解T(x)=0且使用symbolic full-real certificate；bounded integer search只回答bounded問題；非零fiber不是kernel。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性映射的核"]},"variant":{"family_id":"qf-mod33-kernel-membership","seed":202611022008,"parameters":{"matrix":[[1,8],[2,16]],"vector":[-8,1],"domain":"Exact_R2_matrix_kernel","operation":"kernel-membership","semantic_template":"qf-mod33-kernel-membership/{\"matrix\":[[1,8],[2,16]],\"vector\":[-8,1]}"}},"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-r02-linear-map-kernel-0009","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-kernel"],"title_zh":"線性映射的核","stem_zh":"同一精確2×2線性映射T:R²到R²。操作=kernel-membership；資料={\"matrix\":[[1,9],[2,18]],\"vector\":[-9,1]}。請依 [vector,image,in_kernel,boundary] 欄序回答。","answer":{"kind":"expression","value":"([-9, 1], [0, 0], 是, kernel_target_is_zero_vector)","explanation_zh":"Wave5第二層核對=([-9, 1], [0, 0], 是, kernel_target_is_zero_vector)；依完整明示資料重算；kernel只解T(x)=0且使用symbolic full-real certificate；bounded integer search只回答bounded問題；非零fiber不是kernel。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性映射的核"]},"variant":{"family_id":"qf-mod33-kernel-membership","seed":202611022009,"parameters":{"matrix":[[1,9],[2,18]],"vector":[-9,1],"domain":"Exact_R2_matrix_kernel","operation":"kernel-membership","semantic_template":"qf-mod33-kernel-membership/{\"matrix\":[[1,9],[2,18]],\"vector\":[-9,1]}"}},"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-r02-linear-map-kernel-0010","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-kernel"],"title_zh":"線性映射的核","stem_zh":"同一精確2×2線性映射T:R²到R²。操作=kernel-membership；資料={\"matrix\":[[1,10],[2,20]],\"vector\":[-10,1]}。請依 [vector,image,in_kernel,boundary] 欄序回答。","answer":{"kind":"expression","value":"([-10, 1], [0, 0], 是, kernel_target_is_zero_vector)","explanation_zh":"Wave5第二層核對=([-10, 1], [0, 0], 是, kernel_target_is_zero_vector)；依完整明示資料重算；kernel只解T(x)=0且使用symbolic full-real certificate；bounded integer search只回答bounded問題；非零fiber不是kernel。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性映射的核"]},"variant":{"family_id":"qf-mod33-kernel-membership","seed":202611022010,"parameters":{"matrix":[[1,10],[2,20]],"vector":[-10,1],"domain":"Exact_R2_matrix_kernel","operation":"kernel-membership","semantic_template":"qf-mod33-kernel-membership/{\"matrix\":[[1,10],[2,20]],\"vector\":[-10,1]}"}},"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-r02-linear-map-kernel-0011","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-kernel"],"title_zh":"線性映射的核","stem_zh":"同一精確2×2線性映射T:R²到R²。操作=kernel-membership；資料={\"matrix\":[[1,11],[2,22]],\"vector\":[-11,1]}。請依 [vector,image,in_kernel,boundary] 欄序回答。","answer":{"kind":"expression","value":"([-11, 1], [0, 0], 是, kernel_target_is_zero_vector)","explanation_zh":"Wave5第二層核對=([-11, 1], [0, 0], 是, kernel_target_is_zero_vector)；依完整明示資料重算；kernel只解T(x)=0且使用symbolic full-real certificate；bounded integer search只回答bounded問題；非零fiber不是kernel。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性映射的核"]},"variant":{"family_id":"qf-mod33-kernel-membership","seed":202611022011,"parameters":{"matrix":[[1,11],[2,22]],"vector":[-11,1],"domain":"Exact_R2_matrix_kernel","operation":"kernel-membership","semantic_template":"qf-mod33-kernel-membership/{\"matrix\":[[1,11],[2,22]],\"vector\":[-11,1]}"}},"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-r02-linear-map-kernel-0012","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-kernel"],"title_zh":"線性映射的核","stem_zh":"同一精確2×2線性映射T:R²到R²。操作=kernel-membership；資料={\"matrix\":[[1,12],[2,24]],\"vector\":[-12,1]}。請依 [vector,image,in_kernel,boundary] 欄序回答。","answer":{"kind":"expression","value":"([-12, 1], [0, 0], 是, kernel_target_is_zero_vector)","explanation_zh":"Wave5第二層核對=([-12, 1], [0, 0], 是, kernel_target_is_zero_vector)；依完整明示資料重算；kernel只解T(x)=0且使用symbolic full-real certificate；bounded integer search只回答bounded問題；非零fiber不是kernel。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性映射的核"]},"variant":{"family_id":"qf-mod33-kernel-membership","seed":202611022012,"parameters":{"matrix":[[1,12],[2,24]],"vector":[-12,1],"domain":"Exact_R2_matrix_kernel","operation":"kernel-membership","semantic_template":"qf-mod33-kernel-membership/{\"matrix\":[[1,12],[2,24]],\"vector\":[-12,1]}"}},"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-r02-linear-map-kernel-0013","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-kernel"],"title_zh":"線性映射的核","stem_zh":"同一精確2×2線性映射T:R²到R²。操作=kernel-membership；資料={\"matrix\":[[1,13],[2,26]],\"vector\":[-13,1]}。請依 [vector,image,in_kernel,boundary] 欄序回答。","answer":{"kind":"expression","value":"([-13, 1], [0, 0], 是, kernel_target_is_zero_vector)","explanation_zh":"Wave5第二層核對=([-13, 1], [0, 0], 是, kernel_target_is_zero_vector)；依完整明示資料重算；kernel只解T(x)=0且使用symbolic full-real certificate；bounded integer search只回答bounded問題；非零fiber不是kernel。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性映射的核"]},"variant":{"family_id":"qf-mod33-kernel-membership","seed":202611022013,"parameters":{"matrix":[[1,13],[2,26]],"vector":[-13,1],"domain":"Exact_R2_matrix_kernel","operation":"kernel-membership","semantic_template":"qf-mod33-kernel-membership/{\"matrix\":[[1,13],[2,26]],\"vector\":[-13,1]}"}},"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-r02-linear-map-kernel-0014","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-kernel"],"title_zh":"線性映射的核","stem_zh":"同一精確2×2線性映射T:R²到R²。操作=kernel-membership；資料={\"matrix\":[[1,14],[2,28]],\"vector\":[-14,1]}。請依 [vector,image,in_kernel,boundary] 欄序回答。","answer":{"kind":"expression","value":"([-14, 1], [0, 0], 是, kernel_target_is_zero_vector)","explanation_zh":"Wave5第二層核對=([-14, 1], [0, 0], 是, kernel_target_is_zero_vector)；依完整明示資料重算；kernel只解T(x)=0且使用symbolic full-real certificate；bounded integer search只回答bounded問題；非零fiber不是kernel。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性映射的核"]},"variant":{"family_id":"qf-mod33-kernel-membership","seed":202611022014,"parameters":{"matrix":[[1,14],[2,28]],"vector":[-14,1],"domain":"Exact_R2_matrix_kernel","operation":"kernel-membership","semantic_template":"qf-mod33-kernel-membership/{\"matrix\":[[1,14],[2,28]],\"vector\":[-14,1]}"}},"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-r02-linear-map-kernel-0015","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-kernel"],"title_zh":"線性映射的核","stem_zh":"同一精確2×2線性映射T:R²到R²。操作=kernel-membership；資料={\"matrix\":[[1,15],[2,30]],\"vector\":[-15,1]}。請依 [vector,image,in_kernel,boundary] 欄序回答。","answer":{"kind":"expression","value":"([-15, 1], [0, 0], 是, kernel_target_is_zero_vector)","explanation_zh":"Wave5第二層核對=([-15, 1], [0, 0], 是, kernel_target_is_zero_vector)；依完整明示資料重算；kernel只解T(x)=0且使用symbolic full-real certificate；bounded integer search只回答bounded問題；非零fiber不是kernel。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性映射的核"]},"variant":{"family_id":"qf-mod33-kernel-membership","seed":202611022015,"parameters":{"matrix":[[1,15],[2,30]],"vector":[-15,1],"domain":"Exact_R2_matrix_kernel","operation":"kernel-membership","semantic_template":"qf-mod33-kernel-membership/{\"matrix\":[[1,15],[2,30]],\"vector\":[-15,1]}"}},"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-r02-linear-map-kernel-0016","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-kernel"],"title_zh":"線性映射的核","stem_zh":"同一精確2×2線性映射T:R²到R²。操作=kernel-membership；資料={\"matrix\":[[1,16],[2,32]],\"vector\":[-16,1]}。請依 [vector,image,in_kernel,boundary] 欄序回答。","answer":{"kind":"expression","value":"([-16, 1], [0, 0], 是, kernel_target_is_zero_vector)","explanation_zh":"Wave5第二層核對=([-16, 1], [0, 0], 是, kernel_target_is_zero_vector)；依完整明示資料重算；kernel只解T(x)=0且使用symbolic full-real certificate；bounded integer search只回答bounded問題；非零fiber不是kernel。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性映射的核"]},"variant":{"family_id":"qf-mod33-kernel-membership","seed":202611022016,"parameters":{"matrix":[[1,16],[2,32]],"vector":[-16,1],"domain":"Exact_R2_matrix_kernel","operation":"kernel-membership","semantic_template":"qf-mod33-kernel-membership/{\"matrix\":[[1,16],[2,32]],\"vector\":[-16,1]}"}},"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-r02-linear-map-kernel-0017","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-kernel"],"title_zh":"線性映射的核","stem_zh":"同一精確2×2線性映射T:R²到R²。操作=kernel-membership；資料={\"matrix\":[[1,17],[2,34]],\"vector\":[-17,1]}。請依 [vector,image,in_kernel,boundary] 欄序回答。","answer":{"kind":"expression","value":"([-17, 1], [0, 0], 是, kernel_target_is_zero_vector)","explanation_zh":"Wave5第二層核對=([-17, 1], [0, 0], 是, kernel_target_is_zero_vector)；依完整明示資料重算；kernel只解T(x)=0且使用symbolic full-real certificate；bounded integer search只回答bounded問題；非零fiber不是kernel。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性映射的核"]},"variant":{"family_id":"qf-mod33-kernel-membership","seed":202611022017,"parameters":{"matrix":[[1,17],[2,34]],"vector":[-17,1],"domain":"Exact_R2_matrix_kernel","operation":"kernel-membership","semantic_template":"qf-mod33-kernel-membership/{\"matrix\":[[1,17],[2,34]],\"vector\":[-17,1]}"}},"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-r02-linear-map-kernel-0018","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-kernel"],"title_zh":"線性映射的核","stem_zh":"同一精確2×2線性映射T:R²到R²。操作=kernel-membership；資料={\"matrix\":[[1,18],[2,36]],\"vector\":[-18,1]}。請依 [vector,image,in_kernel,boundary] 欄序回答。","answer":{"kind":"expression","value":"([-18, 1], [0, 0], 是, kernel_target_is_zero_vector)","explanation_zh":"Wave5第二層核對=([-18, 1], [0, 0], 是, kernel_target_is_zero_vector)；依完整明示資料重算；kernel只解T(x)=0且使用symbolic full-real certificate；bounded integer search只回答bounded問題；非零fiber不是kernel。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性映射的核"]},"variant":{"family_id":"qf-mod33-kernel-membership","seed":202611022018,"parameters":{"matrix":[[1,18],[2,36]],"vector":[-18,1],"domain":"Exact_R2_matrix_kernel","operation":"kernel-membership","semantic_template":"qf-mod33-kernel-membership/{\"matrix\":[[1,18],[2,36]],\"vector\":[-18,1]}"}},"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-r02-linear-map-kernel-0019","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-kernel"],"title_zh":"線性映射的核","stem_zh":"同一精確2×2線性映射T:R²到R²。操作=kernel-membership；資料={\"matrix\":[[1,19],[2,38]],\"vector\":[-19,1]}。請依 [vector,image,in_kernel,boundary] 欄序回答。","answer":{"kind":"expression","value":"([-19, 1], [0, 0], 是, kernel_target_is_zero_vector)","explanation_zh":"Wave5第二層核對=([-19, 1], [0, 0], 是, kernel_target_is_zero_vector)；依完整明示資料重算；kernel只解T(x)=0且使用symbolic full-real certificate；bounded integer search只回答bounded問題；非零fiber不是kernel。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性映射的核"]},"variant":{"family_id":"qf-mod33-kernel-membership","seed":202611022019,"parameters":{"matrix":[[1,19],[2,38]],"vector":[-19,1],"domain":"Exact_R2_matrix_kernel","operation":"kernel-membership","semantic_template":"qf-mod33-kernel-membership/{\"matrix\":[[1,19],[2,38]],\"vector\":[-19,1]}"}},"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-r02-linear-map-kernel-0020","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-kernel"],"title_zh":"線性映射的核","stem_zh":"同一精確2×2線性映射T:R²到R²。操作=kernel-membership；資料={\"matrix\":[[1,20],[2,40]],\"vector\":[-20,1]}。請依 [vector,image,in_kernel,boundary] 欄序回答。","answer":{"kind":"expression","value":"([-20, 1], [0, 0], 是, kernel_target_is_zero_vector)","explanation_zh":"Wave5第二層核對=([-20, 1], [0, 0], 是, kernel_target_is_zero_vector)；依完整明示資料重算；kernel只解T(x)=0且使用symbolic full-real certificate；bounded integer search只回答bounded問題；非零fiber不是kernel。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性映射的核"]},"variant":{"family_id":"qf-mod33-kernel-membership","seed":202611022020,"parameters":{"matrix":[[1,20],[2,40]],"vector":[-20,1],"domain":"Exact_R2_matrix_kernel","operation":"kernel-membership","semantic_template":"qf-mod33-kernel-membership/{\"matrix\":[[1,20],[2,40]],\"vector\":[-20,1]}"}},"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-r02-linear-map-kernel-0021","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-kernel"],"title_zh":"線性映射的核","stem_zh":"同一精確2×2線性映射T:R²到R²。操作=kernel-membership；資料={\"matrix\":[[1,21],[2,42]],\"vector\":[-21,1]}。請依 [vector,image,in_kernel,boundary] 欄序回答。","answer":{"kind":"expression","value":"([-21, 1], [0, 0], 是, kernel_target_is_zero_vector)","explanation_zh":"Wave5第二層核對=([-21, 1], [0, 0], 是, kernel_target_is_zero_vector)；依完整明示資料重算；kernel只解T(x)=0且使用symbolic full-real certificate；bounded integer search只回答bounded問題；非零fiber不是kernel。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性映射的核"]},"variant":{"family_id":"qf-mod33-kernel-membership","seed":202611022021,"parameters":{"matrix":[[1,21],[2,42]],"vector":[-21,1],"domain":"Exact_R2_matrix_kernel","operation":"kernel-membership","semantic_template":"qf-mod33-kernel-membership/{\"matrix\":[[1,21],[2,42]],\"vector\":[-21,1]}"}},"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-r02-linear-map-kernel-0022","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-kernel"],"title_zh":"線性映射的核","stem_zh":"同一精確2×2線性映射T:R²到R²。操作=kernel-membership；資料={\"matrix\":[[1,22],[2,44]],\"vector\":[-22,1]}。請依 [vector,image,in_kernel,boundary] 欄序回答。","answer":{"kind":"expression","value":"([-22, 1], [0, 0], 是, kernel_target_is_zero_vector)","explanation_zh":"Wave5第二層核對=([-22, 1], [0, 0], 是, kernel_target_is_zero_vector)；依完整明示資料重算；kernel只解T(x)=0且使用symbolic full-real certificate；bounded integer search只回答bounded問題；非零fiber不是kernel。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性映射的核"]},"variant":{"family_id":"qf-mod33-kernel-membership","seed":202611022022,"parameters":{"matrix":[[1,22],[2,44]],"vector":[-22,1],"domain":"Exact_R2_matrix_kernel","operation":"kernel-membership","semantic_template":"qf-mod33-kernel-membership/{\"matrix\":[[1,22],[2,44]],\"vector\":[-22,1]}"}},"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-r02-linear-map-kernel-0023","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-kernel"],"title_zh":"線性映射的核","stem_zh":"同一精確2×2線性映射T:R²到R²。操作=kernel-membership；資料={\"matrix\":[[1,23],[2,46]],\"vector\":[-23,1]}。請依 [vector,image,in_kernel,boundary] 欄序回答。","answer":{"kind":"expression","value":"([-23, 1], [0, 0], 是, kernel_target_is_zero_vector)","explanation_zh":"Wave5第二層核對=([-23, 1], [0, 0], 是, kernel_target_is_zero_vector)；依完整明示資料重算；kernel只解T(x)=0且使用symbolic full-real certificate；bounded integer search只回答bounded問題；非零fiber不是kernel。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性映射的核"]},"variant":{"family_id":"qf-mod33-kernel-membership","seed":202611022023,"parameters":{"matrix":[[1,23],[2,46]],"vector":[-23,1],"domain":"Exact_R2_matrix_kernel","operation":"kernel-membership","semantic_template":"qf-mod33-kernel-membership/{\"matrix\":[[1,23],[2,46]],\"vector\":[-23,1]}"}},"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-r02-linear-map-kernel-0024","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-kernel"],"title_zh":"線性映射的核","stem_zh":"同一精確2×2線性映射T:R²到R²。操作=kernel-membership；資料={\"matrix\":[[1,24],[2,48]],\"vector\":[-24,1]}。請依 [vector,image,in_kernel,boundary] 欄序回答。","answer":{"kind":"expression","value":"([-24, 1], [0, 0], 是, kernel_target_is_zero_vector)","explanation_zh":"Wave5第二層核對=([-24, 1], [0, 0], 是, kernel_target_is_zero_vector)；依完整明示資料重算；kernel只解T(x)=0且使用symbolic full-real certificate；bounded integer search只回答bounded問題；非零fiber不是kernel。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性映射的核"]},"variant":{"family_id":"qf-mod33-kernel-membership","seed":202611022024,"parameters":{"matrix":[[1,24],[2,48]],"vector":[-24,1],"domain":"Exact_R2_matrix_kernel","operation":"kernel-membership","semantic_template":"qf-mod33-kernel-membership/{\"matrix\":[[1,24],[2,48]],\"vector\":[-24,1]}"}},"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-r02-linear-map-kernel-0025","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-kernel"],"title_zh":"線性映射的核","stem_zh":"同一精確2×2線性映射T:R²到R²。操作=kernel-membership；資料={\"matrix\":[[1,25],[2,50]],\"vector\":[-25,1]}。請依 [vector,image,in_kernel,boundary] 欄序回答。","answer":{"kind":"expression","value":"([-25, 1], [0, 0], 是, kernel_target_is_zero_vector)","explanation_zh":"Wave5第二層核對=([-25, 1], [0, 0], 是, kernel_target_is_zero_vector)；依完整明示資料重算；kernel只解T(x)=0且使用symbolic full-real certificate；bounded integer search只回答bounded問題；非零fiber不是kernel。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性映射的核"]},"variant":{"family_id":"qf-mod33-kernel-membership","seed":202611022025,"parameters":{"matrix":[[1,25],[2,50]],"vector":[-25,1],"domain":"Exact_R2_matrix_kernel","operation":"kernel-membership","semantic_template":"qf-mod33-kernel-membership/{\"matrix\":[[1,25],[2,50]],\"vector\":[-25,1]}"}},"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-r02-linear-map-kernel-0026","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-kernel"],"title_zh":"線性映射的核","stem_zh":"同一精確2×2線性映射T:R²到R²。操作=kernel-basis；資料={\"matrix\":[[1,1],[2,2]]}。請依 [matrix,rank,kernel_basis,nullity,certificate] 欄序回答。","answer":{"kind":"expression","value":"([[1, 1], [2, 2]], 1, [[1, -1]], 1, symbolic_full_real_kernel)","explanation_zh":"Wave5第二層核對=([[1, 1], [2, 2]], 1, [[1, -1]], 1, symbolic_full_real_kernel)；依完整明示資料重算；kernel只解T(x)=0且使用symbolic full-real certificate；bounded integer search只回答bounded問題；非零fiber不是kernel。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性映射的核"]},"variant":{"family_id":"qf-mod33-kernel-basis","seed":202611022026,"parameters":{"matrix":[[1,1],[2,2]],"domain":"Exact_R2_matrix_kernel","operation":"kernel-basis","semantic_template":"qf-mod33-kernel-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-r02-linear-map-kernel-0027","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-kernel"],"title_zh":"線性映射的核","stem_zh":"同一精確2×2線性映射T:R²到R²。操作=kernel-basis；資料={\"matrix\":[[1,2],[2,4]]}。請依 [matrix,rank,kernel_basis,nullity,certificate] 欄序回答。","answer":{"kind":"expression","value":"([[1, 2], [2, 4]], 1, [[2, -1]], 1, symbolic_full_real_kernel)","explanation_zh":"Wave5第二層核對=([[1, 2], [2, 4]], 1, [[2, -1]], 1, symbolic_full_real_kernel)；依完整明示資料重算；kernel只解T(x)=0且使用symbolic full-real certificate；bounded integer search只回答bounded問題；非零fiber不是kernel。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性映射的核"]},"variant":{"family_id":"qf-mod33-kernel-basis","seed":202611022027,"parameters":{"matrix":[[1,2],[2,4]],"domain":"Exact_R2_matrix_kernel","operation":"kernel-basis","semantic_template":"qf-mod33-kernel-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-r02-linear-map-kernel-0028","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-kernel"],"title_zh":"線性映射的核","stem_zh":"同一精確2×2線性映射T:R²到R²。操作=kernel-basis；資料={\"matrix\":[[1,3],[2,6]]}。請依 [matrix,rank,kernel_basis,nullity,certificate] 欄序回答。","answer":{"kind":"expression","value":"([[1, 3], [2, 6]], 1, [[3, -1]], 1, symbolic_full_real_kernel)","explanation_zh":"Wave5第二層核對=([[1, 3], [2, 6]], 1, [[3, -1]], 1, symbolic_full_real_kernel)；依完整明示資料重算；kernel只解T(x)=0且使用symbolic full-real certificate；bounded integer search只回答bounded問題；非零fiber不是kernel。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性映射的核"]},"variant":{"family_id":"qf-mod33-kernel-basis","seed":202611022028,"parameters":{"matrix":[[1,3],[2,6]],"domain":"Exact_R2_matrix_kernel","operation":"kernel-basis","semantic_template":"qf-mod33-kernel-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-r02-linear-map-kernel-0029","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-kernel"],"title_zh":"線性映射的核","stem_zh":"同一精確2×2線性映射T:R²到R²。操作=kernel-basis；資料={\"matrix\":[[1,4],[2,8]]}。請依 [matrix,rank,kernel_basis,nullity,certificate] 欄序回答。","answer":{"kind":"expression","value":"([[1, 4], [2, 8]], 1, [[4, -1]], 1, symbolic_full_real_kernel)","explanation_zh":"Wave5第二層核對=([[1, 4], [2, 8]], 1, [[4, -1]], 1, symbolic_full_real_kernel)；依完整明示資料重算；kernel只解T(x)=0且使用symbolic full-real certificate；bounded integer search只回答bounded問題；非零fiber不是kernel。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性映射的核"]},"variant":{"family_id":"qf-mod33-kernel-basis","seed":202611022029,"parameters":{"matrix":[[1,4],[2,8]],"domain":"Exact_R2_matrix_kernel","operation":"kernel-basis","semantic_template":"qf-mod33-kernel-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-r02-linear-map-kernel-0030","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-kernel"],"title_zh":"線性映射的核","stem_zh":"同一精確2×2線性映射T:R²到R²。操作=kernel-basis；資料={\"matrix\":[[1,5],[2,10]]}。請依 [matrix,rank,kernel_basis,nullity,certificate] 欄序回答。","answer":{"kind":"expression","value":"([[1, 5], [2, 10]], 1, [[5, -1]], 1, symbolic_full_real_kernel)","explanation_zh":"Wave5第二層核對=([[1, 5], [2, 10]], 1, [[5, -1]], 1, symbolic_full_real_kernel)；依完整明示資料重算；kernel只解T(x)=0且使用symbolic full-real certificate；bounded integer search只回答bounded問題；非零fiber不是kernel。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性映射的核"]},"variant":{"family_id":"qf-mod33-kernel-basis","seed":202611022030,"parameters":{"matrix":[[1,5],[2,10]],"domain":"Exact_R2_matrix_kernel","operation":"kernel-basis","semantic_template":"qf-mod33-kernel-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-r02-linear-map-kernel-0031","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-kernel"],"title_zh":"線性映射的核","stem_zh":"同一精確2×2線性映射T:R²到R²。操作=kernel-basis；資料={\"matrix\":[[1,6],[2,12]]}。請依 [matrix,rank,kernel_basis,nullity,certificate] 欄序回答。","answer":{"kind":"expression","value":"([[1, 6], [2, 12]], 1, [[6, -1]], 1, symbolic_full_real_kernel)","explanation_zh":"Wave5第二層核對=([[1, 6], [2, 12]], 1, [[6, -1]], 1, symbolic_full_real_kernel)；依完整明示資料重算；kernel只解T(x)=0且使用symbolic full-real certificate；bounded integer search只回答bounded問題；非零fiber不是kernel。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性映射的核"]},"variant":{"family_id":"qf-mod33-kernel-basis","seed":202611022031,"parameters":{"matrix":[[1,6],[2,12]],"domain":"Exact_R2_matrix_kernel","operation":"kernel-basis","semantic_template":"qf-mod33-kernel-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-r02-linear-map-kernel-0032","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-kernel"],"title_zh":"線性映射的核","stem_zh":"同一精確2×2線性映射T:R²到R²。操作=kernel-basis；資料={\"matrix\":[[1,7],[2,14]]}。請依 [matrix,rank,kernel_basis,nullity,certificate] 欄序回答。","answer":{"kind":"expression","value":"([[1, 7], [2, 14]], 1, [[7, -1]], 1, symbolic_full_real_kernel)","explanation_zh":"Wave5第二層核對=([[1, 7], [2, 14]], 1, [[7, -1]], 1, symbolic_full_real_kernel)；依完整明示資料重算；kernel只解T(x)=0且使用symbolic full-real certificate；bounded integer search只回答bounded問題；非零fiber不是kernel。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性映射的核"]},"variant":{"family_id":"qf-mod33-kernel-basis","seed":202611022032,"parameters":{"matrix":[[1,7],[2,14]],"domain":"Exact_R2_matrix_kernel","operation":"kernel-basis","semantic_template":"qf-mod33-kernel-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-r02-linear-map-kernel-0033","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-kernel"],"title_zh":"線性映射的核","stem_zh":"同一精確2×2線性映射T:R²到R²。操作=kernel-basis；資料={\"matrix\":[[1,8],[2,16]]}。請依 [matrix,rank,kernel_basis,nullity,certificate] 欄序回答。","answer":{"kind":"expression","value":"([[1, 8], [2, 16]], 1, [[8, -1]], 1, symbolic_full_real_kernel)","explanation_zh":"Wave5第二層核對=([[1, 8], [2, 16]], 1, [[8, -1]], 1, symbolic_full_real_kernel)；依完整明示資料重算；kernel只解T(x)=0且使用symbolic full-real certificate；bounded integer search只回答bounded問題；非零fiber不是kernel。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性映射的核"]},"variant":{"family_id":"qf-mod33-kernel-basis","seed":202611022033,"parameters":{"matrix":[[1,8],[2,16]],"domain":"Exact_R2_matrix_kernel","operation":"kernel-basis","semantic_template":"qf-mod33-kernel-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-r02-linear-map-kernel-0034","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-kernel"],"title_zh":"線性映射的核","stem_zh":"同一精確2×2線性映射T:R²到R²。操作=kernel-basis；資料={\"matrix\":[[1,9],[2,18]]}。請依 [matrix,rank,kernel_basis,nullity,certificate] 欄序回答。","answer":{"kind":"expression","value":"([[1, 9], [2, 18]], 1, [[9, -1]], 1, symbolic_full_real_kernel)","explanation_zh":"Wave5第二層核對=([[1, 9], [2, 18]], 1, [[9, -1]], 1, symbolic_full_real_kernel)；依完整明示資料重算；kernel只解T(x)=0且使用symbolic full-real certificate；bounded integer search只回答bounded問題；非零fiber不是kernel。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性映射的核"]},"variant":{"family_id":"qf-mod33-kernel-basis","seed":202611022034,"parameters":{"matrix":[[1,9],[2,18]],"domain":"Exact_R2_matrix_kernel","operation":"kernel-basis","semantic_template":"qf-mod33-kernel-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-r02-linear-map-kernel-0035","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-kernel"],"title_zh":"線性映射的核","stem_zh":"同一精確2×2線性映射T:R²到R²。操作=kernel-basis；資料={\"matrix\":[[1,10],[2,20]]}。請依 [matrix,rank,kernel_basis,nullity,certificate] 欄序回答。","answer":{"kind":"expression","value":"([[1, 10], [2, 20]], 1, [[10, -1]], 1, symbolic_full_real_kernel)","explanation_zh":"Wave5第二層核對=([[1, 10], [2, 20]], 1, [[10, -1]], 1, symbolic_full_real_kernel)；依完整明示資料重算；kernel只解T(x)=0且使用symbolic full-real certificate；bounded integer search只回答bounded問題；非零fiber不是kernel。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性映射的核"]},"variant":{"family_id":"qf-mod33-kernel-basis","seed":202611022035,"parameters":{"matrix":[[1,10],[2,20]],"domain":"Exact_R2_matrix_kernel","operation":"kernel-basis","semantic_template":"qf-mod33-kernel-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-r02-linear-map-kernel-0036","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-kernel"],"title_zh":"線性映射的核","stem_zh":"同一精確2×2線性映射T:R²到R²。操作=kernel-basis；資料={\"matrix\":[[1,11],[2,22]]}。請依 [matrix,rank,kernel_basis,nullity,certificate] 欄序回答。","answer":{"kind":"expression","value":"([[1, 11], [2, 22]], 1, [[11, -1]], 1, symbolic_full_real_kernel)","explanation_zh":"Wave5第二層核對=([[1, 11], [2, 22]], 1, [[11, -1]], 1, symbolic_full_real_kernel)；依完整明示資料重算；kernel只解T(x)=0且使用symbolic full-real certificate；bounded integer search只回答bounded問題；非零fiber不是kernel。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性映射的核"]},"variant":{"family_id":"qf-mod33-kernel-basis","seed":202611022036,"parameters":{"matrix":[[1,11],[2,22]],"domain":"Exact_R2_matrix_kernel","operation":"kernel-basis","semantic_template":"qf-mod33-kernel-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-r02-linear-map-kernel-0037","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-kernel"],"title_zh":"線性映射的核","stem_zh":"同一精確2×2線性映射T:R²到R²。操作=kernel-basis；資料={\"matrix\":[[1,12],[2,24]]}。請依 [matrix,rank,kernel_basis,nullity,certificate] 欄序回答。","answer":{"kind":"expression","value":"([[1, 12], [2, 24]], 1, [[12, -1]], 1, symbolic_full_real_kernel)","explanation_zh":"Wave5第二層核對=([[1, 12], [2, 24]], 1, [[12, -1]], 1, symbolic_full_real_kernel)；依完整明示資料重算；kernel只解T(x)=0且使用symbolic full-real certificate；bounded integer search只回答bounded問題；非零fiber不是kernel。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性映射的核"]},"variant":{"family_id":"qf-mod33-kernel-basis","seed":202611022037,"parameters":{"matrix":[[1,12],[2,24]],"domain":"Exact_R2_matrix_kernel","operation":"kernel-basis","semantic_template":"qf-mod33-kernel-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-r02-linear-map-kernel-0038","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-kernel"],"title_zh":"線性映射的核","stem_zh":"同一精確2×2線性映射T:R²到R²。操作=kernel-basis；資料={\"matrix\":[[1,13],[2,26]]}。請依 [matrix,rank,kernel_basis,nullity,certificate] 欄序回答。","answer":{"kind":"expression","value":"([[1, 13], [2, 26]], 1, [[13, -1]], 1, symbolic_full_real_kernel)","explanation_zh":"Wave5第二層核對=([[1, 13], [2, 26]], 1, [[13, -1]], 1, symbolic_full_real_kernel)；依完整明示資料重算；kernel只解T(x)=0且使用symbolic full-real certificate；bounded integer search只回答bounded問題；非零fiber不是kernel。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性映射的核"]},"variant":{"family_id":"qf-mod33-kernel-basis","seed":202611022038,"parameters":{"matrix":[[1,13],[2,26]],"domain":"Exact_R2_matrix_kernel","operation":"kernel-basis","semantic_template":"qf-mod33-kernel-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-r02-linear-map-kernel-0039","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-kernel"],"title_zh":"線性映射的核","stem_zh":"同一精確2×2線性映射T:R²到R²。操作=kernel-basis；資料={\"matrix\":[[1,14],[2,28]]}。請依 [matrix,rank,kernel_basis,nullity,certificate] 欄序回答。","answer":{"kind":"expression","value":"([[1, 14], [2, 28]], 1, [[14, -1]], 1, symbolic_full_real_kernel)","explanation_zh":"Wave5第二層核對=([[1, 14], [2, 28]], 1, [[14, -1]], 1, symbolic_full_real_kernel)；依完整明示資料重算；kernel只解T(x)=0且使用symbolic full-real certificate；bounded integer search只回答bounded問題；非零fiber不是kernel。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性映射的核"]},"variant":{"family_id":"qf-mod33-kernel-basis","seed":202611022039,"parameters":{"matrix":[[1,14],[2,28]],"domain":"Exact_R2_matrix_kernel","operation":"kernel-basis","semantic_template":"qf-mod33-kernel-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-r02-linear-map-kernel-0040","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-kernel"],"title_zh":"線性映射的核","stem_zh":"同一精確2×2線性映射T:R²到R²。操作=kernel-basis；資料={\"matrix\":[[1,15],[2,30]]}。請依 [matrix,rank,kernel_basis,nullity,certificate] 欄序回答。","answer":{"kind":"expression","value":"([[1, 15], [2, 30]], 1, [[15, -1]], 1, symbolic_full_real_kernel)","explanation_zh":"Wave5第二層核對=([[1, 15], [2, 30]], 1, [[15, -1]], 1, symbolic_full_real_kernel)；依完整明示資料重算；kernel只解T(x)=0且使用symbolic full-real certificate；bounded integer search只回答bounded問題；非零fiber不是kernel。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性映射的核"]},"variant":{"family_id":"qf-mod33-kernel-basis","seed":202611022040,"parameters":{"matrix":[[1,15],[2,30]],"domain":"Exact_R2_matrix_kernel","operation":"kernel-basis","semantic_template":"qf-mod33-kernel-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-r02-linear-map-kernel-0041","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-kernel"],"title_zh":"線性映射的核","stem_zh":"同一精確2×2線性映射T:R²到R²。操作=kernel-basis；資料={\"matrix\":[[1,16],[2,32]]}。請依 [matrix,rank,kernel_basis,nullity,certificate] 欄序回答。","answer":{"kind":"expression","value":"([[1, 16], [2, 32]], 1, [[16, -1]], 1, symbolic_full_real_kernel)","explanation_zh":"Wave5第二層核對=([[1, 16], [2, 32]], 1, [[16, -1]], 1, symbolic_full_real_kernel)；依完整明示資料重算；kernel只解T(x)=0且使用symbolic full-real certificate；bounded integer search只回答bounded問題；非零fiber不是kernel。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性映射的核"]},"variant":{"family_id":"qf-mod33-kernel-basis","seed":202611022041,"parameters":{"matrix":[[1,16],[2,32]],"domain":"Exact_R2_matrix_kernel","operation":"kernel-basis","semantic_template":"qf-mod33-kernel-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-r02-linear-map-kernel-0042","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-kernel"],"title_zh":"線性映射的核","stem_zh":"同一精確2×2線性映射T:R²到R²。操作=kernel-basis；資料={\"matrix\":[[1,17],[2,34]]}。請依 [matrix,rank,kernel_basis,nullity,certificate] 欄序回答。","answer":{"kind":"expression","value":"([[1, 17], [2, 34]], 1, [[17, -1]], 1, symbolic_full_real_kernel)","explanation_zh":"Wave5第二層核對=([[1, 17], [2, 34]], 1, [[17, -1]], 1, symbolic_full_real_kernel)；依完整明示資料重算；kernel只解T(x)=0且使用symbolic full-real certificate；bounded integer search只回答bounded問題；非零fiber不是kernel。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性映射的核"]},"variant":{"family_id":"qf-mod33-kernel-basis","seed":202611022042,"parameters":{"matrix":[[1,17],[2,34]],"domain":"Exact_R2_matrix_kernel","operation":"kernel-basis","semantic_template":"qf-mod33-kernel-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-r02-linear-map-kernel-0043","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-kernel"],"title_zh":"線性映射的核","stem_zh":"同一精確2×2線性映射T:R²到R²。操作=kernel-basis；資料={\"matrix\":[[1,18],[2,36]]}。請依 [matrix,rank,kernel_basis,nullity,certificate] 欄序回答。","answer":{"kind":"expression","value":"([[1, 18], [2, 36]], 1, [[18, -1]], 1, symbolic_full_real_kernel)","explanation_zh":"Wave5第二層核對=([[1, 18], [2, 36]], 1, [[18, -1]], 1, symbolic_full_real_kernel)；依完整明示資料重算；kernel只解T(x)=0且使用symbolic full-real certificate；bounded integer search只回答bounded問題；非零fiber不是kernel。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性映射的核"]},"variant":{"family_id":"qf-mod33-kernel-basis","seed":202611022043,"parameters":{"matrix":[[1,18],[2,36]],"domain":"Exact_R2_matrix_kernel","operation":"kernel-basis","semantic_template":"qf-mod33-kernel-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-r02-linear-map-kernel-0044","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-kernel"],"title_zh":"線性映射的核","stem_zh":"同一精確2×2線性映射T:R²到R²。操作=kernel-basis；資料={\"matrix\":[[1,19],[2,38]]}。請依 [matrix,rank,kernel_basis,nullity,certificate] 欄序回答。","answer":{"kind":"expression","value":"([[1, 19], [2, 38]], 1, [[19, -1]], 1, symbolic_full_real_kernel)","explanation_zh":"Wave5第二層核對=([[1, 19], [2, 38]], 1, [[19, -1]], 1, symbolic_full_real_kernel)；依完整明示資料重算；kernel只解T(x)=0且使用symbolic full-real certificate；bounded integer search只回答bounded問題；非零fiber不是kernel。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性映射的核"]},"variant":{"family_id":"qf-mod33-kernel-basis","seed":202611022044,"parameters":{"matrix":[[1,19],[2,38]],"domain":"Exact_R2_matrix_kernel","operation":"kernel-basis","semantic_template":"qf-mod33-kernel-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-r02-linear-map-kernel-0045","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-kernel"],"title_zh":"線性映射的核","stem_zh":"同一精確2×2線性映射T:R²到R²。操作=kernel-basis；資料={\"matrix\":[[1,20],[2,40]]}。請依 [matrix,rank,kernel_basis,nullity,certificate] 欄序回答。","answer":{"kind":"expression","value":"([[1, 20], [2, 40]], 1, [[20, -1]], 1, symbolic_full_real_kernel)","explanation_zh":"Wave5第二層核對=([[1, 20], [2, 40]], 1, [[20, -1]], 1, symbolic_full_real_kernel)；依完整明示資料重算；kernel只解T(x)=0且使用symbolic full-real certificate；bounded integer search只回答bounded問題；非零fiber不是kernel。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性映射的核"]},"variant":{"family_id":"qf-mod33-kernel-basis","seed":202611022045,"parameters":{"matrix":[[1,20],[2,40]],"domain":"Exact_R2_matrix_kernel","operation":"kernel-basis","semantic_template":"qf-mod33-kernel-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-r02-linear-map-kernel-0046","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-kernel"],"title_zh":"線性映射的核","stem_zh":"同一精確2×2線性映射T:R²到R²。操作=kernel-basis；資料={\"matrix\":[[1,21],[2,42]]}。請依 [matrix,rank,kernel_basis,nullity,certificate] 欄序回答。","answer":{"kind":"expression","value":"([[1, 21], [2, 42]], 1, [[21, -1]], 1, symbolic_full_real_kernel)","explanation_zh":"Wave5第二層核對=([[1, 21], [2, 42]], 1, [[21, -1]], 1, symbolic_full_real_kernel)；依完整明示資料重算；kernel只解T(x)=0且使用symbolic full-real certificate；bounded integer search只回答bounded問題；非零fiber不是kernel。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性映射的核"]},"variant":{"family_id":"qf-mod33-kernel-basis","seed":202611022046,"parameters":{"matrix":[[1,21],[2,42]],"domain":"Exact_R2_matrix_kernel","operation":"kernel-basis","semantic_template":"qf-mod33-kernel-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-r02-linear-map-kernel-0047","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-kernel"],"title_zh":"線性映射的核","stem_zh":"同一精確2×2線性映射T:R²到R²。操作=kernel-basis；資料={\"matrix\":[[1,22],[2,44]]}。請依 [matrix,rank,kernel_basis,nullity,certificate] 欄序回答。","answer":{"kind":"expression","value":"([[1, 22], [2, 44]], 1, [[22, -1]], 1, symbolic_full_real_kernel)","explanation_zh":"Wave5第二層核對=([[1, 22], [2, 44]], 1, [[22, -1]], 1, symbolic_full_real_kernel)；依完整明示資料重算；kernel只解T(x)=0且使用symbolic full-real certificate；bounded integer search只回答bounded問題；非零fiber不是kernel。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性映射的核"]},"variant":{"family_id":"qf-mod33-kernel-basis","seed":202611022047,"parameters":{"matrix":[[1,22],[2,44]],"domain":"Exact_R2_matrix_kernel","operation":"kernel-basis","semantic_template":"qf-mod33-kernel-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-r02-linear-map-kernel-0048","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-kernel"],"title_zh":"線性映射的核","stem_zh":"同一精確2×2線性映射T:R²到R²。操作=kernel-basis；資料={\"matrix\":[[1,23],[2,46]]}。請依 [matrix,rank,kernel_basis,nullity,certificate] 欄序回答。","answer":{"kind":"expression","value":"([[1, 23], [2, 46]], 1, [[23, -1]], 1, symbolic_full_real_kernel)","explanation_zh":"Wave5第二層核對=([[1, 23], [2, 46]], 1, [[23, -1]], 1, symbolic_full_real_kernel)；依完整明示資料重算；kernel只解T(x)=0且使用symbolic full-real certificate；bounded integer search只回答bounded問題；非零fiber不是kernel。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性映射的核"]},"variant":{"family_id":"qf-mod33-kernel-basis","seed":202611022048,"parameters":{"matrix":[[1,23],[2,46]],"domain":"Exact_R2_matrix_kernel","operation":"kernel-basis","semantic_template":"qf-mod33-kernel-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-r02-linear-map-kernel-0049","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-kernel"],"title_zh":"線性映射的核","stem_zh":"同一精確2×2線性映射T:R²到R²。操作=kernel-basis；資料={\"matrix\":[[1,24],[2,48]]}。請依 [matrix,rank,kernel_basis,nullity,certificate] 欄序回答。","answer":{"kind":"expression","value":"([[1, 24], [2, 48]], 1, [[24, -1]], 1, symbolic_full_real_kernel)","explanation_zh":"Wave5第二層核對=([[1, 24], [2, 48]], 1, [[24, -1]], 1, symbolic_full_real_kernel)；依完整明示資料重算；kernel只解T(x)=0且使用symbolic full-real certificate；bounded integer search只回答bounded問題；非零fiber不是kernel。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性映射的核"]},"variant":{"family_id":"qf-mod33-kernel-basis","seed":202611022049,"parameters":{"matrix":[[1,24],[2,48]],"domain":"Exact_R2_matrix_kernel","operation":"kernel-basis","semantic_template":"qf-mod33-kernel-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-r02-linear-map-kernel-0050","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-kernel"],"title_zh":"線性映射的核","stem_zh":"同一精確2×2線性映射T:R²到R²。操作=kernel-basis；資料={\"matrix\":[[1,25],[2,50]]}。請依 [matrix,rank,kernel_basis,nullity,certificate] 欄序回答。","answer":{"kind":"expression","value":"([[1, 25], [2, 50]], 1, [[25, -1]], 1, symbolic_full_real_kernel)","explanation_zh":"Wave5第二層核對=([[1, 25], [2, 50]], 1, [[25, -1]], 1, symbolic_full_real_kernel)；依完整明示資料重算；kernel只解T(x)=0且使用symbolic full-real certificate；bounded integer search只回答bounded問題；非零fiber不是kernel。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性映射的核"]},"variant":{"family_id":"qf-mod33-kernel-basis","seed":202611022050,"parameters":{"matrix":[[1,25],[2,50]],"domain":"Exact_R2_matrix_kernel","operation":"kernel-basis","semantic_template":"qf-mod33-kernel-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-r02-linear-map-kernel-0051","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-kernel"],"title_zh":"線性映射的核","stem_zh":"同一精確2×2線性映射T:R²到R²。操作=kernel-nullity；資料={\"matrix\":[[1,1],[2,2]]}。請依 [rank,nullity,kernel_basis] 欄序回答。","answer":{"kind":"expression","value":"(1, 1, [[1, -1]])","explanation_zh":"Wave5第二層核對=(1, 1, [[1, -1]])；依完整明示資料重算；kernel只解T(x)=0且使用symbolic full-real certificate；bounded integer search只回答bounded問題；非零fiber不是kernel。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性映射的核"]},"variant":{"family_id":"qf-mod33-kernel-nullity","seed":202611022051,"parameters":{"matrix":[[1,1],[2,2]],"domain":"Exact_R2_matrix_kernel","operation":"kernel-nullity","semantic_template":"qf-mod33-kernel-nullity/{\"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-r02-linear-map-kernel-0052","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-kernel"],"title_zh":"線性映射的核","stem_zh":"同一精確2×2線性映射T:R²到R²。操作=kernel-nullity；資料={\"matrix\":[[1,2],[2,4]]}。請依 [rank,nullity,kernel_basis] 欄序回答。","answer":{"kind":"expression","value":"(1, 1, [[2, -1]])","explanation_zh":"Wave5第二層核對=(1, 1, [[2, -1]])；依完整明示資料重算；kernel只解T(x)=0且使用symbolic full-real certificate；bounded integer search只回答bounded問題；非零fiber不是kernel。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性映射的核"]},"variant":{"family_id":"qf-mod33-kernel-nullity","seed":202611022052,"parameters":{"matrix":[[1,2],[2,4]],"domain":"Exact_R2_matrix_kernel","operation":"kernel-nullity","semantic_template":"qf-mod33-kernel-nullity/{\"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-r02-linear-map-kernel-0053","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-kernel"],"title_zh":"線性映射的核","stem_zh":"同一精確2×2線性映射T:R²到R²。操作=kernel-nullity；資料={\"matrix\":[[1,3],[2,6]]}。請依 [rank,nullity,kernel_basis] 欄序回答。","answer":{"kind":"expression","value":"(1, 1, [[3, -1]])","explanation_zh":"Wave5第二層核對=(1, 1, [[3, -1]])；依完整明示資料重算；kernel只解T(x)=0且使用symbolic full-real certificate；bounded integer search只回答bounded問題；非零fiber不是kernel。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性映射的核"]},"variant":{"family_id":"qf-mod33-kernel-nullity","seed":202611022053,"parameters":{"matrix":[[1,3],[2,6]],"domain":"Exact_R2_matrix_kernel","operation":"kernel-nullity","semantic_template":"qf-mod33-kernel-nullity/{\"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-r02-linear-map-kernel-0054","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-kernel"],"title_zh":"線性映射的核","stem_zh":"同一精確2×2線性映射T:R²到R²。操作=kernel-nullity；資料={\"matrix\":[[1,4],[2,8]]}。請依 [rank,nullity,kernel_basis] 欄序回答。","answer":{"kind":"expression","value":"(1, 1, [[4, -1]])","explanation_zh":"Wave5第二層核對=(1, 1, [[4, -1]])；依完整明示資料重算；kernel只解T(x)=0且使用symbolic full-real certificate；bounded integer search只回答bounded問題；非零fiber不是kernel。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性映射的核"]},"variant":{"family_id":"qf-mod33-kernel-nullity","seed":202611022054,"parameters":{"matrix":[[1,4],[2,8]],"domain":"Exact_R2_matrix_kernel","operation":"kernel-nullity","semantic_template":"qf-mod33-kernel-nullity/{\"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-r02-linear-map-kernel-0055","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-kernel"],"title_zh":"線性映射的核","stem_zh":"同一精確2×2線性映射T:R²到R²。操作=kernel-nullity；資料={\"matrix\":[[1,5],[2,10]]}。請依 [rank,nullity,kernel_basis] 欄序回答。","answer":{"kind":"expression","value":"(1, 1, [[5, -1]])","explanation_zh":"Wave5第二層核對=(1, 1, [[5, -1]])；依完整明示資料重算；kernel只解T(x)=0且使用symbolic full-real certificate；bounded integer search只回答bounded問題；非零fiber不是kernel。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性映射的核"]},"variant":{"family_id":"qf-mod33-kernel-nullity","seed":202611022055,"parameters":{"matrix":[[1,5],[2,10]],"domain":"Exact_R2_matrix_kernel","operation":"kernel-nullity","semantic_template":"qf-mod33-kernel-nullity/{\"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-r02-linear-map-kernel-0056","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-kernel"],"title_zh":"線性映射的核","stem_zh":"同一精確2×2線性映射T:R²到R²。操作=kernel-nullity；資料={\"matrix\":[[1,6],[2,12]]}。請依 [rank,nullity,kernel_basis] 欄序回答。","answer":{"kind":"expression","value":"(1, 1, [[6, -1]])","explanation_zh":"Wave5第二層核對=(1, 1, [[6, -1]])；依完整明示資料重算；kernel只解T(x)=0且使用symbolic full-real certificate；bounded integer search只回答bounded問題；非零fiber不是kernel。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性映射的核"]},"variant":{"family_id":"qf-mod33-kernel-nullity","seed":202611022056,"parameters":{"matrix":[[1,6],[2,12]],"domain":"Exact_R2_matrix_kernel","operation":"kernel-nullity","semantic_template":"qf-mod33-kernel-nullity/{\"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-r02-linear-map-kernel-0057","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-kernel"],"title_zh":"線性映射的核","stem_zh":"同一精確2×2線性映射T:R²到R²。操作=kernel-nullity；資料={\"matrix\":[[1,7],[2,14]]}。請依 [rank,nullity,kernel_basis] 欄序回答。","answer":{"kind":"expression","value":"(1, 1, [[7, -1]])","explanation_zh":"Wave5第二層核對=(1, 1, [[7, -1]])；依完整明示資料重算；kernel只解T(x)=0且使用symbolic full-real certificate；bounded integer search只回答bounded問題；非零fiber不是kernel。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性映射的核"]},"variant":{"family_id":"qf-mod33-kernel-nullity","seed":202611022057,"parameters":{"matrix":[[1,7],[2,14]],"domain":"Exact_R2_matrix_kernel","operation":"kernel-nullity","semantic_template":"qf-mod33-kernel-nullity/{\"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-r02-linear-map-kernel-0058","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-kernel"],"title_zh":"線性映射的核","stem_zh":"同一精確2×2線性映射T:R²到R²。操作=kernel-nullity；資料={\"matrix\":[[1,8],[2,16]]}。請依 [rank,nullity,kernel_basis] 欄序回答。","answer":{"kind":"expression","value":"(1, 1, [[8, -1]])","explanation_zh":"Wave5第二層核對=(1, 1, [[8, -1]])；依完整明示資料重算；kernel只解T(x)=0且使用symbolic full-real certificate；bounded integer search只回答bounded問題；非零fiber不是kernel。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性映射的核"]},"variant":{"family_id":"qf-mod33-kernel-nullity","seed":202611022058,"parameters":{"matrix":[[1,8],[2,16]],"domain":"Exact_R2_matrix_kernel","operation":"kernel-nullity","semantic_template":"qf-mod33-kernel-nullity/{\"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-r02-linear-map-kernel-0059","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-kernel"],"title_zh":"線性映射的核","stem_zh":"同一精確2×2線性映射T:R²到R²。操作=kernel-nullity；資料={\"matrix\":[[1,9],[2,18]]}。請依 [rank,nullity,kernel_basis] 欄序回答。","answer":{"kind":"expression","value":"(1, 1, [[9, -1]])","explanation_zh":"Wave5第二層核對=(1, 1, [[9, -1]])；依完整明示資料重算；kernel只解T(x)=0且使用symbolic full-real certificate；bounded integer search只回答bounded問題；非零fiber不是kernel。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性映射的核"]},"variant":{"family_id":"qf-mod33-kernel-nullity","seed":202611022059,"parameters":{"matrix":[[1,9],[2,18]],"domain":"Exact_R2_matrix_kernel","operation":"kernel-nullity","semantic_template":"qf-mod33-kernel-nullity/{\"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-r02-linear-map-kernel-0060","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-kernel"],"title_zh":"線性映射的核","stem_zh":"同一精確2×2線性映射T:R²到R²。操作=kernel-nullity；資料={\"matrix\":[[1,10],[2,20]]}。請依 [rank,nullity,kernel_basis] 欄序回答。","answer":{"kind":"expression","value":"(1, 1, [[10, -1]])","explanation_zh":"Wave5第二層核對=(1, 1, [[10, -1]])；依完整明示資料重算；kernel只解T(x)=0且使用symbolic full-real certificate；bounded integer search只回答bounded問題；非零fiber不是kernel。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性映射的核"]},"variant":{"family_id":"qf-mod33-kernel-nullity","seed":202611022060,"parameters":{"matrix":[[1,10],[2,20]],"domain":"Exact_R2_matrix_kernel","operation":"kernel-nullity","semantic_template":"qf-mod33-kernel-nullity/{\"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-r02-linear-map-kernel-0061","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-kernel"],"title_zh":"線性映射的核","stem_zh":"同一精確2×2線性映射T:R²到R²。操作=kernel-nullity；資料={\"matrix\":[[1,11],[2,22]]}。請依 [rank,nullity,kernel_basis] 欄序回答。","answer":{"kind":"expression","value":"(1, 1, [[11, -1]])","explanation_zh":"Wave5第二層核對=(1, 1, [[11, -1]])；依完整明示資料重算；kernel只解T(x)=0且使用symbolic full-real certificate；bounded integer search只回答bounded問題；非零fiber不是kernel。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性映射的核"]},"variant":{"family_id":"qf-mod33-kernel-nullity","seed":202611022061,"parameters":{"matrix":[[1,11],[2,22]],"domain":"Exact_R2_matrix_kernel","operation":"kernel-nullity","semantic_template":"qf-mod33-kernel-nullity/{\"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-r02-linear-map-kernel-0062","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-kernel"],"title_zh":"線性映射的核","stem_zh":"同一精確2×2線性映射T:R²到R²。操作=kernel-nullity；資料={\"matrix\":[[1,12],[2,24]]}。請依 [rank,nullity,kernel_basis] 欄序回答。","answer":{"kind":"expression","value":"(1, 1, [[12, -1]])","explanation_zh":"Wave5第二層核對=(1, 1, [[12, -1]])；依完整明示資料重算；kernel只解T(x)=0且使用symbolic full-real certificate；bounded integer search只回答bounded問題；非零fiber不是kernel。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性映射的核"]},"variant":{"family_id":"qf-mod33-kernel-nullity","seed":202611022062,"parameters":{"matrix":[[1,12],[2,24]],"domain":"Exact_R2_matrix_kernel","operation":"kernel-nullity","semantic_template":"qf-mod33-kernel-nullity/{\"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-r02-linear-map-kernel-0063","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-kernel"],"title_zh":"線性映射的核","stem_zh":"同一精確2×2線性映射T:R²到R²。操作=kernel-nullity；資料={\"matrix\":[[1,13],[2,26]]}。請依 [rank,nullity,kernel_basis] 欄序回答。","answer":{"kind":"expression","value":"(1, 1, [[13, -1]])","explanation_zh":"Wave5第二層核對=(1, 1, [[13, -1]])；依完整明示資料重算；kernel只解T(x)=0且使用symbolic full-real certificate；bounded integer search只回答bounded問題；非零fiber不是kernel。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性映射的核"]},"variant":{"family_id":"qf-mod33-kernel-nullity","seed":202611022063,"parameters":{"matrix":[[1,13],[2,26]],"domain":"Exact_R2_matrix_kernel","operation":"kernel-nullity","semantic_template":"qf-mod33-kernel-nullity/{\"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-r02-linear-map-kernel-0064","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-kernel"],"title_zh":"線性映射的核","stem_zh":"同一精確2×2線性映射T:R²到R²。操作=kernel-nullity；資料={\"matrix\":[[1,14],[2,28]]}。請依 [rank,nullity,kernel_basis] 欄序回答。","answer":{"kind":"expression","value":"(1, 1, [[14, -1]])","explanation_zh":"Wave5第二層核對=(1, 1, [[14, -1]])；依完整明示資料重算；kernel只解T(x)=0且使用symbolic full-real certificate；bounded integer search只回答bounded問題；非零fiber不是kernel。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性映射的核"]},"variant":{"family_id":"qf-mod33-kernel-nullity","seed":202611022064,"parameters":{"matrix":[[1,14],[2,28]],"domain":"Exact_R2_matrix_kernel","operation":"kernel-nullity","semantic_template":"qf-mod33-kernel-nullity/{\"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-r02-linear-map-kernel-0065","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-kernel"],"title_zh":"線性映射的核","stem_zh":"同一精確2×2線性映射T:R²到R²。操作=kernel-nullity；資料={\"matrix\":[[1,15],[2,30]]}。請依 [rank,nullity,kernel_basis] 欄序回答。","answer":{"kind":"expression","value":"(1, 1, [[15, -1]])","explanation_zh":"Wave5第二層核對=(1, 1, [[15, -1]])；依完整明示資料重算；kernel只解T(x)=0且使用symbolic full-real certificate；bounded integer search只回答bounded問題；非零fiber不是kernel。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性映射的核"]},"variant":{"family_id":"qf-mod33-kernel-nullity","seed":202611022065,"parameters":{"matrix":[[1,15],[2,30]],"domain":"Exact_R2_matrix_kernel","operation":"kernel-nullity","semantic_template":"qf-mod33-kernel-nullity/{\"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-r02-linear-map-kernel-0066","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-kernel"],"title_zh":"線性映射的核","stem_zh":"同一精確2×2線性映射T:R²到R²。操作=kernel-nullity；資料={\"matrix\":[[1,16],[2,32]]}。請依 [rank,nullity,kernel_basis] 欄序回答。","answer":{"kind":"expression","value":"(1, 1, [[16, -1]])","explanation_zh":"Wave5第二層核對=(1, 1, [[16, -1]])；依完整明示資料重算；kernel只解T(x)=0且使用symbolic full-real certificate；bounded integer search只回答bounded問題；非零fiber不是kernel。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性映射的核"]},"variant":{"family_id":"qf-mod33-kernel-nullity","seed":202611022066,"parameters":{"matrix":[[1,16],[2,32]],"domain":"Exact_R2_matrix_kernel","operation":"kernel-nullity","semantic_template":"qf-mod33-kernel-nullity/{\"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-r02-linear-map-kernel-0067","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-kernel"],"title_zh":"線性映射的核","stem_zh":"同一精確2×2線性映射T:R²到R²。操作=kernel-nullity；資料={\"matrix\":[[1,17],[2,34]]}。請依 [rank,nullity,kernel_basis] 欄序回答。","answer":{"kind":"expression","value":"(1, 1, [[17, -1]])","explanation_zh":"Wave5第二層核對=(1, 1, [[17, -1]])；依完整明示資料重算；kernel只解T(x)=0且使用symbolic full-real certificate；bounded integer search只回答bounded問題；非零fiber不是kernel。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性映射的核"]},"variant":{"family_id":"qf-mod33-kernel-nullity","seed":202611022067,"parameters":{"matrix":[[1,17],[2,34]],"domain":"Exact_R2_matrix_kernel","operation":"kernel-nullity","semantic_template":"qf-mod33-kernel-nullity/{\"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-r02-linear-map-kernel-0068","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-kernel"],"title_zh":"線性映射的核","stem_zh":"同一精確2×2線性映射T:R²到R²。操作=kernel-nullity；資料={\"matrix\":[[1,18],[2,36]]}。請依 [rank,nullity,kernel_basis] 欄序回答。","answer":{"kind":"expression","value":"(1, 1, [[18, -1]])","explanation_zh":"Wave5第二層核對=(1, 1, [[18, -1]])；依完整明示資料重算；kernel只解T(x)=0且使用symbolic full-real certificate；bounded integer search只回答bounded問題；非零fiber不是kernel。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性映射的核"]},"variant":{"family_id":"qf-mod33-kernel-nullity","seed":202611022068,"parameters":{"matrix":[[1,18],[2,36]],"domain":"Exact_R2_matrix_kernel","operation":"kernel-nullity","semantic_template":"qf-mod33-kernel-nullity/{\"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-r02-linear-map-kernel-0069","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-kernel"],"title_zh":"線性映射的核","stem_zh":"同一精確2×2線性映射T:R²到R²。操作=kernel-nullity；資料={\"matrix\":[[1,19],[2,38]]}。請依 [rank,nullity,kernel_basis] 欄序回答。","answer":{"kind":"expression","value":"(1, 1, [[19, -1]])","explanation_zh":"Wave5第二層核對=(1, 1, [[19, -1]])；依完整明示資料重算；kernel只解T(x)=0且使用symbolic full-real certificate；bounded integer search只回答bounded問題；非零fiber不是kernel。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性映射的核"]},"variant":{"family_id":"qf-mod33-kernel-nullity","seed":202611022069,"parameters":{"matrix":[[1,19],[2,38]],"domain":"Exact_R2_matrix_kernel","operation":"kernel-nullity","semantic_template":"qf-mod33-kernel-nullity/{\"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-r02-linear-map-kernel-0070","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-kernel"],"title_zh":"線性映射的核","stem_zh":"同一精確2×2線性映射T:R²到R²。操作=kernel-nullity；資料={\"matrix\":[[1,20],[2,40]]}。請依 [rank,nullity,kernel_basis] 欄序回答。","answer":{"kind":"expression","value":"(1, 1, [[20, -1]])","explanation_zh":"Wave5第二層核對=(1, 1, [[20, -1]])；依完整明示資料重算；kernel只解T(x)=0且使用symbolic full-real certificate；bounded integer search只回答bounded問題；非零fiber不是kernel。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性映射的核"]},"variant":{"family_id":"qf-mod33-kernel-nullity","seed":202611022070,"parameters":{"matrix":[[1,20],[2,40]],"domain":"Exact_R2_matrix_kernel","operation":"kernel-nullity","semantic_template":"qf-mod33-kernel-nullity/{\"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-r02-linear-map-kernel-0071","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-kernel"],"title_zh":"線性映射的核","stem_zh":"同一精確2×2線性映射T:R²到R²。操作=kernel-nullity；資料={\"matrix\":[[1,21],[2,42]]}。請依 [rank,nullity,kernel_basis] 欄序回答。","answer":{"kind":"expression","value":"(1, 1, [[21, -1]])","explanation_zh":"Wave5第二層核對=(1, 1, [[21, -1]])；依完整明示資料重算；kernel只解T(x)=0且使用symbolic full-real certificate；bounded integer search只回答bounded問題；非零fiber不是kernel。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性映射的核"]},"variant":{"family_id":"qf-mod33-kernel-nullity","seed":202611022071,"parameters":{"matrix":[[1,21],[2,42]],"domain":"Exact_R2_matrix_kernel","operation":"kernel-nullity","semantic_template":"qf-mod33-kernel-nullity/{\"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-r02-linear-map-kernel-0072","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-kernel"],"title_zh":"線性映射的核","stem_zh":"同一精確2×2線性映射T:R²到R²。操作=kernel-nullity；資料={\"matrix\":[[1,22],[2,44]]}。請依 [rank,nullity,kernel_basis] 欄序回答。","answer":{"kind":"expression","value":"(1, 1, [[22, -1]])","explanation_zh":"Wave5第二層核對=(1, 1, [[22, -1]])；依完整明示資料重算；kernel只解T(x)=0且使用symbolic full-real certificate；bounded integer search只回答bounded問題；非零fiber不是kernel。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性映射的核"]},"variant":{"family_id":"qf-mod33-kernel-nullity","seed":202611022072,"parameters":{"matrix":[[1,22],[2,44]],"domain":"Exact_R2_matrix_kernel","operation":"kernel-nullity","semantic_template":"qf-mod33-kernel-nullity/{\"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-r02-linear-map-kernel-0073","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-kernel"],"title_zh":"線性映射的核","stem_zh":"同一精確2×2線性映射T:R²到R²。操作=kernel-nullity；資料={\"matrix\":[[1,23],[2,46]]}。請依 [rank,nullity,kernel_basis] 欄序回答。","answer":{"kind":"expression","value":"(1, 1, [[23, -1]])","explanation_zh":"Wave5第二層核對=(1, 1, [[23, -1]])；依完整明示資料重算；kernel只解T(x)=0且使用symbolic full-real certificate；bounded integer search只回答bounded問題；非零fiber不是kernel。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性映射的核"]},"variant":{"family_id":"qf-mod33-kernel-nullity","seed":202611022073,"parameters":{"matrix":[[1,23],[2,46]],"domain":"Exact_R2_matrix_kernel","operation":"kernel-nullity","semantic_template":"qf-mod33-kernel-nullity/{\"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-r02-linear-map-kernel-0074","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-kernel"],"title_zh":"線性映射的核","stem_zh":"同一精確2×2線性映射T:R²到R²。操作=kernel-nullity；資料={\"matrix\":[[1,24],[2,48]]}。請依 [rank,nullity,kernel_basis] 欄序回答。","answer":{"kind":"expression","value":"(1, 1, [[24, -1]])","explanation_zh":"Wave5第二層核對=(1, 1, [[24, -1]])；依完整明示資料重算；kernel只解T(x)=0且使用symbolic full-real certificate；bounded integer search只回答bounded問題；非零fiber不是kernel。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性映射的核"]},"variant":{"family_id":"qf-mod33-kernel-nullity","seed":202611022074,"parameters":{"matrix":[[1,24],[2,48]],"domain":"Exact_R2_matrix_kernel","operation":"kernel-nullity","semantic_template":"qf-mod33-kernel-nullity/{\"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-r02-linear-map-kernel-0075","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-kernel"],"title_zh":"線性映射的核","stem_zh":"同一精確2×2線性映射T:R²到R²。操作=kernel-nullity；資料={\"matrix\":[[1,25],[2,50]]}。請依 [rank,nullity,kernel_basis] 欄序回答。","answer":{"kind":"expression","value":"(1, 1, [[25, -1]])","explanation_zh":"Wave5第二層核對=(1, 1, [[25, -1]])；依完整明示資料重算；kernel只解T(x)=0且使用symbolic full-real certificate；bounded integer search只回答bounded問題；非零fiber不是kernel。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性映射的核"]},"variant":{"family_id":"qf-mod33-kernel-nullity","seed":202611022075,"parameters":{"matrix":[[1,25],[2,50]],"domain":"Exact_R2_matrix_kernel","operation":"kernel-nullity","semantic_template":"qf-mod33-kernel-nullity/{\"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-r02-linear-map-kernel-0076","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-kernel"],"title_zh":"線性映射的核","stem_zh":"同一精確2×2線性映射T:R²到R²。操作=kernel-bounded-search-boundary；資料={\"matrix\":[[1,1],[2,2]],\"bound\":1}。請依 [bound,integer_kernel_points,global_kernel_basis,complete,boundary] 欄序回答。","answer":{"kind":"expression","value":"(1, [[-1, 1], [0, 0], [1, -1]], [[1, -1]], 否, bounded_integer_search_not_full_real_kernel)","explanation_zh":"Wave5第二層核對=(1, [[-1, 1], [0, 0], [1, -1]], [[1, -1]], 否, bounded_integer_search_not_full_real_kernel)；依完整明示資料重算；kernel只解T(x)=0且使用symbolic full-real certificate；bounded integer search只回答bounded問題；非零fiber不是kernel。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性映射的核"]},"variant":{"family_id":"qf-mod33-kernel-bounded-search-boundary","seed":202611022076,"parameters":{"matrix":[[1,1],[2,2]],"bound":1,"domain":"Exact_R2_matrix_kernel","operation":"kernel-bounded-search-boundary","semantic_template":"qf-mod33-kernel-bounded-search-boundary/{\"bound\":1,\"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-r02-linear-map-kernel-0077","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-kernel"],"title_zh":"線性映射的核","stem_zh":"同一精確2×2線性映射T:R²到R²。操作=kernel-bounded-search-boundary；資料={\"matrix\":[[1,2],[2,4]],\"bound\":2}。請依 [bound,integer_kernel_points,global_kernel_basis,complete,boundary] 欄序回答。","answer":{"kind":"expression","value":"(2, [[-2, 1], [0, 0], [2, -1]], [[2, -1]], 否, bounded_integer_search_not_full_real_kernel)","explanation_zh":"Wave5第二層核對=(2, [[-2, 1], [0, 0], [2, -1]], [[2, -1]], 否, bounded_integer_search_not_full_real_kernel)；依完整明示資料重算；kernel只解T(x)=0且使用symbolic full-real certificate；bounded integer search只回答bounded問題；非零fiber不是kernel。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性映射的核"]},"variant":{"family_id":"qf-mod33-kernel-bounded-search-boundary","seed":202611022077,"parameters":{"matrix":[[1,2],[2,4]],"bound":2,"domain":"Exact_R2_matrix_kernel","operation":"kernel-bounded-search-boundary","semantic_template":"qf-mod33-kernel-bounded-search-boundary/{\"bound\":2,\"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-r02-linear-map-kernel-0078","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-kernel"],"title_zh":"線性映射的核","stem_zh":"同一精確2×2線性映射T:R²到R²。操作=kernel-bounded-search-boundary；資料={\"matrix\":[[1,3],[2,6]],\"bound\":3}。請依 [bound,integer_kernel_points,global_kernel_basis,complete,boundary] 欄序回答。","answer":{"kind":"expression","value":"(3, [[-3, 1], [0, 0], [3, -1]], [[3, -1]], 否, bounded_integer_search_not_full_real_kernel)","explanation_zh":"Wave5第二層核對=(3, [[-3, 1], [0, 0], [3, -1]], [[3, -1]], 否, bounded_integer_search_not_full_real_kernel)；依完整明示資料重算；kernel只解T(x)=0且使用symbolic full-real certificate；bounded integer search只回答bounded問題；非零fiber不是kernel。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性映射的核"]},"variant":{"family_id":"qf-mod33-kernel-bounded-search-boundary","seed":202611022078,"parameters":{"matrix":[[1,3],[2,6]],"bound":3,"domain":"Exact_R2_matrix_kernel","operation":"kernel-bounded-search-boundary","semantic_template":"qf-mod33-kernel-bounded-search-boundary/{\"bound\":3,\"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-r02-linear-map-kernel-0079","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-kernel"],"title_zh":"線性映射的核","stem_zh":"同一精確2×2線性映射T:R²到R²。操作=kernel-bounded-search-boundary；資料={\"matrix\":[[1,4],[2,8]],\"bound\":4}。請依 [bound,integer_kernel_points,global_kernel_basis,complete,boundary] 欄序回答。","answer":{"kind":"expression","value":"(4, [[-4, 1], [0, 0], [4, -1]], [[4, -1]], 否, bounded_integer_search_not_full_real_kernel)","explanation_zh":"Wave5第二層核對=(4, [[-4, 1], [0, 0], [4, -1]], [[4, -1]], 否, bounded_integer_search_not_full_real_kernel)；依完整明示資料重算；kernel只解T(x)=0且使用symbolic full-real certificate；bounded integer search只回答bounded問題；非零fiber不是kernel。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性映射的核"]},"variant":{"family_id":"qf-mod33-kernel-bounded-search-boundary","seed":202611022079,"parameters":{"matrix":[[1,4],[2,8]],"bound":4,"domain":"Exact_R2_matrix_kernel","operation":"kernel-bounded-search-boundary","semantic_template":"qf-mod33-kernel-bounded-search-boundary/{\"bound\":4,\"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-r02-linear-map-kernel-0080","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-kernel"],"title_zh":"線性映射的核","stem_zh":"同一精確2×2線性映射T:R²到R²。操作=kernel-bounded-search-boundary；資料={\"matrix\":[[1,5],[2,10]],\"bound\":5}。請依 [bound,integer_kernel_points,global_kernel_basis,complete,boundary] 欄序回答。","answer":{"kind":"expression","value":"(5, [[-5, 1], [0, 0], [5, -1]], [[5, -1]], 否, bounded_integer_search_not_full_real_kernel)","explanation_zh":"Wave5第二層核對=(5, [[-5, 1], [0, 0], [5, -1]], [[5, -1]], 否, bounded_integer_search_not_full_real_kernel)；依完整明示資料重算；kernel只解T(x)=0且使用symbolic full-real certificate；bounded integer search只回答bounded問題；非零fiber不是kernel。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性映射的核"]},"variant":{"family_id":"qf-mod33-kernel-bounded-search-boundary","seed":202611022080,"parameters":{"matrix":[[1,5],[2,10]],"bound":5,"domain":"Exact_R2_matrix_kernel","operation":"kernel-bounded-search-boundary","semantic_template":"qf-mod33-kernel-bounded-search-boundary/{\"bound\":5,\"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-r02-linear-map-kernel-0081","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-kernel"],"title_zh":"線性映射的核","stem_zh":"同一精確2×2線性映射T:R²到R²。操作=kernel-bounded-search-boundary；資料={\"matrix\":[[1,6],[2,12]],\"bound\":1}。請依 [bound,integer_kernel_points,global_kernel_basis,complete,boundary] 欄序回答。","answer":{"kind":"expression","value":"(1, [[0, 0]], [[6, -1]], 否, bounded_integer_search_not_full_real_kernel)","explanation_zh":"Wave5第二層核對=(1, [[0, 0]], [[6, -1]], 否, bounded_integer_search_not_full_real_kernel)；依完整明示資料重算；kernel只解T(x)=0且使用symbolic full-real certificate；bounded integer search只回答bounded問題；非零fiber不是kernel。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性映射的核"]},"variant":{"family_id":"qf-mod33-kernel-bounded-search-boundary","seed":202611022081,"parameters":{"matrix":[[1,6],[2,12]],"bound":1,"domain":"Exact_R2_matrix_kernel","operation":"kernel-bounded-search-boundary","semantic_template":"qf-mod33-kernel-bounded-search-boundary/{\"bound\":1,\"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-r02-linear-map-kernel-0082","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-kernel"],"title_zh":"線性映射的核","stem_zh":"同一精確2×2線性映射T:R²到R²。操作=kernel-bounded-search-boundary；資料={\"matrix\":[[1,7],[2,14]],\"bound\":2}。請依 [bound,integer_kernel_points,global_kernel_basis,complete,boundary] 欄序回答。","answer":{"kind":"expression","value":"(2, [[0, 0]], [[7, -1]], 否, bounded_integer_search_not_full_real_kernel)","explanation_zh":"Wave5第二層核對=(2, [[0, 0]], [[7, -1]], 否, bounded_integer_search_not_full_real_kernel)；依完整明示資料重算；kernel只解T(x)=0且使用symbolic full-real certificate；bounded integer search只回答bounded問題；非零fiber不是kernel。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性映射的核"]},"variant":{"family_id":"qf-mod33-kernel-bounded-search-boundary","seed":202611022082,"parameters":{"matrix":[[1,7],[2,14]],"bound":2,"domain":"Exact_R2_matrix_kernel","operation":"kernel-bounded-search-boundary","semantic_template":"qf-mod33-kernel-bounded-search-boundary/{\"bound\":2,\"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-r02-linear-map-kernel-0083","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-kernel"],"title_zh":"線性映射的核","stem_zh":"同一精確2×2線性映射T:R²到R²。操作=kernel-bounded-search-boundary；資料={\"matrix\":[[1,8],[2,16]],\"bound\":3}。請依 [bound,integer_kernel_points,global_kernel_basis,complete,boundary] 欄序回答。","answer":{"kind":"expression","value":"(3, [[0, 0]], [[8, -1]], 否, bounded_integer_search_not_full_real_kernel)","explanation_zh":"Wave5第二層核對=(3, [[0, 0]], [[8, -1]], 否, bounded_integer_search_not_full_real_kernel)；依完整明示資料重算；kernel只解T(x)=0且使用symbolic full-real certificate；bounded integer search只回答bounded問題；非零fiber不是kernel。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性映射的核"]},"variant":{"family_id":"qf-mod33-kernel-bounded-search-boundary","seed":202611022083,"parameters":{"matrix":[[1,8],[2,16]],"bound":3,"domain":"Exact_R2_matrix_kernel","operation":"kernel-bounded-search-boundary","semantic_template":"qf-mod33-kernel-bounded-search-boundary/{\"bound\":3,\"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-r02-linear-map-kernel-0084","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-kernel"],"title_zh":"線性映射的核","stem_zh":"同一精確2×2線性映射T:R²到R²。操作=kernel-bounded-search-boundary；資料={\"matrix\":[[1,9],[2,18]],\"bound\":4}。請依 [bound,integer_kernel_points,global_kernel_basis,complete,boundary] 欄序回答。","answer":{"kind":"expression","value":"(4, [[0, 0]], [[9, -1]], 否, bounded_integer_search_not_full_real_kernel)","explanation_zh":"Wave5第二層核對=(4, [[0, 0]], [[9, -1]], 否, bounded_integer_search_not_full_real_kernel)；依完整明示資料重算；kernel只解T(x)=0且使用symbolic full-real certificate；bounded integer search只回答bounded問題；非零fiber不是kernel。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性映射的核"]},"variant":{"family_id":"qf-mod33-kernel-bounded-search-boundary","seed":202611022084,"parameters":{"matrix":[[1,9],[2,18]],"bound":4,"domain":"Exact_R2_matrix_kernel","operation":"kernel-bounded-search-boundary","semantic_template":"qf-mod33-kernel-bounded-search-boundary/{\"bound\":4,\"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-r02-linear-map-kernel-0085","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-kernel"],"title_zh":"線性映射的核","stem_zh":"同一精確2×2線性映射T:R²到R²。操作=kernel-bounded-search-boundary；資料={\"matrix\":[[1,10],[2,20]],\"bound\":5}。請依 [bound,integer_kernel_points,global_kernel_basis,complete,boundary] 欄序回答。","answer":{"kind":"expression","value":"(5, [[0, 0]], [[10, -1]], 否, bounded_integer_search_not_full_real_kernel)","explanation_zh":"Wave5第二層核對=(5, [[0, 0]], [[10, -1]], 否, bounded_integer_search_not_full_real_kernel)；依完整明示資料重算；kernel只解T(x)=0且使用symbolic full-real certificate；bounded integer search只回答bounded問題；非零fiber不是kernel。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性映射的核"]},"variant":{"family_id":"qf-mod33-kernel-bounded-search-boundary","seed":202611022085,"parameters":{"matrix":[[1,10],[2,20]],"bound":5,"domain":"Exact_R2_matrix_kernel","operation":"kernel-bounded-search-boundary","semantic_template":"qf-mod33-kernel-bounded-search-boundary/{\"bound\":5,\"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-r02-linear-map-kernel-0086","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-kernel"],"title_zh":"線性映射的核","stem_zh":"同一精確2×2線性映射T:R²到R²。操作=kernel-bounded-search-boundary；資料={\"matrix\":[[1,11],[2,22]],\"bound\":1}。請依 [bound,integer_kernel_points,global_kernel_basis,complete,boundary] 欄序回答。","answer":{"kind":"expression","value":"(1, [[0, 0]], [[11, -1]], 否, bounded_integer_search_not_full_real_kernel)","explanation_zh":"Wave5第二層核對=(1, [[0, 0]], [[11, -1]], 否, bounded_integer_search_not_full_real_kernel)；依完整明示資料重算；kernel只解T(x)=0且使用symbolic full-real certificate；bounded integer search只回答bounded問題；非零fiber不是kernel。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性映射的核"]},"variant":{"family_id":"qf-mod33-kernel-bounded-search-boundary","seed":202611022086,"parameters":{"matrix":[[1,11],[2,22]],"bound":1,"domain":"Exact_R2_matrix_kernel","operation":"kernel-bounded-search-boundary","semantic_template":"qf-mod33-kernel-bounded-search-boundary/{\"bound\":1,\"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-r02-linear-map-kernel-0087","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-kernel"],"title_zh":"線性映射的核","stem_zh":"同一精確2×2線性映射T:R²到R²。操作=kernel-bounded-search-boundary；資料={\"matrix\":[[1,12],[2,24]],\"bound\":2}。請依 [bound,integer_kernel_points,global_kernel_basis,complete,boundary] 欄序回答。","answer":{"kind":"expression","value":"(2, [[0, 0]], [[12, -1]], 否, bounded_integer_search_not_full_real_kernel)","explanation_zh":"Wave5第二層核對=(2, [[0, 0]], [[12, -1]], 否, bounded_integer_search_not_full_real_kernel)；依完整明示資料重算；kernel只解T(x)=0且使用symbolic full-real certificate；bounded integer search只回答bounded問題；非零fiber不是kernel。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性映射的核"]},"variant":{"family_id":"qf-mod33-kernel-bounded-search-boundary","seed":202611022087,"parameters":{"matrix":[[1,12],[2,24]],"bound":2,"domain":"Exact_R2_matrix_kernel","operation":"kernel-bounded-search-boundary","semantic_template":"qf-mod33-kernel-bounded-search-boundary/{\"bound\":2,\"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-r02-linear-map-kernel-0088","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-kernel"],"title_zh":"線性映射的核","stem_zh":"同一精確2×2線性映射T:R²到R²。操作=kernel-bounded-search-boundary；資料={\"matrix\":[[1,13],[2,26]],\"bound\":3}。請依 [bound,integer_kernel_points,global_kernel_basis,complete,boundary] 欄序回答。","answer":{"kind":"expression","value":"(3, [[0, 0]], [[13, -1]], 否, bounded_integer_search_not_full_real_kernel)","explanation_zh":"Wave5第二層核對=(3, [[0, 0]], [[13, -1]], 否, bounded_integer_search_not_full_real_kernel)；依完整明示資料重算；kernel只解T(x)=0且使用symbolic full-real certificate；bounded integer search只回答bounded問題；非零fiber不是kernel。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性映射的核"]},"variant":{"family_id":"qf-mod33-kernel-bounded-search-boundary","seed":202611022088,"parameters":{"matrix":[[1,13],[2,26]],"bound":3,"domain":"Exact_R2_matrix_kernel","operation":"kernel-bounded-search-boundary","semantic_template":"qf-mod33-kernel-bounded-search-boundary/{\"bound\":3,\"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-r02-linear-map-kernel-0089","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-kernel"],"title_zh":"線性映射的核","stem_zh":"同一精確2×2線性映射T:R²到R²。操作=kernel-bounded-search-boundary；資料={\"matrix\":[[1,14],[2,28]],\"bound\":4}。請依 [bound,integer_kernel_points,global_kernel_basis,complete,boundary] 欄序回答。","answer":{"kind":"expression","value":"(4, [[0, 0]], [[14, -1]], 否, bounded_integer_search_not_full_real_kernel)","explanation_zh":"Wave5第二層核對=(4, [[0, 0]], [[14, -1]], 否, bounded_integer_search_not_full_real_kernel)；依完整明示資料重算；kernel只解T(x)=0且使用symbolic full-real certificate；bounded integer search只回答bounded問題；非零fiber不是kernel。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性映射的核"]},"variant":{"family_id":"qf-mod33-kernel-bounded-search-boundary","seed":202611022089,"parameters":{"matrix":[[1,14],[2,28]],"bound":4,"domain":"Exact_R2_matrix_kernel","operation":"kernel-bounded-search-boundary","semantic_template":"qf-mod33-kernel-bounded-search-boundary/{\"bound\":4,\"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-r02-linear-map-kernel-0090","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-kernel"],"title_zh":"線性映射的核","stem_zh":"同一精確2×2線性映射T:R²到R²。操作=kernel-bounded-search-boundary；資料={\"matrix\":[[1,15],[2,30]],\"bound\":5}。請依 [bound,integer_kernel_points,global_kernel_basis,complete,boundary] 欄序回答。","answer":{"kind":"expression","value":"(5, [[0, 0]], [[15, -1]], 否, bounded_integer_search_not_full_real_kernel)","explanation_zh":"Wave5第二層核對=(5, [[0, 0]], [[15, -1]], 否, bounded_integer_search_not_full_real_kernel)；依完整明示資料重算；kernel只解T(x)=0且使用symbolic full-real certificate；bounded integer search只回答bounded問題；非零fiber不是kernel。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性映射的核"]},"variant":{"family_id":"qf-mod33-kernel-bounded-search-boundary","seed":202611022090,"parameters":{"matrix":[[1,15],[2,30]],"bound":5,"domain":"Exact_R2_matrix_kernel","operation":"kernel-bounded-search-boundary","semantic_template":"qf-mod33-kernel-bounded-search-boundary/{\"bound\":5,\"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-r02-linear-map-kernel-0091","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-kernel"],"title_zh":"線性映射的核","stem_zh":"同一精確2×2線性映射T:R²到R²。操作=kernel-bounded-search-boundary；資料={\"matrix\":[[1,16],[2,32]],\"bound\":1}。請依 [bound,integer_kernel_points,global_kernel_basis,complete,boundary] 欄序回答。","answer":{"kind":"expression","value":"(1, [[0, 0]], [[16, -1]], 否, bounded_integer_search_not_full_real_kernel)","explanation_zh":"Wave5第二層核對=(1, [[0, 0]], [[16, -1]], 否, bounded_integer_search_not_full_real_kernel)；依完整明示資料重算；kernel只解T(x)=0且使用symbolic full-real certificate；bounded integer search只回答bounded問題；非零fiber不是kernel。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性映射的核"]},"variant":{"family_id":"qf-mod33-kernel-bounded-search-boundary","seed":202611022091,"parameters":{"matrix":[[1,16],[2,32]],"bound":1,"domain":"Exact_R2_matrix_kernel","operation":"kernel-bounded-search-boundary","semantic_template":"qf-mod33-kernel-bounded-search-boundary/{\"bound\":1,\"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-r02-linear-map-kernel-0092","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-kernel"],"title_zh":"線性映射的核","stem_zh":"同一精確2×2線性映射T:R²到R²。操作=kernel-bounded-search-boundary；資料={\"matrix\":[[1,17],[2,34]],\"bound\":2}。請依 [bound,integer_kernel_points,global_kernel_basis,complete,boundary] 欄序回答。","answer":{"kind":"expression","value":"(2, [[0, 0]], [[17, -1]], 否, bounded_integer_search_not_full_real_kernel)","explanation_zh":"Wave5第二層核對=(2, [[0, 0]], [[17, -1]], 否, bounded_integer_search_not_full_real_kernel)；依完整明示資料重算；kernel只解T(x)=0且使用symbolic full-real certificate；bounded integer search只回答bounded問題；非零fiber不是kernel。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性映射的核"]},"variant":{"family_id":"qf-mod33-kernel-bounded-search-boundary","seed":202611022092,"parameters":{"matrix":[[1,17],[2,34]],"bound":2,"domain":"Exact_R2_matrix_kernel","operation":"kernel-bounded-search-boundary","semantic_template":"qf-mod33-kernel-bounded-search-boundary/{\"bound\":2,\"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-r02-linear-map-kernel-0093","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-kernel"],"title_zh":"線性映射的核","stem_zh":"同一精確2×2線性映射T:R²到R²。操作=kernel-bounded-search-boundary；資料={\"matrix\":[[1,18],[2,36]],\"bound\":3}。請依 [bound,integer_kernel_points,global_kernel_basis,complete,boundary] 欄序回答。","answer":{"kind":"expression","value":"(3, [[0, 0]], [[18, -1]], 否, bounded_integer_search_not_full_real_kernel)","explanation_zh":"Wave5第二層核對=(3, [[0, 0]], [[18, -1]], 否, bounded_integer_search_not_full_real_kernel)；依完整明示資料重算；kernel只解T(x)=0且使用symbolic full-real certificate；bounded integer search只回答bounded問題；非零fiber不是kernel。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性映射的核"]},"variant":{"family_id":"qf-mod33-kernel-bounded-search-boundary","seed":202611022093,"parameters":{"matrix":[[1,18],[2,36]],"bound":3,"domain":"Exact_R2_matrix_kernel","operation":"kernel-bounded-search-boundary","semantic_template":"qf-mod33-kernel-bounded-search-boundary/{\"bound\":3,\"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-r02-linear-map-kernel-0094","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-kernel"],"title_zh":"線性映射的核","stem_zh":"同一精確2×2線性映射T:R²到R²。操作=kernel-bounded-search-boundary；資料={\"matrix\":[[1,19],[2,38]],\"bound\":4}。請依 [bound,integer_kernel_points,global_kernel_basis,complete,boundary] 欄序回答。","answer":{"kind":"expression","value":"(4, [[0, 0]], [[19, -1]], 否, bounded_integer_search_not_full_real_kernel)","explanation_zh":"Wave5第二層核對=(4, [[0, 0]], [[19, -1]], 否, bounded_integer_search_not_full_real_kernel)；依完整明示資料重算；kernel只解T(x)=0且使用symbolic full-real certificate；bounded integer search只回答bounded問題；非零fiber不是kernel。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性映射的核"]},"variant":{"family_id":"qf-mod33-kernel-bounded-search-boundary","seed":202611022094,"parameters":{"matrix":[[1,19],[2,38]],"bound":4,"domain":"Exact_R2_matrix_kernel","operation":"kernel-bounded-search-boundary","semantic_template":"qf-mod33-kernel-bounded-search-boundary/{\"bound\":4,\"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-r02-linear-map-kernel-0095","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-kernel"],"title_zh":"線性映射的核","stem_zh":"同一精確2×2線性映射T:R²到R²。操作=kernel-bounded-search-boundary；資料={\"matrix\":[[1,20],[2,40]],\"bound\":5}。請依 [bound,integer_kernel_points,global_kernel_basis,complete,boundary] 欄序回答。","answer":{"kind":"expression","value":"(5, [[0, 0]], [[20, -1]], 否, bounded_integer_search_not_full_real_kernel)","explanation_zh":"Wave5第二層核對=(5, [[0, 0]], [[20, -1]], 否, bounded_integer_search_not_full_real_kernel)；依完整明示資料重算；kernel只解T(x)=0且使用symbolic full-real certificate；bounded integer search只回答bounded問題；非零fiber不是kernel。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性映射的核"]},"variant":{"family_id":"qf-mod33-kernel-bounded-search-boundary","seed":202611022095,"parameters":{"matrix":[[1,20],[2,40]],"bound":5,"domain":"Exact_R2_matrix_kernel","operation":"kernel-bounded-search-boundary","semantic_template":"qf-mod33-kernel-bounded-search-boundary/{\"bound\":5,\"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-r02-linear-map-kernel-0096","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-kernel"],"title_zh":"線性映射的核","stem_zh":"同一精確2×2線性映射T:R²到R²。操作=kernel-bounded-search-boundary；資料={\"matrix\":[[1,21],[2,42]],\"bound\":1}。請依 [bound,integer_kernel_points,global_kernel_basis,complete,boundary] 欄序回答。","answer":{"kind":"expression","value":"(1, [[0, 0]], [[21, -1]], 否, bounded_integer_search_not_full_real_kernel)","explanation_zh":"Wave5第二層核對=(1, [[0, 0]], [[21, -1]], 否, bounded_integer_search_not_full_real_kernel)；依完整明示資料重算；kernel只解T(x)=0且使用symbolic full-real certificate；bounded integer search只回答bounded問題；非零fiber不是kernel。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性映射的核"]},"variant":{"family_id":"qf-mod33-kernel-bounded-search-boundary","seed":202611022096,"parameters":{"matrix":[[1,21],[2,42]],"bound":1,"domain":"Exact_R2_matrix_kernel","operation":"kernel-bounded-search-boundary","semantic_template":"qf-mod33-kernel-bounded-search-boundary/{\"bound\":1,\"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-r02-linear-map-kernel-0097","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-kernel"],"title_zh":"線性映射的核","stem_zh":"同一精確2×2線性映射T:R²到R²。操作=kernel-bounded-search-boundary；資料={\"matrix\":[[1,22],[2,44]],\"bound\":2}。請依 [bound,integer_kernel_points,global_kernel_basis,complete,boundary] 欄序回答。","answer":{"kind":"expression","value":"(2, [[0, 0]], [[22, -1]], 否, bounded_integer_search_not_full_real_kernel)","explanation_zh":"Wave5第二層核對=(2, [[0, 0]], [[22, -1]], 否, bounded_integer_search_not_full_real_kernel)；依完整明示資料重算；kernel只解T(x)=0且使用symbolic full-real certificate；bounded integer search只回答bounded問題；非零fiber不是kernel。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性映射的核"]},"variant":{"family_id":"qf-mod33-kernel-bounded-search-boundary","seed":202611022097,"parameters":{"matrix":[[1,22],[2,44]],"bound":2,"domain":"Exact_R2_matrix_kernel","operation":"kernel-bounded-search-boundary","semantic_template":"qf-mod33-kernel-bounded-search-boundary/{\"bound\":2,\"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-r02-linear-map-kernel-0098","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-kernel"],"title_zh":"線性映射的核","stem_zh":"同一精確2×2線性映射T:R²到R²。操作=kernel-bounded-search-boundary；資料={\"matrix\":[[1,23],[2,46]],\"bound\":3}。請依 [bound,integer_kernel_points,global_kernel_basis,complete,boundary] 欄序回答。","answer":{"kind":"expression","value":"(3, [[0, 0]], [[23, -1]], 否, bounded_integer_search_not_full_real_kernel)","explanation_zh":"Wave5第二層核對=(3, [[0, 0]], [[23, -1]], 否, bounded_integer_search_not_full_real_kernel)；依完整明示資料重算；kernel只解T(x)=0且使用symbolic full-real certificate；bounded integer search只回答bounded問題；非零fiber不是kernel。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性映射的核"]},"variant":{"family_id":"qf-mod33-kernel-bounded-search-boundary","seed":202611022098,"parameters":{"matrix":[[1,23],[2,46]],"bound":3,"domain":"Exact_R2_matrix_kernel","operation":"kernel-bounded-search-boundary","semantic_template":"qf-mod33-kernel-bounded-search-boundary/{\"bound\":3,\"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-r02-linear-map-kernel-0099","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-kernel"],"title_zh":"線性映射的核","stem_zh":"同一精確2×2線性映射T:R²到R²。操作=kernel-bounded-search-boundary；資料={\"matrix\":[[1,24],[2,48]],\"bound\":4}。請依 [bound,integer_kernel_points,global_kernel_basis,complete,boundary] 欄序回答。","answer":{"kind":"expression","value":"(4, [[0, 0]], [[24, -1]], 否, bounded_integer_search_not_full_real_kernel)","explanation_zh":"Wave5第二層核對=(4, [[0, 0]], [[24, -1]], 否, bounded_integer_search_not_full_real_kernel)；依完整明示資料重算；kernel只解T(x)=0且使用symbolic full-real certificate；bounded integer search只回答bounded問題；非零fiber不是kernel。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性映射的核"]},"variant":{"family_id":"qf-mod33-kernel-bounded-search-boundary","seed":202611022099,"parameters":{"matrix":[[1,24],[2,48]],"bound":4,"domain":"Exact_R2_matrix_kernel","operation":"kernel-bounded-search-boundary","semantic_template":"qf-mod33-kernel-bounded-search-boundary/{\"bound\":4,\"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-r02-linear-map-kernel-0100","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-map-kernel"],"title_zh":"線性映射的核","stem_zh":"同一精確2×2線性映射T:R²到R²。操作=kernel-bounded-search-boundary；資料={\"matrix\":[[1,25],[2,50]],\"bound\":5}。請依 [bound,integer_kernel_points,global_kernel_basis,complete,boundary] 欄序回答。","answer":{"kind":"expression","value":"(5, [[0, 0]], [[25, -1]], 否, bounded_integer_search_not_full_real_kernel)","explanation_zh":"Wave5第二層核對=(5, [[0, 0]], [[25, -1]], 否, bounded_integer_search_not_full_real_kernel)；依完整明示資料重算；kernel只解T(x)=0且使用symbolic full-real certificate；bounded integer search只回答bounded問題；非零fiber不是kernel。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性映射的核"]},"variant":{"family_id":"qf-mod33-kernel-bounded-search-boundary","seed":202611022100,"parameters":{"matrix":[[1,25],[2,50]],"bound":5,"domain":"Exact_R2_matrix_kernel","operation":"kernel-bounded-search-boundary","semantic_template":"qf-mod33-kernel-bounded-search-boundary/{\"bound\":5,\"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"}}