{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r06-euclidean-length-0001","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-length"],"title_zh":"有向位移與平方L2長度","stem_zh":"等單位正交L2；同一實平面與長度單位；P=[0, 0]、Q=[0, 0]。依序求 (Qx−Px, Qy−Py, L2(P,Q)²)，前兩項保留方向符號。","answer":{"kind":"expression","value":"(0, 0, 0)","explanation_zh":"有向位移與平方長度=(0, 0, 0)。兩座標差帶符號，平方和非負；實際L2取非負主根，不把有向分量當距離。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["有向位移與平方L2長度"]},"variant":{"family_id":"qf-density12-length-displacement-square","seed":202610126001,"parameters":{"P":[0,0],"Q":[0,0],"frame":"orthogonal_equal_unit","operation":"density12_length_displacement_square","domain":"integer_euclidean_plane","semantic_template":"qf-density12-length-displacement-square/[\"0\",\"0\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r06-euclidean-length-0002","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-length"],"title_zh":"有向位移與平方L2長度","stem_zh":"等單位正交L2；同一實平面與長度單位；P=[-1, 0]、Q=[-12, -2]。依序求 (Qx−Px, Qy−Py, L2(P,Q)²)，前兩項保留方向符號。","answer":{"kind":"expression","value":"(-11, -2, 125)","explanation_zh":"有向位移與平方長度=(-11, -2, 125)。兩座標差帶符號，平方和非負；實際L2取非負主根，不把有向分量當距離。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["有向位移與平方L2長度"]},"variant":{"family_id":"qf-density12-length-displacement-square","seed":202610126002,"parameters":{"P":[-1,0],"Q":[-12,-2],"frame":"orthogonal_equal_unit","operation":"density12_length_displacement_square","domain":"integer_euclidean_plane","semantic_template":"qf-density12-length-displacement-square/[\"-11\",\"-2\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r06-euclidean-length-0003","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-length"],"title_zh":"有向位移與平方L2長度","stem_zh":"等單位正交L2；同一實平面與長度單位；P=[0, -1]、Q=[-10, 2]。依序求 (Qx−Px, Qy−Py, L2(P,Q)²)，前兩項保留方向符號。","answer":{"kind":"expression","value":"(-10, 3, 109)","explanation_zh":"有向位移與平方長度=(-10, 3, 109)。兩座標差帶符號，平方和非負；實際L2取非負主根，不把有向分量當距離。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["有向位移與平方L2長度"]},"variant":{"family_id":"qf-density12-length-displacement-square","seed":202610126003,"parameters":{"P":[0,-1],"Q":[-10,2],"frame":"orthogonal_equal_unit","operation":"density12_length_displacement_square","domain":"integer_euclidean_plane","semantic_template":"qf-density12-length-displacement-square/[\"-10\",\"3\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r06-euclidean-length-0004","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-length"],"title_zh":"有向位移與平方L2長度","stem_zh":"等單位正交L2；同一實平面與長度單位；P=[1, 1]、Q=[-8, -3]。依序求 (Qx−Px, Qy−Py, L2(P,Q)²)，前兩項保留方向符號。","answer":{"kind":"expression","value":"(-9, -4, 97)","explanation_zh":"有向位移與平方長度=(-9, -4, 97)。兩座標差帶符號，平方和非負；實際L2取非負主根，不把有向分量當距離。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["有向位移與平方L2長度"]},"variant":{"family_id":"qf-density12-length-displacement-square","seed":202610126004,"parameters":{"P":[1,1],"Q":[-8,-3],"frame":"orthogonal_equal_unit","operation":"density12_length_displacement_square","domain":"integer_euclidean_plane","semantic_template":"qf-density12-length-displacement-square/[\"-9\",\"-4\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r06-euclidean-length-0005","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-length"],"title_zh":"有向位移與平方L2長度","stem_zh":"等單位正交L2；同一實平面與長度單位；P=[-2, 0]、Q=[-10, 5]。依序求 (Qx−Px, Qy−Py, L2(P,Q)²)，前兩項保留方向符號。","answer":{"kind":"expression","value":"(-8, 5, 89)","explanation_zh":"有向位移與平方長度=(-8, 5, 89)。兩座標差帶符號，平方和非負；實際L2取非負主根，不把有向分量當距離。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["有向位移與平方L2長度"]},"variant":{"family_id":"qf-density12-length-displacement-square","seed":202610126005,"parameters":{"P":[-2,0],"Q":[-10,5],"frame":"orthogonal_equal_unit","operation":"density12_length_displacement_square","domain":"integer_euclidean_plane","semantic_template":"qf-density12-length-displacement-square/[\"-8\",\"5\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r06-euclidean-length-0006","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-length"],"title_zh":"有向位移與平方L2長度","stem_zh":"等單位正交L2；同一實平面與長度單位；P=[-1, -1]、Q=[-8, -2]。依序求 (Qx−Px, Qy−Py, L2(P,Q)²)，前兩項保留方向符號。","answer":{"kind":"expression","value":"(-7, -1, 50)","explanation_zh":"有向位移與平方長度=(-7, -1, 50)。兩座標差帶符號，平方和非負；實際L2取非負主根，不把有向分量當距離。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["有向位移與平方L2長度"]},"variant":{"family_id":"qf-density12-length-displacement-square","seed":202610126006,"parameters":{"P":[-1,-1],"Q":[-8,-2],"frame":"orthogonal_equal_unit","operation":"density12_length_displacement_square","domain":"integer_euclidean_plane","semantic_template":"qf-density12-length-displacement-square/[\"-7\",\"-1\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r06-euclidean-length-0007","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-length"],"title_zh":"有向位移與平方L2長度","stem_zh":"等單位正交L2；同一實平面與長度單位；P=[0, 1]、Q=[-6, 3]。依序求 (Qx−Px, Qy−Py, L2(P,Q)²)，前兩項保留方向符號。","answer":{"kind":"expression","value":"(-6, 2, 40)","explanation_zh":"有向位移與平方長度=(-6, 2, 40)。兩座標差帶符號，平方和非負；實際L2取非負主根，不把有向分量當距離。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["有向位移與平方L2長度"]},"variant":{"family_id":"qf-density12-length-displacement-square","seed":202610126007,"parameters":{"P":[0,1],"Q":[-6,3],"frame":"orthogonal_equal_unit","operation":"density12_length_displacement_square","domain":"integer_euclidean_plane","semantic_template":"qf-density12-length-displacement-square/[\"-6\",\"2\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r06-euclidean-length-0008","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-length"],"title_zh":"有向位移與平方L2長度","stem_zh":"等單位正交L2；同一實平面與長度單位；P=[1, 0]、Q=[-4, -3]。依序求 (Qx−Px, Qy−Py, L2(P,Q)²)，前兩項保留方向符號。","answer":{"kind":"expression","value":"(-5, -3, 34)","explanation_zh":"有向位移與平方長度=(-5, -3, 34)。兩座標差帶符號，平方和非負；實際L2取非負主根，不把有向分量當距離。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["有向位移與平方L2長度"]},"variant":{"family_id":"qf-density12-length-displacement-square","seed":202610126008,"parameters":{"P":[1,0],"Q":[-4,-3],"frame":"orthogonal_equal_unit","operation":"density12_length_displacement_square","domain":"integer_euclidean_plane","semantic_template":"qf-density12-length-displacement-square/[\"-5\",\"-3\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r06-euclidean-length-0009","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-length"],"title_zh":"有向位移與平方L2長度","stem_zh":"等單位正交L2；同一實平面與長度單位；P=[-2, -1]、Q=[-6, 3]。依序求 (Qx−Px, Qy−Py, L2(P,Q)²)，前兩項保留方向符號。","answer":{"kind":"expression","value":"(-4, 4, 32)","explanation_zh":"有向位移與平方長度=(-4, 4, 32)。兩座標差帶符號，平方和非負；實際L2取非負主根，不把有向分量當距離。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["有向位移與平方L2長度"]},"variant":{"family_id":"qf-density12-length-displacement-square","seed":202610126009,"parameters":{"P":[-2,-1],"Q":[-6,3],"frame":"orthogonal_equal_unit","operation":"density12_length_displacement_square","domain":"integer_euclidean_plane","semantic_template":"qf-density12-length-displacement-square/[\"-4\",\"4\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r06-euclidean-length-0010","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-length"],"title_zh":"有向位移與平方L2長度","stem_zh":"等單位正交L2；同一實平面與長度單位；P=[-1, 1]、Q=[-4, -4]。依序求 (Qx−Px, Qy−Py, L2(P,Q)²)，前兩項保留方向符號。","answer":{"kind":"expression","value":"(-3, -5, 34)","explanation_zh":"有向位移與平方長度=(-3, -5, 34)。兩座標差帶符號，平方和非負；實際L2取非負主根，不把有向分量當距離。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["有向位移與平方L2長度"]},"variant":{"family_id":"qf-density12-length-displacement-square","seed":202610126010,"parameters":{"P":[-1,1],"Q":[-4,-4],"frame":"orthogonal_equal_unit","operation":"density12_length_displacement_square","domain":"integer_euclidean_plane","semantic_template":"qf-density12-length-displacement-square/[\"-3\",\"-5\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r06-euclidean-length-0011","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-length"],"title_zh":"有向位移與平方L2長度","stem_zh":"等單位正交L2；同一實平面與長度單位；P=[0, 0]、Q=[-2, 1]。依序求 (Qx−Px, Qy−Py, L2(P,Q)²)，前兩項保留方向符號。","answer":{"kind":"expression","value":"(-2, 1, 5)","explanation_zh":"有向位移與平方長度=(-2, 1, 5)。兩座標差帶符號，平方和非負；實際L2取非負主根，不把有向分量當距離。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["有向位移與平方L2長度"]},"variant":{"family_id":"qf-density12-length-displacement-square","seed":202610126011,"parameters":{"P":[0,0],"Q":[-2,1],"frame":"orthogonal_equal_unit","operation":"density12_length_displacement_square","domain":"integer_euclidean_plane","semantic_template":"qf-density12-length-displacement-square/[\"-2\",\"1\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r06-euclidean-length-0012","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-length"],"title_zh":"有向位移與平方L2長度","stem_zh":"等單位正交L2；同一實平面與長度單位；P=[1, -1]、Q=[0, -3]。依序求 (Qx−Px, Qy−Py, L2(P,Q)²)，前兩項保留方向符號。","answer":{"kind":"expression","value":"(-1, -2, 5)","explanation_zh":"有向位移與平方長度=(-1, -2, 5)。兩座標差帶符號，平方和非負；實際L2取非負主根，不把有向分量當距離。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["有向位移與平方L2長度"]},"variant":{"family_id":"qf-density12-length-displacement-square","seed":202610126012,"parameters":{"P":[1,-1],"Q":[0,-3],"frame":"orthogonal_equal_unit","operation":"density12_length_displacement_square","domain":"integer_euclidean_plane","semantic_template":"qf-density12-length-displacement-square/[\"-1\",\"-2\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r06-euclidean-length-0013","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-length"],"title_zh":"有向位移與平方L2長度","stem_zh":"等單位正交L2；同一實平面與長度單位；P=[-2, 1]、Q=[-2, 4]。依序求 (Qx−Px, Qy−Py, L2(P,Q)²)，前兩項保留方向符號。","answer":{"kind":"expression","value":"(0, 3, 9)","explanation_zh":"有向位移與平方長度=(0, 3, 9)。兩座標差帶符號，平方和非負；實際L2取非負主根，不把有向分量當距離。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["有向位移與平方L2長度"]},"variant":{"family_id":"qf-density12-length-displacement-square","seed":202610126013,"parameters":{"P":[-2,1],"Q":[-2,4],"frame":"orthogonal_equal_unit","operation":"density12_length_displacement_square","domain":"integer_euclidean_plane","semantic_template":"qf-density12-length-displacement-square/[\"0\",\"3\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r06-euclidean-length-0014","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-length"],"title_zh":"有向位移與平方L2長度","stem_zh":"等單位正交L2；同一實平面與長度單位；P=[-1, 0]、Q=[0, -4]。依序求 (Qx−Px, Qy−Py, L2(P,Q)²)，前兩項保留方向符號。","answer":{"kind":"expression","value":"(1, -4, 17)","explanation_zh":"有向位移與平方長度=(1, -4, 17)。兩座標差帶符號，平方和非負；實際L2取非負主根，不把有向分量當距離。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["有向位移與平方L2長度"]},"variant":{"family_id":"qf-density12-length-displacement-square","seed":202610126014,"parameters":{"P":[-1,0],"Q":[0,-4],"frame":"orthogonal_equal_unit","operation":"density12_length_displacement_square","domain":"integer_euclidean_plane","semantic_template":"qf-density12-length-displacement-square/[\"1\",\"-4\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r06-euclidean-length-0015","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-length"],"title_zh":"有向位移與平方L2長度","stem_zh":"等單位正交L2；同一實平面與長度單位；P=[0, -1]、Q=[2, 4]。依序求 (Qx−Px, Qy−Py, L2(P,Q)²)，前兩項保留方向符號。","answer":{"kind":"expression","value":"(2, 5, 29)","explanation_zh":"有向位移與平方長度=(2, 5, 29)。兩座標差帶符號，平方和非負；實際L2取非負主根，不把有向分量當距離。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["有向位移與平方L2長度"]},"variant":{"family_id":"qf-density12-length-displacement-square","seed":202610126015,"parameters":{"P":[0,-1],"Q":[2,4],"frame":"orthogonal_equal_unit","operation":"density12_length_displacement_square","domain":"integer_euclidean_plane","semantic_template":"qf-density12-length-displacement-square/[\"2\",\"5\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r06-euclidean-length-0016","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-length"],"title_zh":"有向位移與平方L2長度","stem_zh":"等單位正交L2；同一實平面與長度單位；P=[1, 1]、Q=[4, 0]。依序求 (Qx−Px, Qy−Py, L2(P,Q)²)，前兩項保留方向符號。","answer":{"kind":"expression","value":"(3, -1, 10)","explanation_zh":"有向位移與平方長度=(3, -1, 10)。兩座標差帶符號，平方和非負；實際L2取非負主根，不把有向分量當距離。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["有向位移與平方L2長度"]},"variant":{"family_id":"qf-density12-length-displacement-square","seed":202610126016,"parameters":{"P":[1,1],"Q":[4,0],"frame":"orthogonal_equal_unit","operation":"density12_length_displacement_square","domain":"integer_euclidean_plane","semantic_template":"qf-density12-length-displacement-square/[\"3\",\"-1\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r06-euclidean-length-0017","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-length"],"title_zh":"有向位移與平方L2長度","stem_zh":"等單位正交L2；同一實平面與長度單位；P=[-2, 0]、Q=[2, 2]。依序求 (Qx−Px, Qy−Py, L2(P,Q)²)，前兩項保留方向符號。","answer":{"kind":"expression","value":"(4, 2, 20)","explanation_zh":"有向位移與平方長度=(4, 2, 20)。兩座標差帶符號，平方和非負；實際L2取非負主根，不把有向分量當距離。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["有向位移與平方L2長度"]},"variant":{"family_id":"qf-density12-length-displacement-square","seed":202610126017,"parameters":{"P":[-2,0],"Q":[2,2],"frame":"orthogonal_equal_unit","operation":"density12_length_displacement_square","domain":"integer_euclidean_plane","semantic_template":"qf-density12-length-displacement-square/[\"4\",\"2\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r06-euclidean-length-0018","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-length"],"title_zh":"有向位移與平方L2長度","stem_zh":"等單位正交L2；同一實平面與長度單位；P=[-1, -1]、Q=[4, -4]。依序求 (Qx−Px, Qy−Py, L2(P,Q)²)，前兩項保留方向符號。","answer":{"kind":"expression","value":"(5, -3, 34)","explanation_zh":"有向位移與平方長度=(5, -3, 34)。兩座標差帶符號，平方和非負；實際L2取非負主根，不把有向分量當距離。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["有向位移與平方L2長度"]},"variant":{"family_id":"qf-density12-length-displacement-square","seed":202610126018,"parameters":{"P":[-1,-1],"Q":[4,-4],"frame":"orthogonal_equal_unit","operation":"density12_length_displacement_square","domain":"integer_euclidean_plane","semantic_template":"qf-density12-length-displacement-square/[\"5\",\"-3\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r06-euclidean-length-0019","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-length"],"title_zh":"有向位移與平方L2長度","stem_zh":"等單位正交L2；同一實平面與長度單位；P=[0, 1]、Q=[6, 5]。依序求 (Qx−Px, Qy−Py, L2(P,Q)²)，前兩項保留方向符號。","answer":{"kind":"expression","value":"(6, 4, 52)","explanation_zh":"有向位移與平方長度=(6, 4, 52)。兩座標差帶符號，平方和非負；實際L2取非負主根，不把有向分量當距離。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["有向位移與平方L2長度"]},"variant":{"family_id":"qf-density12-length-displacement-square","seed":202610126019,"parameters":{"P":[0,1],"Q":[6,5],"frame":"orthogonal_equal_unit","operation":"density12_length_displacement_square","domain":"integer_euclidean_plane","semantic_template":"qf-density12-length-displacement-square/[\"6\",\"4\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r06-euclidean-length-0020","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-length"],"title_zh":"有向位移與平方L2長度","stem_zh":"等單位正交L2；同一實平面與長度單位；P=[1, 0]、Q=[8, -5]。依序求 (Qx−Px, Qy−Py, L2(P,Q)²)，前兩項保留方向符號。","answer":{"kind":"expression","value":"(7, -5, 74)","explanation_zh":"有向位移與平方長度=(7, -5, 74)。兩座標差帶符號，平方和非負；實際L2取非負主根，不把有向分量當距離。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["有向位移與平方L2長度"]},"variant":{"family_id":"qf-density12-length-displacement-square","seed":202610126020,"parameters":{"P":[1,0],"Q":[8,-5],"frame":"orthogonal_equal_unit","operation":"density12_length_displacement_square","domain":"integer_euclidean_plane","semantic_template":"qf-density12-length-displacement-square/[\"7\",\"-5\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r06-euclidean-length-0021","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-length"],"title_zh":"有向位移與平方L2長度","stem_zh":"等單位正交L2；同一實平面與長度單位；P=[-2, -1]、Q=[6, 0]。依序求 (Qx−Px, Qy−Py, L2(P,Q)²)，前兩項保留方向符號。","answer":{"kind":"expression","value":"(8, 1, 65)","explanation_zh":"有向位移與平方長度=(8, 1, 65)。兩座標差帶符號，平方和非負；實際L2取非負主根，不把有向分量當距離。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["有向位移與平方L2長度"]},"variant":{"family_id":"qf-density12-length-displacement-square","seed":202610126021,"parameters":{"P":[-2,-1],"Q":[6,0],"frame":"orthogonal_equal_unit","operation":"density12_length_displacement_square","domain":"integer_euclidean_plane","semantic_template":"qf-density12-length-displacement-square/[\"8\",\"1\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r06-euclidean-length-0022","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-length"],"title_zh":"有向位移與平方L2長度","stem_zh":"等單位正交L2；同一實平面與長度單位；P=[-1, 1]、Q=[8, -1]。依序求 (Qx−Px, Qy−Py, L2(P,Q)²)，前兩項保留方向符號。","answer":{"kind":"expression","value":"(9, -2, 85)","explanation_zh":"有向位移與平方長度=(9, -2, 85)。兩座標差帶符號，平方和非負；實際L2取非負主根，不把有向分量當距離。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["有向位移與平方L2長度"]},"variant":{"family_id":"qf-density12-length-displacement-square","seed":202610126022,"parameters":{"P":[-1,1],"Q":[8,-1],"frame":"orthogonal_equal_unit","operation":"density12_length_displacement_square","domain":"integer_euclidean_plane","semantic_template":"qf-density12-length-displacement-square/[\"9\",\"-2\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r06-euclidean-length-0023","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-length"],"title_zh":"有向位移與平方L2長度","stem_zh":"等單位正交L2；同一實平面與長度單位；P=[0, 0]、Q=[10, 3]。依序求 (Qx−Px, Qy−Py, L2(P,Q)²)，前兩項保留方向符號。","answer":{"kind":"expression","value":"(10, 3, 109)","explanation_zh":"有向位移與平方長度=(10, 3, 109)。兩座標差帶符號，平方和非負；實際L2取非負主根，不把有向分量當距離。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["有向位移與平方L2長度"]},"variant":{"family_id":"qf-density12-length-displacement-square","seed":202610126023,"parameters":{"P":[0,0],"Q":[10,3],"frame":"orthogonal_equal_unit","operation":"density12_length_displacement_square","domain":"integer_euclidean_plane","semantic_template":"qf-density12-length-displacement-square/[\"10\",\"3\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r06-euclidean-length-0024","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-length"],"title_zh":"有向位移與平方L2長度","stem_zh":"等單位正交L2；同一實平面與長度單位；P=[1, -1]、Q=[12, -5]。依序求 (Qx−Px, Qy−Py, L2(P,Q)²)，前兩項保留方向符號。","answer":{"kind":"expression","value":"(11, -4, 137)","explanation_zh":"有向位移與平方長度=(11, -4, 137)。兩座標差帶符號，平方和非負；實際L2取非負主根，不把有向分量當距離。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["有向位移與平方L2長度"]},"variant":{"family_id":"qf-density12-length-displacement-square","seed":202610126024,"parameters":{"P":[1,-1],"Q":[12,-5],"frame":"orthogonal_equal_unit","operation":"density12_length_displacement_square","domain":"integer_euclidean_plane","semantic_template":"qf-density12-length-displacement-square/[\"11\",\"-4\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r06-euclidean-length-0025","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-length"],"title_zh":"有向位移與平方L2長度","stem_zh":"等單位正交L2；同一實平面與長度單位；P=[-2, 1]、Q=[10, 6]。依序求 (Qx−Px, Qy−Py, L2(P,Q)²)，前兩項保留方向符號。","answer":{"kind":"expression","value":"(12, 5, 169)","explanation_zh":"有向位移與平方長度=(12, 5, 169)。兩座標差帶符號，平方和非負；實際L2取非負主根，不把有向分量當距離。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["有向位移與平方L2長度"]},"variant":{"family_id":"qf-density12-length-displacement-square","seed":202610126025,"parameters":{"P":[-2,1],"Q":[10,6],"frame":"orthogonal_equal_unit","operation":"density12_length_displacement_square","domain":"integer_euclidean_plane","semantic_template":"qf-density12-length-displacement-square/[\"12\",\"5\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r06-euclidean-length-0026","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-length"],"title_zh":"兩段有序整數主距離","stem_zh":"等單位正交L2；同一實平面與長度單位；點P、Q、R依序為 [[0, 0], [0, 0], [0, 0]]，兩段平方和均為完全平方。依序求 (d(P,Q), d(Q,R)) 的整數主距離。","answer":{"kind":"expression","value":"(0, 0)","explanation_zh":"兩段整數主距離=(0, 0)。分別求精確平方和及非負主根，保留兩段次序；重合點給零，不改用有號位移。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["兩段有序整數主距離"]},"variant":{"family_id":"qf-density12-length-integer-distance-pair","seed":202610126026,"parameters":{"points":[[0,0],[0,0],[0,0]],"frame":"orthogonal_equal_unit","operation":"density12_length_integer_distance_pair","domain":"integer_euclidean_plane","semantic_template":"qf-density12-length-integer-distance-pair/[\"0\",\"0\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r06-euclidean-length-0027","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-length"],"title_zh":"兩段有序整數主距離","stem_zh":"等單位正交L2；同一實平面與長度單位；點P、Q、R依序為 [[0, 0], [3, 4], [3, 4]]，兩段平方和均為完全平方。依序求 (d(P,Q), d(Q,R)) 的整數主距離。","answer":{"kind":"expression","value":"(5, 0)","explanation_zh":"兩段整數主距離=(5, 0)。分別求精確平方和及非負主根，保留兩段次序；重合點給零，不改用有號位移。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["兩段有序整數主距離"]},"variant":{"family_id":"qf-density12-length-integer-distance-pair","seed":202610126027,"parameters":{"points":[[0,0],[3,4],[3,4]],"frame":"orthogonal_equal_unit","operation":"density12_length_integer_distance_pair","domain":"integer_euclidean_plane","semantic_template":"qf-density12-length-integer-distance-pair/[\"25\",\"0\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r06-euclidean-length-0028","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-length"],"title_zh":"兩段有序整數主距離","stem_zh":"等單位正交L2；同一實平面與長度單位；點P、Q、R依序為 [[0, 0], [0, 0], [5, 12]]，兩段平方和均為完全平方。依序求 (d(P,Q), d(Q,R)) 的整數主距離。","answer":{"kind":"expression","value":"(0, 13)","explanation_zh":"兩段整數主距離=(0, 13)。分別求精確平方和及非負主根，保留兩段次序；重合點給零，不改用有號位移。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["兩段有序整數主距離"]},"variant":{"family_id":"qf-density12-length-integer-distance-pair","seed":202610126028,"parameters":{"points":[[0,0],[0,0],[5,12]],"frame":"orthogonal_equal_unit","operation":"density12_length_integer_distance_pair","domain":"integer_euclidean_plane","semantic_template":"qf-density12-length-integer-distance-pair/[\"0\",\"169\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r06-euclidean-length-0029","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-length"],"title_zh":"兩段有序整數主距離","stem_zh":"等單位正交L2；同一實平面與長度單位；點P、Q、R依序為 [[0, 0], [3, 4], [8, 16]]，兩段平方和均為完全平方。依序求 (d(P,Q), d(Q,R)) 的整數主距離。","answer":{"kind":"expression","value":"(5, 13)","explanation_zh":"兩段整數主距離=(5, 13)。分別求精確平方和及非負主根，保留兩段次序；重合點給零，不改用有號位移。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["兩段有序整數主距離"]},"variant":{"family_id":"qf-density12-length-integer-distance-pair","seed":202610126029,"parameters":{"points":[[0,0],[3,4],[8,16]],"frame":"orthogonal_equal_unit","operation":"density12_length_integer_distance_pair","domain":"integer_euclidean_plane","semantic_template":"qf-density12-length-integer-distance-pair/[\"25\",\"169\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r06-euclidean-length-0030","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-length"],"title_zh":"兩段有序整數主距離","stem_zh":"等單位正交L2；同一實平面與長度單位；點P、Q、R依序為 [[0, 0], [6, 8], [11, 20]]，兩段平方和均為完全平方。依序求 (d(P,Q), d(Q,R)) 的整數主距離。","answer":{"kind":"expression","value":"(10, 13)","explanation_zh":"兩段整數主距離=(10, 13)。分別求精確平方和及非負主根，保留兩段次序；重合點給零，不改用有號位移。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["兩段有序整數主距離"]},"variant":{"family_id":"qf-density12-length-integer-distance-pair","seed":202610126030,"parameters":{"points":[[0,0],[6,8],[11,20]],"frame":"orthogonal_equal_unit","operation":"density12_length_integer_distance_pair","domain":"integer_euclidean_plane","semantic_template":"qf-density12-length-integer-distance-pair/[\"100\",\"169\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r06-euclidean-length-0031","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-length"],"title_zh":"兩段有序整數主距離","stem_zh":"等單位正交L2；同一實平面與長度單位；點P、Q、R依序為 [[0, 0], [9, 12], [14, 24]]，兩段平方和均為完全平方。依序求 (d(P,Q), d(Q,R)) 的整數主距離。","answer":{"kind":"expression","value":"(15, 13)","explanation_zh":"兩段整數主距離=(15, 13)。分別求精確平方和及非負主根，保留兩段次序；重合點給零，不改用有號位移。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["兩段有序整數主距離"]},"variant":{"family_id":"qf-density12-length-integer-distance-pair","seed":202610126031,"parameters":{"points":[[0,0],[9,12],[14,24]],"frame":"orthogonal_equal_unit","operation":"density12_length_integer_distance_pair","domain":"integer_euclidean_plane","semantic_template":"qf-density12-length-integer-distance-pair/[\"225\",\"169\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r06-euclidean-length-0032","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-length"],"title_zh":"兩段有序整數主距離","stem_zh":"等單位正交L2；同一實平面與長度單位；點P、Q、R依序為 [[0, 0], [12, 16], [17, 28]]，兩段平方和均為完全平方。依序求 (d(P,Q), d(Q,R)) 的整數主距離。","answer":{"kind":"expression","value":"(20, 13)","explanation_zh":"兩段整數主距離=(20, 13)。分別求精確平方和及非負主根，保留兩段次序；重合點給零，不改用有號位移。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["兩段有序整數主距離"]},"variant":{"family_id":"qf-density12-length-integer-distance-pair","seed":202610126032,"parameters":{"points":[[0,0],[12,16],[17,28]],"frame":"orthogonal_equal_unit","operation":"density12_length_integer_distance_pair","domain":"integer_euclidean_plane","semantic_template":"qf-density12-length-integer-distance-pair/[\"400\",\"169\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r06-euclidean-length-0033","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-length"],"title_zh":"兩段有序整數主距離","stem_zh":"等單位正交L2；同一實平面與長度單位；點P、Q、R依序為 [[0, 0], [15, 20], [20, 32]]，兩段平方和均為完全平方。依序求 (d(P,Q), d(Q,R)) 的整數主距離。","answer":{"kind":"expression","value":"(25, 13)","explanation_zh":"兩段整數主距離=(25, 13)。分別求精確平方和及非負主根，保留兩段次序；重合點給零，不改用有號位移。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["兩段有序整數主距離"]},"variant":{"family_id":"qf-density12-length-integer-distance-pair","seed":202610126033,"parameters":{"points":[[0,0],[15,20],[20,32]],"frame":"orthogonal_equal_unit","operation":"density12_length_integer_distance_pair","domain":"integer_euclidean_plane","semantic_template":"qf-density12-length-integer-distance-pair/[\"625\",\"169\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r06-euclidean-length-0034","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-length"],"title_zh":"兩段有序整數主距離","stem_zh":"等單位正交L2；同一實平面與長度單位；點P、Q、R依序為 [[0, 0], [18, 24], [23, 36]]，兩段平方和均為完全平方。依序求 (d(P,Q), d(Q,R)) 的整數主距離。","answer":{"kind":"expression","value":"(30, 13)","explanation_zh":"兩段整數主距離=(30, 13)。分別求精確平方和及非負主根，保留兩段次序；重合點給零，不改用有號位移。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["兩段有序整數主距離"]},"variant":{"family_id":"qf-density12-length-integer-distance-pair","seed":202610126034,"parameters":{"points":[[0,0],[18,24],[23,36]],"frame":"orthogonal_equal_unit","operation":"density12_length_integer_distance_pair","domain":"integer_euclidean_plane","semantic_template":"qf-density12-length-integer-distance-pair/[\"900\",\"169\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r06-euclidean-length-0035","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-length"],"title_zh":"兩段有序整數主距離","stem_zh":"等單位正交L2；同一實平面與長度單位；點P、Q、R依序為 [[0, 0], [21, 28], [26, 40]]，兩段平方和均為完全平方。依序求 (d(P,Q), d(Q,R)) 的整數主距離。","answer":{"kind":"expression","value":"(35, 13)","explanation_zh":"兩段整數主距離=(35, 13)。分別求精確平方和及非負主根，保留兩段次序；重合點給零，不改用有號位移。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["兩段有序整數主距離"]},"variant":{"family_id":"qf-density12-length-integer-distance-pair","seed":202610126035,"parameters":{"points":[[0,0],[21,28],[26,40]],"frame":"orthogonal_equal_unit","operation":"density12_length_integer_distance_pair","domain":"integer_euclidean_plane","semantic_template":"qf-density12-length-integer-distance-pair/[\"1225\",\"169\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r06-euclidean-length-0036","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-length"],"title_zh":"兩段有序整數主距離","stem_zh":"等單位正交L2；同一實平面與長度單位；點P、Q、R依序為 [[0, 0], [24, 32], [29, 44]]，兩段平方和均為完全平方。依序求 (d(P,Q), d(Q,R)) 的整數主距離。","answer":{"kind":"expression","value":"(40, 13)","explanation_zh":"兩段整數主距離=(40, 13)。分別求精確平方和及非負主根，保留兩段次序；重合點給零，不改用有號位移。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["兩段有序整數主距離"]},"variant":{"family_id":"qf-density12-length-integer-distance-pair","seed":202610126036,"parameters":{"points":[[0,0],[24,32],[29,44]],"frame":"orthogonal_equal_unit","operation":"density12_length_integer_distance_pair","domain":"integer_euclidean_plane","semantic_template":"qf-density12-length-integer-distance-pair/[\"1600\",\"169\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r06-euclidean-length-0037","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-length"],"title_zh":"兩段有序整數主距離","stem_zh":"等單位正交L2；同一實平面與長度單位；點P、Q、R依序為 [[0, 0], [27, 36], [32, 48]]，兩段平方和均為完全平方。依序求 (d(P,Q), d(Q,R)) 的整數主距離。","answer":{"kind":"expression","value":"(45, 13)","explanation_zh":"兩段整數主距離=(45, 13)。分別求精確平方和及非負主根，保留兩段次序；重合點給零，不改用有號位移。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["兩段有序整數主距離"]},"variant":{"family_id":"qf-density12-length-integer-distance-pair","seed":202610126037,"parameters":{"points":[[0,0],[27,36],[32,48]],"frame":"orthogonal_equal_unit","operation":"density12_length_integer_distance_pair","domain":"integer_euclidean_plane","semantic_template":"qf-density12-length-integer-distance-pair/[\"2025\",\"169\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r06-euclidean-length-0038","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-length"],"title_zh":"兩段有序整數主距離","stem_zh":"等單位正交L2；同一實平面與長度單位；點P、Q、R依序為 [[0, 0], [30, 40], [35, 52]]，兩段平方和均為完全平方。依序求 (d(P,Q), d(Q,R)) 的整數主距離。","answer":{"kind":"expression","value":"(50, 13)","explanation_zh":"兩段整數主距離=(50, 13)。分別求精確平方和及非負主根，保留兩段次序；重合點給零，不改用有號位移。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["兩段有序整數主距離"]},"variant":{"family_id":"qf-density12-length-integer-distance-pair","seed":202610126038,"parameters":{"points":[[0,0],[30,40],[35,52]],"frame":"orthogonal_equal_unit","operation":"density12_length_integer_distance_pair","domain":"integer_euclidean_plane","semantic_template":"qf-density12-length-integer-distance-pair/[\"2500\",\"169\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r06-euclidean-length-0039","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-length"],"title_zh":"兩段有序整數主距離","stem_zh":"等單位正交L2；同一實平面與長度單位；點P、Q、R依序為 [[0, 0], [33, 44], [38, 56]]，兩段平方和均為完全平方。依序求 (d(P,Q), d(Q,R)) 的整數主距離。","answer":{"kind":"expression","value":"(55, 13)","explanation_zh":"兩段整數主距離=(55, 13)。分別求精確平方和及非負主根，保留兩段次序；重合點給零，不改用有號位移。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["兩段有序整數主距離"]},"variant":{"family_id":"qf-density12-length-integer-distance-pair","seed":202610126039,"parameters":{"points":[[0,0],[33,44],[38,56]],"frame":"orthogonal_equal_unit","operation":"density12_length_integer_distance_pair","domain":"integer_euclidean_plane","semantic_template":"qf-density12-length-integer-distance-pair/[\"3025\",\"169\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r06-euclidean-length-0040","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-length"],"title_zh":"兩段有序整數主距離","stem_zh":"等單位正交L2；同一實平面與長度單位；點P、Q、R依序為 [[0, 0], [36, 48], [41, 60]]，兩段平方和均為完全平方。依序求 (d(P,Q), d(Q,R)) 的整數主距離。","answer":{"kind":"expression","value":"(60, 13)","explanation_zh":"兩段整數主距離=(60, 13)。分別求精確平方和及非負主根，保留兩段次序；重合點給零，不改用有號位移。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["兩段有序整數主距離"]},"variant":{"family_id":"qf-density12-length-integer-distance-pair","seed":202610126040,"parameters":{"points":[[0,0],[36,48],[41,60]],"frame":"orthogonal_equal_unit","operation":"density12_length_integer_distance_pair","domain":"integer_euclidean_plane","semantic_template":"qf-density12-length-integer-distance-pair/[\"3600\",\"169\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r06-euclidean-length-0041","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-length"],"title_zh":"兩段有序整數主距離","stem_zh":"等單位正交L2；同一實平面與長度單位；點P、Q、R依序為 [[0, 0], [39, 52], [44, 64]]，兩段平方和均為完全平方。依序求 (d(P,Q), d(Q,R)) 的整數主距離。","answer":{"kind":"expression","value":"(65, 13)","explanation_zh":"兩段整數主距離=(65, 13)。分別求精確平方和及非負主根，保留兩段次序；重合點給零，不改用有號位移。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["兩段有序整數主距離"]},"variant":{"family_id":"qf-density12-length-integer-distance-pair","seed":202610126041,"parameters":{"points":[[0,0],[39,52],[44,64]],"frame":"orthogonal_equal_unit","operation":"density12_length_integer_distance_pair","domain":"integer_euclidean_plane","semantic_template":"qf-density12-length-integer-distance-pair/[\"4225\",\"169\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r06-euclidean-length-0042","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-length"],"title_zh":"兩段有序整數主距離","stem_zh":"等單位正交L2；同一實平面與長度單位；點P、Q、R依序為 [[0, 0], [42, 56], [47, 68]]，兩段平方和均為完全平方。依序求 (d(P,Q), d(Q,R)) 的整數主距離。","answer":{"kind":"expression","value":"(70, 13)","explanation_zh":"兩段整數主距離=(70, 13)。分別求精確平方和及非負主根，保留兩段次序；重合點給零，不改用有號位移。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["兩段有序整數主距離"]},"variant":{"family_id":"qf-density12-length-integer-distance-pair","seed":202610126042,"parameters":{"points":[[0,0],[42,56],[47,68]],"frame":"orthogonal_equal_unit","operation":"density12_length_integer_distance_pair","domain":"integer_euclidean_plane","semantic_template":"qf-density12-length-integer-distance-pair/[\"4900\",\"169\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r06-euclidean-length-0043","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-length"],"title_zh":"兩段有序整數主距離","stem_zh":"等單位正交L2；同一實平面與長度單位；點P、Q、R依序為 [[0, 0], [45, 60], [50, 72]]，兩段平方和均為完全平方。依序求 (d(P,Q), d(Q,R)) 的整數主距離。","answer":{"kind":"expression","value":"(75, 13)","explanation_zh":"兩段整數主距離=(75, 13)。分別求精確平方和及非負主根，保留兩段次序；重合點給零，不改用有號位移。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["兩段有序整數主距離"]},"variant":{"family_id":"qf-density12-length-integer-distance-pair","seed":202610126043,"parameters":{"points":[[0,0],[45,60],[50,72]],"frame":"orthogonal_equal_unit","operation":"density12_length_integer_distance_pair","domain":"integer_euclidean_plane","semantic_template":"qf-density12-length-integer-distance-pair/[\"5625\",\"169\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r06-euclidean-length-0044","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-length"],"title_zh":"兩段有序整數主距離","stem_zh":"等單位正交L2；同一實平面與長度單位；點P、Q、R依序為 [[0, 0], [48, 64], [53, 76]]，兩段平方和均為完全平方。依序求 (d(P,Q), d(Q,R)) 的整數主距離。","answer":{"kind":"expression","value":"(80, 13)","explanation_zh":"兩段整數主距離=(80, 13)。分別求精確平方和及非負主根，保留兩段次序；重合點給零，不改用有號位移。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["兩段有序整數主距離"]},"variant":{"family_id":"qf-density12-length-integer-distance-pair","seed":202610126044,"parameters":{"points":[[0,0],[48,64],[53,76]],"frame":"orthogonal_equal_unit","operation":"density12_length_integer_distance_pair","domain":"integer_euclidean_plane","semantic_template":"qf-density12-length-integer-distance-pair/[\"6400\",\"169\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r06-euclidean-length-0045","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-length"],"title_zh":"兩段有序整數主距離","stem_zh":"等單位正交L2；同一實平面與長度單位；點P、Q、R依序為 [[0, 0], [51, 68], [56, 80]]，兩段平方和均為完全平方。依序求 (d(P,Q), d(Q,R)) 的整數主距離。","answer":{"kind":"expression","value":"(85, 13)","explanation_zh":"兩段整數主距離=(85, 13)。分別求精確平方和及非負主根，保留兩段次序；重合點給零，不改用有號位移。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["兩段有序整數主距離"]},"variant":{"family_id":"qf-density12-length-integer-distance-pair","seed":202610126045,"parameters":{"points":[[0,0],[51,68],[56,80]],"frame":"orthogonal_equal_unit","operation":"density12_length_integer_distance_pair","domain":"integer_euclidean_plane","semantic_template":"qf-density12-length-integer-distance-pair/[\"7225\",\"169\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r06-euclidean-length-0046","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-length"],"title_zh":"兩段有序整數主距離","stem_zh":"等單位正交L2；同一實平面與長度單位；點P、Q、R依序為 [[0, 0], [54, 72], [59, 84]]，兩段平方和均為完全平方。依序求 (d(P,Q), d(Q,R)) 的整數主距離。","answer":{"kind":"expression","value":"(90, 13)","explanation_zh":"兩段整數主距離=(90, 13)。分別求精確平方和及非負主根，保留兩段次序；重合點給零，不改用有號位移。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["兩段有序整數主距離"]},"variant":{"family_id":"qf-density12-length-integer-distance-pair","seed":202610126046,"parameters":{"points":[[0,0],[54,72],[59,84]],"frame":"orthogonal_equal_unit","operation":"density12_length_integer_distance_pair","domain":"integer_euclidean_plane","semantic_template":"qf-density12-length-integer-distance-pair/[\"8100\",\"169\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r06-euclidean-length-0047","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-length"],"title_zh":"兩段有序整數主距離","stem_zh":"等單位正交L2；同一實平面與長度單位；點P、Q、R依序為 [[0, 0], [57, 76], [62, 88]]，兩段平方和均為完全平方。依序求 (d(P,Q), d(Q,R)) 的整數主距離。","answer":{"kind":"expression","value":"(95, 13)","explanation_zh":"兩段整數主距離=(95, 13)。分別求精確平方和及非負主根，保留兩段次序；重合點給零，不改用有號位移。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["兩段有序整數主距離"]},"variant":{"family_id":"qf-density12-length-integer-distance-pair","seed":202610126047,"parameters":{"points":[[0,0],[57,76],[62,88]],"frame":"orthogonal_equal_unit","operation":"density12_length_integer_distance_pair","domain":"integer_euclidean_plane","semantic_template":"qf-density12-length-integer-distance-pair/[\"9025\",\"169\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r06-euclidean-length-0048","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-length"],"title_zh":"兩段有序整數主距離","stem_zh":"等單位正交L2；同一實平面與長度單位；點P、Q、R依序為 [[0, 0], [60, 80], [65, 92]]，兩段平方和均為完全平方。依序求 (d(P,Q), d(Q,R)) 的整數主距離。","answer":{"kind":"expression","value":"(100, 13)","explanation_zh":"兩段整數主距離=(100, 13)。分別求精確平方和及非負主根，保留兩段次序；重合點給零，不改用有號位移。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["兩段有序整數主距離"]},"variant":{"family_id":"qf-density12-length-integer-distance-pair","seed":202610126048,"parameters":{"points":[[0,0],[60,80],[65,92]],"frame":"orthogonal_equal_unit","operation":"density12_length_integer_distance_pair","domain":"integer_euclidean_plane","semantic_template":"qf-density12-length-integer-distance-pair/[\"10000\",\"169\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r06-euclidean-length-0049","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-length"],"title_zh":"兩段有序整數主距離","stem_zh":"等單位正交L2；同一實平面與長度單位；點P、Q、R依序為 [[0, 0], [63, 84], [68, 96]]，兩段平方和均為完全平方。依序求 (d(P,Q), d(Q,R)) 的整數主距離。","answer":{"kind":"expression","value":"(105, 13)","explanation_zh":"兩段整數主距離=(105, 13)。分別求精確平方和及非負主根，保留兩段次序；重合點給零，不改用有號位移。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["兩段有序整數主距離"]},"variant":{"family_id":"qf-density12-length-integer-distance-pair","seed":202610126049,"parameters":{"points":[[0,0],[63,84],[68,96]],"frame":"orthogonal_equal_unit","operation":"density12_length_integer_distance_pair","domain":"integer_euclidean_plane","semantic_template":"qf-density12-length-integer-distance-pair/[\"11025\",\"169\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r06-euclidean-length-0050","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-length"],"title_zh":"兩段有序整數主距離","stem_zh":"等單位正交L2；同一實平面與長度單位；點P、Q、R依序為 [[0, 0], [66, 88], [71, 100]]，兩段平方和均為完全平方。依序求 (d(P,Q), d(Q,R)) 的整數主距離。","answer":{"kind":"expression","value":"(110, 13)","explanation_zh":"兩段整數主距離=(110, 13)。分別求精確平方和及非負主根，保留兩段次序；重合點給零，不改用有號位移。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["兩段有序整數主距離"]},"variant":{"family_id":"qf-density12-length-integer-distance-pair","seed":202610126050,"parameters":{"points":[[0,0],[66,88],[71,100]],"frame":"orthogonal_equal_unit","operation":"density12_length_integer_distance_pair","domain":"integer_euclidean_plane","semantic_template":"qf-density12-length-integer-distance-pair/[\"12100\",\"169\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r06-euclidean-length-0051","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-length"],"title_zh":"中點座標及兩段平方","stem_zh":"等單位正交L2；同一實平面與長度單位；P=[0, 0]、Q=[0, 0]，M是中點。依序求 (Mx, My, d(P,M)², d(M,Q)²)，四項均用最簡正分母分數，後兩項明示為平方。","answer":{"kind":"expression","value":"(0/1, 0/1, 0/1, 0/1)","explanation_zh":"中點與兩段平方=(0/1, 0/1, 0/1, 0/1)。座標平均給中點，每段平方為全段平方的四分之一；平方量不是距離本身，實際長度取非負主根。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["中點座標及兩段平方"]},"variant":{"family_id":"qf-density12-length-midpoint-equidistance","seed":202610126051,"parameters":{"P":[0,0],"Q":[0,0],"frame":"orthogonal_equal_unit","operation":"density12_length_midpoint_equidistance","domain":"integer_euclidean_plane","semantic_template":"qf-density12-length-midpoint-equidistance/[\"0/1\",\"0/1\",\"0\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r06-euclidean-length-0052","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-length"],"title_zh":"中點座標及兩段平方","stem_zh":"等單位正交L2；同一實平面與長度單位；P=[0, 0]、Q=[1, 0]，M是中點。依序求 (Mx, My, d(P,M)², d(M,Q)²)，四項均用最簡正分母分數，後兩項明示為平方。","answer":{"kind":"expression","value":"(1/2, 0/1, 1/4, 1/4)","explanation_zh":"中點與兩段平方=(1/2, 0/1, 1/4, 1/4)。座標平均給中點，每段平方為全段平方的四分之一；平方量不是距離本身，實際長度取非負主根。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["中點座標及兩段平方"]},"variant":{"family_id":"qf-density12-length-midpoint-equidistance","seed":202610126052,"parameters":{"P":[0,0],"Q":[1,0],"frame":"orthogonal_equal_unit","operation":"density12_length_midpoint_equidistance","domain":"integer_euclidean_plane","semantic_template":"qf-density12-length-midpoint-equidistance/[\"1/2\",\"0/1\",\"1\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r06-euclidean-length-0053","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-length"],"title_zh":"中點座標及兩段平方","stem_zh":"等單位正交L2；同一實平面與長度單位；P=[-2, 1]、Q=[3, 4]，M是中點。依序求 (Mx, My, d(P,M)², d(M,Q)²)，四項均用最簡正分母分數，後兩項明示為平方。","answer":{"kind":"expression","value":"(1/2, 5/2, 17/2, 17/2)","explanation_zh":"中點與兩段平方=(1/2, 5/2, 17/2, 17/2)。座標平均給中點，每段平方為全段平方的四分之一；平方量不是距離本身，實際長度取非負主根。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["中點座標及兩段平方"]},"variant":{"family_id":"qf-density12-length-midpoint-equidistance","seed":202610126053,"parameters":{"P":[-2,1],"Q":[3,4],"frame":"orthogonal_equal_unit","operation":"density12_length_midpoint_equidistance","domain":"integer_euclidean_plane","semantic_template":"qf-density12-length-midpoint-equidistance/[\"1/2\",\"5/2\",\"34\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r06-euclidean-length-0054","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-length"],"title_zh":"中點座標及兩段平方","stem_zh":"等單位正交L2；同一實平面與長度單位；P=[-3, 1]、Q=[4, 5]，M是中點。依序求 (Mx, My, d(P,M)², d(M,Q)²)，四項均用最簡正分母分數，後兩項明示為平方。","answer":{"kind":"expression","value":"(1/2, 3/1, 65/4, 65/4)","explanation_zh":"中點與兩段平方=(1/2, 3/1, 65/4, 65/4)。座標平均給中點，每段平方為全段平方的四分之一；平方量不是距離本身，實際長度取非負主根。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["中點座標及兩段平方"]},"variant":{"family_id":"qf-density12-length-midpoint-equidistance","seed":202610126054,"parameters":{"P":[-3,1],"Q":[4,5],"frame":"orthogonal_equal_unit","operation":"density12_length_midpoint_equidistance","domain":"integer_euclidean_plane","semantic_template":"qf-density12-length-midpoint-equidistance/[\"1/2\",\"3/1\",\"65\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r06-euclidean-length-0055","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-length"],"title_zh":"中點座標及兩段平方","stem_zh":"等單位正交L2；同一實平面與長度單位；P=[-4, 1]、Q=[5, 6]，M是中點。依序求 (Mx, My, d(P,M)², d(M,Q)²)，四項均用最簡正分母分數，後兩項明示為平方。","answer":{"kind":"expression","value":"(1/2, 7/2, 53/2, 53/2)","explanation_zh":"中點與兩段平方=(1/2, 7/2, 53/2, 53/2)。座標平均給中點，每段平方為全段平方的四分之一；平方量不是距離本身，實際長度取非負主根。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["中點座標及兩段平方"]},"variant":{"family_id":"qf-density12-length-midpoint-equidistance","seed":202610126055,"parameters":{"P":[-4,1],"Q":[5,6],"frame":"orthogonal_equal_unit","operation":"density12_length_midpoint_equidistance","domain":"integer_euclidean_plane","semantic_template":"qf-density12-length-midpoint-equidistance/[\"1/2\",\"7/2\",\"106\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r06-euclidean-length-0056","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-length"],"title_zh":"中點座標及兩段平方","stem_zh":"等單位正交L2；同一實平面與長度單位；P=[-5, 1]、Q=[6, 7]，M是中點。依序求 (Mx, My, d(P,M)², d(M,Q)²)，四項均用最簡正分母分數，後兩項明示為平方。","answer":{"kind":"expression","value":"(1/2, 4/1, 157/4, 157/4)","explanation_zh":"中點與兩段平方=(1/2, 4/1, 157/4, 157/4)。座標平均給中點，每段平方為全段平方的四分之一；平方量不是距離本身，實際長度取非負主根。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["中點座標及兩段平方"]},"variant":{"family_id":"qf-density12-length-midpoint-equidistance","seed":202610126056,"parameters":{"P":[-5,1],"Q":[6,7],"frame":"orthogonal_equal_unit","operation":"density12_length_midpoint_equidistance","domain":"integer_euclidean_plane","semantic_template":"qf-density12-length-midpoint-equidistance/[\"1/2\",\"4/1\",\"157\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r06-euclidean-length-0057","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-length"],"title_zh":"中點座標及兩段平方","stem_zh":"等單位正交L2；同一實平面與長度單位；P=[-6, 1]、Q=[7, 8]，M是中點。依序求 (Mx, My, d(P,M)², d(M,Q)²)，四項均用最簡正分母分數，後兩項明示為平方。","answer":{"kind":"expression","value":"(1/2, 9/2, 109/2, 109/2)","explanation_zh":"中點與兩段平方=(1/2, 9/2, 109/2, 109/2)。座標平均給中點，每段平方為全段平方的四分之一；平方量不是距離本身，實際長度取非負主根。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["中點座標及兩段平方"]},"variant":{"family_id":"qf-density12-length-midpoint-equidistance","seed":202610126057,"parameters":{"P":[-6,1],"Q":[7,8],"frame":"orthogonal_equal_unit","operation":"density12_length_midpoint_equidistance","domain":"integer_euclidean_plane","semantic_template":"qf-density12-length-midpoint-equidistance/[\"1/2\",\"9/2\",\"218\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r06-euclidean-length-0058","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-length"],"title_zh":"中點座標及兩段平方","stem_zh":"等單位正交L2；同一實平面與長度單位；P=[-7, 1]、Q=[8, 9]，M是中點。依序求 (Mx, My, d(P,M)², d(M,Q)²)，四項均用最簡正分母分數，後兩項明示為平方。","answer":{"kind":"expression","value":"(1/2, 5/1, 289/4, 289/4)","explanation_zh":"中點與兩段平方=(1/2, 5/1, 289/4, 289/4)。座標平均給中點，每段平方為全段平方的四分之一；平方量不是距離本身，實際長度取非負主根。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["中點座標及兩段平方"]},"variant":{"family_id":"qf-density12-length-midpoint-equidistance","seed":202610126058,"parameters":{"P":[-7,1],"Q":[8,9],"frame":"orthogonal_equal_unit","operation":"density12_length_midpoint_equidistance","domain":"integer_euclidean_plane","semantic_template":"qf-density12-length-midpoint-equidistance/[\"1/2\",\"5/1\",\"289\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r06-euclidean-length-0059","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-length"],"title_zh":"中點座標及兩段平方","stem_zh":"等單位正交L2；同一實平面與長度單位；P=[-8, 1]、Q=[9, 10]，M是中點。依序求 (Mx, My, d(P,M)², d(M,Q)²)，四項均用最簡正分母分數，後兩項明示為平方。","answer":{"kind":"expression","value":"(1/2, 11/2, 185/2, 185/2)","explanation_zh":"中點與兩段平方=(1/2, 11/2, 185/2, 185/2)。座標平均給中點，每段平方為全段平方的四分之一；平方量不是距離本身，實際長度取非負主根。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["中點座標及兩段平方"]},"variant":{"family_id":"qf-density12-length-midpoint-equidistance","seed":202610126059,"parameters":{"P":[-8,1],"Q":[9,10],"frame":"orthogonal_equal_unit","operation":"density12_length_midpoint_equidistance","domain":"integer_euclidean_plane","semantic_template":"qf-density12-length-midpoint-equidistance/[\"1/2\",\"11/2\",\"370\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r06-euclidean-length-0060","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-length"],"title_zh":"中點座標及兩段平方","stem_zh":"等單位正交L2；同一實平面與長度單位；P=[-9, 1]、Q=[10, 11]，M是中點。依序求 (Mx, My, d(P,M)², d(M,Q)²)，四項均用最簡正分母分數，後兩項明示為平方。","answer":{"kind":"expression","value":"(1/2, 6/1, 461/4, 461/4)","explanation_zh":"中點與兩段平方=(1/2, 6/1, 461/4, 461/4)。座標平均給中點，每段平方為全段平方的四分之一；平方量不是距離本身，實際長度取非負主根。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["中點座標及兩段平方"]},"variant":{"family_id":"qf-density12-length-midpoint-equidistance","seed":202610126060,"parameters":{"P":[-9,1],"Q":[10,11],"frame":"orthogonal_equal_unit","operation":"density12_length_midpoint_equidistance","domain":"integer_euclidean_plane","semantic_template":"qf-density12-length-midpoint-equidistance/[\"1/2\",\"6/1\",\"461\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r06-euclidean-length-0061","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-length"],"title_zh":"中點座標及兩段平方","stem_zh":"等單位正交L2；同一實平面與長度單位；P=[-10, 1]、Q=[11, 12]，M是中點。依序求 (Mx, My, d(P,M)², d(M,Q)²)，四項均用最簡正分母分數，後兩項明示為平方。","answer":{"kind":"expression","value":"(1/2, 13/2, 281/2, 281/2)","explanation_zh":"中點與兩段平方=(1/2, 13/2, 281/2, 281/2)。座標平均給中點，每段平方為全段平方的四分之一；平方量不是距離本身，實際長度取非負主根。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["中點座標及兩段平方"]},"variant":{"family_id":"qf-density12-length-midpoint-equidistance","seed":202610126061,"parameters":{"P":[-10,1],"Q":[11,12],"frame":"orthogonal_equal_unit","operation":"density12_length_midpoint_equidistance","domain":"integer_euclidean_plane","semantic_template":"qf-density12-length-midpoint-equidistance/[\"1/2\",\"13/2\",\"562\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r06-euclidean-length-0062","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-length"],"title_zh":"中點座標及兩段平方","stem_zh":"等單位正交L2；同一實平面與長度單位；P=[-11, 1]、Q=[12, 13]，M是中點。依序求 (Mx, My, d(P,M)², d(M,Q)²)，四項均用最簡正分母分數，後兩項明示為平方。","answer":{"kind":"expression","value":"(1/2, 7/1, 673/4, 673/4)","explanation_zh":"中點與兩段平方=(1/2, 7/1, 673/4, 673/4)。座標平均給中點，每段平方為全段平方的四分之一；平方量不是距離本身，實際長度取非負主根。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["中點座標及兩段平方"]},"variant":{"family_id":"qf-density12-length-midpoint-equidistance","seed":202610126062,"parameters":{"P":[-11,1],"Q":[12,13],"frame":"orthogonal_equal_unit","operation":"density12_length_midpoint_equidistance","domain":"integer_euclidean_plane","semantic_template":"qf-density12-length-midpoint-equidistance/[\"1/2\",\"7/1\",\"673\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r06-euclidean-length-0063","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-length"],"title_zh":"中點座標及兩段平方","stem_zh":"等單位正交L2；同一實平面與長度單位；P=[-12, 1]、Q=[13, 14]，M是中點。依序求 (Mx, My, d(P,M)², d(M,Q)²)，四項均用最簡正分母分數，後兩項明示為平方。","answer":{"kind":"expression","value":"(1/2, 15/2, 397/2, 397/2)","explanation_zh":"中點與兩段平方=(1/2, 15/2, 397/2, 397/2)。座標平均給中點，每段平方為全段平方的四分之一；平方量不是距離本身，實際長度取非負主根。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["中點座標及兩段平方"]},"variant":{"family_id":"qf-density12-length-midpoint-equidistance","seed":202610126063,"parameters":{"P":[-12,1],"Q":[13,14],"frame":"orthogonal_equal_unit","operation":"density12_length_midpoint_equidistance","domain":"integer_euclidean_plane","semantic_template":"qf-density12-length-midpoint-equidistance/[\"1/2\",\"15/2\",\"794\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r06-euclidean-length-0064","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-length"],"title_zh":"中點座標及兩段平方","stem_zh":"等單位正交L2；同一實平面與長度單位；P=[-13, 1]、Q=[14, 15]，M是中點。依序求 (Mx, My, d(P,M)², d(M,Q)²)，四項均用最簡正分母分數，後兩項明示為平方。","answer":{"kind":"expression","value":"(1/2, 8/1, 925/4, 925/4)","explanation_zh":"中點與兩段平方=(1/2, 8/1, 925/4, 925/4)。座標平均給中點，每段平方為全段平方的四分之一；平方量不是距離本身，實際長度取非負主根。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["中點座標及兩段平方"]},"variant":{"family_id":"qf-density12-length-midpoint-equidistance","seed":202610126064,"parameters":{"P":[-13,1],"Q":[14,15],"frame":"orthogonal_equal_unit","operation":"density12_length_midpoint_equidistance","domain":"integer_euclidean_plane","semantic_template":"qf-density12-length-midpoint-equidistance/[\"1/2\",\"8/1\",\"925\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r06-euclidean-length-0065","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-length"],"title_zh":"中點座標及兩段平方","stem_zh":"等單位正交L2；同一實平面與長度單位；P=[-14, 1]、Q=[15, 16]，M是中點。依序求 (Mx, My, d(P,M)², d(M,Q)²)，四項均用最簡正分母分數，後兩項明示為平方。","answer":{"kind":"expression","value":"(1/2, 17/2, 533/2, 533/2)","explanation_zh":"中點與兩段平方=(1/2, 17/2, 533/2, 533/2)。座標平均給中點，每段平方為全段平方的四分之一；平方量不是距離本身，實際長度取非負主根。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["中點座標及兩段平方"]},"variant":{"family_id":"qf-density12-length-midpoint-equidistance","seed":202610126065,"parameters":{"P":[-14,1],"Q":[15,16],"frame":"orthogonal_equal_unit","operation":"density12_length_midpoint_equidistance","domain":"integer_euclidean_plane","semantic_template":"qf-density12-length-midpoint-equidistance/[\"1/2\",\"17/2\",\"1066\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r06-euclidean-length-0066","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-length"],"title_zh":"中點座標及兩段平方","stem_zh":"等單位正交L2；同一實平面與長度單位；P=[-15, 1]、Q=[16, 17]，M是中點。依序求 (Mx, My, d(P,M)², d(M,Q)²)，四項均用最簡正分母分數，後兩項明示為平方。","answer":{"kind":"expression","value":"(1/2, 9/1, 1217/4, 1217/4)","explanation_zh":"中點與兩段平方=(1/2, 9/1, 1217/4, 1217/4)。座標平均給中點，每段平方為全段平方的四分之一；平方量不是距離本身，實際長度取非負主根。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["中點座標及兩段平方"]},"variant":{"family_id":"qf-density12-length-midpoint-equidistance","seed":202610126066,"parameters":{"P":[-15,1],"Q":[16,17],"frame":"orthogonal_equal_unit","operation":"density12_length_midpoint_equidistance","domain":"integer_euclidean_plane","semantic_template":"qf-density12-length-midpoint-equidistance/[\"1/2\",\"9/1\",\"1217\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r06-euclidean-length-0067","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-length"],"title_zh":"中點座標及兩段平方","stem_zh":"等單位正交L2；同一實平面與長度單位；P=[-16, 1]、Q=[17, 18]，M是中點。依序求 (Mx, My, d(P,M)², d(M,Q)²)，四項均用最簡正分母分數，後兩項明示為平方。","answer":{"kind":"expression","value":"(1/2, 19/2, 689/2, 689/2)","explanation_zh":"中點與兩段平方=(1/2, 19/2, 689/2, 689/2)。座標平均給中點，每段平方為全段平方的四分之一；平方量不是距離本身，實際長度取非負主根。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["中點座標及兩段平方"]},"variant":{"family_id":"qf-density12-length-midpoint-equidistance","seed":202610126067,"parameters":{"P":[-16,1],"Q":[17,18],"frame":"orthogonal_equal_unit","operation":"density12_length_midpoint_equidistance","domain":"integer_euclidean_plane","semantic_template":"qf-density12-length-midpoint-equidistance/[\"1/2\",\"19/2\",\"1378\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r06-euclidean-length-0068","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-length"],"title_zh":"中點座標及兩段平方","stem_zh":"等單位正交L2；同一實平面與長度單位；P=[-17, 1]、Q=[18, 19]，M是中點。依序求 (Mx, My, d(P,M)², d(M,Q)²)，四項均用最簡正分母分數，後兩項明示為平方。","answer":{"kind":"expression","value":"(1/2, 10/1, 1549/4, 1549/4)","explanation_zh":"中點與兩段平方=(1/2, 10/1, 1549/4, 1549/4)。座標平均給中點，每段平方為全段平方的四分之一；平方量不是距離本身，實際長度取非負主根。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["中點座標及兩段平方"]},"variant":{"family_id":"qf-density12-length-midpoint-equidistance","seed":202610126068,"parameters":{"P":[-17,1],"Q":[18,19],"frame":"orthogonal_equal_unit","operation":"density12_length_midpoint_equidistance","domain":"integer_euclidean_plane","semantic_template":"qf-density12-length-midpoint-equidistance/[\"1/2\",\"10/1\",\"1549\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r06-euclidean-length-0069","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-length"],"title_zh":"中點座標及兩段平方","stem_zh":"等單位正交L2；同一實平面與長度單位；P=[-18, 1]、Q=[19, 20]，M是中點。依序求 (Mx, My, d(P,M)², d(M,Q)²)，四項均用最簡正分母分數，後兩項明示為平方。","answer":{"kind":"expression","value":"(1/2, 21/2, 865/2, 865/2)","explanation_zh":"中點與兩段平方=(1/2, 21/2, 865/2, 865/2)。座標平均給中點，每段平方為全段平方的四分之一；平方量不是距離本身，實際長度取非負主根。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["中點座標及兩段平方"]},"variant":{"family_id":"qf-density12-length-midpoint-equidistance","seed":202610126069,"parameters":{"P":[-18,1],"Q":[19,20],"frame":"orthogonal_equal_unit","operation":"density12_length_midpoint_equidistance","domain":"integer_euclidean_plane","semantic_template":"qf-density12-length-midpoint-equidistance/[\"1/2\",\"21/2\",\"1730\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r06-euclidean-length-0070","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-length"],"title_zh":"中點座標及兩段平方","stem_zh":"等單位正交L2；同一實平面與長度單位；P=[-19, 1]、Q=[20, 21]，M是中點。依序求 (Mx, My, d(P,M)², d(M,Q)²)，四項均用最簡正分母分數，後兩項明示為平方。","answer":{"kind":"expression","value":"(1/2, 11/1, 1921/4, 1921/4)","explanation_zh":"中點與兩段平方=(1/2, 11/1, 1921/4, 1921/4)。座標平均給中點，每段平方為全段平方的四分之一；平方量不是距離本身，實際長度取非負主根。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["中點座標及兩段平方"]},"variant":{"family_id":"qf-density12-length-midpoint-equidistance","seed":202610126070,"parameters":{"P":[-19,1],"Q":[20,21],"frame":"orthogonal_equal_unit","operation":"density12_length_midpoint_equidistance","domain":"integer_euclidean_plane","semantic_template":"qf-density12-length-midpoint-equidistance/[\"1/2\",\"11/1\",\"1921\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r06-euclidean-length-0071","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-length"],"title_zh":"中點座標及兩段平方","stem_zh":"等單位正交L2；同一實平面與長度單位；P=[-20, 1]、Q=[21, 22]，M是中點。依序求 (Mx, My, d(P,M)², d(M,Q)²)，四項均用最簡正分母分數，後兩項明示為平方。","answer":{"kind":"expression","value":"(1/2, 23/2, 1061/2, 1061/2)","explanation_zh":"中點與兩段平方=(1/2, 23/2, 1061/2, 1061/2)。座標平均給中點，每段平方為全段平方的四分之一；平方量不是距離本身，實際長度取非負主根。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["中點座標及兩段平方"]},"variant":{"family_id":"qf-density12-length-midpoint-equidistance","seed":202610126071,"parameters":{"P":[-20,1],"Q":[21,22],"frame":"orthogonal_equal_unit","operation":"density12_length_midpoint_equidistance","domain":"integer_euclidean_plane","semantic_template":"qf-density12-length-midpoint-equidistance/[\"1/2\",\"23/2\",\"2122\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r06-euclidean-length-0072","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-length"],"title_zh":"中點座標及兩段平方","stem_zh":"等單位正交L2；同一實平面與長度單位；P=[-21, 1]、Q=[22, 23]，M是中點。依序求 (Mx, My, d(P,M)², d(M,Q)²)，四項均用最簡正分母分數，後兩項明示為平方。","answer":{"kind":"expression","value":"(1/2, 12/1, 2333/4, 2333/4)","explanation_zh":"中點與兩段平方=(1/2, 12/1, 2333/4, 2333/4)。座標平均給中點，每段平方為全段平方的四分之一；平方量不是距離本身，實際長度取非負主根。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["中點座標及兩段平方"]},"variant":{"family_id":"qf-density12-length-midpoint-equidistance","seed":202610126072,"parameters":{"P":[-21,1],"Q":[22,23],"frame":"orthogonal_equal_unit","operation":"density12_length_midpoint_equidistance","domain":"integer_euclidean_plane","semantic_template":"qf-density12-length-midpoint-equidistance/[\"1/2\",\"12/1\",\"2333\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r06-euclidean-length-0073","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-length"],"title_zh":"中點座標及兩段平方","stem_zh":"等單位正交L2；同一實平面與長度單位；P=[-22, 1]、Q=[23, 24]，M是中點。依序求 (Mx, My, d(P,M)², d(M,Q)²)，四項均用最簡正分母分數，後兩項明示為平方。","answer":{"kind":"expression","value":"(1/2, 25/2, 1277/2, 1277/2)","explanation_zh":"中點與兩段平方=(1/2, 25/2, 1277/2, 1277/2)。座標平均給中點，每段平方為全段平方的四分之一；平方量不是距離本身，實際長度取非負主根。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["中點座標及兩段平方"]},"variant":{"family_id":"qf-density12-length-midpoint-equidistance","seed":202610126073,"parameters":{"P":[-22,1],"Q":[23,24],"frame":"orthogonal_equal_unit","operation":"density12_length_midpoint_equidistance","domain":"integer_euclidean_plane","semantic_template":"qf-density12-length-midpoint-equidistance/[\"1/2\",\"25/2\",\"2554\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r06-euclidean-length-0074","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-length"],"title_zh":"中點座標及兩段平方","stem_zh":"等單位正交L2；同一實平面與長度單位；P=[-23, 1]、Q=[24, 25]，M是中點。依序求 (Mx, My, d(P,M)², d(M,Q)²)，四項均用最簡正分母分數，後兩項明示為平方。","answer":{"kind":"expression","value":"(1/2, 13/1, 2785/4, 2785/4)","explanation_zh":"中點與兩段平方=(1/2, 13/1, 2785/4, 2785/4)。座標平均給中點，每段平方為全段平方的四分之一；平方量不是距離本身，實際長度取非負主根。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["中點座標及兩段平方"]},"variant":{"family_id":"qf-density12-length-midpoint-equidistance","seed":202610126074,"parameters":{"P":[-23,1],"Q":[24,25],"frame":"orthogonal_equal_unit","operation":"density12_length_midpoint_equidistance","domain":"integer_euclidean_plane","semantic_template":"qf-density12-length-midpoint-equidistance/[\"1/2\",\"13/1\",\"2785\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r06-euclidean-length-0075","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-length"],"title_zh":"中點座標及兩段平方","stem_zh":"等單位正交L2；同一實平面與長度單位；P=[-24, 1]、Q=[25, 26]，M是中點。依序求 (Mx, My, d(P,M)², d(M,Q)²)，四項均用最簡正分母分數，後兩項明示為平方。","answer":{"kind":"expression","value":"(1/2, 27/2, 1513/2, 1513/2)","explanation_zh":"中點與兩段平方=(1/2, 27/2, 1513/2, 1513/2)。座標平均給中點，每段平方為全段平方的四分之一；平方量不是距離本身，實際長度取非負主根。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["中點座標及兩段平方"]},"variant":{"family_id":"qf-density12-length-midpoint-equidistance","seed":202610126075,"parameters":{"P":[-24,1],"Q":[25,26],"frame":"orthogonal_equal_unit","operation":"density12_length_midpoint_equidistance","domain":"integer_euclidean_plane","semantic_template":"qf-density12-length-midpoint-equidistance/[\"1/2\",\"27/2\",\"3026\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r06-euclidean-length-0076","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-length"],"title_zh":"軸上路徑和與直線差","stem_zh":"等單位正交L2；同一實平面與長度單位；P=(0,0)、Q=(0,0)、R=(0,0)。依序答 (P經Q到R的路徑和, P到R直線長, 路徑−直線差, 是否相等)。","answer":{"kind":"expression","value":"(0, 0, 0, 是)","explanation_zh":"軸路徑與直線差=(0, 0, 0, 是)。各軸段用絕對差即非負主根；路徑和不當直線長，Q在段外可有正缺陷。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["軸上路徑和與直線差"]},"variant":{"family_id":"qf-density12-length-triangle-defect","seed":202610126076,"parameters":{"x":0,"y":0,"z":0,"frame":"orthogonal_equal_unit","operation":"density12_length_triangle_defect","domain":"integer_euclidean_axis","semantic_template":"qf-density12-length-triangle-defect/[\"0\",\"0\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r06-euclidean-length-0077","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-length"],"title_zh":"軸上路徑和與直線差","stem_zh":"等單位正交L2；同一實平面與長度單位；P=(0,0)、Q=(1,0)、R=(2,0)。依序答 (P經Q到R的路徑和, P到R直線長, 路徑−直線差, 是否相等)。","answer":{"kind":"expression","value":"(2, 2, 0, 是)","explanation_zh":"軸路徑與直線差=(2, 2, 0, 是)。各軸段用絕對差即非負主根；路徑和不當直線長，Q在段外可有正缺陷。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["軸上路徑和與直線差"]},"variant":{"family_id":"qf-density12-length-triangle-defect","seed":202610126077,"parameters":{"x":0,"y":1,"z":2,"frame":"orthogonal_equal_unit","operation":"density12_length_triangle_defect","domain":"integer_euclidean_axis","semantic_template":"qf-density12-length-triangle-defect/[\"2\",\"2\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r06-euclidean-length-0078","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-length"],"title_zh":"軸上路徑和與直線差","stem_zh":"等單位正交L2；同一實平面與長度單位；P=(0,0)、Q=(-1,0)、R=(2,0)。依序答 (P經Q到R的路徑和, P到R直線長, 路徑−直線差, 是否相等)。","answer":{"kind":"expression","value":"(4, 2, 2, 否)","explanation_zh":"軸路徑與直線差=(4, 2, 2, 否)。各軸段用絕對差即非負主根；路徑和不當直線長，Q在段外可有正缺陷。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["軸上路徑和與直線差"]},"variant":{"family_id":"qf-density12-length-triangle-defect","seed":202610126078,"parameters":{"x":0,"y":-1,"z":2,"frame":"orthogonal_equal_unit","operation":"density12_length_triangle_defect","domain":"integer_euclidean_axis","semantic_template":"qf-density12-length-triangle-defect/[\"4\",\"2\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r06-euclidean-length-0079","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-length"],"title_zh":"軸上路徑和與直線差","stem_zh":"等單位正交L2；同一實平面與長度單位；P=(0,0)、Q=(-3,0)、R=(3,0)。依序答 (P經Q到R的路徑和, P到R直線長, 路徑−直線差, 是否相等)。","answer":{"kind":"expression","value":"(9, 3, 6, 否)","explanation_zh":"軸路徑與直線差=(9, 3, 6, 否)。各軸段用絕對差即非負主根；路徑和不當直線長，Q在段外可有正缺陷。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["軸上路徑和與直線差"]},"variant":{"family_id":"qf-density12-length-triangle-defect","seed":202610126079,"parameters":{"x":0,"y":-3,"z":3,"frame":"orthogonal_equal_unit","operation":"density12_length_triangle_defect","domain":"integer_euclidean_axis","semantic_template":"qf-density12-length-triangle-defect/[\"9\",\"3\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r06-euclidean-length-0080","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-length"],"title_zh":"軸上路徑和與直線差","stem_zh":"等單位正交L2；同一實平面與長度單位；P=(0,0)、Q=(2,0)、R=(4,0)。依序答 (P經Q到R的路徑和, P到R直線長, 路徑−直線差, 是否相等)。","answer":{"kind":"expression","value":"(4, 4, 0, 是)","explanation_zh":"軸路徑與直線差=(4, 4, 0, 是)。各軸段用絕對差即非負主根；路徑和不當直線長，Q在段外可有正缺陷。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["軸上路徑和與直線差"]},"variant":{"family_id":"qf-density12-length-triangle-defect","seed":202610126080,"parameters":{"x":0,"y":2,"z":4,"frame":"orthogonal_equal_unit","operation":"density12_length_triangle_defect","domain":"integer_euclidean_axis","semantic_template":"qf-density12-length-triangle-defect/[\"4\",\"4\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r06-euclidean-length-0081","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-length"],"title_zh":"軸上路徑和與直線差","stem_zh":"等單位正交L2；同一實平面與長度單位；P=(0,0)、Q=(-5,0)、R=(5,0)。依序答 (P經Q到R的路徑和, P到R直線長, 路徑−直線差, 是否相等)。","answer":{"kind":"expression","value":"(15, 5, 10, 否)","explanation_zh":"軸路徑與直線差=(15, 5, 10, 否)。各軸段用絕對差即非負主根；路徑和不當直線長，Q在段外可有正缺陷。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["軸上路徑和與直線差"]},"variant":{"family_id":"qf-density12-length-triangle-defect","seed":202610126081,"parameters":{"x":0,"y":-5,"z":5,"frame":"orthogonal_equal_unit","operation":"density12_length_triangle_defect","domain":"integer_euclidean_axis","semantic_template":"qf-density12-length-triangle-defect/[\"15\",\"5\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r06-euclidean-length-0082","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-length"],"title_zh":"軸上路徑和與直線差","stem_zh":"等單位正交L2；同一實平面與長度單位；P=(0,0)、Q=(3,0)、R=(6,0)。依序答 (P經Q到R的路徑和, P到R直線長, 路徑−直線差, 是否相等)。","answer":{"kind":"expression","value":"(6, 6, 0, 是)","explanation_zh":"軸路徑與直線差=(6, 6, 0, 是)。各軸段用絕對差即非負主根；路徑和不當直線長，Q在段外可有正缺陷。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["軸上路徑和與直線差"]},"variant":{"family_id":"qf-density12-length-triangle-defect","seed":202610126082,"parameters":{"x":0,"y":3,"z":6,"frame":"orthogonal_equal_unit","operation":"density12_length_triangle_defect","domain":"integer_euclidean_axis","semantic_template":"qf-density12-length-triangle-defect/[\"6\",\"6\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r06-euclidean-length-0083","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-length"],"title_zh":"軸上路徑和與直線差","stem_zh":"等單位正交L2；同一實平面與長度單位；P=(0,0)、Q=(-7,0)、R=(7,0)。依序答 (P經Q到R的路徑和, P到R直線長, 路徑−直線差, 是否相等)。","answer":{"kind":"expression","value":"(21, 7, 14, 否)","explanation_zh":"軸路徑與直線差=(21, 7, 14, 否)。各軸段用絕對差即非負主根；路徑和不當直線長，Q在段外可有正缺陷。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["軸上路徑和與直線差"]},"variant":{"family_id":"qf-density12-length-triangle-defect","seed":202610126083,"parameters":{"x":0,"y":-7,"z":7,"frame":"orthogonal_equal_unit","operation":"density12_length_triangle_defect","domain":"integer_euclidean_axis","semantic_template":"qf-density12-length-triangle-defect/[\"21\",\"7\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r06-euclidean-length-0084","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-length"],"title_zh":"軸上路徑和與直線差","stem_zh":"等單位正交L2；同一實平面與長度單位；P=(0,0)、Q=(4,0)、R=(8,0)。依序答 (P經Q到R的路徑和, P到R直線長, 路徑−直線差, 是否相等)。","answer":{"kind":"expression","value":"(8, 8, 0, 是)","explanation_zh":"軸路徑與直線差=(8, 8, 0, 是)。各軸段用絕對差即非負主根；路徑和不當直線長，Q在段外可有正缺陷。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["軸上路徑和與直線差"]},"variant":{"family_id":"qf-density12-length-triangle-defect","seed":202610126084,"parameters":{"x":0,"y":4,"z":8,"frame":"orthogonal_equal_unit","operation":"density12_length_triangle_defect","domain":"integer_euclidean_axis","semantic_template":"qf-density12-length-triangle-defect/[\"8\",\"8\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r06-euclidean-length-0085","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-length"],"title_zh":"軸上路徑和與直線差","stem_zh":"等單位正交L2；同一實平面與長度單位；P=(0,0)、Q=(-9,0)、R=(9,0)。依序答 (P經Q到R的路徑和, P到R直線長, 路徑−直線差, 是否相等)。","answer":{"kind":"expression","value":"(27, 9, 18, 否)","explanation_zh":"軸路徑與直線差=(27, 9, 18, 否)。各軸段用絕對差即非負主根；路徑和不當直線長，Q在段外可有正缺陷。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["軸上路徑和與直線差"]},"variant":{"family_id":"qf-density12-length-triangle-defect","seed":202610126085,"parameters":{"x":0,"y":-9,"z":9,"frame":"orthogonal_equal_unit","operation":"density12_length_triangle_defect","domain":"integer_euclidean_axis","semantic_template":"qf-density12-length-triangle-defect/[\"27\",\"9\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r06-euclidean-length-0086","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-length"],"title_zh":"軸上路徑和與直線差","stem_zh":"等單位正交L2；同一實平面與長度單位；P=(0,0)、Q=(5,0)、R=(10,0)。依序答 (P經Q到R的路徑和, P到R直線長, 路徑−直線差, 是否相等)。","answer":{"kind":"expression","value":"(10, 10, 0, 是)","explanation_zh":"軸路徑與直線差=(10, 10, 0, 是)。各軸段用絕對差即非負主根；路徑和不當直線長，Q在段外可有正缺陷。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["軸上路徑和與直線差"]},"variant":{"family_id":"qf-density12-length-triangle-defect","seed":202610126086,"parameters":{"x":0,"y":5,"z":10,"frame":"orthogonal_equal_unit","operation":"density12_length_triangle_defect","domain":"integer_euclidean_axis","semantic_template":"qf-density12-length-triangle-defect/[\"10\",\"10\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r06-euclidean-length-0087","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-length"],"title_zh":"軸上路徑和與直線差","stem_zh":"等單位正交L2；同一實平面與長度單位；P=(0,0)、Q=(-11,0)、R=(11,0)。依序答 (P經Q到R的路徑和, P到R直線長, 路徑−直線差, 是否相等)。","answer":{"kind":"expression","value":"(33, 11, 22, 否)","explanation_zh":"軸路徑與直線差=(33, 11, 22, 否)。各軸段用絕對差即非負主根；路徑和不當直線長，Q在段外可有正缺陷。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["軸上路徑和與直線差"]},"variant":{"family_id":"qf-density12-length-triangle-defect","seed":202610126087,"parameters":{"x":0,"y":-11,"z":11,"frame":"orthogonal_equal_unit","operation":"density12_length_triangle_defect","domain":"integer_euclidean_axis","semantic_template":"qf-density12-length-triangle-defect/[\"33\",\"11\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r06-euclidean-length-0088","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-length"],"title_zh":"軸上路徑和與直線差","stem_zh":"等單位正交L2；同一實平面與長度單位；P=(0,0)、Q=(6,0)、R=(12,0)。依序答 (P經Q到R的路徑和, P到R直線長, 路徑−直線差, 是否相等)。","answer":{"kind":"expression","value":"(12, 12, 0, 是)","explanation_zh":"軸路徑與直線差=(12, 12, 0, 是)。各軸段用絕對差即非負主根；路徑和不當直線長，Q在段外可有正缺陷。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["軸上路徑和與直線差"]},"variant":{"family_id":"qf-density12-length-triangle-defect","seed":202610126088,"parameters":{"x":0,"y":6,"z":12,"frame":"orthogonal_equal_unit","operation":"density12_length_triangle_defect","domain":"integer_euclidean_axis","semantic_template":"qf-density12-length-triangle-defect/[\"12\",\"12\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r06-euclidean-length-0089","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-length"],"title_zh":"軸上路徑和與直線差","stem_zh":"等單位正交L2；同一實平面與長度單位；P=(0,0)、Q=(-13,0)、R=(13,0)。依序答 (P經Q到R的路徑和, P到R直線長, 路徑−直線差, 是否相等)。","answer":{"kind":"expression","value":"(39, 13, 26, 否)","explanation_zh":"軸路徑與直線差=(39, 13, 26, 否)。各軸段用絕對差即非負主根；路徑和不當直線長，Q在段外可有正缺陷。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["軸上路徑和與直線差"]},"variant":{"family_id":"qf-density12-length-triangle-defect","seed":202610126089,"parameters":{"x":0,"y":-13,"z":13,"frame":"orthogonal_equal_unit","operation":"density12_length_triangle_defect","domain":"integer_euclidean_axis","semantic_template":"qf-density12-length-triangle-defect/[\"39\",\"13\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r06-euclidean-length-0090","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-length"],"title_zh":"軸上路徑和與直線差","stem_zh":"等單位正交L2；同一實平面與長度單位；P=(0,0)、Q=(7,0)、R=(14,0)。依序答 (P經Q到R的路徑和, P到R直線長, 路徑−直線差, 是否相等)。","answer":{"kind":"expression","value":"(14, 14, 0, 是)","explanation_zh":"軸路徑與直線差=(14, 14, 0, 是)。各軸段用絕對差即非負主根；路徑和不當直線長，Q在段外可有正缺陷。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["軸上路徑和與直線差"]},"variant":{"family_id":"qf-density12-length-triangle-defect","seed":202610126090,"parameters":{"x":0,"y":7,"z":14,"frame":"orthogonal_equal_unit","operation":"density12_length_triangle_defect","domain":"integer_euclidean_axis","semantic_template":"qf-density12-length-triangle-defect/[\"14\",\"14\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r06-euclidean-length-0091","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-length"],"title_zh":"軸上路徑和與直線差","stem_zh":"等單位正交L2；同一實平面與長度單位；P=(0,0)、Q=(-15,0)、R=(15,0)。依序答 (P經Q到R的路徑和, P到R直線長, 路徑−直線差, 是否相等)。","answer":{"kind":"expression","value":"(45, 15, 30, 否)","explanation_zh":"軸路徑與直線差=(45, 15, 30, 否)。各軸段用絕對差即非負主根；路徑和不當直線長，Q在段外可有正缺陷。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["軸上路徑和與直線差"]},"variant":{"family_id":"qf-density12-length-triangle-defect","seed":202610126091,"parameters":{"x":0,"y":-15,"z":15,"frame":"orthogonal_equal_unit","operation":"density12_length_triangle_defect","domain":"integer_euclidean_axis","semantic_template":"qf-density12-length-triangle-defect/[\"45\",\"15\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r06-euclidean-length-0092","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-length"],"title_zh":"軸上路徑和與直線差","stem_zh":"等單位正交L2；同一實平面與長度單位；P=(0,0)、Q=(8,0)、R=(16,0)。依序答 (P經Q到R的路徑和, P到R直線長, 路徑−直線差, 是否相等)。","answer":{"kind":"expression","value":"(16, 16, 0, 是)","explanation_zh":"軸路徑與直線差=(16, 16, 0, 是)。各軸段用絕對差即非負主根；路徑和不當直線長，Q在段外可有正缺陷。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["軸上路徑和與直線差"]},"variant":{"family_id":"qf-density12-length-triangle-defect","seed":202610126092,"parameters":{"x":0,"y":8,"z":16,"frame":"orthogonal_equal_unit","operation":"density12_length_triangle_defect","domain":"integer_euclidean_axis","semantic_template":"qf-density12-length-triangle-defect/[\"16\",\"16\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r06-euclidean-length-0093","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-length"],"title_zh":"軸上路徑和與直線差","stem_zh":"等單位正交L2；同一實平面與長度單位；P=(0,0)、Q=(-17,0)、R=(17,0)。依序答 (P經Q到R的路徑和, P到R直線長, 路徑−直線差, 是否相等)。","answer":{"kind":"expression","value":"(51, 17, 34, 否)","explanation_zh":"軸路徑與直線差=(51, 17, 34, 否)。各軸段用絕對差即非負主根；路徑和不當直線長，Q在段外可有正缺陷。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["軸上路徑和與直線差"]},"variant":{"family_id":"qf-density12-length-triangle-defect","seed":202610126093,"parameters":{"x":0,"y":-17,"z":17,"frame":"orthogonal_equal_unit","operation":"density12_length_triangle_defect","domain":"integer_euclidean_axis","semantic_template":"qf-density12-length-triangle-defect/[\"51\",\"17\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r06-euclidean-length-0094","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-length"],"title_zh":"軸上路徑和與直線差","stem_zh":"等單位正交L2；同一實平面與長度單位；P=(0,0)、Q=(9,0)、R=(18,0)。依序答 (P經Q到R的路徑和, P到R直線長, 路徑−直線差, 是否相等)。","answer":{"kind":"expression","value":"(18, 18, 0, 是)","explanation_zh":"軸路徑與直線差=(18, 18, 0, 是)。各軸段用絕對差即非負主根；路徑和不當直線長，Q在段外可有正缺陷。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["軸上路徑和與直線差"]},"variant":{"family_id":"qf-density12-length-triangle-defect","seed":202610126094,"parameters":{"x":0,"y":9,"z":18,"frame":"orthogonal_equal_unit","operation":"density12_length_triangle_defect","domain":"integer_euclidean_axis","semantic_template":"qf-density12-length-triangle-defect/[\"18\",\"18\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r06-euclidean-length-0095","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-length"],"title_zh":"軸上路徑和與直線差","stem_zh":"等單位正交L2；同一實平面與長度單位；P=(0,0)、Q=(-19,0)、R=(19,0)。依序答 (P經Q到R的路徑和, P到R直線長, 路徑−直線差, 是否相等)。","answer":{"kind":"expression","value":"(57, 19, 38, 否)","explanation_zh":"軸路徑與直線差=(57, 19, 38, 否)。各軸段用絕對差即非負主根；路徑和不當直線長，Q在段外可有正缺陷。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["軸上路徑和與直線差"]},"variant":{"family_id":"qf-density12-length-triangle-defect","seed":202610126095,"parameters":{"x":0,"y":-19,"z":19,"frame":"orthogonal_equal_unit","operation":"density12_length_triangle_defect","domain":"integer_euclidean_axis","semantic_template":"qf-density12-length-triangle-defect/[\"57\",\"19\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r06-euclidean-length-0096","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-length"],"title_zh":"軸上路徑和與直線差","stem_zh":"等單位正交L2；同一實平面與長度單位；P=(0,0)、Q=(10,0)、R=(20,0)。依序答 (P經Q到R的路徑和, P到R直線長, 路徑−直線差, 是否相等)。","answer":{"kind":"expression","value":"(20, 20, 0, 是)","explanation_zh":"軸路徑與直線差=(20, 20, 0, 是)。各軸段用絕對差即非負主根；路徑和不當直線長，Q在段外可有正缺陷。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["軸上路徑和與直線差"]},"variant":{"family_id":"qf-density12-length-triangle-defect","seed":202610126096,"parameters":{"x":0,"y":10,"z":20,"frame":"orthogonal_equal_unit","operation":"density12_length_triangle_defect","domain":"integer_euclidean_axis","semantic_template":"qf-density12-length-triangle-defect/[\"20\",\"20\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r06-euclidean-length-0097","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-length"],"title_zh":"軸上路徑和與直線差","stem_zh":"等單位正交L2；同一實平面與長度單位；P=(0,0)、Q=(-21,0)、R=(21,0)。依序答 (P經Q到R的路徑和, P到R直線長, 路徑−直線差, 是否相等)。","answer":{"kind":"expression","value":"(63, 21, 42, 否)","explanation_zh":"軸路徑與直線差=(63, 21, 42, 否)。各軸段用絕對差即非負主根；路徑和不當直線長，Q在段外可有正缺陷。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["軸上路徑和與直線差"]},"variant":{"family_id":"qf-density12-length-triangle-defect","seed":202610126097,"parameters":{"x":0,"y":-21,"z":21,"frame":"orthogonal_equal_unit","operation":"density12_length_triangle_defect","domain":"integer_euclidean_axis","semantic_template":"qf-density12-length-triangle-defect/[\"63\",\"21\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r06-euclidean-length-0098","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-length"],"title_zh":"軸上路徑和與直線差","stem_zh":"等單位正交L2；同一實平面與長度單位；P=(0,0)、Q=(11,0)、R=(22,0)。依序答 (P經Q到R的路徑和, P到R直線長, 路徑−直線差, 是否相等)。","answer":{"kind":"expression","value":"(22, 22, 0, 是)","explanation_zh":"軸路徑與直線差=(22, 22, 0, 是)。各軸段用絕對差即非負主根；路徑和不當直線長，Q在段外可有正缺陷。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["軸上路徑和與直線差"]},"variant":{"family_id":"qf-density12-length-triangle-defect","seed":202610126098,"parameters":{"x":0,"y":11,"z":22,"frame":"orthogonal_equal_unit","operation":"density12_length_triangle_defect","domain":"integer_euclidean_axis","semantic_template":"qf-density12-length-triangle-defect/[\"22\",\"22\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r06-euclidean-length-0099","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-length"],"title_zh":"軸上路徑和與直線差","stem_zh":"等單位正交L2；同一實平面與長度單位；P=(0,0)、Q=(-23,0)、R=(23,0)。依序答 (P經Q到R的路徑和, P到R直線長, 路徑−直線差, 是否相等)。","answer":{"kind":"expression","value":"(69, 23, 46, 否)","explanation_zh":"軸路徑與直線差=(69, 23, 46, 否)。各軸段用絕對差即非負主根；路徑和不當直線長，Q在段外可有正缺陷。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["軸上路徑和與直線差"]},"variant":{"family_id":"qf-density12-length-triangle-defect","seed":202610126099,"parameters":{"x":0,"y":-23,"z":23,"frame":"orthogonal_equal_unit","operation":"density12_length_triangle_defect","domain":"integer_euclidean_axis","semantic_template":"qf-density12-length-triangle-defect/[\"69\",\"23\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r06-euclidean-length-0100","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-length"],"title_zh":"軸上路徑和與直線差","stem_zh":"等單位正交L2；同一實平面與長度單位；P=(0,0)、Q=(12,0)、R=(24,0)。依序答 (P經Q到R的路徑和, P到R直線長, 路徑−直線差, 是否相等)。","answer":{"kind":"expression","value":"(24, 24, 0, 是)","explanation_zh":"軸路徑與直線差=(24, 24, 0, 是)。各軸段用絕對差即非負主根；路徑和不當直線長，Q在段外可有正缺陷。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["軸上路徑和與直線差"]},"variant":{"family_id":"qf-density12-length-triangle-defect","seed":202610126100,"parameters":{"x":0,"y":12,"z":24,"frame":"orthogonal_equal_unit","operation":"density12_length_triangle_defect","domain":"integer_euclidean_axis","semantic_template":"qf-density12-length-triangle-defect/[\"24\",\"24\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}