{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r07-quotient-vector-space-0001","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-vector-space"],"title_zh":"商向量空間","stem_zh":"有限維實向量空間、明示子空間基底與exact membership。操作=quotient-coset-equivalence；資料={\"ambient_dimension\":3,\"basis_w\":[[1,1,0]],\"v\":[2,2,3],\"u\":[4,4,3]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([-2, -2, 0], 是, v_plus_W_equals_u_plus_W_iff_difference_in_W, quotient_elements_are_cosets_not_unique_representatives)","explanation_zh":"Wave5封閉層核對=([-2, -2, 0], 是, v_plus_W_equals_u_plus_W_iff_difference_in_W, quotient_elements_are_cosets_not_unique_representatives)；依完整明示資料獨立重算；quotient元素是coset且代表元不唯一；[v]=[u] iff v-u在W；加法與scalar multiplication須與代表元無關；dim公式限有限維且零coset是W。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["商向量空間"]},"variant":{"family_id":"qf-mod37-quotient-coset-equivalence","seed":202611067001,"parameters":{"ambient_dimension":3,"basis_w":[[1,1,0]],"v":[2,2,3],"u":[4,4,3],"domain":"Finite_dimensional_real_quotient_space","operation":"quotient-coset-equivalence","semantic_template":"qf-mod37-quotient-coset-equivalence/{\"ambient_dimension\":3,\"basis_w\":[[1,1,0]],\"u\":[4,4,3],\"v\":[2,2,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r07-quotient-vector-space-0002","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-vector-space"],"title_zh":"商向量空間","stem_zh":"有限維實向量空間、明示子空間基底與exact membership。操作=quotient-coset-equivalence；資料={\"ambient_dimension\":3,\"basis_w\":[[1,2,0]],\"v\":[3,2,3],\"u\":[5,6,3]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([-2, -4, 0], 是, v_plus_W_equals_u_plus_W_iff_difference_in_W, quotient_elements_are_cosets_not_unique_representatives)","explanation_zh":"Wave5封閉層核對=([-2, -4, 0], 是, v_plus_W_equals_u_plus_W_iff_difference_in_W, quotient_elements_are_cosets_not_unique_representatives)；依完整明示資料獨立重算；quotient元素是coset且代表元不唯一；[v]=[u] iff v-u在W；加法與scalar multiplication須與代表元無關；dim公式限有限維且零coset是W。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["商向量空間"]},"variant":{"family_id":"qf-mod37-quotient-coset-equivalence","seed":202611067002,"parameters":{"ambient_dimension":3,"basis_w":[[1,2,0]],"v":[3,2,3],"u":[5,6,3],"domain":"Finite_dimensional_real_quotient_space","operation":"quotient-coset-equivalence","semantic_template":"qf-mod37-quotient-coset-equivalence/{\"ambient_dimension\":3,\"basis_w\":[[1,2,0]],\"u\":[5,6,3],\"v\":[3,2,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r07-quotient-vector-space-0003","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-vector-space"],"title_zh":"商向量空間","stem_zh":"有限維實向量空間、明示子空間基底與exact membership。操作=quotient-coset-equivalence；資料={\"ambient_dimension\":3,\"basis_w\":[[1,3,0]],\"v\":[4,2,3],\"u\":[6,8,3]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([-2, -6, 0], 是, v_plus_W_equals_u_plus_W_iff_difference_in_W, quotient_elements_are_cosets_not_unique_representatives)","explanation_zh":"Wave5封閉層核對=([-2, -6, 0], 是, v_plus_W_equals_u_plus_W_iff_difference_in_W, quotient_elements_are_cosets_not_unique_representatives)；依完整明示資料獨立重算；quotient元素是coset且代表元不唯一；[v]=[u] iff v-u在W；加法與scalar multiplication須與代表元無關；dim公式限有限維且零coset是W。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["商向量空間"]},"variant":{"family_id":"qf-mod37-quotient-coset-equivalence","seed":202611067003,"parameters":{"ambient_dimension":3,"basis_w":[[1,3,0]],"v":[4,2,3],"u":[6,8,3],"domain":"Finite_dimensional_real_quotient_space","operation":"quotient-coset-equivalence","semantic_template":"qf-mod37-quotient-coset-equivalence/{\"ambient_dimension\":3,\"basis_w\":[[1,3,0]],\"u\":[6,8,3],\"v\":[4,2,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r07-quotient-vector-space-0004","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-vector-space"],"title_zh":"商向量空間","stem_zh":"有限維實向量空間、明示子空間基底與exact membership。操作=quotient-coset-equivalence；資料={\"ambient_dimension\":3,\"basis_w\":[[1,4,0]],\"v\":[5,2,3],\"u\":[7,10,3]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([-2, -8, 0], 是, v_plus_W_equals_u_plus_W_iff_difference_in_W, quotient_elements_are_cosets_not_unique_representatives)","explanation_zh":"Wave5封閉層核對=([-2, -8, 0], 是, v_plus_W_equals_u_plus_W_iff_difference_in_W, quotient_elements_are_cosets_not_unique_representatives)；依完整明示資料獨立重算；quotient元素是coset且代表元不唯一；[v]=[u] iff v-u在W；加法與scalar multiplication須與代表元無關；dim公式限有限維且零coset是W。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["商向量空間"]},"variant":{"family_id":"qf-mod37-quotient-coset-equivalence","seed":202611067004,"parameters":{"ambient_dimension":3,"basis_w":[[1,4,0]],"v":[5,2,3],"u":[7,10,3],"domain":"Finite_dimensional_real_quotient_space","operation":"quotient-coset-equivalence","semantic_template":"qf-mod37-quotient-coset-equivalence/{\"ambient_dimension\":3,\"basis_w\":[[1,4,0]],\"u\":[7,10,3],\"v\":[5,2,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r07-quotient-vector-space-0005","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-vector-space"],"title_zh":"商向量空間","stem_zh":"有限維實向量空間、明示子空間基底與exact membership。操作=quotient-coset-equivalence；資料={\"ambient_dimension\":3,\"basis_w\":[[1,5,0]],\"v\":[6,2,3],\"u\":[8,12,3]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([-2, -10, 0], 是, v_plus_W_equals_u_plus_W_iff_difference_in_W, quotient_elements_are_cosets_not_unique_representatives)","explanation_zh":"Wave5封閉層核對=([-2, -10, 0], 是, v_plus_W_equals_u_plus_W_iff_difference_in_W, quotient_elements_are_cosets_not_unique_representatives)；依完整明示資料獨立重算；quotient元素是coset且代表元不唯一；[v]=[u] iff v-u在W；加法與scalar multiplication須與代表元無關；dim公式限有限維且零coset是W。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["商向量空間"]},"variant":{"family_id":"qf-mod37-quotient-coset-equivalence","seed":202611067005,"parameters":{"ambient_dimension":3,"basis_w":[[1,5,0]],"v":[6,2,3],"u":[8,12,3],"domain":"Finite_dimensional_real_quotient_space","operation":"quotient-coset-equivalence","semantic_template":"qf-mod37-quotient-coset-equivalence/{\"ambient_dimension\":3,\"basis_w\":[[1,5,0]],\"u\":[8,12,3],\"v\":[6,2,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r07-quotient-vector-space-0006","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-vector-space"],"title_zh":"商向量空間","stem_zh":"有限維實向量空間、明示子空間基底與exact membership。操作=quotient-coset-equivalence；資料={\"ambient_dimension\":3,\"basis_w\":[[1,6,0]],\"v\":[7,2,3],\"u\":[9,14,3]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([-2, -12, 0], 是, v_plus_W_equals_u_plus_W_iff_difference_in_W, quotient_elements_are_cosets_not_unique_representatives)","explanation_zh":"Wave5封閉層核對=([-2, -12, 0], 是, v_plus_W_equals_u_plus_W_iff_difference_in_W, quotient_elements_are_cosets_not_unique_representatives)；依完整明示資料獨立重算；quotient元素是coset且代表元不唯一；[v]=[u] iff v-u在W；加法與scalar multiplication須與代表元無關；dim公式限有限維且零coset是W。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["商向量空間"]},"variant":{"family_id":"qf-mod37-quotient-coset-equivalence","seed":202611067006,"parameters":{"ambient_dimension":3,"basis_w":[[1,6,0]],"v":[7,2,3],"u":[9,14,3],"domain":"Finite_dimensional_real_quotient_space","operation":"quotient-coset-equivalence","semantic_template":"qf-mod37-quotient-coset-equivalence/{\"ambient_dimension\":3,\"basis_w\":[[1,6,0]],\"u\":[9,14,3],\"v\":[7,2,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r07-quotient-vector-space-0007","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-vector-space"],"title_zh":"商向量空間","stem_zh":"有限維實向量空間、明示子空間基底與exact membership。操作=quotient-coset-equivalence；資料={\"ambient_dimension\":3,\"basis_w\":[[1,7,0]],\"v\":[8,2,3],\"u\":[10,16,3]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([-2, -14, 0], 是, v_plus_W_equals_u_plus_W_iff_difference_in_W, quotient_elements_are_cosets_not_unique_representatives)","explanation_zh":"Wave5封閉層核對=([-2, -14, 0], 是, v_plus_W_equals_u_plus_W_iff_difference_in_W, quotient_elements_are_cosets_not_unique_representatives)；依完整明示資料獨立重算；quotient元素是coset且代表元不唯一；[v]=[u] iff v-u在W；加法與scalar multiplication須與代表元無關；dim公式限有限維且零coset是W。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["商向量空間"]},"variant":{"family_id":"qf-mod37-quotient-coset-equivalence","seed":202611067007,"parameters":{"ambient_dimension":3,"basis_w":[[1,7,0]],"v":[8,2,3],"u":[10,16,3],"domain":"Finite_dimensional_real_quotient_space","operation":"quotient-coset-equivalence","semantic_template":"qf-mod37-quotient-coset-equivalence/{\"ambient_dimension\":3,\"basis_w\":[[1,7,0]],\"u\":[10,16,3],\"v\":[8,2,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r07-quotient-vector-space-0008","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-vector-space"],"title_zh":"商向量空間","stem_zh":"有限維實向量空間、明示子空間基底與exact membership。操作=quotient-coset-equivalence；資料={\"ambient_dimension\":3,\"basis_w\":[[1,8,0]],\"v\":[9,2,3],\"u\":[11,18,3]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([-2, -16, 0], 是, v_plus_W_equals_u_plus_W_iff_difference_in_W, quotient_elements_are_cosets_not_unique_representatives)","explanation_zh":"Wave5封閉層核對=([-2, -16, 0], 是, v_plus_W_equals_u_plus_W_iff_difference_in_W, quotient_elements_are_cosets_not_unique_representatives)；依完整明示資料獨立重算；quotient元素是coset且代表元不唯一；[v]=[u] iff v-u在W；加法與scalar multiplication須與代表元無關；dim公式限有限維且零coset是W。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["商向量空間"]},"variant":{"family_id":"qf-mod37-quotient-coset-equivalence","seed":202611067008,"parameters":{"ambient_dimension":3,"basis_w":[[1,8,0]],"v":[9,2,3],"u":[11,18,3],"domain":"Finite_dimensional_real_quotient_space","operation":"quotient-coset-equivalence","semantic_template":"qf-mod37-quotient-coset-equivalence/{\"ambient_dimension\":3,\"basis_w\":[[1,8,0]],\"u\":[11,18,3],\"v\":[9,2,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r07-quotient-vector-space-0009","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-vector-space"],"title_zh":"商向量空間","stem_zh":"有限維實向量空間、明示子空間基底與exact membership。操作=quotient-coset-equivalence；資料={\"ambient_dimension\":3,\"basis_w\":[[1,9,0]],\"v\":[10,2,3],\"u\":[12,20,3]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([-2, -18, 0], 是, v_plus_W_equals_u_plus_W_iff_difference_in_W, quotient_elements_are_cosets_not_unique_representatives)","explanation_zh":"Wave5封閉層核對=([-2, -18, 0], 是, v_plus_W_equals_u_plus_W_iff_difference_in_W, quotient_elements_are_cosets_not_unique_representatives)；依完整明示資料獨立重算；quotient元素是coset且代表元不唯一；[v]=[u] iff v-u在W；加法與scalar multiplication須與代表元無關；dim公式限有限維且零coset是W。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["商向量空間"]},"variant":{"family_id":"qf-mod37-quotient-coset-equivalence","seed":202611067009,"parameters":{"ambient_dimension":3,"basis_w":[[1,9,0]],"v":[10,2,3],"u":[12,20,3],"domain":"Finite_dimensional_real_quotient_space","operation":"quotient-coset-equivalence","semantic_template":"qf-mod37-quotient-coset-equivalence/{\"ambient_dimension\":3,\"basis_w\":[[1,9,0]],\"u\":[12,20,3],\"v\":[10,2,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r07-quotient-vector-space-0010","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-vector-space"],"title_zh":"商向量空間","stem_zh":"有限維實向量空間、明示子空間基底與exact membership。操作=quotient-coset-equivalence；資料={\"ambient_dimension\":3,\"basis_w\":[[1,10,0]],\"v\":[11,2,3],\"u\":[13,22,3]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([-2, -20, 0], 是, v_plus_W_equals_u_plus_W_iff_difference_in_W, quotient_elements_are_cosets_not_unique_representatives)","explanation_zh":"Wave5封閉層核對=([-2, -20, 0], 是, v_plus_W_equals_u_plus_W_iff_difference_in_W, quotient_elements_are_cosets_not_unique_representatives)；依完整明示資料獨立重算；quotient元素是coset且代表元不唯一；[v]=[u] iff v-u在W；加法與scalar multiplication須與代表元無關；dim公式限有限維且零coset是W。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["商向量空間"]},"variant":{"family_id":"qf-mod37-quotient-coset-equivalence","seed":202611067010,"parameters":{"ambient_dimension":3,"basis_w":[[1,10,0]],"v":[11,2,3],"u":[13,22,3],"domain":"Finite_dimensional_real_quotient_space","operation":"quotient-coset-equivalence","semantic_template":"qf-mod37-quotient-coset-equivalence/{\"ambient_dimension\":3,\"basis_w\":[[1,10,0]],\"u\":[13,22,3],\"v\":[11,2,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r07-quotient-vector-space-0011","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-vector-space"],"title_zh":"商向量空間","stem_zh":"有限維實向量空間、明示子空間基底與exact membership。操作=quotient-coset-equivalence；資料={\"ambient_dimension\":3,\"basis_w\":[[1,11,0]],\"v\":[12,2,3],\"u\":[14,24,3]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([-2, -22, 0], 是, v_plus_W_equals_u_plus_W_iff_difference_in_W, quotient_elements_are_cosets_not_unique_representatives)","explanation_zh":"Wave5封閉層核對=([-2, -22, 0], 是, v_plus_W_equals_u_plus_W_iff_difference_in_W, quotient_elements_are_cosets_not_unique_representatives)；依完整明示資料獨立重算；quotient元素是coset且代表元不唯一；[v]=[u] iff v-u在W；加法與scalar multiplication須與代表元無關；dim公式限有限維且零coset是W。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["商向量空間"]},"variant":{"family_id":"qf-mod37-quotient-coset-equivalence","seed":202611067011,"parameters":{"ambient_dimension":3,"basis_w":[[1,11,0]],"v":[12,2,3],"u":[14,24,3],"domain":"Finite_dimensional_real_quotient_space","operation":"quotient-coset-equivalence","semantic_template":"qf-mod37-quotient-coset-equivalence/{\"ambient_dimension\":3,\"basis_w\":[[1,11,0]],\"u\":[14,24,3],\"v\":[12,2,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r07-quotient-vector-space-0012","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-vector-space"],"title_zh":"商向量空間","stem_zh":"有限維實向量空間、明示子空間基底與exact membership。操作=quotient-coset-equivalence；資料={\"ambient_dimension\":3,\"basis_w\":[[1,12,0]],\"v\":[13,2,3],\"u\":[15,26,3]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([-2, -24, 0], 是, v_plus_W_equals_u_plus_W_iff_difference_in_W, quotient_elements_are_cosets_not_unique_representatives)","explanation_zh":"Wave5封閉層核對=([-2, -24, 0], 是, v_plus_W_equals_u_plus_W_iff_difference_in_W, quotient_elements_are_cosets_not_unique_representatives)；依完整明示資料獨立重算；quotient元素是coset且代表元不唯一；[v]=[u] iff v-u在W；加法與scalar multiplication須與代表元無關；dim公式限有限維且零coset是W。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["商向量空間"]},"variant":{"family_id":"qf-mod37-quotient-coset-equivalence","seed":202611067012,"parameters":{"ambient_dimension":3,"basis_w":[[1,12,0]],"v":[13,2,3],"u":[15,26,3],"domain":"Finite_dimensional_real_quotient_space","operation":"quotient-coset-equivalence","semantic_template":"qf-mod37-quotient-coset-equivalence/{\"ambient_dimension\":3,\"basis_w\":[[1,12,0]],\"u\":[15,26,3],\"v\":[13,2,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r07-quotient-vector-space-0013","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-vector-space"],"title_zh":"商向量空間","stem_zh":"有限維實向量空間、明示子空間基底與exact membership。操作=quotient-coset-equivalence；資料={\"ambient_dimension\":3,\"basis_w\":[[1,13,0]],\"v\":[14,2,3],\"u\":[16,28,3]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([-2, -26, 0], 是, v_plus_W_equals_u_plus_W_iff_difference_in_W, quotient_elements_are_cosets_not_unique_representatives)","explanation_zh":"Wave5封閉層核對=([-2, -26, 0], 是, v_plus_W_equals_u_plus_W_iff_difference_in_W, quotient_elements_are_cosets_not_unique_representatives)；依完整明示資料獨立重算；quotient元素是coset且代表元不唯一；[v]=[u] iff v-u在W；加法與scalar multiplication須與代表元無關；dim公式限有限維且零coset是W。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["商向量空間"]},"variant":{"family_id":"qf-mod37-quotient-coset-equivalence","seed":202611067013,"parameters":{"ambient_dimension":3,"basis_w":[[1,13,0]],"v":[14,2,3],"u":[16,28,3],"domain":"Finite_dimensional_real_quotient_space","operation":"quotient-coset-equivalence","semantic_template":"qf-mod37-quotient-coset-equivalence/{\"ambient_dimension\":3,\"basis_w\":[[1,13,0]],\"u\":[16,28,3],\"v\":[14,2,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r07-quotient-vector-space-0014","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-vector-space"],"title_zh":"商向量空間","stem_zh":"有限維實向量空間、明示子空間基底與exact membership。操作=quotient-coset-equivalence；資料={\"ambient_dimension\":3,\"basis_w\":[[1,14,0]],\"v\":[15,2,3],\"u\":[17,30,3]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([-2, -28, 0], 是, v_plus_W_equals_u_plus_W_iff_difference_in_W, quotient_elements_are_cosets_not_unique_representatives)","explanation_zh":"Wave5封閉層核對=([-2, -28, 0], 是, v_plus_W_equals_u_plus_W_iff_difference_in_W, quotient_elements_are_cosets_not_unique_representatives)；依完整明示資料獨立重算；quotient元素是coset且代表元不唯一；[v]=[u] iff v-u在W；加法與scalar multiplication須與代表元無關；dim公式限有限維且零coset是W。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["商向量空間"]},"variant":{"family_id":"qf-mod37-quotient-coset-equivalence","seed":202611067014,"parameters":{"ambient_dimension":3,"basis_w":[[1,14,0]],"v":[15,2,3],"u":[17,30,3],"domain":"Finite_dimensional_real_quotient_space","operation":"quotient-coset-equivalence","semantic_template":"qf-mod37-quotient-coset-equivalence/{\"ambient_dimension\":3,\"basis_w\":[[1,14,0]],\"u\":[17,30,3],\"v\":[15,2,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r07-quotient-vector-space-0015","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-vector-space"],"title_zh":"商向量空間","stem_zh":"有限維實向量空間、明示子空間基底與exact membership。操作=quotient-coset-equivalence；資料={\"ambient_dimension\":3,\"basis_w\":[[1,15,0]],\"v\":[16,2,3],\"u\":[18,32,3]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([-2, -30, 0], 是, v_plus_W_equals_u_plus_W_iff_difference_in_W, quotient_elements_are_cosets_not_unique_representatives)","explanation_zh":"Wave5封閉層核對=([-2, -30, 0], 是, v_plus_W_equals_u_plus_W_iff_difference_in_W, quotient_elements_are_cosets_not_unique_representatives)；依完整明示資料獨立重算；quotient元素是coset且代表元不唯一；[v]=[u] iff v-u在W；加法與scalar multiplication須與代表元無關；dim公式限有限維且零coset是W。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["商向量空間"]},"variant":{"family_id":"qf-mod37-quotient-coset-equivalence","seed":202611067015,"parameters":{"ambient_dimension":3,"basis_w":[[1,15,0]],"v":[16,2,3],"u":[18,32,3],"domain":"Finite_dimensional_real_quotient_space","operation":"quotient-coset-equivalence","semantic_template":"qf-mod37-quotient-coset-equivalence/{\"ambient_dimension\":3,\"basis_w\":[[1,15,0]],\"u\":[18,32,3],\"v\":[16,2,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r07-quotient-vector-space-0016","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-vector-space"],"title_zh":"商向量空間","stem_zh":"有限維實向量空間、明示子空間基底與exact membership。操作=quotient-coset-equivalence；資料={\"ambient_dimension\":3,\"basis_w\":[[1,16,0]],\"v\":[17,2,3],\"u\":[19,34,3]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([-2, -32, 0], 是, v_plus_W_equals_u_plus_W_iff_difference_in_W, quotient_elements_are_cosets_not_unique_representatives)","explanation_zh":"Wave5封閉層核對=([-2, -32, 0], 是, v_plus_W_equals_u_plus_W_iff_difference_in_W, quotient_elements_are_cosets_not_unique_representatives)；依完整明示資料獨立重算；quotient元素是coset且代表元不唯一；[v]=[u] iff v-u在W；加法與scalar multiplication須與代表元無關；dim公式限有限維且零coset是W。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["商向量空間"]},"variant":{"family_id":"qf-mod37-quotient-coset-equivalence","seed":202611067016,"parameters":{"ambient_dimension":3,"basis_w":[[1,16,0]],"v":[17,2,3],"u":[19,34,3],"domain":"Finite_dimensional_real_quotient_space","operation":"quotient-coset-equivalence","semantic_template":"qf-mod37-quotient-coset-equivalence/{\"ambient_dimension\":3,\"basis_w\":[[1,16,0]],\"u\":[19,34,3],\"v\":[17,2,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r07-quotient-vector-space-0017","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-vector-space"],"title_zh":"商向量空間","stem_zh":"有限維實向量空間、明示子空間基底與exact membership。操作=quotient-coset-equivalence；資料={\"ambient_dimension\":3,\"basis_w\":[[1,17,0]],\"v\":[18,2,3],\"u\":[20,36,3]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([-2, -34, 0], 是, v_plus_W_equals_u_plus_W_iff_difference_in_W, quotient_elements_are_cosets_not_unique_representatives)","explanation_zh":"Wave5封閉層核對=([-2, -34, 0], 是, v_plus_W_equals_u_plus_W_iff_difference_in_W, quotient_elements_are_cosets_not_unique_representatives)；依完整明示資料獨立重算；quotient元素是coset且代表元不唯一；[v]=[u] iff v-u在W；加法與scalar multiplication須與代表元無關；dim公式限有限維且零coset是W。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["商向量空間"]},"variant":{"family_id":"qf-mod37-quotient-coset-equivalence","seed":202611067017,"parameters":{"ambient_dimension":3,"basis_w":[[1,17,0]],"v":[18,2,3],"u":[20,36,3],"domain":"Finite_dimensional_real_quotient_space","operation":"quotient-coset-equivalence","semantic_template":"qf-mod37-quotient-coset-equivalence/{\"ambient_dimension\":3,\"basis_w\":[[1,17,0]],\"u\":[20,36,3],\"v\":[18,2,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r07-quotient-vector-space-0018","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-vector-space"],"title_zh":"商向量空間","stem_zh":"有限維實向量空間、明示子空間基底與exact membership。操作=quotient-coset-equivalence；資料={\"ambient_dimension\":3,\"basis_w\":[[1,18,0]],\"v\":[19,2,3],\"u\":[21,38,3]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([-2, -36, 0], 是, v_plus_W_equals_u_plus_W_iff_difference_in_W, quotient_elements_are_cosets_not_unique_representatives)","explanation_zh":"Wave5封閉層核對=([-2, -36, 0], 是, v_plus_W_equals_u_plus_W_iff_difference_in_W, quotient_elements_are_cosets_not_unique_representatives)；依完整明示資料獨立重算；quotient元素是coset且代表元不唯一；[v]=[u] iff v-u在W；加法與scalar multiplication須與代表元無關；dim公式限有限維且零coset是W。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["商向量空間"]},"variant":{"family_id":"qf-mod37-quotient-coset-equivalence","seed":202611067018,"parameters":{"ambient_dimension":3,"basis_w":[[1,18,0]],"v":[19,2,3],"u":[21,38,3],"domain":"Finite_dimensional_real_quotient_space","operation":"quotient-coset-equivalence","semantic_template":"qf-mod37-quotient-coset-equivalence/{\"ambient_dimension\":3,\"basis_w\":[[1,18,0]],\"u\":[21,38,3],\"v\":[19,2,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r07-quotient-vector-space-0019","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-vector-space"],"title_zh":"商向量空間","stem_zh":"有限維實向量空間、明示子空間基底與exact membership。操作=quotient-coset-equivalence；資料={\"ambient_dimension\":3,\"basis_w\":[[1,19,0]],\"v\":[20,2,3],\"u\":[22,40,3]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([-2, -38, 0], 是, v_plus_W_equals_u_plus_W_iff_difference_in_W, quotient_elements_are_cosets_not_unique_representatives)","explanation_zh":"Wave5封閉層核對=([-2, -38, 0], 是, v_plus_W_equals_u_plus_W_iff_difference_in_W, quotient_elements_are_cosets_not_unique_representatives)；依完整明示資料獨立重算；quotient元素是coset且代表元不唯一；[v]=[u] iff v-u在W；加法與scalar multiplication須與代表元無關；dim公式限有限維且零coset是W。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["商向量空間"]},"variant":{"family_id":"qf-mod37-quotient-coset-equivalence","seed":202611067019,"parameters":{"ambient_dimension":3,"basis_w":[[1,19,0]],"v":[20,2,3],"u":[22,40,3],"domain":"Finite_dimensional_real_quotient_space","operation":"quotient-coset-equivalence","semantic_template":"qf-mod37-quotient-coset-equivalence/{\"ambient_dimension\":3,\"basis_w\":[[1,19,0]],\"u\":[22,40,3],\"v\":[20,2,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r07-quotient-vector-space-0020","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-vector-space"],"title_zh":"商向量空間","stem_zh":"有限維實向量空間、明示子空間基底與exact membership。操作=quotient-coset-equivalence；資料={\"ambient_dimension\":3,\"basis_w\":[[1,20,0]],\"v\":[21,2,3],\"u\":[23,42,3]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([-2, -40, 0], 是, v_plus_W_equals_u_plus_W_iff_difference_in_W, quotient_elements_are_cosets_not_unique_representatives)","explanation_zh":"Wave5封閉層核對=([-2, -40, 0], 是, v_plus_W_equals_u_plus_W_iff_difference_in_W, quotient_elements_are_cosets_not_unique_representatives)；依完整明示資料獨立重算；quotient元素是coset且代表元不唯一；[v]=[u] iff v-u在W；加法與scalar multiplication須與代表元無關；dim公式限有限維且零coset是W。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["商向量空間"]},"variant":{"family_id":"qf-mod37-quotient-coset-equivalence","seed":202611067020,"parameters":{"ambient_dimension":3,"basis_w":[[1,20,0]],"v":[21,2,3],"u":[23,42,3],"domain":"Finite_dimensional_real_quotient_space","operation":"quotient-coset-equivalence","semantic_template":"qf-mod37-quotient-coset-equivalence/{\"ambient_dimension\":3,\"basis_w\":[[1,20,0]],\"u\":[23,42,3],\"v\":[21,2,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r07-quotient-vector-space-0021","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-vector-space"],"title_zh":"商向量空間","stem_zh":"有限維實向量空間、明示子空間基底與exact membership。操作=quotient-coset-equivalence；資料={\"ambient_dimension\":3,\"basis_w\":[[1,21,0]],\"v\":[22,2,3],\"u\":[24,44,3]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([-2, -42, 0], 是, v_plus_W_equals_u_plus_W_iff_difference_in_W, quotient_elements_are_cosets_not_unique_representatives)","explanation_zh":"Wave5封閉層核對=([-2, -42, 0], 是, v_plus_W_equals_u_plus_W_iff_difference_in_W, quotient_elements_are_cosets_not_unique_representatives)；依完整明示資料獨立重算；quotient元素是coset且代表元不唯一；[v]=[u] iff v-u在W；加法與scalar multiplication須與代表元無關；dim公式限有限維且零coset是W。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["商向量空間"]},"variant":{"family_id":"qf-mod37-quotient-coset-equivalence","seed":202611067021,"parameters":{"ambient_dimension":3,"basis_w":[[1,21,0]],"v":[22,2,3],"u":[24,44,3],"domain":"Finite_dimensional_real_quotient_space","operation":"quotient-coset-equivalence","semantic_template":"qf-mod37-quotient-coset-equivalence/{\"ambient_dimension\":3,\"basis_w\":[[1,21,0]],\"u\":[24,44,3],\"v\":[22,2,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r07-quotient-vector-space-0022","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-vector-space"],"title_zh":"商向量空間","stem_zh":"有限維實向量空間、明示子空間基底與exact membership。操作=quotient-coset-equivalence；資料={\"ambient_dimension\":3,\"basis_w\":[[1,22,0]],\"v\":[23,2,3],\"u\":[25,46,3]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([-2, -44, 0], 是, v_plus_W_equals_u_plus_W_iff_difference_in_W, quotient_elements_are_cosets_not_unique_representatives)","explanation_zh":"Wave5封閉層核對=([-2, -44, 0], 是, v_plus_W_equals_u_plus_W_iff_difference_in_W, quotient_elements_are_cosets_not_unique_representatives)；依完整明示資料獨立重算；quotient元素是coset且代表元不唯一；[v]=[u] iff v-u在W；加法與scalar multiplication須與代表元無關；dim公式限有限維且零coset是W。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["商向量空間"]},"variant":{"family_id":"qf-mod37-quotient-coset-equivalence","seed":202611067022,"parameters":{"ambient_dimension":3,"basis_w":[[1,22,0]],"v":[23,2,3],"u":[25,46,3],"domain":"Finite_dimensional_real_quotient_space","operation":"quotient-coset-equivalence","semantic_template":"qf-mod37-quotient-coset-equivalence/{\"ambient_dimension\":3,\"basis_w\":[[1,22,0]],\"u\":[25,46,3],\"v\":[23,2,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r07-quotient-vector-space-0023","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-vector-space"],"title_zh":"商向量空間","stem_zh":"有限維實向量空間、明示子空間基底與exact membership。操作=quotient-coset-equivalence；資料={\"ambient_dimension\":3,\"basis_w\":[[1,23,0]],\"v\":[24,2,3],\"u\":[26,48,3]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([-2, -46, 0], 是, v_plus_W_equals_u_plus_W_iff_difference_in_W, quotient_elements_are_cosets_not_unique_representatives)","explanation_zh":"Wave5封閉層核對=([-2, -46, 0], 是, v_plus_W_equals_u_plus_W_iff_difference_in_W, quotient_elements_are_cosets_not_unique_representatives)；依完整明示資料獨立重算；quotient元素是coset且代表元不唯一；[v]=[u] iff v-u在W；加法與scalar multiplication須與代表元無關；dim公式限有限維且零coset是W。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["商向量空間"]},"variant":{"family_id":"qf-mod37-quotient-coset-equivalence","seed":202611067023,"parameters":{"ambient_dimension":3,"basis_w":[[1,23,0]],"v":[24,2,3],"u":[26,48,3],"domain":"Finite_dimensional_real_quotient_space","operation":"quotient-coset-equivalence","semantic_template":"qf-mod37-quotient-coset-equivalence/{\"ambient_dimension\":3,\"basis_w\":[[1,23,0]],\"u\":[26,48,3],\"v\":[24,2,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r07-quotient-vector-space-0024","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-vector-space"],"title_zh":"商向量空間","stem_zh":"有限維實向量空間、明示子空間基底與exact membership。操作=quotient-coset-equivalence；資料={\"ambient_dimension\":3,\"basis_w\":[[1,24,0]],\"v\":[25,2,3],\"u\":[27,50,3]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([-2, -48, 0], 是, v_plus_W_equals_u_plus_W_iff_difference_in_W, quotient_elements_are_cosets_not_unique_representatives)","explanation_zh":"Wave5封閉層核對=([-2, -48, 0], 是, v_plus_W_equals_u_plus_W_iff_difference_in_W, quotient_elements_are_cosets_not_unique_representatives)；依完整明示資料獨立重算；quotient元素是coset且代表元不唯一；[v]=[u] iff v-u在W；加法與scalar multiplication須與代表元無關；dim公式限有限維且零coset是W。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["商向量空間"]},"variant":{"family_id":"qf-mod37-quotient-coset-equivalence","seed":202611067024,"parameters":{"ambient_dimension":3,"basis_w":[[1,24,0]],"v":[25,2,3],"u":[27,50,3],"domain":"Finite_dimensional_real_quotient_space","operation":"quotient-coset-equivalence","semantic_template":"qf-mod37-quotient-coset-equivalence/{\"ambient_dimension\":3,\"basis_w\":[[1,24,0]],\"u\":[27,50,3],\"v\":[25,2,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r07-quotient-vector-space-0025","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-vector-space"],"title_zh":"商向量空間","stem_zh":"有限維實向量空間、明示子空間基底與exact membership。操作=quotient-coset-equivalence；資料={\"ambient_dimension\":3,\"basis_w\":[[1,25,0]],\"v\":[26,2,3],\"u\":[28,52,3]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([-2, -50, 0], 是, v_plus_W_equals_u_plus_W_iff_difference_in_W, quotient_elements_are_cosets_not_unique_representatives)","explanation_zh":"Wave5封閉層核對=([-2, -50, 0], 是, v_plus_W_equals_u_plus_W_iff_difference_in_W, quotient_elements_are_cosets_not_unique_representatives)；依完整明示資料獨立重算；quotient元素是coset且代表元不唯一；[v]=[u] iff v-u在W；加法與scalar multiplication須與代表元無關；dim公式限有限維且零coset是W。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["商向量空間"]},"variant":{"family_id":"qf-mod37-quotient-coset-equivalence","seed":202611067025,"parameters":{"ambient_dimension":3,"basis_w":[[1,25,0]],"v":[26,2,3],"u":[28,52,3],"domain":"Finite_dimensional_real_quotient_space","operation":"quotient-coset-equivalence","semantic_template":"qf-mod37-quotient-coset-equivalence/{\"ambient_dimension\":3,\"basis_w\":[[1,25,0]],\"u\":[28,52,3],\"v\":[26,2,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r07-quotient-vector-space-0026","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-vector-space"],"title_zh":"商向量空間","stem_zh":"有限維實向量空間、明示子空間基底與exact membership。操作=quotient-coset-operations；資料={\"ambient_dimension\":3,\"basis_w\":[[1,1,0]],\"v\":[2,2,3],\"u\":[3,1,4],\"scalar\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([5, 3, 7], [2, 2, 3], [v]+[u]=[v+u], lambda[v]=[lambda v], representatives_only_not_unique_quotient_elements)","explanation_zh":"Wave5封閉層核對=([5, 3, 7], [2, 2, 3], [v]+[u]=[v+u], lambda[v]=[lambda v], representatives_only_not_unique_quotient_elements)；依完整明示資料獨立重算；quotient元素是coset且代表元不唯一；[v]=[u] iff v-u在W；加法與scalar multiplication須與代表元無關；dim公式限有限維且零coset是W。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["商向量空間"]},"variant":{"family_id":"qf-mod37-quotient-coset-operations","seed":202611067026,"parameters":{"ambient_dimension":3,"basis_w":[[1,1,0]],"v":[2,2,3],"u":[3,1,4],"scalar":1,"domain":"Finite_dimensional_real_quotient_space","operation":"quotient-coset-operations","semantic_template":"qf-mod37-quotient-coset-operations/{\"ambient_dimension\":3,\"basis_w\":[[1,1,0]],\"scalar\":1,\"u\":[3,1,4],\"v\":[2,2,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r07-quotient-vector-space-0027","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-vector-space"],"title_zh":"商向量空間","stem_zh":"有限維實向量空間、明示子空間基底與exact membership。操作=quotient-coset-operations；資料={\"ambient_dimension\":3,\"basis_w\":[[1,2,0]],\"v\":[3,2,3],\"u\":[3,2,4],\"scalar\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([6, 4, 7], [6, 4, 6], [v]+[u]=[v+u], lambda[v]=[lambda v], representatives_only_not_unique_quotient_elements)","explanation_zh":"Wave5封閉層核對=([6, 4, 7], [6, 4, 6], [v]+[u]=[v+u], lambda[v]=[lambda v], representatives_only_not_unique_quotient_elements)；依完整明示資料獨立重算；quotient元素是coset且代表元不唯一；[v]=[u] iff v-u在W；加法與scalar multiplication須與代表元無關；dim公式限有限維且零coset是W。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["商向量空間"]},"variant":{"family_id":"qf-mod37-quotient-coset-operations","seed":202611067027,"parameters":{"ambient_dimension":3,"basis_w":[[1,2,0]],"v":[3,2,3],"u":[3,2,4],"scalar":2,"domain":"Finite_dimensional_real_quotient_space","operation":"quotient-coset-operations","semantic_template":"qf-mod37-quotient-coset-operations/{\"ambient_dimension\":3,\"basis_w\":[[1,2,0]],\"scalar\":2,\"u\":[3,2,4],\"v\":[3,2,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r07-quotient-vector-space-0028","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-vector-space"],"title_zh":"商向量空間","stem_zh":"有限維實向量空間、明示子空間基底與exact membership。操作=quotient-coset-operations；資料={\"ambient_dimension\":3,\"basis_w\":[[1,3,0]],\"v\":[4,2,3],\"u\":[3,3,4],\"scalar\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([7, 5, 7], [12, 6, 9], [v]+[u]=[v+u], lambda[v]=[lambda v], representatives_only_not_unique_quotient_elements)","explanation_zh":"Wave5封閉層核對=([7, 5, 7], [12, 6, 9], [v]+[u]=[v+u], lambda[v]=[lambda v], representatives_only_not_unique_quotient_elements)；依完整明示資料獨立重算；quotient元素是coset且代表元不唯一；[v]=[u] iff v-u在W；加法與scalar multiplication須與代表元無關；dim公式限有限維且零coset是W。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["商向量空間"]},"variant":{"family_id":"qf-mod37-quotient-coset-operations","seed":202611067028,"parameters":{"ambient_dimension":3,"basis_w":[[1,3,0]],"v":[4,2,3],"u":[3,3,4],"scalar":3,"domain":"Finite_dimensional_real_quotient_space","operation":"quotient-coset-operations","semantic_template":"qf-mod37-quotient-coset-operations/{\"ambient_dimension\":3,\"basis_w\":[[1,3,0]],\"scalar\":3,\"u\":[3,3,4],\"v\":[4,2,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r07-quotient-vector-space-0029","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-vector-space"],"title_zh":"商向量空間","stem_zh":"有限維實向量空間、明示子空間基底與exact membership。操作=quotient-coset-operations；資料={\"ambient_dimension\":3,\"basis_w\":[[1,4,0]],\"v\":[5,2,3],\"u\":[3,4,4],\"scalar\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([8, 6, 7], [20, 8, 12], [v]+[u]=[v+u], lambda[v]=[lambda v], representatives_only_not_unique_quotient_elements)","explanation_zh":"Wave5封閉層核對=([8, 6, 7], [20, 8, 12], [v]+[u]=[v+u], lambda[v]=[lambda v], representatives_only_not_unique_quotient_elements)；依完整明示資料獨立重算；quotient元素是coset且代表元不唯一；[v]=[u] iff v-u在W；加法與scalar multiplication須與代表元無關；dim公式限有限維且零coset是W。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["商向量空間"]},"variant":{"family_id":"qf-mod37-quotient-coset-operations","seed":202611067029,"parameters":{"ambient_dimension":3,"basis_w":[[1,4,0]],"v":[5,2,3],"u":[3,4,4],"scalar":4,"domain":"Finite_dimensional_real_quotient_space","operation":"quotient-coset-operations","semantic_template":"qf-mod37-quotient-coset-operations/{\"ambient_dimension\":3,\"basis_w\":[[1,4,0]],\"scalar\":4,\"u\":[3,4,4],\"v\":[5,2,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r07-quotient-vector-space-0030","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-vector-space"],"title_zh":"商向量空間","stem_zh":"有限維實向量空間、明示子空間基底與exact membership。操作=quotient-coset-operations；資料={\"ambient_dimension\":3,\"basis_w\":[[1,5,0]],\"v\":[6,2,3],\"u\":[3,5,4],\"scalar\":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([9, 7, 7], [30, 10, 15], [v]+[u]=[v+u], lambda[v]=[lambda v], representatives_only_not_unique_quotient_elements)","explanation_zh":"Wave5封閉層核對=([9, 7, 7], [30, 10, 15], [v]+[u]=[v+u], lambda[v]=[lambda v], representatives_only_not_unique_quotient_elements)；依完整明示資料獨立重算；quotient元素是coset且代表元不唯一；[v]=[u] iff v-u在W；加法與scalar multiplication須與代表元無關；dim公式限有限維且零coset是W。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["商向量空間"]},"variant":{"family_id":"qf-mod37-quotient-coset-operations","seed":202611067030,"parameters":{"ambient_dimension":3,"basis_w":[[1,5,0]],"v":[6,2,3],"u":[3,5,4],"scalar":5,"domain":"Finite_dimensional_real_quotient_space","operation":"quotient-coset-operations","semantic_template":"qf-mod37-quotient-coset-operations/{\"ambient_dimension\":3,\"basis_w\":[[1,5,0]],\"scalar\":5,\"u\":[3,5,4],\"v\":[6,2,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r07-quotient-vector-space-0031","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-vector-space"],"title_zh":"商向量空間","stem_zh":"有限維實向量空間、明示子空間基底與exact membership。操作=quotient-coset-operations；資料={\"ambient_dimension\":3,\"basis_w\":[[1,6,0]],\"v\":[7,2,3],\"u\":[3,6,4],\"scalar\":6}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([10, 8, 7], [42, 12, 18], [v]+[u]=[v+u], lambda[v]=[lambda v], representatives_only_not_unique_quotient_elements)","explanation_zh":"Wave5封閉層核對=([10, 8, 7], [42, 12, 18], [v]+[u]=[v+u], lambda[v]=[lambda v], representatives_only_not_unique_quotient_elements)；依完整明示資料獨立重算；quotient元素是coset且代表元不唯一；[v]=[u] iff v-u在W；加法與scalar multiplication須與代表元無關；dim公式限有限維且零coset是W。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["商向量空間"]},"variant":{"family_id":"qf-mod37-quotient-coset-operations","seed":202611067031,"parameters":{"ambient_dimension":3,"basis_w":[[1,6,0]],"v":[7,2,3],"u":[3,6,4],"scalar":6,"domain":"Finite_dimensional_real_quotient_space","operation":"quotient-coset-operations","semantic_template":"qf-mod37-quotient-coset-operations/{\"ambient_dimension\":3,\"basis_w\":[[1,6,0]],\"scalar\":6,\"u\":[3,6,4],\"v\":[7,2,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r07-quotient-vector-space-0032","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-vector-space"],"title_zh":"商向量空間","stem_zh":"有限維實向量空間、明示子空間基底與exact membership。操作=quotient-coset-operations；資料={\"ambient_dimension\":3,\"basis_w\":[[1,7,0]],\"v\":[8,2,3],\"u\":[3,7,4],\"scalar\":7}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([11, 9, 7], [56, 14, 21], [v]+[u]=[v+u], lambda[v]=[lambda v], representatives_only_not_unique_quotient_elements)","explanation_zh":"Wave5封閉層核對=([11, 9, 7], [56, 14, 21], [v]+[u]=[v+u], lambda[v]=[lambda v], representatives_only_not_unique_quotient_elements)；依完整明示資料獨立重算；quotient元素是coset且代表元不唯一；[v]=[u] iff v-u在W；加法與scalar multiplication須與代表元無關；dim公式限有限維且零coset是W。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["商向量空間"]},"variant":{"family_id":"qf-mod37-quotient-coset-operations","seed":202611067032,"parameters":{"ambient_dimension":3,"basis_w":[[1,7,0]],"v":[8,2,3],"u":[3,7,4],"scalar":7,"domain":"Finite_dimensional_real_quotient_space","operation":"quotient-coset-operations","semantic_template":"qf-mod37-quotient-coset-operations/{\"ambient_dimension\":3,\"basis_w\":[[1,7,0]],\"scalar\":7,\"u\":[3,7,4],\"v\":[8,2,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r07-quotient-vector-space-0033","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-vector-space"],"title_zh":"商向量空間","stem_zh":"有限維實向量空間、明示子空間基底與exact membership。操作=quotient-coset-operations；資料={\"ambient_dimension\":3,\"basis_w\":[[1,8,0]],\"v\":[9,2,3],\"u\":[3,8,4],\"scalar\":8}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([12, 10, 7], [72, 16, 24], [v]+[u]=[v+u], lambda[v]=[lambda v], representatives_only_not_unique_quotient_elements)","explanation_zh":"Wave5封閉層核對=([12, 10, 7], [72, 16, 24], [v]+[u]=[v+u], lambda[v]=[lambda v], representatives_only_not_unique_quotient_elements)；依完整明示資料獨立重算；quotient元素是coset且代表元不唯一；[v]=[u] iff v-u在W；加法與scalar multiplication須與代表元無關；dim公式限有限維且零coset是W。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["商向量空間"]},"variant":{"family_id":"qf-mod37-quotient-coset-operations","seed":202611067033,"parameters":{"ambient_dimension":3,"basis_w":[[1,8,0]],"v":[9,2,3],"u":[3,8,4],"scalar":8,"domain":"Finite_dimensional_real_quotient_space","operation":"quotient-coset-operations","semantic_template":"qf-mod37-quotient-coset-operations/{\"ambient_dimension\":3,\"basis_w\":[[1,8,0]],\"scalar\":8,\"u\":[3,8,4],\"v\":[9,2,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r07-quotient-vector-space-0034","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-vector-space"],"title_zh":"商向量空間","stem_zh":"有限維實向量空間、明示子空間基底與exact membership。操作=quotient-coset-operations；資料={\"ambient_dimension\":3,\"basis_w\":[[1,9,0]],\"v\":[10,2,3],\"u\":[3,9,4],\"scalar\":9}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([13, 11, 7], [90, 18, 27], [v]+[u]=[v+u], lambda[v]=[lambda v], representatives_only_not_unique_quotient_elements)","explanation_zh":"Wave5封閉層核對=([13, 11, 7], [90, 18, 27], [v]+[u]=[v+u], lambda[v]=[lambda v], representatives_only_not_unique_quotient_elements)；依完整明示資料獨立重算；quotient元素是coset且代表元不唯一；[v]=[u] iff v-u在W；加法與scalar multiplication須與代表元無關；dim公式限有限維且零coset是W。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["商向量空間"]},"variant":{"family_id":"qf-mod37-quotient-coset-operations","seed":202611067034,"parameters":{"ambient_dimension":3,"basis_w":[[1,9,0]],"v":[10,2,3],"u":[3,9,4],"scalar":9,"domain":"Finite_dimensional_real_quotient_space","operation":"quotient-coset-operations","semantic_template":"qf-mod37-quotient-coset-operations/{\"ambient_dimension\":3,\"basis_w\":[[1,9,0]],\"scalar\":9,\"u\":[3,9,4],\"v\":[10,2,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r07-quotient-vector-space-0035","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-vector-space"],"title_zh":"商向量空間","stem_zh":"有限維實向量空間、明示子空間基底與exact membership。操作=quotient-coset-operations；資料={\"ambient_dimension\":3,\"basis_w\":[[1,10,0]],\"v\":[11,2,3],\"u\":[3,10,4],\"scalar\":10}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([14, 12, 7], [110, 20, 30], [v]+[u]=[v+u], lambda[v]=[lambda v], representatives_only_not_unique_quotient_elements)","explanation_zh":"Wave5封閉層核對=([14, 12, 7], [110, 20, 30], [v]+[u]=[v+u], lambda[v]=[lambda v], representatives_only_not_unique_quotient_elements)；依完整明示資料獨立重算；quotient元素是coset且代表元不唯一；[v]=[u] iff v-u在W；加法與scalar multiplication須與代表元無關；dim公式限有限維且零coset是W。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["商向量空間"]},"variant":{"family_id":"qf-mod37-quotient-coset-operations","seed":202611067035,"parameters":{"ambient_dimension":3,"basis_w":[[1,10,0]],"v":[11,2,3],"u":[3,10,4],"scalar":10,"domain":"Finite_dimensional_real_quotient_space","operation":"quotient-coset-operations","semantic_template":"qf-mod37-quotient-coset-operations/{\"ambient_dimension\":3,\"basis_w\":[[1,10,0]],\"scalar\":10,\"u\":[3,10,4],\"v\":[11,2,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r07-quotient-vector-space-0036","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-vector-space"],"title_zh":"商向量空間","stem_zh":"有限維實向量空間、明示子空間基底與exact membership。操作=quotient-coset-operations；資料={\"ambient_dimension\":3,\"basis_w\":[[1,11,0]],\"v\":[12,2,3],\"u\":[3,11,4],\"scalar\":11}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([15, 13, 7], [132, 22, 33], [v]+[u]=[v+u], lambda[v]=[lambda v], representatives_only_not_unique_quotient_elements)","explanation_zh":"Wave5封閉層核對=([15, 13, 7], [132, 22, 33], [v]+[u]=[v+u], lambda[v]=[lambda v], representatives_only_not_unique_quotient_elements)；依完整明示資料獨立重算；quotient元素是coset且代表元不唯一；[v]=[u] iff v-u在W；加法與scalar multiplication須與代表元無關；dim公式限有限維且零coset是W。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["商向量空間"]},"variant":{"family_id":"qf-mod37-quotient-coset-operations","seed":202611067036,"parameters":{"ambient_dimension":3,"basis_w":[[1,11,0]],"v":[12,2,3],"u":[3,11,4],"scalar":11,"domain":"Finite_dimensional_real_quotient_space","operation":"quotient-coset-operations","semantic_template":"qf-mod37-quotient-coset-operations/{\"ambient_dimension\":3,\"basis_w\":[[1,11,0]],\"scalar\":11,\"u\":[3,11,4],\"v\":[12,2,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r07-quotient-vector-space-0037","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-vector-space"],"title_zh":"商向量空間","stem_zh":"有限維實向量空間、明示子空間基底與exact membership。操作=quotient-coset-operations；資料={\"ambient_dimension\":3,\"basis_w\":[[1,12,0]],\"v\":[13,2,3],\"u\":[3,12,4],\"scalar\":12}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([16, 14, 7], [156, 24, 36], [v]+[u]=[v+u], lambda[v]=[lambda v], representatives_only_not_unique_quotient_elements)","explanation_zh":"Wave5封閉層核對=([16, 14, 7], [156, 24, 36], [v]+[u]=[v+u], lambda[v]=[lambda v], representatives_only_not_unique_quotient_elements)；依完整明示資料獨立重算；quotient元素是coset且代表元不唯一；[v]=[u] iff v-u在W；加法與scalar multiplication須與代表元無關；dim公式限有限維且零coset是W。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["商向量空間"]},"variant":{"family_id":"qf-mod37-quotient-coset-operations","seed":202611067037,"parameters":{"ambient_dimension":3,"basis_w":[[1,12,0]],"v":[13,2,3],"u":[3,12,4],"scalar":12,"domain":"Finite_dimensional_real_quotient_space","operation":"quotient-coset-operations","semantic_template":"qf-mod37-quotient-coset-operations/{\"ambient_dimension\":3,\"basis_w\":[[1,12,0]],\"scalar\":12,\"u\":[3,12,4],\"v\":[13,2,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r07-quotient-vector-space-0038","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-vector-space"],"title_zh":"商向量空間","stem_zh":"有限維實向量空間、明示子空間基底與exact membership。操作=quotient-coset-operations；資料={\"ambient_dimension\":3,\"basis_w\":[[1,13,0]],\"v\":[14,2,3],\"u\":[3,13,4],\"scalar\":13}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([17, 15, 7], [182, 26, 39], [v]+[u]=[v+u], lambda[v]=[lambda v], representatives_only_not_unique_quotient_elements)","explanation_zh":"Wave5封閉層核對=([17, 15, 7], [182, 26, 39], [v]+[u]=[v+u], lambda[v]=[lambda v], representatives_only_not_unique_quotient_elements)；依完整明示資料獨立重算；quotient元素是coset且代表元不唯一；[v]=[u] iff v-u在W；加法與scalar multiplication須與代表元無關；dim公式限有限維且零coset是W。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["商向量空間"]},"variant":{"family_id":"qf-mod37-quotient-coset-operations","seed":202611067038,"parameters":{"ambient_dimension":3,"basis_w":[[1,13,0]],"v":[14,2,3],"u":[3,13,4],"scalar":13,"domain":"Finite_dimensional_real_quotient_space","operation":"quotient-coset-operations","semantic_template":"qf-mod37-quotient-coset-operations/{\"ambient_dimension\":3,\"basis_w\":[[1,13,0]],\"scalar\":13,\"u\":[3,13,4],\"v\":[14,2,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r07-quotient-vector-space-0039","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-vector-space"],"title_zh":"商向量空間","stem_zh":"有限維實向量空間、明示子空間基底與exact membership。操作=quotient-coset-operations；資料={\"ambient_dimension\":3,\"basis_w\":[[1,14,0]],\"v\":[15,2,3],\"u\":[3,14,4],\"scalar\":14}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([18, 16, 7], [210, 28, 42], [v]+[u]=[v+u], lambda[v]=[lambda v], representatives_only_not_unique_quotient_elements)","explanation_zh":"Wave5封閉層核對=([18, 16, 7], [210, 28, 42], [v]+[u]=[v+u], lambda[v]=[lambda v], representatives_only_not_unique_quotient_elements)；依完整明示資料獨立重算；quotient元素是coset且代表元不唯一；[v]=[u] iff v-u在W；加法與scalar multiplication須與代表元無關；dim公式限有限維且零coset是W。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["商向量空間"]},"variant":{"family_id":"qf-mod37-quotient-coset-operations","seed":202611067039,"parameters":{"ambient_dimension":3,"basis_w":[[1,14,0]],"v":[15,2,3],"u":[3,14,4],"scalar":14,"domain":"Finite_dimensional_real_quotient_space","operation":"quotient-coset-operations","semantic_template":"qf-mod37-quotient-coset-operations/{\"ambient_dimension\":3,\"basis_w\":[[1,14,0]],\"scalar\":14,\"u\":[3,14,4],\"v\":[15,2,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r07-quotient-vector-space-0040","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-vector-space"],"title_zh":"商向量空間","stem_zh":"有限維實向量空間、明示子空間基底與exact membership。操作=quotient-coset-operations；資料={\"ambient_dimension\":3,\"basis_w\":[[1,15,0]],\"v\":[16,2,3],\"u\":[3,15,4],\"scalar\":15}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([19, 17, 7], [240, 30, 45], [v]+[u]=[v+u], lambda[v]=[lambda v], representatives_only_not_unique_quotient_elements)","explanation_zh":"Wave5封閉層核對=([19, 17, 7], [240, 30, 45], [v]+[u]=[v+u], lambda[v]=[lambda v], representatives_only_not_unique_quotient_elements)；依完整明示資料獨立重算；quotient元素是coset且代表元不唯一；[v]=[u] iff v-u在W；加法與scalar multiplication須與代表元無關；dim公式限有限維且零coset是W。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["商向量空間"]},"variant":{"family_id":"qf-mod37-quotient-coset-operations","seed":202611067040,"parameters":{"ambient_dimension":3,"basis_w":[[1,15,0]],"v":[16,2,3],"u":[3,15,4],"scalar":15,"domain":"Finite_dimensional_real_quotient_space","operation":"quotient-coset-operations","semantic_template":"qf-mod37-quotient-coset-operations/{\"ambient_dimension\":3,\"basis_w\":[[1,15,0]],\"scalar\":15,\"u\":[3,15,4],\"v\":[16,2,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r07-quotient-vector-space-0041","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-vector-space"],"title_zh":"商向量空間","stem_zh":"有限維實向量空間、明示子空間基底與exact membership。操作=quotient-coset-operations；資料={\"ambient_dimension\":3,\"basis_w\":[[1,16,0]],\"v\":[17,2,3],\"u\":[3,16,4],\"scalar\":16}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([20, 18, 7], [272, 32, 48], [v]+[u]=[v+u], lambda[v]=[lambda v], representatives_only_not_unique_quotient_elements)","explanation_zh":"Wave5封閉層核對=([20, 18, 7], [272, 32, 48], [v]+[u]=[v+u], lambda[v]=[lambda v], representatives_only_not_unique_quotient_elements)；依完整明示資料獨立重算；quotient元素是coset且代表元不唯一；[v]=[u] iff v-u在W；加法與scalar multiplication須與代表元無關；dim公式限有限維且零coset是W。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["商向量空間"]},"variant":{"family_id":"qf-mod37-quotient-coset-operations","seed":202611067041,"parameters":{"ambient_dimension":3,"basis_w":[[1,16,0]],"v":[17,2,3],"u":[3,16,4],"scalar":16,"domain":"Finite_dimensional_real_quotient_space","operation":"quotient-coset-operations","semantic_template":"qf-mod37-quotient-coset-operations/{\"ambient_dimension\":3,\"basis_w\":[[1,16,0]],\"scalar\":16,\"u\":[3,16,4],\"v\":[17,2,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r07-quotient-vector-space-0042","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-vector-space"],"title_zh":"商向量空間","stem_zh":"有限維實向量空間、明示子空間基底與exact membership。操作=quotient-coset-operations；資料={\"ambient_dimension\":3,\"basis_w\":[[1,17,0]],\"v\":[18,2,3],\"u\":[3,17,4],\"scalar\":17}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([21, 19, 7], [306, 34, 51], [v]+[u]=[v+u], lambda[v]=[lambda v], representatives_only_not_unique_quotient_elements)","explanation_zh":"Wave5封閉層核對=([21, 19, 7], [306, 34, 51], [v]+[u]=[v+u], lambda[v]=[lambda v], representatives_only_not_unique_quotient_elements)；依完整明示資料獨立重算；quotient元素是coset且代表元不唯一；[v]=[u] iff v-u在W；加法與scalar multiplication須與代表元無關；dim公式限有限維且零coset是W。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["商向量空間"]},"variant":{"family_id":"qf-mod37-quotient-coset-operations","seed":202611067042,"parameters":{"ambient_dimension":3,"basis_w":[[1,17,0]],"v":[18,2,3],"u":[3,17,4],"scalar":17,"domain":"Finite_dimensional_real_quotient_space","operation":"quotient-coset-operations","semantic_template":"qf-mod37-quotient-coset-operations/{\"ambient_dimension\":3,\"basis_w\":[[1,17,0]],\"scalar\":17,\"u\":[3,17,4],\"v\":[18,2,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r07-quotient-vector-space-0043","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-vector-space"],"title_zh":"商向量空間","stem_zh":"有限維實向量空間、明示子空間基底與exact membership。操作=quotient-coset-operations；資料={\"ambient_dimension\":3,\"basis_w\":[[1,18,0]],\"v\":[19,2,3],\"u\":[3,18,4],\"scalar\":18}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([22, 20, 7], [342, 36, 54], [v]+[u]=[v+u], lambda[v]=[lambda v], representatives_only_not_unique_quotient_elements)","explanation_zh":"Wave5封閉層核對=([22, 20, 7], [342, 36, 54], [v]+[u]=[v+u], lambda[v]=[lambda v], representatives_only_not_unique_quotient_elements)；依完整明示資料獨立重算；quotient元素是coset且代表元不唯一；[v]=[u] iff v-u在W；加法與scalar multiplication須與代表元無關；dim公式限有限維且零coset是W。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["商向量空間"]},"variant":{"family_id":"qf-mod37-quotient-coset-operations","seed":202611067043,"parameters":{"ambient_dimension":3,"basis_w":[[1,18,0]],"v":[19,2,3],"u":[3,18,4],"scalar":18,"domain":"Finite_dimensional_real_quotient_space","operation":"quotient-coset-operations","semantic_template":"qf-mod37-quotient-coset-operations/{\"ambient_dimension\":3,\"basis_w\":[[1,18,0]],\"scalar\":18,\"u\":[3,18,4],\"v\":[19,2,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r07-quotient-vector-space-0044","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-vector-space"],"title_zh":"商向量空間","stem_zh":"有限維實向量空間、明示子空間基底與exact membership。操作=quotient-coset-operations；資料={\"ambient_dimension\":3,\"basis_w\":[[1,19,0]],\"v\":[20,2,3],\"u\":[3,19,4],\"scalar\":19}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([23, 21, 7], [380, 38, 57], [v]+[u]=[v+u], lambda[v]=[lambda v], representatives_only_not_unique_quotient_elements)","explanation_zh":"Wave5封閉層核對=([23, 21, 7], [380, 38, 57], [v]+[u]=[v+u], lambda[v]=[lambda v], representatives_only_not_unique_quotient_elements)；依完整明示資料獨立重算；quotient元素是coset且代表元不唯一；[v]=[u] iff v-u在W；加法與scalar multiplication須與代表元無關；dim公式限有限維且零coset是W。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["商向量空間"]},"variant":{"family_id":"qf-mod37-quotient-coset-operations","seed":202611067044,"parameters":{"ambient_dimension":3,"basis_w":[[1,19,0]],"v":[20,2,3],"u":[3,19,4],"scalar":19,"domain":"Finite_dimensional_real_quotient_space","operation":"quotient-coset-operations","semantic_template":"qf-mod37-quotient-coset-operations/{\"ambient_dimension\":3,\"basis_w\":[[1,19,0]],\"scalar\":19,\"u\":[3,19,4],\"v\":[20,2,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r07-quotient-vector-space-0045","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-vector-space"],"title_zh":"商向量空間","stem_zh":"有限維實向量空間、明示子空間基底與exact membership。操作=quotient-coset-operations；資料={\"ambient_dimension\":3,\"basis_w\":[[1,20,0]],\"v\":[21,2,3],\"u\":[3,20,4],\"scalar\":20}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([24, 22, 7], [420, 40, 60], [v]+[u]=[v+u], lambda[v]=[lambda v], representatives_only_not_unique_quotient_elements)","explanation_zh":"Wave5封閉層核對=([24, 22, 7], [420, 40, 60], [v]+[u]=[v+u], lambda[v]=[lambda v], representatives_only_not_unique_quotient_elements)；依完整明示資料獨立重算；quotient元素是coset且代表元不唯一；[v]=[u] iff v-u在W；加法與scalar multiplication須與代表元無關；dim公式限有限維且零coset是W。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["商向量空間"]},"variant":{"family_id":"qf-mod37-quotient-coset-operations","seed":202611067045,"parameters":{"ambient_dimension":3,"basis_w":[[1,20,0]],"v":[21,2,3],"u":[3,20,4],"scalar":20,"domain":"Finite_dimensional_real_quotient_space","operation":"quotient-coset-operations","semantic_template":"qf-mod37-quotient-coset-operations/{\"ambient_dimension\":3,\"basis_w\":[[1,20,0]],\"scalar\":20,\"u\":[3,20,4],\"v\":[21,2,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r07-quotient-vector-space-0046","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-vector-space"],"title_zh":"商向量空間","stem_zh":"有限維實向量空間、明示子空間基底與exact membership。操作=quotient-coset-operations；資料={\"ambient_dimension\":3,\"basis_w\":[[1,21,0]],\"v\":[22,2,3],\"u\":[3,21,4],\"scalar\":21}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([25, 23, 7], [462, 42, 63], [v]+[u]=[v+u], lambda[v]=[lambda v], representatives_only_not_unique_quotient_elements)","explanation_zh":"Wave5封閉層核對=([25, 23, 7], [462, 42, 63], [v]+[u]=[v+u], lambda[v]=[lambda v], representatives_only_not_unique_quotient_elements)；依完整明示資料獨立重算；quotient元素是coset且代表元不唯一；[v]=[u] iff v-u在W；加法與scalar multiplication須與代表元無關；dim公式限有限維且零coset是W。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["商向量空間"]},"variant":{"family_id":"qf-mod37-quotient-coset-operations","seed":202611067046,"parameters":{"ambient_dimension":3,"basis_w":[[1,21,0]],"v":[22,2,3],"u":[3,21,4],"scalar":21,"domain":"Finite_dimensional_real_quotient_space","operation":"quotient-coset-operations","semantic_template":"qf-mod37-quotient-coset-operations/{\"ambient_dimension\":3,\"basis_w\":[[1,21,0]],\"scalar\":21,\"u\":[3,21,4],\"v\":[22,2,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r07-quotient-vector-space-0047","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-vector-space"],"title_zh":"商向量空間","stem_zh":"有限維實向量空間、明示子空間基底與exact membership。操作=quotient-coset-operations；資料={\"ambient_dimension\":3,\"basis_w\":[[1,22,0]],\"v\":[23,2,3],\"u\":[3,22,4],\"scalar\":22}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([26, 24, 7], [506, 44, 66], [v]+[u]=[v+u], lambda[v]=[lambda v], representatives_only_not_unique_quotient_elements)","explanation_zh":"Wave5封閉層核對=([26, 24, 7], [506, 44, 66], [v]+[u]=[v+u], lambda[v]=[lambda v], representatives_only_not_unique_quotient_elements)；依完整明示資料獨立重算；quotient元素是coset且代表元不唯一；[v]=[u] iff v-u在W；加法與scalar multiplication須與代表元無關；dim公式限有限維且零coset是W。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["商向量空間"]},"variant":{"family_id":"qf-mod37-quotient-coset-operations","seed":202611067047,"parameters":{"ambient_dimension":3,"basis_w":[[1,22,0]],"v":[23,2,3],"u":[3,22,4],"scalar":22,"domain":"Finite_dimensional_real_quotient_space","operation":"quotient-coset-operations","semantic_template":"qf-mod37-quotient-coset-operations/{\"ambient_dimension\":3,\"basis_w\":[[1,22,0]],\"scalar\":22,\"u\":[3,22,4],\"v\":[23,2,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r07-quotient-vector-space-0048","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-vector-space"],"title_zh":"商向量空間","stem_zh":"有限維實向量空間、明示子空間基底與exact membership。操作=quotient-coset-operations；資料={\"ambient_dimension\":3,\"basis_w\":[[1,23,0]],\"v\":[24,2,3],\"u\":[3,23,4],\"scalar\":23}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([27, 25, 7], [552, 46, 69], [v]+[u]=[v+u], lambda[v]=[lambda v], representatives_only_not_unique_quotient_elements)","explanation_zh":"Wave5封閉層核對=([27, 25, 7], [552, 46, 69], [v]+[u]=[v+u], lambda[v]=[lambda v], representatives_only_not_unique_quotient_elements)；依完整明示資料獨立重算；quotient元素是coset且代表元不唯一；[v]=[u] iff v-u在W；加法與scalar multiplication須與代表元無關；dim公式限有限維且零coset是W。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["商向量空間"]},"variant":{"family_id":"qf-mod37-quotient-coset-operations","seed":202611067048,"parameters":{"ambient_dimension":3,"basis_w":[[1,23,0]],"v":[24,2,3],"u":[3,23,4],"scalar":23,"domain":"Finite_dimensional_real_quotient_space","operation":"quotient-coset-operations","semantic_template":"qf-mod37-quotient-coset-operations/{\"ambient_dimension\":3,\"basis_w\":[[1,23,0]],\"scalar\":23,\"u\":[3,23,4],\"v\":[24,2,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r07-quotient-vector-space-0049","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-vector-space"],"title_zh":"商向量空間","stem_zh":"有限維實向量空間、明示子空間基底與exact membership。操作=quotient-coset-operations；資料={\"ambient_dimension\":3,\"basis_w\":[[1,24,0]],\"v\":[25,2,3],\"u\":[3,24,4],\"scalar\":24}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([28, 26, 7], [600, 48, 72], [v]+[u]=[v+u], lambda[v]=[lambda v], representatives_only_not_unique_quotient_elements)","explanation_zh":"Wave5封閉層核對=([28, 26, 7], [600, 48, 72], [v]+[u]=[v+u], lambda[v]=[lambda v], representatives_only_not_unique_quotient_elements)；依完整明示資料獨立重算；quotient元素是coset且代表元不唯一；[v]=[u] iff v-u在W；加法與scalar multiplication須與代表元無關；dim公式限有限維且零coset是W。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["商向量空間"]},"variant":{"family_id":"qf-mod37-quotient-coset-operations","seed":202611067049,"parameters":{"ambient_dimension":3,"basis_w":[[1,24,0]],"v":[25,2,3],"u":[3,24,4],"scalar":24,"domain":"Finite_dimensional_real_quotient_space","operation":"quotient-coset-operations","semantic_template":"qf-mod37-quotient-coset-operations/{\"ambient_dimension\":3,\"basis_w\":[[1,24,0]],\"scalar\":24,\"u\":[3,24,4],\"v\":[25,2,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r07-quotient-vector-space-0050","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-vector-space"],"title_zh":"商向量空間","stem_zh":"有限維實向量空間、明示子空間基底與exact membership。操作=quotient-coset-operations；資料={\"ambient_dimension\":3,\"basis_w\":[[1,25,0]],\"v\":[26,2,3],\"u\":[3,25,4],\"scalar\":25}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([29, 27, 7], [650, 50, 75], [v]+[u]=[v+u], lambda[v]=[lambda v], representatives_only_not_unique_quotient_elements)","explanation_zh":"Wave5封閉層核對=([29, 27, 7], [650, 50, 75], [v]+[u]=[v+u], lambda[v]=[lambda v], representatives_only_not_unique_quotient_elements)；依完整明示資料獨立重算；quotient元素是coset且代表元不唯一；[v]=[u] iff v-u在W；加法與scalar multiplication須與代表元無關；dim公式限有限維且零coset是W。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["商向量空間"]},"variant":{"family_id":"qf-mod37-quotient-coset-operations","seed":202611067050,"parameters":{"ambient_dimension":3,"basis_w":[[1,25,0]],"v":[26,2,3],"u":[3,25,4],"scalar":25,"domain":"Finite_dimensional_real_quotient_space","operation":"quotient-coset-operations","semantic_template":"qf-mod37-quotient-coset-operations/{\"ambient_dimension\":3,\"basis_w\":[[1,25,0]],\"scalar\":25,\"u\":[3,25,4],\"v\":[26,2,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r07-quotient-vector-space-0051","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-vector-space"],"title_zh":"商向量空間","stem_zh":"有限維實向量空間、明示子空間基底與exact membership。操作=quotient-representative-independence；資料={\"ambient_dimension\":3,\"basis_w\":[[1,1,0]],\"v\":[2,2,3],\"u\":[3,1,4],\"shift_v_coefficients\":[2],\"shift_u_coefficients\":[-1],\"scalar\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(是, 是, 是, 是, 是, addition_and_scalar_multiplication_independent_of_representatives)","explanation_zh":"Wave5封閉層核對=(是, 是, 是, 是, 是, addition_and_scalar_multiplication_independent_of_representatives)；依完整明示資料獨立重算；quotient元素是coset且代表元不唯一；[v]=[u] iff v-u在W；加法與scalar multiplication須與代表元無關；dim公式限有限維且零coset是W。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["商向量空間"]},"variant":{"family_id":"qf-mod37-quotient-representative-independence","seed":202611067051,"parameters":{"ambient_dimension":3,"basis_w":[[1,1,0]],"v":[2,2,3],"u":[3,1,4],"shift_v_coefficients":[2],"shift_u_coefficients":[-1],"scalar":1,"domain":"Finite_dimensional_real_quotient_space","operation":"quotient-representative-independence","semantic_template":"qf-mod37-quotient-representative-independence/{\"ambient_dimension\":3,\"basis_w\":[[1,1,0]],\"scalar\":1,\"shift_u_coefficients\":[-1],\"shift_v_coefficients\":[2],\"u\":[3,1,4],\"v\":[2,2,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r07-quotient-vector-space-0052","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-vector-space"],"title_zh":"商向量空間","stem_zh":"有限維實向量空間、明示子空間基底與exact membership。操作=quotient-representative-independence；資料={\"ambient_dimension\":3,\"basis_w\":[[1,2,0]],\"v\":[3,2,3],\"u\":[3,2,4],\"shift_v_coefficients\":[3],\"shift_u_coefficients\":[-2],\"scalar\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(是, 是, 是, 是, 是, addition_and_scalar_multiplication_independent_of_representatives)","explanation_zh":"Wave5封閉層核對=(是, 是, 是, 是, 是, addition_and_scalar_multiplication_independent_of_representatives)；依完整明示資料獨立重算；quotient元素是coset且代表元不唯一；[v]=[u] iff v-u在W；加法與scalar multiplication須與代表元無關；dim公式限有限維且零coset是W。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["商向量空間"]},"variant":{"family_id":"qf-mod37-quotient-representative-independence","seed":202611067052,"parameters":{"ambient_dimension":3,"basis_w":[[1,2,0]],"v":[3,2,3],"u":[3,2,4],"shift_v_coefficients":[3],"shift_u_coefficients":[-2],"scalar":2,"domain":"Finite_dimensional_real_quotient_space","operation":"quotient-representative-independence","semantic_template":"qf-mod37-quotient-representative-independence/{\"ambient_dimension\":3,\"basis_w\":[[1,2,0]],\"scalar\":2,\"shift_u_coefficients\":[-2],\"shift_v_coefficients\":[3],\"u\":[3,2,4],\"v\":[3,2,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r07-quotient-vector-space-0053","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-vector-space"],"title_zh":"商向量空間","stem_zh":"有限維實向量空間、明示子空間基底與exact membership。操作=quotient-representative-independence；資料={\"ambient_dimension\":3,\"basis_w\":[[1,3,0]],\"v\":[4,2,3],\"u\":[3,3,4],\"shift_v_coefficients\":[4],\"shift_u_coefficients\":[-3],\"scalar\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(是, 是, 是, 是, 是, addition_and_scalar_multiplication_independent_of_representatives)","explanation_zh":"Wave5封閉層核對=(是, 是, 是, 是, 是, addition_and_scalar_multiplication_independent_of_representatives)；依完整明示資料獨立重算；quotient元素是coset且代表元不唯一；[v]=[u] iff v-u在W；加法與scalar multiplication須與代表元無關；dim公式限有限維且零coset是W。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["商向量空間"]},"variant":{"family_id":"qf-mod37-quotient-representative-independence","seed":202611067053,"parameters":{"ambient_dimension":3,"basis_w":[[1,3,0]],"v":[4,2,3],"u":[3,3,4],"shift_v_coefficients":[4],"shift_u_coefficients":[-3],"scalar":3,"domain":"Finite_dimensional_real_quotient_space","operation":"quotient-representative-independence","semantic_template":"qf-mod37-quotient-representative-independence/{\"ambient_dimension\":3,\"basis_w\":[[1,3,0]],\"scalar\":3,\"shift_u_coefficients\":[-3],\"shift_v_coefficients\":[4],\"u\":[3,3,4],\"v\":[4,2,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r07-quotient-vector-space-0054","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-vector-space"],"title_zh":"商向量空間","stem_zh":"有限維實向量空間、明示子空間基底與exact membership。操作=quotient-representative-independence；資料={\"ambient_dimension\":3,\"basis_w\":[[1,4,0]],\"v\":[5,2,3],\"u\":[3,4,4],\"shift_v_coefficients\":[5],\"shift_u_coefficients\":[-4],\"scalar\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(是, 是, 是, 是, 是, addition_and_scalar_multiplication_independent_of_representatives)","explanation_zh":"Wave5封閉層核對=(是, 是, 是, 是, 是, addition_and_scalar_multiplication_independent_of_representatives)；依完整明示資料獨立重算；quotient元素是coset且代表元不唯一；[v]=[u] iff v-u在W；加法與scalar multiplication須與代表元無關；dim公式限有限維且零coset是W。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["商向量空間"]},"variant":{"family_id":"qf-mod37-quotient-representative-independence","seed":202611067054,"parameters":{"ambient_dimension":3,"basis_w":[[1,4,0]],"v":[5,2,3],"u":[3,4,4],"shift_v_coefficients":[5],"shift_u_coefficients":[-4],"scalar":4,"domain":"Finite_dimensional_real_quotient_space","operation":"quotient-representative-independence","semantic_template":"qf-mod37-quotient-representative-independence/{\"ambient_dimension\":3,\"basis_w\":[[1,4,0]],\"scalar\":4,\"shift_u_coefficients\":[-4],\"shift_v_coefficients\":[5],\"u\":[3,4,4],\"v\":[5,2,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r07-quotient-vector-space-0055","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-vector-space"],"title_zh":"商向量空間","stem_zh":"有限維實向量空間、明示子空間基底與exact membership。操作=quotient-representative-independence；資料={\"ambient_dimension\":3,\"basis_w\":[[1,5,0]],\"v\":[6,2,3],\"u\":[3,5,4],\"shift_v_coefficients\":[6],\"shift_u_coefficients\":[-5],\"scalar\":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(是, 是, 是, 是, 是, addition_and_scalar_multiplication_independent_of_representatives)","explanation_zh":"Wave5封閉層核對=(是, 是, 是, 是, 是, addition_and_scalar_multiplication_independent_of_representatives)；依完整明示資料獨立重算；quotient元素是coset且代表元不唯一；[v]=[u] iff v-u在W；加法與scalar multiplication須與代表元無關；dim公式限有限維且零coset是W。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["商向量空間"]},"variant":{"family_id":"qf-mod37-quotient-representative-independence","seed":202611067055,"parameters":{"ambient_dimension":3,"basis_w":[[1,5,0]],"v":[6,2,3],"u":[3,5,4],"shift_v_coefficients":[6],"shift_u_coefficients":[-5],"scalar":5,"domain":"Finite_dimensional_real_quotient_space","operation":"quotient-representative-independence","semantic_template":"qf-mod37-quotient-representative-independence/{\"ambient_dimension\":3,\"basis_w\":[[1,5,0]],\"scalar\":5,\"shift_u_coefficients\":[-5],\"shift_v_coefficients\":[6],\"u\":[3,5,4],\"v\":[6,2,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r07-quotient-vector-space-0056","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-vector-space"],"title_zh":"商向量空間","stem_zh":"有限維實向量空間、明示子空間基底與exact membership。操作=quotient-representative-independence；資料={\"ambient_dimension\":3,\"basis_w\":[[1,6,0]],\"v\":[7,2,3],\"u\":[3,6,4],\"shift_v_coefficients\":[7],\"shift_u_coefficients\":[-6],\"scalar\":6}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(是, 是, 是, 是, 是, addition_and_scalar_multiplication_independent_of_representatives)","explanation_zh":"Wave5封閉層核對=(是, 是, 是, 是, 是, addition_and_scalar_multiplication_independent_of_representatives)；依完整明示資料獨立重算；quotient元素是coset且代表元不唯一；[v]=[u] iff v-u在W；加法與scalar multiplication須與代表元無關；dim公式限有限維且零coset是W。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["商向量空間"]},"variant":{"family_id":"qf-mod37-quotient-representative-independence","seed":202611067056,"parameters":{"ambient_dimension":3,"basis_w":[[1,6,0]],"v":[7,2,3],"u":[3,6,4],"shift_v_coefficients":[7],"shift_u_coefficients":[-6],"scalar":6,"domain":"Finite_dimensional_real_quotient_space","operation":"quotient-representative-independence","semantic_template":"qf-mod37-quotient-representative-independence/{\"ambient_dimension\":3,\"basis_w\":[[1,6,0]],\"scalar\":6,\"shift_u_coefficients\":[-6],\"shift_v_coefficients\":[7],\"u\":[3,6,4],\"v\":[7,2,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r07-quotient-vector-space-0057","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-vector-space"],"title_zh":"商向量空間","stem_zh":"有限維實向量空間、明示子空間基底與exact membership。操作=quotient-representative-independence；資料={\"ambient_dimension\":3,\"basis_w\":[[1,7,0]],\"v\":[8,2,3],\"u\":[3,7,4],\"shift_v_coefficients\":[8],\"shift_u_coefficients\":[-7],\"scalar\":7}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(是, 是, 是, 是, 是, addition_and_scalar_multiplication_independent_of_representatives)","explanation_zh":"Wave5封閉層核對=(是, 是, 是, 是, 是, addition_and_scalar_multiplication_independent_of_representatives)；依完整明示資料獨立重算；quotient元素是coset且代表元不唯一；[v]=[u] iff v-u在W；加法與scalar multiplication須與代表元無關；dim公式限有限維且零coset是W。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["商向量空間"]},"variant":{"family_id":"qf-mod37-quotient-representative-independence","seed":202611067057,"parameters":{"ambient_dimension":3,"basis_w":[[1,7,0]],"v":[8,2,3],"u":[3,7,4],"shift_v_coefficients":[8],"shift_u_coefficients":[-7],"scalar":7,"domain":"Finite_dimensional_real_quotient_space","operation":"quotient-representative-independence","semantic_template":"qf-mod37-quotient-representative-independence/{\"ambient_dimension\":3,\"basis_w\":[[1,7,0]],\"scalar\":7,\"shift_u_coefficients\":[-7],\"shift_v_coefficients\":[8],\"u\":[3,7,4],\"v\":[8,2,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r07-quotient-vector-space-0058","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-vector-space"],"title_zh":"商向量空間","stem_zh":"有限維實向量空間、明示子空間基底與exact membership。操作=quotient-representative-independence；資料={\"ambient_dimension\":3,\"basis_w\":[[1,8,0]],\"v\":[9,2,3],\"u\":[3,8,4],\"shift_v_coefficients\":[9],\"shift_u_coefficients\":[-8],\"scalar\":8}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(是, 是, 是, 是, 是, addition_and_scalar_multiplication_independent_of_representatives)","explanation_zh":"Wave5封閉層核對=(是, 是, 是, 是, 是, addition_and_scalar_multiplication_independent_of_representatives)；依完整明示資料獨立重算；quotient元素是coset且代表元不唯一；[v]=[u] iff v-u在W；加法與scalar multiplication須與代表元無關；dim公式限有限維且零coset是W。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["商向量空間"]},"variant":{"family_id":"qf-mod37-quotient-representative-independence","seed":202611067058,"parameters":{"ambient_dimension":3,"basis_w":[[1,8,0]],"v":[9,2,3],"u":[3,8,4],"shift_v_coefficients":[9],"shift_u_coefficients":[-8],"scalar":8,"domain":"Finite_dimensional_real_quotient_space","operation":"quotient-representative-independence","semantic_template":"qf-mod37-quotient-representative-independence/{\"ambient_dimension\":3,\"basis_w\":[[1,8,0]],\"scalar\":8,\"shift_u_coefficients\":[-8],\"shift_v_coefficients\":[9],\"u\":[3,8,4],\"v\":[9,2,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r07-quotient-vector-space-0059","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-vector-space"],"title_zh":"商向量空間","stem_zh":"有限維實向量空間、明示子空間基底與exact membership。操作=quotient-representative-independence；資料={\"ambient_dimension\":3,\"basis_w\":[[1,9,0]],\"v\":[10,2,3],\"u\":[3,9,4],\"shift_v_coefficients\":[10],\"shift_u_coefficients\":[-9],\"scalar\":9}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(是, 是, 是, 是, 是, addition_and_scalar_multiplication_independent_of_representatives)","explanation_zh":"Wave5封閉層核對=(是, 是, 是, 是, 是, addition_and_scalar_multiplication_independent_of_representatives)；依完整明示資料獨立重算；quotient元素是coset且代表元不唯一；[v]=[u] iff v-u在W；加法與scalar multiplication須與代表元無關；dim公式限有限維且零coset是W。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["商向量空間"]},"variant":{"family_id":"qf-mod37-quotient-representative-independence","seed":202611067059,"parameters":{"ambient_dimension":3,"basis_w":[[1,9,0]],"v":[10,2,3],"u":[3,9,4],"shift_v_coefficients":[10],"shift_u_coefficients":[-9],"scalar":9,"domain":"Finite_dimensional_real_quotient_space","operation":"quotient-representative-independence","semantic_template":"qf-mod37-quotient-representative-independence/{\"ambient_dimension\":3,\"basis_w\":[[1,9,0]],\"scalar\":9,\"shift_u_coefficients\":[-9],\"shift_v_coefficients\":[10],\"u\":[3,9,4],\"v\":[10,2,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r07-quotient-vector-space-0060","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-vector-space"],"title_zh":"商向量空間","stem_zh":"有限維實向量空間、明示子空間基底與exact membership。操作=quotient-representative-independence；資料={\"ambient_dimension\":3,\"basis_w\":[[1,10,0]],\"v\":[11,2,3],\"u\":[3,10,4],\"shift_v_coefficients\":[11],\"shift_u_coefficients\":[-10],\"scalar\":10}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(是, 是, 是, 是, 是, addition_and_scalar_multiplication_independent_of_representatives)","explanation_zh":"Wave5封閉層核對=(是, 是, 是, 是, 是, addition_and_scalar_multiplication_independent_of_representatives)；依完整明示資料獨立重算；quotient元素是coset且代表元不唯一；[v]=[u] iff v-u在W；加法與scalar multiplication須與代表元無關；dim公式限有限維且零coset是W。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["商向量空間"]},"variant":{"family_id":"qf-mod37-quotient-representative-independence","seed":202611067060,"parameters":{"ambient_dimension":3,"basis_w":[[1,10,0]],"v":[11,2,3],"u":[3,10,4],"shift_v_coefficients":[11],"shift_u_coefficients":[-10],"scalar":10,"domain":"Finite_dimensional_real_quotient_space","operation":"quotient-representative-independence","semantic_template":"qf-mod37-quotient-representative-independence/{\"ambient_dimension\":3,\"basis_w\":[[1,10,0]],\"scalar\":10,\"shift_u_coefficients\":[-10],\"shift_v_coefficients\":[11],\"u\":[3,10,4],\"v\":[11,2,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r07-quotient-vector-space-0061","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-vector-space"],"title_zh":"商向量空間","stem_zh":"有限維實向量空間、明示子空間基底與exact membership。操作=quotient-representative-independence；資料={\"ambient_dimension\":3,\"basis_w\":[[1,11,0]],\"v\":[12,2,3],\"u\":[3,11,4],\"shift_v_coefficients\":[12],\"shift_u_coefficients\":[-11],\"scalar\":11}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(是, 是, 是, 是, 是, addition_and_scalar_multiplication_independent_of_representatives)","explanation_zh":"Wave5封閉層核對=(是, 是, 是, 是, 是, addition_and_scalar_multiplication_independent_of_representatives)；依完整明示資料獨立重算；quotient元素是coset且代表元不唯一；[v]=[u] iff v-u在W；加法與scalar multiplication須與代表元無關；dim公式限有限維且零coset是W。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["商向量空間"]},"variant":{"family_id":"qf-mod37-quotient-representative-independence","seed":202611067061,"parameters":{"ambient_dimension":3,"basis_w":[[1,11,0]],"v":[12,2,3],"u":[3,11,4],"shift_v_coefficients":[12],"shift_u_coefficients":[-11],"scalar":11,"domain":"Finite_dimensional_real_quotient_space","operation":"quotient-representative-independence","semantic_template":"qf-mod37-quotient-representative-independence/{\"ambient_dimension\":3,\"basis_w\":[[1,11,0]],\"scalar\":11,\"shift_u_coefficients\":[-11],\"shift_v_coefficients\":[12],\"u\":[3,11,4],\"v\":[12,2,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r07-quotient-vector-space-0062","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-vector-space"],"title_zh":"商向量空間","stem_zh":"有限維實向量空間、明示子空間基底與exact membership。操作=quotient-representative-independence；資料={\"ambient_dimension\":3,\"basis_w\":[[1,12,0]],\"v\":[13,2,3],\"u\":[3,12,4],\"shift_v_coefficients\":[13],\"shift_u_coefficients\":[-12],\"scalar\":12}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(是, 是, 是, 是, 是, addition_and_scalar_multiplication_independent_of_representatives)","explanation_zh":"Wave5封閉層核對=(是, 是, 是, 是, 是, addition_and_scalar_multiplication_independent_of_representatives)；依完整明示資料獨立重算；quotient元素是coset且代表元不唯一；[v]=[u] iff v-u在W；加法與scalar multiplication須與代表元無關；dim公式限有限維且零coset是W。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["商向量空間"]},"variant":{"family_id":"qf-mod37-quotient-representative-independence","seed":202611067062,"parameters":{"ambient_dimension":3,"basis_w":[[1,12,0]],"v":[13,2,3],"u":[3,12,4],"shift_v_coefficients":[13],"shift_u_coefficients":[-12],"scalar":12,"domain":"Finite_dimensional_real_quotient_space","operation":"quotient-representative-independence","semantic_template":"qf-mod37-quotient-representative-independence/{\"ambient_dimension\":3,\"basis_w\":[[1,12,0]],\"scalar\":12,\"shift_u_coefficients\":[-12],\"shift_v_coefficients\":[13],\"u\":[3,12,4],\"v\":[13,2,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r07-quotient-vector-space-0063","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-vector-space"],"title_zh":"商向量空間","stem_zh":"有限維實向量空間、明示子空間基底與exact membership。操作=quotient-representative-independence；資料={\"ambient_dimension\":3,\"basis_w\":[[1,13,0]],\"v\":[14,2,3],\"u\":[3,13,4],\"shift_v_coefficients\":[14],\"shift_u_coefficients\":[-13],\"scalar\":13}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(是, 是, 是, 是, 是, addition_and_scalar_multiplication_independent_of_representatives)","explanation_zh":"Wave5封閉層核對=(是, 是, 是, 是, 是, addition_and_scalar_multiplication_independent_of_representatives)；依完整明示資料獨立重算；quotient元素是coset且代表元不唯一；[v]=[u] iff v-u在W；加法與scalar multiplication須與代表元無關；dim公式限有限維且零coset是W。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["商向量空間"]},"variant":{"family_id":"qf-mod37-quotient-representative-independence","seed":202611067063,"parameters":{"ambient_dimension":3,"basis_w":[[1,13,0]],"v":[14,2,3],"u":[3,13,4],"shift_v_coefficients":[14],"shift_u_coefficients":[-13],"scalar":13,"domain":"Finite_dimensional_real_quotient_space","operation":"quotient-representative-independence","semantic_template":"qf-mod37-quotient-representative-independence/{\"ambient_dimension\":3,\"basis_w\":[[1,13,0]],\"scalar\":13,\"shift_u_coefficients\":[-13],\"shift_v_coefficients\":[14],\"u\":[3,13,4],\"v\":[14,2,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r07-quotient-vector-space-0064","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-vector-space"],"title_zh":"商向量空間","stem_zh":"有限維實向量空間、明示子空間基底與exact membership。操作=quotient-representative-independence；資料={\"ambient_dimension\":3,\"basis_w\":[[1,14,0]],\"v\":[15,2,3],\"u\":[3,14,4],\"shift_v_coefficients\":[15],\"shift_u_coefficients\":[-14],\"scalar\":14}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(是, 是, 是, 是, 是, addition_and_scalar_multiplication_independent_of_representatives)","explanation_zh":"Wave5封閉層核對=(是, 是, 是, 是, 是, addition_and_scalar_multiplication_independent_of_representatives)；依完整明示資料獨立重算；quotient元素是coset且代表元不唯一；[v]=[u] iff v-u在W；加法與scalar multiplication須與代表元無關；dim公式限有限維且零coset是W。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["商向量空間"]},"variant":{"family_id":"qf-mod37-quotient-representative-independence","seed":202611067064,"parameters":{"ambient_dimension":3,"basis_w":[[1,14,0]],"v":[15,2,3],"u":[3,14,4],"shift_v_coefficients":[15],"shift_u_coefficients":[-14],"scalar":14,"domain":"Finite_dimensional_real_quotient_space","operation":"quotient-representative-independence","semantic_template":"qf-mod37-quotient-representative-independence/{\"ambient_dimension\":3,\"basis_w\":[[1,14,0]],\"scalar\":14,\"shift_u_coefficients\":[-14],\"shift_v_coefficients\":[15],\"u\":[3,14,4],\"v\":[15,2,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r07-quotient-vector-space-0065","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-vector-space"],"title_zh":"商向量空間","stem_zh":"有限維實向量空間、明示子空間基底與exact membership。操作=quotient-representative-independence；資料={\"ambient_dimension\":3,\"basis_w\":[[1,15,0]],\"v\":[16,2,3],\"u\":[3,15,4],\"shift_v_coefficients\":[16],\"shift_u_coefficients\":[-15],\"scalar\":15}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(是, 是, 是, 是, 是, addition_and_scalar_multiplication_independent_of_representatives)","explanation_zh":"Wave5封閉層核對=(是, 是, 是, 是, 是, addition_and_scalar_multiplication_independent_of_representatives)；依完整明示資料獨立重算；quotient元素是coset且代表元不唯一；[v]=[u] iff v-u在W；加法與scalar multiplication須與代表元無關；dim公式限有限維且零coset是W。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["商向量空間"]},"variant":{"family_id":"qf-mod37-quotient-representative-independence","seed":202611067065,"parameters":{"ambient_dimension":3,"basis_w":[[1,15,0]],"v":[16,2,3],"u":[3,15,4],"shift_v_coefficients":[16],"shift_u_coefficients":[-15],"scalar":15,"domain":"Finite_dimensional_real_quotient_space","operation":"quotient-representative-independence","semantic_template":"qf-mod37-quotient-representative-independence/{\"ambient_dimension\":3,\"basis_w\":[[1,15,0]],\"scalar\":15,\"shift_u_coefficients\":[-15],\"shift_v_coefficients\":[16],\"u\":[3,15,4],\"v\":[16,2,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r07-quotient-vector-space-0066","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-vector-space"],"title_zh":"商向量空間","stem_zh":"有限維實向量空間、明示子空間基底與exact membership。操作=quotient-representative-independence；資料={\"ambient_dimension\":3,\"basis_w\":[[1,16,0]],\"v\":[17,2,3],\"u\":[3,16,4],\"shift_v_coefficients\":[17],\"shift_u_coefficients\":[-16],\"scalar\":16}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(是, 是, 是, 是, 是, addition_and_scalar_multiplication_independent_of_representatives)","explanation_zh":"Wave5封閉層核對=(是, 是, 是, 是, 是, addition_and_scalar_multiplication_independent_of_representatives)；依完整明示資料獨立重算；quotient元素是coset且代表元不唯一；[v]=[u] iff v-u在W；加法與scalar multiplication須與代表元無關；dim公式限有限維且零coset是W。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["商向量空間"]},"variant":{"family_id":"qf-mod37-quotient-representative-independence","seed":202611067066,"parameters":{"ambient_dimension":3,"basis_w":[[1,16,0]],"v":[17,2,3],"u":[3,16,4],"shift_v_coefficients":[17],"shift_u_coefficients":[-16],"scalar":16,"domain":"Finite_dimensional_real_quotient_space","operation":"quotient-representative-independence","semantic_template":"qf-mod37-quotient-representative-independence/{\"ambient_dimension\":3,\"basis_w\":[[1,16,0]],\"scalar\":16,\"shift_u_coefficients\":[-16],\"shift_v_coefficients\":[17],\"u\":[3,16,4],\"v\":[17,2,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r07-quotient-vector-space-0067","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-vector-space"],"title_zh":"商向量空間","stem_zh":"有限維實向量空間、明示子空間基底與exact membership。操作=quotient-representative-independence；資料={\"ambient_dimension\":3,\"basis_w\":[[1,17,0]],\"v\":[18,2,3],\"u\":[3,17,4],\"shift_v_coefficients\":[18],\"shift_u_coefficients\":[-17],\"scalar\":17}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(是, 是, 是, 是, 是, addition_and_scalar_multiplication_independent_of_representatives)","explanation_zh":"Wave5封閉層核對=(是, 是, 是, 是, 是, addition_and_scalar_multiplication_independent_of_representatives)；依完整明示資料獨立重算；quotient元素是coset且代表元不唯一；[v]=[u] iff v-u在W；加法與scalar multiplication須與代表元無關；dim公式限有限維且零coset是W。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["商向量空間"]},"variant":{"family_id":"qf-mod37-quotient-representative-independence","seed":202611067067,"parameters":{"ambient_dimension":3,"basis_w":[[1,17,0]],"v":[18,2,3],"u":[3,17,4],"shift_v_coefficients":[18],"shift_u_coefficients":[-17],"scalar":17,"domain":"Finite_dimensional_real_quotient_space","operation":"quotient-representative-independence","semantic_template":"qf-mod37-quotient-representative-independence/{\"ambient_dimension\":3,\"basis_w\":[[1,17,0]],\"scalar\":17,\"shift_u_coefficients\":[-17],\"shift_v_coefficients\":[18],\"u\":[3,17,4],\"v\":[18,2,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r07-quotient-vector-space-0068","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-vector-space"],"title_zh":"商向量空間","stem_zh":"有限維實向量空間、明示子空間基底與exact membership。操作=quotient-representative-independence；資料={\"ambient_dimension\":3,\"basis_w\":[[1,18,0]],\"v\":[19,2,3],\"u\":[3,18,4],\"shift_v_coefficients\":[19],\"shift_u_coefficients\":[-18],\"scalar\":18}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(是, 是, 是, 是, 是, addition_and_scalar_multiplication_independent_of_representatives)","explanation_zh":"Wave5封閉層核對=(是, 是, 是, 是, 是, addition_and_scalar_multiplication_independent_of_representatives)；依完整明示資料獨立重算；quotient元素是coset且代表元不唯一；[v]=[u] iff v-u在W；加法與scalar multiplication須與代表元無關；dim公式限有限維且零coset是W。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["商向量空間"]},"variant":{"family_id":"qf-mod37-quotient-representative-independence","seed":202611067068,"parameters":{"ambient_dimension":3,"basis_w":[[1,18,0]],"v":[19,2,3],"u":[3,18,4],"shift_v_coefficients":[19],"shift_u_coefficients":[-18],"scalar":18,"domain":"Finite_dimensional_real_quotient_space","operation":"quotient-representative-independence","semantic_template":"qf-mod37-quotient-representative-independence/{\"ambient_dimension\":3,\"basis_w\":[[1,18,0]],\"scalar\":18,\"shift_u_coefficients\":[-18],\"shift_v_coefficients\":[19],\"u\":[3,18,4],\"v\":[19,2,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r07-quotient-vector-space-0069","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-vector-space"],"title_zh":"商向量空間","stem_zh":"有限維實向量空間、明示子空間基底與exact membership。操作=quotient-representative-independence；資料={\"ambient_dimension\":3,\"basis_w\":[[1,19,0]],\"v\":[20,2,3],\"u\":[3,19,4],\"shift_v_coefficients\":[20],\"shift_u_coefficients\":[-19],\"scalar\":19}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(是, 是, 是, 是, 是, addition_and_scalar_multiplication_independent_of_representatives)","explanation_zh":"Wave5封閉層核對=(是, 是, 是, 是, 是, addition_and_scalar_multiplication_independent_of_representatives)；依完整明示資料獨立重算；quotient元素是coset且代表元不唯一；[v]=[u] iff v-u在W；加法與scalar multiplication須與代表元無關；dim公式限有限維且零coset是W。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["商向量空間"]},"variant":{"family_id":"qf-mod37-quotient-representative-independence","seed":202611067069,"parameters":{"ambient_dimension":3,"basis_w":[[1,19,0]],"v":[20,2,3],"u":[3,19,4],"shift_v_coefficients":[20],"shift_u_coefficients":[-19],"scalar":19,"domain":"Finite_dimensional_real_quotient_space","operation":"quotient-representative-independence","semantic_template":"qf-mod37-quotient-representative-independence/{\"ambient_dimension\":3,\"basis_w\":[[1,19,0]],\"scalar\":19,\"shift_u_coefficients\":[-19],\"shift_v_coefficients\":[20],\"u\":[3,19,4],\"v\":[20,2,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r07-quotient-vector-space-0070","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-vector-space"],"title_zh":"商向量空間","stem_zh":"有限維實向量空間、明示子空間基底與exact membership。操作=quotient-representative-independence；資料={\"ambient_dimension\":3,\"basis_w\":[[1,20,0]],\"v\":[21,2,3],\"u\":[3,20,4],\"shift_v_coefficients\":[21],\"shift_u_coefficients\":[-20],\"scalar\":20}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(是, 是, 是, 是, 是, addition_and_scalar_multiplication_independent_of_representatives)","explanation_zh":"Wave5封閉層核對=(是, 是, 是, 是, 是, addition_and_scalar_multiplication_independent_of_representatives)；依完整明示資料獨立重算；quotient元素是coset且代表元不唯一；[v]=[u] iff v-u在W；加法與scalar multiplication須與代表元無關；dim公式限有限維且零coset是W。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["商向量空間"]},"variant":{"family_id":"qf-mod37-quotient-representative-independence","seed":202611067070,"parameters":{"ambient_dimension":3,"basis_w":[[1,20,0]],"v":[21,2,3],"u":[3,20,4],"shift_v_coefficients":[21],"shift_u_coefficients":[-20],"scalar":20,"domain":"Finite_dimensional_real_quotient_space","operation":"quotient-representative-independence","semantic_template":"qf-mod37-quotient-representative-independence/{\"ambient_dimension\":3,\"basis_w\":[[1,20,0]],\"scalar\":20,\"shift_u_coefficients\":[-20],\"shift_v_coefficients\":[21],\"u\":[3,20,4],\"v\":[21,2,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r07-quotient-vector-space-0071","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-vector-space"],"title_zh":"商向量空間","stem_zh":"有限維實向量空間、明示子空間基底與exact membership。操作=quotient-representative-independence；資料={\"ambient_dimension\":3,\"basis_w\":[[1,21,0]],\"v\":[22,2,3],\"u\":[3,21,4],\"shift_v_coefficients\":[22],\"shift_u_coefficients\":[-21],\"scalar\":21}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(是, 是, 是, 是, 是, addition_and_scalar_multiplication_independent_of_representatives)","explanation_zh":"Wave5封閉層核對=(是, 是, 是, 是, 是, addition_and_scalar_multiplication_independent_of_representatives)；依完整明示資料獨立重算；quotient元素是coset且代表元不唯一；[v]=[u] iff v-u在W；加法與scalar multiplication須與代表元無關；dim公式限有限維且零coset是W。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["商向量空間"]},"variant":{"family_id":"qf-mod37-quotient-representative-independence","seed":202611067071,"parameters":{"ambient_dimension":3,"basis_w":[[1,21,0]],"v":[22,2,3],"u":[3,21,4],"shift_v_coefficients":[22],"shift_u_coefficients":[-21],"scalar":21,"domain":"Finite_dimensional_real_quotient_space","operation":"quotient-representative-independence","semantic_template":"qf-mod37-quotient-representative-independence/{\"ambient_dimension\":3,\"basis_w\":[[1,21,0]],\"scalar\":21,\"shift_u_coefficients\":[-21],\"shift_v_coefficients\":[22],\"u\":[3,21,4],\"v\":[22,2,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r07-quotient-vector-space-0072","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-vector-space"],"title_zh":"商向量空間","stem_zh":"有限維實向量空間、明示子空間基底與exact membership。操作=quotient-representative-independence；資料={\"ambient_dimension\":3,\"basis_w\":[[1,22,0]],\"v\":[23,2,3],\"u\":[3,22,4],\"shift_v_coefficients\":[23],\"shift_u_coefficients\":[-22],\"scalar\":22}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(是, 是, 是, 是, 是, addition_and_scalar_multiplication_independent_of_representatives)","explanation_zh":"Wave5封閉層核對=(是, 是, 是, 是, 是, addition_and_scalar_multiplication_independent_of_representatives)；依完整明示資料獨立重算；quotient元素是coset且代表元不唯一；[v]=[u] iff v-u在W；加法與scalar multiplication須與代表元無關；dim公式限有限維且零coset是W。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["商向量空間"]},"variant":{"family_id":"qf-mod37-quotient-representative-independence","seed":202611067072,"parameters":{"ambient_dimension":3,"basis_w":[[1,22,0]],"v":[23,2,3],"u":[3,22,4],"shift_v_coefficients":[23],"shift_u_coefficients":[-22],"scalar":22,"domain":"Finite_dimensional_real_quotient_space","operation":"quotient-representative-independence","semantic_template":"qf-mod37-quotient-representative-independence/{\"ambient_dimension\":3,\"basis_w\":[[1,22,0]],\"scalar\":22,\"shift_u_coefficients\":[-22],\"shift_v_coefficients\":[23],\"u\":[3,22,4],\"v\":[23,2,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r07-quotient-vector-space-0073","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-vector-space"],"title_zh":"商向量空間","stem_zh":"有限維實向量空間、明示子空間基底與exact membership。操作=quotient-representative-independence；資料={\"ambient_dimension\":3,\"basis_w\":[[1,23,0]],\"v\":[24,2,3],\"u\":[3,23,4],\"shift_v_coefficients\":[24],\"shift_u_coefficients\":[-23],\"scalar\":23}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(是, 是, 是, 是, 是, addition_and_scalar_multiplication_independent_of_representatives)","explanation_zh":"Wave5封閉層核對=(是, 是, 是, 是, 是, addition_and_scalar_multiplication_independent_of_representatives)；依完整明示資料獨立重算；quotient元素是coset且代表元不唯一；[v]=[u] iff v-u在W；加法與scalar multiplication須與代表元無關；dim公式限有限維且零coset是W。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["商向量空間"]},"variant":{"family_id":"qf-mod37-quotient-representative-independence","seed":202611067073,"parameters":{"ambient_dimension":3,"basis_w":[[1,23,0]],"v":[24,2,3],"u":[3,23,4],"shift_v_coefficients":[24],"shift_u_coefficients":[-23],"scalar":23,"domain":"Finite_dimensional_real_quotient_space","operation":"quotient-representative-independence","semantic_template":"qf-mod37-quotient-representative-independence/{\"ambient_dimension\":3,\"basis_w\":[[1,23,0]],\"scalar\":23,\"shift_u_coefficients\":[-23],\"shift_v_coefficients\":[24],\"u\":[3,23,4],\"v\":[24,2,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r07-quotient-vector-space-0074","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-vector-space"],"title_zh":"商向量空間","stem_zh":"有限維實向量空間、明示子空間基底與exact membership。操作=quotient-representative-independence；資料={\"ambient_dimension\":3,\"basis_w\":[[1,24,0]],\"v\":[25,2,3],\"u\":[3,24,4],\"shift_v_coefficients\":[25],\"shift_u_coefficients\":[-24],\"scalar\":24}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(是, 是, 是, 是, 是, addition_and_scalar_multiplication_independent_of_representatives)","explanation_zh":"Wave5封閉層核對=(是, 是, 是, 是, 是, addition_and_scalar_multiplication_independent_of_representatives)；依完整明示資料獨立重算；quotient元素是coset且代表元不唯一；[v]=[u] iff v-u在W；加法與scalar multiplication須與代表元無關；dim公式限有限維且零coset是W。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["商向量空間"]},"variant":{"family_id":"qf-mod37-quotient-representative-independence","seed":202611067074,"parameters":{"ambient_dimension":3,"basis_w":[[1,24,0]],"v":[25,2,3],"u":[3,24,4],"shift_v_coefficients":[25],"shift_u_coefficients":[-24],"scalar":24,"domain":"Finite_dimensional_real_quotient_space","operation":"quotient-representative-independence","semantic_template":"qf-mod37-quotient-representative-independence/{\"ambient_dimension\":3,\"basis_w\":[[1,24,0]],\"scalar\":24,\"shift_u_coefficients\":[-24],\"shift_v_coefficients\":[25],\"u\":[3,24,4],\"v\":[25,2,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r07-quotient-vector-space-0075","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-vector-space"],"title_zh":"商向量空間","stem_zh":"有限維實向量空間、明示子空間基底與exact membership。操作=quotient-representative-independence；資料={\"ambient_dimension\":3,\"basis_w\":[[1,25,0]],\"v\":[26,2,3],\"u\":[3,25,4],\"shift_v_coefficients\":[26],\"shift_u_coefficients\":[-25],\"scalar\":25}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(是, 是, 是, 是, 是, addition_and_scalar_multiplication_independent_of_representatives)","explanation_zh":"Wave5封閉層核對=(是, 是, 是, 是, 是, addition_and_scalar_multiplication_independent_of_representatives)；依完整明示資料獨立重算；quotient元素是coset且代表元不唯一；[v]=[u] iff v-u在W；加法與scalar multiplication須與代表元無關；dim公式限有限維且零coset是W。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["商向量空間"]},"variant":{"family_id":"qf-mod37-quotient-representative-independence","seed":202611067075,"parameters":{"ambient_dimension":3,"basis_w":[[1,25,0]],"v":[26,2,3],"u":[3,25,4],"shift_v_coefficients":[26],"shift_u_coefficients":[-25],"scalar":25,"domain":"Finite_dimensional_real_quotient_space","operation":"quotient-representative-independence","semantic_template":"qf-mod37-quotient-representative-independence/{\"ambient_dimension\":3,\"basis_w\":[[1,25,0]],\"scalar\":25,\"shift_u_coefficients\":[-25],\"shift_v_coefficients\":[26],\"u\":[3,25,4],\"v\":[26,2,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r07-quotient-vector-space-0076","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-vector-space"],"title_zh":"商向量空間","stem_zh":"有限維實向量空間、明示子空間基底與exact membership。操作=quotient-dimension-formula；資料={\"ambient_dimension\":3,\"basis_w\":[[1,1,0]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 1, 2, 是, finite_dimensional_dimension_formula_and_zero_coset_is_W)","explanation_zh":"Wave5封閉層核對=(3, 1, 2, 是, finite_dimensional_dimension_formula_and_zero_coset_is_W)；依完整明示資料獨立重算；quotient元素是coset且代表元不唯一；[v]=[u] iff v-u在W；加法與scalar multiplication須與代表元無關；dim公式限有限維且零coset是W。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["商向量空間"]},"variant":{"family_id":"qf-mod37-quotient-dimension-formula","seed":202611067076,"parameters":{"ambient_dimension":3,"basis_w":[[1,1,0]],"domain":"Finite_dimensional_real_quotient_space","operation":"quotient-dimension-formula","semantic_template":"qf-mod37-quotient-dimension-formula/{\"ambient_dimension\":3,\"basis_w\":[[1,1,0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r07-quotient-vector-space-0077","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-vector-space"],"title_zh":"商向量空間","stem_zh":"有限維實向量空間、明示子空間基底與exact membership。操作=quotient-dimension-formula；資料={\"ambient_dimension\":3,\"basis_w\":[[1,2,0]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 1, 2, 是, finite_dimensional_dimension_formula_and_zero_coset_is_W)","explanation_zh":"Wave5封閉層核對=(3, 1, 2, 是, finite_dimensional_dimension_formula_and_zero_coset_is_W)；依完整明示資料獨立重算；quotient元素是coset且代表元不唯一；[v]=[u] iff v-u在W；加法與scalar multiplication須與代表元無關；dim公式限有限維且零coset是W。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["商向量空間"]},"variant":{"family_id":"qf-mod37-quotient-dimension-formula","seed":202611067077,"parameters":{"ambient_dimension":3,"basis_w":[[1,2,0]],"domain":"Finite_dimensional_real_quotient_space","operation":"quotient-dimension-formula","semantic_template":"qf-mod37-quotient-dimension-formula/{\"ambient_dimension\":3,\"basis_w\":[[1,2,0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r07-quotient-vector-space-0078","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-vector-space"],"title_zh":"商向量空間","stem_zh":"有限維實向量空間、明示子空間基底與exact membership。操作=quotient-dimension-formula；資料={\"ambient_dimension\":3,\"basis_w\":[[1,3,0]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 1, 2, 是, finite_dimensional_dimension_formula_and_zero_coset_is_W)","explanation_zh":"Wave5封閉層核對=(3, 1, 2, 是, finite_dimensional_dimension_formula_and_zero_coset_is_W)；依完整明示資料獨立重算；quotient元素是coset且代表元不唯一；[v]=[u] iff v-u在W；加法與scalar multiplication須與代表元無關；dim公式限有限維且零coset是W。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["商向量空間"]},"variant":{"family_id":"qf-mod37-quotient-dimension-formula","seed":202611067078,"parameters":{"ambient_dimension":3,"basis_w":[[1,3,0]],"domain":"Finite_dimensional_real_quotient_space","operation":"quotient-dimension-formula","semantic_template":"qf-mod37-quotient-dimension-formula/{\"ambient_dimension\":3,\"basis_w\":[[1,3,0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r07-quotient-vector-space-0079","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-vector-space"],"title_zh":"商向量空間","stem_zh":"有限維實向量空間、明示子空間基底與exact membership。操作=quotient-dimension-formula；資料={\"ambient_dimension\":3,\"basis_w\":[[1,4,0]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 1, 2, 是, finite_dimensional_dimension_formula_and_zero_coset_is_W)","explanation_zh":"Wave5封閉層核對=(3, 1, 2, 是, finite_dimensional_dimension_formula_and_zero_coset_is_W)；依完整明示資料獨立重算；quotient元素是coset且代表元不唯一；[v]=[u] iff v-u在W；加法與scalar multiplication須與代表元無關；dim公式限有限維且零coset是W。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["商向量空間"]},"variant":{"family_id":"qf-mod37-quotient-dimension-formula","seed":202611067079,"parameters":{"ambient_dimension":3,"basis_w":[[1,4,0]],"domain":"Finite_dimensional_real_quotient_space","operation":"quotient-dimension-formula","semantic_template":"qf-mod37-quotient-dimension-formula/{\"ambient_dimension\":3,\"basis_w\":[[1,4,0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r07-quotient-vector-space-0080","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-vector-space"],"title_zh":"商向量空間","stem_zh":"有限維實向量空間、明示子空間基底與exact membership。操作=quotient-dimension-formula；資料={\"ambient_dimension\":3,\"basis_w\":[[1,5,0]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 1, 2, 是, finite_dimensional_dimension_formula_and_zero_coset_is_W)","explanation_zh":"Wave5封閉層核對=(3, 1, 2, 是, finite_dimensional_dimension_formula_and_zero_coset_is_W)；依完整明示資料獨立重算；quotient元素是coset且代表元不唯一；[v]=[u] iff v-u在W；加法與scalar multiplication須與代表元無關；dim公式限有限維且零coset是W。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["商向量空間"]},"variant":{"family_id":"qf-mod37-quotient-dimension-formula","seed":202611067080,"parameters":{"ambient_dimension":3,"basis_w":[[1,5,0]],"domain":"Finite_dimensional_real_quotient_space","operation":"quotient-dimension-formula","semantic_template":"qf-mod37-quotient-dimension-formula/{\"ambient_dimension\":3,\"basis_w\":[[1,5,0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r07-quotient-vector-space-0081","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-vector-space"],"title_zh":"商向量空間","stem_zh":"有限維實向量空間、明示子空間基底與exact membership。操作=quotient-dimension-formula；資料={\"ambient_dimension\":3,\"basis_w\":[[1,6,0]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 1, 2, 是, finite_dimensional_dimension_formula_and_zero_coset_is_W)","explanation_zh":"Wave5封閉層核對=(3, 1, 2, 是, finite_dimensional_dimension_formula_and_zero_coset_is_W)；依完整明示資料獨立重算；quotient元素是coset且代表元不唯一；[v]=[u] iff v-u在W；加法與scalar multiplication須與代表元無關；dim公式限有限維且零coset是W。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["商向量空間"]},"variant":{"family_id":"qf-mod37-quotient-dimension-formula","seed":202611067081,"parameters":{"ambient_dimension":3,"basis_w":[[1,6,0]],"domain":"Finite_dimensional_real_quotient_space","operation":"quotient-dimension-formula","semantic_template":"qf-mod37-quotient-dimension-formula/{\"ambient_dimension\":3,\"basis_w\":[[1,6,0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r07-quotient-vector-space-0082","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-vector-space"],"title_zh":"商向量空間","stem_zh":"有限維實向量空間、明示子空間基底與exact membership。操作=quotient-dimension-formula；資料={\"ambient_dimension\":3,\"basis_w\":[[1,7,0]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 1, 2, 是, finite_dimensional_dimension_formula_and_zero_coset_is_W)","explanation_zh":"Wave5封閉層核對=(3, 1, 2, 是, finite_dimensional_dimension_formula_and_zero_coset_is_W)；依完整明示資料獨立重算；quotient元素是coset且代表元不唯一；[v]=[u] iff v-u在W；加法與scalar multiplication須與代表元無關；dim公式限有限維且零coset是W。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["商向量空間"]},"variant":{"family_id":"qf-mod37-quotient-dimension-formula","seed":202611067082,"parameters":{"ambient_dimension":3,"basis_w":[[1,7,0]],"domain":"Finite_dimensional_real_quotient_space","operation":"quotient-dimension-formula","semantic_template":"qf-mod37-quotient-dimension-formula/{\"ambient_dimension\":3,\"basis_w\":[[1,7,0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r07-quotient-vector-space-0083","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-vector-space"],"title_zh":"商向量空間","stem_zh":"有限維實向量空間、明示子空間基底與exact membership。操作=quotient-dimension-formula；資料={\"ambient_dimension\":3,\"basis_w\":[[1,8,0]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 1, 2, 是, finite_dimensional_dimension_formula_and_zero_coset_is_W)","explanation_zh":"Wave5封閉層核對=(3, 1, 2, 是, finite_dimensional_dimension_formula_and_zero_coset_is_W)；依完整明示資料獨立重算；quotient元素是coset且代表元不唯一；[v]=[u] iff v-u在W；加法與scalar multiplication須與代表元無關；dim公式限有限維且零coset是W。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["商向量空間"]},"variant":{"family_id":"qf-mod37-quotient-dimension-formula","seed":202611067083,"parameters":{"ambient_dimension":3,"basis_w":[[1,8,0]],"domain":"Finite_dimensional_real_quotient_space","operation":"quotient-dimension-formula","semantic_template":"qf-mod37-quotient-dimension-formula/{\"ambient_dimension\":3,\"basis_w\":[[1,8,0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r07-quotient-vector-space-0084","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-vector-space"],"title_zh":"商向量空間","stem_zh":"有限維實向量空間、明示子空間基底與exact membership。操作=quotient-dimension-formula；資料={\"ambient_dimension\":3,\"basis_w\":[[1,9,0]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 1, 2, 是, finite_dimensional_dimension_formula_and_zero_coset_is_W)","explanation_zh":"Wave5封閉層核對=(3, 1, 2, 是, finite_dimensional_dimension_formula_and_zero_coset_is_W)；依完整明示資料獨立重算；quotient元素是coset且代表元不唯一；[v]=[u] iff v-u在W；加法與scalar multiplication須與代表元無關；dim公式限有限維且零coset是W。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["商向量空間"]},"variant":{"family_id":"qf-mod37-quotient-dimension-formula","seed":202611067084,"parameters":{"ambient_dimension":3,"basis_w":[[1,9,0]],"domain":"Finite_dimensional_real_quotient_space","operation":"quotient-dimension-formula","semantic_template":"qf-mod37-quotient-dimension-formula/{\"ambient_dimension\":3,\"basis_w\":[[1,9,0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r07-quotient-vector-space-0085","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-vector-space"],"title_zh":"商向量空間","stem_zh":"有限維實向量空間、明示子空間基底與exact membership。操作=quotient-dimension-formula；資料={\"ambient_dimension\":3,\"basis_w\":[[1,10,0]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 1, 2, 是, finite_dimensional_dimension_formula_and_zero_coset_is_W)","explanation_zh":"Wave5封閉層核對=(3, 1, 2, 是, finite_dimensional_dimension_formula_and_zero_coset_is_W)；依完整明示資料獨立重算；quotient元素是coset且代表元不唯一；[v]=[u] iff v-u在W；加法與scalar multiplication須與代表元無關；dim公式限有限維且零coset是W。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["商向量空間"]},"variant":{"family_id":"qf-mod37-quotient-dimension-formula","seed":202611067085,"parameters":{"ambient_dimension":3,"basis_w":[[1,10,0]],"domain":"Finite_dimensional_real_quotient_space","operation":"quotient-dimension-formula","semantic_template":"qf-mod37-quotient-dimension-formula/{\"ambient_dimension\":3,\"basis_w\":[[1,10,0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r07-quotient-vector-space-0086","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-vector-space"],"title_zh":"商向量空間","stem_zh":"有限維實向量空間、明示子空間基底與exact membership。操作=quotient-dimension-formula；資料={\"ambient_dimension\":3,\"basis_w\":[[1,11,0]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 1, 2, 是, finite_dimensional_dimension_formula_and_zero_coset_is_W)","explanation_zh":"Wave5封閉層核對=(3, 1, 2, 是, finite_dimensional_dimension_formula_and_zero_coset_is_W)；依完整明示資料獨立重算；quotient元素是coset且代表元不唯一；[v]=[u] iff v-u在W；加法與scalar multiplication須與代表元無關；dim公式限有限維且零coset是W。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["商向量空間"]},"variant":{"family_id":"qf-mod37-quotient-dimension-formula","seed":202611067086,"parameters":{"ambient_dimension":3,"basis_w":[[1,11,0]],"domain":"Finite_dimensional_real_quotient_space","operation":"quotient-dimension-formula","semantic_template":"qf-mod37-quotient-dimension-formula/{\"ambient_dimension\":3,\"basis_w\":[[1,11,0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r07-quotient-vector-space-0087","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-vector-space"],"title_zh":"商向量空間","stem_zh":"有限維實向量空間、明示子空間基底與exact membership。操作=quotient-dimension-formula；資料={\"ambient_dimension\":3,\"basis_w\":[[1,12,0]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 1, 2, 是, finite_dimensional_dimension_formula_and_zero_coset_is_W)","explanation_zh":"Wave5封閉層核對=(3, 1, 2, 是, finite_dimensional_dimension_formula_and_zero_coset_is_W)；依完整明示資料獨立重算；quotient元素是coset且代表元不唯一；[v]=[u] iff v-u在W；加法與scalar multiplication須與代表元無關；dim公式限有限維且零coset是W。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["商向量空間"]},"variant":{"family_id":"qf-mod37-quotient-dimension-formula","seed":202611067087,"parameters":{"ambient_dimension":3,"basis_w":[[1,12,0]],"domain":"Finite_dimensional_real_quotient_space","operation":"quotient-dimension-formula","semantic_template":"qf-mod37-quotient-dimension-formula/{\"ambient_dimension\":3,\"basis_w\":[[1,12,0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r07-quotient-vector-space-0088","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-vector-space"],"title_zh":"商向量空間","stem_zh":"有限維實向量空間、明示子空間基底與exact membership。操作=quotient-dimension-formula；資料={\"ambient_dimension\":3,\"basis_w\":[[1,13,0]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 1, 2, 是, finite_dimensional_dimension_formula_and_zero_coset_is_W)","explanation_zh":"Wave5封閉層核對=(3, 1, 2, 是, finite_dimensional_dimension_formula_and_zero_coset_is_W)；依完整明示資料獨立重算；quotient元素是coset且代表元不唯一；[v]=[u] iff v-u在W；加法與scalar multiplication須與代表元無關；dim公式限有限維且零coset是W。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["商向量空間"]},"variant":{"family_id":"qf-mod37-quotient-dimension-formula","seed":202611067088,"parameters":{"ambient_dimension":3,"basis_w":[[1,13,0]],"domain":"Finite_dimensional_real_quotient_space","operation":"quotient-dimension-formula","semantic_template":"qf-mod37-quotient-dimension-formula/{\"ambient_dimension\":3,\"basis_w\":[[1,13,0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r07-quotient-vector-space-0089","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-vector-space"],"title_zh":"商向量空間","stem_zh":"有限維實向量空間、明示子空間基底與exact membership。操作=quotient-dimension-formula；資料={\"ambient_dimension\":3,\"basis_w\":[[1,14,0]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 1, 2, 是, finite_dimensional_dimension_formula_and_zero_coset_is_W)","explanation_zh":"Wave5封閉層核對=(3, 1, 2, 是, finite_dimensional_dimension_formula_and_zero_coset_is_W)；依完整明示資料獨立重算；quotient元素是coset且代表元不唯一；[v]=[u] iff v-u在W；加法與scalar multiplication須與代表元無關；dim公式限有限維且零coset是W。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["商向量空間"]},"variant":{"family_id":"qf-mod37-quotient-dimension-formula","seed":202611067089,"parameters":{"ambient_dimension":3,"basis_w":[[1,14,0]],"domain":"Finite_dimensional_real_quotient_space","operation":"quotient-dimension-formula","semantic_template":"qf-mod37-quotient-dimension-formula/{\"ambient_dimension\":3,\"basis_w\":[[1,14,0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r07-quotient-vector-space-0090","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-vector-space"],"title_zh":"商向量空間","stem_zh":"有限維實向量空間、明示子空間基底與exact membership。操作=quotient-dimension-formula；資料={\"ambient_dimension\":3,\"basis_w\":[[1,15,0]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 1, 2, 是, finite_dimensional_dimension_formula_and_zero_coset_is_W)","explanation_zh":"Wave5封閉層核對=(3, 1, 2, 是, finite_dimensional_dimension_formula_and_zero_coset_is_W)；依完整明示資料獨立重算；quotient元素是coset且代表元不唯一；[v]=[u] iff v-u在W；加法與scalar multiplication須與代表元無關；dim公式限有限維且零coset是W。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["商向量空間"]},"variant":{"family_id":"qf-mod37-quotient-dimension-formula","seed":202611067090,"parameters":{"ambient_dimension":3,"basis_w":[[1,15,0]],"domain":"Finite_dimensional_real_quotient_space","operation":"quotient-dimension-formula","semantic_template":"qf-mod37-quotient-dimension-formula/{\"ambient_dimension\":3,\"basis_w\":[[1,15,0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r07-quotient-vector-space-0091","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-vector-space"],"title_zh":"商向量空間","stem_zh":"有限維實向量空間、明示子空間基底與exact membership。操作=quotient-dimension-formula；資料={\"ambient_dimension\":3,\"basis_w\":[[1,16,0]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 1, 2, 是, finite_dimensional_dimension_formula_and_zero_coset_is_W)","explanation_zh":"Wave5封閉層核對=(3, 1, 2, 是, finite_dimensional_dimension_formula_and_zero_coset_is_W)；依完整明示資料獨立重算；quotient元素是coset且代表元不唯一；[v]=[u] iff v-u在W；加法與scalar multiplication須與代表元無關；dim公式限有限維且零coset是W。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["商向量空間"]},"variant":{"family_id":"qf-mod37-quotient-dimension-formula","seed":202611067091,"parameters":{"ambient_dimension":3,"basis_w":[[1,16,0]],"domain":"Finite_dimensional_real_quotient_space","operation":"quotient-dimension-formula","semantic_template":"qf-mod37-quotient-dimension-formula/{\"ambient_dimension\":3,\"basis_w\":[[1,16,0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r07-quotient-vector-space-0092","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-vector-space"],"title_zh":"商向量空間","stem_zh":"有限維實向量空間、明示子空間基底與exact membership。操作=quotient-dimension-formula；資料={\"ambient_dimension\":3,\"basis_w\":[[1,17,0]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 1, 2, 是, finite_dimensional_dimension_formula_and_zero_coset_is_W)","explanation_zh":"Wave5封閉層核對=(3, 1, 2, 是, finite_dimensional_dimension_formula_and_zero_coset_is_W)；依完整明示資料獨立重算；quotient元素是coset且代表元不唯一；[v]=[u] iff v-u在W；加法與scalar multiplication須與代表元無關；dim公式限有限維且零coset是W。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["商向量空間"]},"variant":{"family_id":"qf-mod37-quotient-dimension-formula","seed":202611067092,"parameters":{"ambient_dimension":3,"basis_w":[[1,17,0]],"domain":"Finite_dimensional_real_quotient_space","operation":"quotient-dimension-formula","semantic_template":"qf-mod37-quotient-dimension-formula/{\"ambient_dimension\":3,\"basis_w\":[[1,17,0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r07-quotient-vector-space-0093","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-vector-space"],"title_zh":"商向量空間","stem_zh":"有限維實向量空間、明示子空間基底與exact membership。操作=quotient-dimension-formula；資料={\"ambient_dimension\":3,\"basis_w\":[[1,18,0]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 1, 2, 是, finite_dimensional_dimension_formula_and_zero_coset_is_W)","explanation_zh":"Wave5封閉層核對=(3, 1, 2, 是, finite_dimensional_dimension_formula_and_zero_coset_is_W)；依完整明示資料獨立重算；quotient元素是coset且代表元不唯一；[v]=[u] iff v-u在W；加法與scalar multiplication須與代表元無關；dim公式限有限維且零coset是W。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["商向量空間"]},"variant":{"family_id":"qf-mod37-quotient-dimension-formula","seed":202611067093,"parameters":{"ambient_dimension":3,"basis_w":[[1,18,0]],"domain":"Finite_dimensional_real_quotient_space","operation":"quotient-dimension-formula","semantic_template":"qf-mod37-quotient-dimension-formula/{\"ambient_dimension\":3,\"basis_w\":[[1,18,0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r07-quotient-vector-space-0094","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-vector-space"],"title_zh":"商向量空間","stem_zh":"有限維實向量空間、明示子空間基底與exact membership。操作=quotient-dimension-formula；資料={\"ambient_dimension\":3,\"basis_w\":[[1,19,0]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 1, 2, 是, finite_dimensional_dimension_formula_and_zero_coset_is_W)","explanation_zh":"Wave5封閉層核對=(3, 1, 2, 是, finite_dimensional_dimension_formula_and_zero_coset_is_W)；依完整明示資料獨立重算；quotient元素是coset且代表元不唯一；[v]=[u] iff v-u在W；加法與scalar multiplication須與代表元無關；dim公式限有限維且零coset是W。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["商向量空間"]},"variant":{"family_id":"qf-mod37-quotient-dimension-formula","seed":202611067094,"parameters":{"ambient_dimension":3,"basis_w":[[1,19,0]],"domain":"Finite_dimensional_real_quotient_space","operation":"quotient-dimension-formula","semantic_template":"qf-mod37-quotient-dimension-formula/{\"ambient_dimension\":3,\"basis_w\":[[1,19,0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r07-quotient-vector-space-0095","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-vector-space"],"title_zh":"商向量空間","stem_zh":"有限維實向量空間、明示子空間基底與exact membership。操作=quotient-dimension-formula；資料={\"ambient_dimension\":3,\"basis_w\":[[1,20,0]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 1, 2, 是, finite_dimensional_dimension_formula_and_zero_coset_is_W)","explanation_zh":"Wave5封閉層核對=(3, 1, 2, 是, finite_dimensional_dimension_formula_and_zero_coset_is_W)；依完整明示資料獨立重算；quotient元素是coset且代表元不唯一；[v]=[u] iff v-u在W；加法與scalar multiplication須與代表元無關；dim公式限有限維且零coset是W。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["商向量空間"]},"variant":{"family_id":"qf-mod37-quotient-dimension-formula","seed":202611067095,"parameters":{"ambient_dimension":3,"basis_w":[[1,20,0]],"domain":"Finite_dimensional_real_quotient_space","operation":"quotient-dimension-formula","semantic_template":"qf-mod37-quotient-dimension-formula/{\"ambient_dimension\":3,\"basis_w\":[[1,20,0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r07-quotient-vector-space-0096","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-vector-space"],"title_zh":"商向量空間","stem_zh":"有限維實向量空間、明示子空間基底與exact membership。操作=quotient-dimension-formula；資料={\"ambient_dimension\":3,\"basis_w\":[[1,21,0]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 1, 2, 是, finite_dimensional_dimension_formula_and_zero_coset_is_W)","explanation_zh":"Wave5封閉層核對=(3, 1, 2, 是, finite_dimensional_dimension_formula_and_zero_coset_is_W)；依完整明示資料獨立重算；quotient元素是coset且代表元不唯一；[v]=[u] iff v-u在W；加法與scalar multiplication須與代表元無關；dim公式限有限維且零coset是W。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["商向量空間"]},"variant":{"family_id":"qf-mod37-quotient-dimension-formula","seed":202611067096,"parameters":{"ambient_dimension":3,"basis_w":[[1,21,0]],"domain":"Finite_dimensional_real_quotient_space","operation":"quotient-dimension-formula","semantic_template":"qf-mod37-quotient-dimension-formula/{\"ambient_dimension\":3,\"basis_w\":[[1,21,0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r07-quotient-vector-space-0097","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-vector-space"],"title_zh":"商向量空間","stem_zh":"有限維實向量空間、明示子空間基底與exact membership。操作=quotient-dimension-formula；資料={\"ambient_dimension\":3,\"basis_w\":[[1,22,0]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 1, 2, 是, finite_dimensional_dimension_formula_and_zero_coset_is_W)","explanation_zh":"Wave5封閉層核對=(3, 1, 2, 是, finite_dimensional_dimension_formula_and_zero_coset_is_W)；依完整明示資料獨立重算；quotient元素是coset且代表元不唯一；[v]=[u] iff v-u在W；加法與scalar multiplication須與代表元無關；dim公式限有限維且零coset是W。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["商向量空間"]},"variant":{"family_id":"qf-mod37-quotient-dimension-formula","seed":202611067097,"parameters":{"ambient_dimension":3,"basis_w":[[1,22,0]],"domain":"Finite_dimensional_real_quotient_space","operation":"quotient-dimension-formula","semantic_template":"qf-mod37-quotient-dimension-formula/{\"ambient_dimension\":3,\"basis_w\":[[1,22,0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r07-quotient-vector-space-0098","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-vector-space"],"title_zh":"商向量空間","stem_zh":"有限維實向量空間、明示子空間基底與exact membership。操作=quotient-dimension-formula；資料={\"ambient_dimension\":3,\"basis_w\":[[1,23,0]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 1, 2, 是, finite_dimensional_dimension_formula_and_zero_coset_is_W)","explanation_zh":"Wave5封閉層核對=(3, 1, 2, 是, finite_dimensional_dimension_formula_and_zero_coset_is_W)；依完整明示資料獨立重算；quotient元素是coset且代表元不唯一；[v]=[u] iff v-u在W；加法與scalar multiplication須與代表元無關；dim公式限有限維且零coset是W。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["商向量空間"]},"variant":{"family_id":"qf-mod37-quotient-dimension-formula","seed":202611067098,"parameters":{"ambient_dimension":3,"basis_w":[[1,23,0]],"domain":"Finite_dimensional_real_quotient_space","operation":"quotient-dimension-formula","semantic_template":"qf-mod37-quotient-dimension-formula/{\"ambient_dimension\":3,\"basis_w\":[[1,23,0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r07-quotient-vector-space-0099","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-vector-space"],"title_zh":"商向量空間","stem_zh":"有限維實向量空間、明示子空間基底與exact membership。操作=quotient-dimension-formula；資料={\"ambient_dimension\":3,\"basis_w\":[[1,24,0]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 1, 2, 是, finite_dimensional_dimension_formula_and_zero_coset_is_W)","explanation_zh":"Wave5封閉層核對=(3, 1, 2, 是, finite_dimensional_dimension_formula_and_zero_coset_is_W)；依完整明示資料獨立重算；quotient元素是coset且代表元不唯一；[v]=[u] iff v-u在W；加法與scalar multiplication須與代表元無關；dim公式限有限維且零coset是W。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["商向量空間"]},"variant":{"family_id":"qf-mod37-quotient-dimension-formula","seed":202611067099,"parameters":{"ambient_dimension":3,"basis_w":[[1,24,0]],"domain":"Finite_dimensional_real_quotient_space","operation":"quotient-dimension-formula","semantic_template":"qf-mod37-quotient-dimension-formula/{\"ambient_dimension\":3,\"basis_w\":[[1,24,0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r07-quotient-vector-space-0100","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-vector-space"],"title_zh":"商向量空間","stem_zh":"有限維實向量空間、明示子空間基底與exact membership。操作=quotient-dimension-formula；資料={\"ambient_dimension\":3,\"basis_w\":[[1,25,0]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 1, 2, 是, finite_dimensional_dimension_formula_and_zero_coset_is_W)","explanation_zh":"Wave5封閉層核對=(3, 1, 2, 是, finite_dimensional_dimension_formula_and_zero_coset_is_W)；依完整明示資料獨立重算；quotient元素是coset且代表元不唯一；[v]=[u] iff v-u在W；加法與scalar multiplication須與代表元無關；dim公式限有限維且零coset是W。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["商向量空間"]},"variant":{"family_id":"qf-mod37-quotient-dimension-formula","seed":202611067100,"parameters":{"ambient_dimension":3,"basis_w":[[1,25,0]],"domain":"Finite_dimensional_real_quotient_space","operation":"quotient-dimension-formula","semantic_template":"qf-mod37-quotient-dimension-formula/{\"ambient_dimension\":3,\"basis_w\":[[1,25,0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}