{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r02-real-number-0001","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","language":"zh-Hant","source_mko_ids":["mko-real-number"],"title_zh":"有序有理端點的中點距離","stem_zh":"有理點嵌入實數的有限切片；a=0/1≤b=0/1。依序求 (中點, 中點−a, b−中點)，用既約正分母分數。","answer":{"kind":"expression","value":"(0/1, 0/1, 0/1)","explanation_zh":"有序中點與兩側距離=(0/1, 0/1, 0/1)。精確加總除二，再分別相減核兩側距離。非實數完備性或無理數分類證明，不用近似小數。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["有序有理端點的中點距離"]},"variant":{"family_id":"qf-density14-real-ordered-midpoint","seed":202610142001,"parameters":{"a":[0,1],"b":[0,1],"domain":"finite_rational_real_slice","operation":"density14_real_ordered_midpoint","semantic_template":"qf-density14-real-ordered-midpoint/[\"0/1\",\"0/1\"]"}},"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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r02-real-number-0002","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","language":"zh-Hant","source_mko_ids":["mko-real-number"],"title_zh":"有序有理端點的中點距離","stem_zh":"有理點嵌入實數的有限切片；a=-1/2≤b=1/2。依序求 (中點, 中點−a, b−中點)，用既約正分母分數。","answer":{"kind":"expression","value":"(0/1, 1/2, 1/2)","explanation_zh":"有序中點與兩側距離=(0/1, 1/2, 1/2)。精確加總除二，再分別相減核兩側距離。非實數完備性或無理數分類證明，不用近似小數。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["有序有理端點的中點距離"]},"variant":{"family_id":"qf-density14-real-ordered-midpoint","seed":202610142002,"parameters":{"a":[-1,2],"b":[1,2],"domain":"finite_rational_real_slice","operation":"density14_real_ordered_midpoint","semantic_template":"qf-density14-real-ordered-midpoint/[\"-1/2\",\"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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r02-real-number-0003","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","language":"zh-Hant","source_mko_ids":["mko-real-number"],"title_zh":"有序有理端點的中點距離","stem_zh":"有理點嵌入實數的有限切片；a=-2/2≤b=3/3。依序求 (中點, 中點−a, b−中點)，用既約正分母分數。","answer":{"kind":"expression","value":"(0/1, 1/1, 1/1)","explanation_zh":"有序中點與兩側距離=(0/1, 1/1, 1/1)。精確加總除二，再分別相減核兩側距離。非實數完備性或無理數分類證明，不用近似小數。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["有序有理端點的中點距離"]},"variant":{"family_id":"qf-density14-real-ordered-midpoint","seed":202610142003,"parameters":{"a":[-2,2],"b":[3,3],"domain":"finite_rational_real_slice","operation":"density14_real_ordered_midpoint","semantic_template":"qf-density14-real-ordered-midpoint/[\"-1/1\",\"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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r02-real-number-0004","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","language":"zh-Hant","source_mko_ids":["mko-real-number"],"title_zh":"有序有理端點的中點距離","stem_zh":"有理點嵌入實數的有限切片；a=-3/2≤b=4/3。依序求 (中點, 中點−a, b−中點)，用既約正分母分數。","answer":{"kind":"expression","value":"(-1/12, 17/12, 17/12)","explanation_zh":"有序中點與兩側距離=(-1/12, 17/12, 17/12)。精確加總除二，再分別相減核兩側距離。非實數完備性或無理數分類證明，不用近似小數。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["有序有理端點的中點距離"]},"variant":{"family_id":"qf-density14-real-ordered-midpoint","seed":202610142004,"parameters":{"a":[-3,2],"b":[4,3],"domain":"finite_rational_real_slice","operation":"density14_real_ordered_midpoint","semantic_template":"qf-density14-real-ordered-midpoint/[\"-3/2\",\"4/3\"]"}},"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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r02-real-number-0005","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","language":"zh-Hant","source_mko_ids":["mko-real-number"],"title_zh":"有序有理端點的中點距離","stem_zh":"有理點嵌入實數的有限切片；a=-4/2≤b=5/3。依序求 (中點, 中點−a, b−中點)，用既約正分母分數。","answer":{"kind":"expression","value":"(-1/6, 11/6, 11/6)","explanation_zh":"有序中點與兩側距離=(-1/6, 11/6, 11/6)。精確加總除二，再分別相減核兩側距離。非實數完備性或無理數分類證明，不用近似小數。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["有序有理端點的中點距離"]},"variant":{"family_id":"qf-density14-real-ordered-midpoint","seed":202610142005,"parameters":{"a":[-4,2],"b":[5,3],"domain":"finite_rational_real_slice","operation":"density14_real_ordered_midpoint","semantic_template":"qf-density14-real-ordered-midpoint/[\"-2/1\",\"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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r02-real-number-0006","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","language":"zh-Hant","source_mko_ids":["mko-real-number"],"title_zh":"有序有理端點的中點距離","stem_zh":"有理點嵌入實數的有限切片；a=-5/2≤b=6/3。依序求 (中點, 中點−a, b−中點)，用既約正分母分數。","answer":{"kind":"expression","value":"(-1/4, 9/4, 9/4)","explanation_zh":"有序中點與兩側距離=(-1/4, 9/4, 9/4)。精確加總除二，再分別相減核兩側距離。非實數完備性或無理數分類證明，不用近似小數。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["有序有理端點的中點距離"]},"variant":{"family_id":"qf-density14-real-ordered-midpoint","seed":202610142006,"parameters":{"a":[-5,2],"b":[6,3],"domain":"finite_rational_real_slice","operation":"density14_real_ordered_midpoint","semantic_template":"qf-density14-real-ordered-midpoint/[\"-5/2\",\"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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r02-real-number-0007","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","language":"zh-Hant","source_mko_ids":["mko-real-number"],"title_zh":"有序有理端點的中點距離","stem_zh":"有理點嵌入實數的有限切片；a=-6/2≤b=7/3。依序求 (中點, 中點−a, b−中點)，用既約正分母分數。","answer":{"kind":"expression","value":"(-1/3, 8/3, 8/3)","explanation_zh":"有序中點與兩側距離=(-1/3, 8/3, 8/3)。精確加總除二，再分別相減核兩側距離。非實數完備性或無理數分類證明，不用近似小數。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["有序有理端點的中點距離"]},"variant":{"family_id":"qf-density14-real-ordered-midpoint","seed":202610142007,"parameters":{"a":[-6,2],"b":[7,3],"domain":"finite_rational_real_slice","operation":"density14_real_ordered_midpoint","semantic_template":"qf-density14-real-ordered-midpoint/[\"-3/1\",\"7/3\"]"}},"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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r02-real-number-0008","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","language":"zh-Hant","source_mko_ids":["mko-real-number"],"title_zh":"有序有理端點的中點距離","stem_zh":"有理點嵌入實數的有限切片；a=-7/2≤b=8/3。依序求 (中點, 中點−a, b−中點)，用既約正分母分數。","answer":{"kind":"expression","value":"(-5/12, 37/12, 37/12)","explanation_zh":"有序中點與兩側距離=(-5/12, 37/12, 37/12)。精確加總除二，再分別相減核兩側距離。非實數完備性或無理數分類證明，不用近似小數。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["有序有理端點的中點距離"]},"variant":{"family_id":"qf-density14-real-ordered-midpoint","seed":202610142008,"parameters":{"a":[-7,2],"b":[8,3],"domain":"finite_rational_real_slice","operation":"density14_real_ordered_midpoint","semantic_template":"qf-density14-real-ordered-midpoint/[\"-7/2\",\"8/3\"]"}},"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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r02-real-number-0009","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","language":"zh-Hant","source_mko_ids":["mko-real-number"],"title_zh":"有序有理端點的中點距離","stem_zh":"有理點嵌入實數的有限切片；a=-8/2≤b=9/3。依序求 (中點, 中點−a, b−中點)，用既約正分母分數。","answer":{"kind":"expression","value":"(-1/2, 7/2, 7/2)","explanation_zh":"有序中點與兩側距離=(-1/2, 7/2, 7/2)。精確加總除二，再分別相減核兩側距離。非實數完備性或無理數分類證明，不用近似小數。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["有序有理端點的中點距離"]},"variant":{"family_id":"qf-density14-real-ordered-midpoint","seed":202610142009,"parameters":{"a":[-8,2],"b":[9,3],"domain":"finite_rational_real_slice","operation":"density14_real_ordered_midpoint","semantic_template":"qf-density14-real-ordered-midpoint/[\"-4/1\",\"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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r02-real-number-0010","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","language":"zh-Hant","source_mko_ids":["mko-real-number"],"title_zh":"有序有理端點的中點距離","stem_zh":"有理點嵌入實數的有限切片；a=-9/2≤b=10/3。依序求 (中點, 中點−a, b−中點)，用既約正分母分數。","answer":{"kind":"expression","value":"(-7/12, 47/12, 47/12)","explanation_zh":"有序中點與兩側距離=(-7/12, 47/12, 47/12)。精確加總除二，再分別相減核兩側距離。非實數完備性或無理數分類證明，不用近似小數。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["有序有理端點的中點距離"]},"variant":{"family_id":"qf-density14-real-ordered-midpoint","seed":202610142010,"parameters":{"a":[-9,2],"b":[10,3],"domain":"finite_rational_real_slice","operation":"density14_real_ordered_midpoint","semantic_template":"qf-density14-real-ordered-midpoint/[\"-9/2\",\"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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r02-real-number-0011","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","language":"zh-Hant","source_mko_ids":["mko-real-number"],"title_zh":"有序有理端點的中點距離","stem_zh":"有理點嵌入實數的有限切片；a=-10/2≤b=11/3。依序求 (中點, 中點−a, b−中點)，用既約正分母分數。","answer":{"kind":"expression","value":"(-2/3, 13/3, 13/3)","explanation_zh":"有序中點與兩側距離=(-2/3, 13/3, 13/3)。精確加總除二，再分別相減核兩側距離。非實數完備性或無理數分類證明，不用近似小數。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["有序有理端點的中點距離"]},"variant":{"family_id":"qf-density14-real-ordered-midpoint","seed":202610142011,"parameters":{"a":[-10,2],"b":[11,3],"domain":"finite_rational_real_slice","operation":"density14_real_ordered_midpoint","semantic_template":"qf-density14-real-ordered-midpoint/[\"-5/1\",\"11/3\"]"}},"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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r02-real-number-0012","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","language":"zh-Hant","source_mko_ids":["mko-real-number"],"title_zh":"有序有理端點的中點距離","stem_zh":"有理點嵌入實數的有限切片；a=-11/2≤b=12/3。依序求 (中點, 中點−a, b−中點)，用既約正分母分數。","answer":{"kind":"expression","value":"(-3/4, 19/4, 19/4)","explanation_zh":"有序中點與兩側距離=(-3/4, 19/4, 19/4)。精確加總除二，再分別相減核兩側距離。非實數完備性或無理數分類證明，不用近似小數。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["有序有理端點的中點距離"]},"variant":{"family_id":"qf-density14-real-ordered-midpoint","seed":202610142012,"parameters":{"a":[-11,2],"b":[12,3],"domain":"finite_rational_real_slice","operation":"density14_real_ordered_midpoint","semantic_template":"qf-density14-real-ordered-midpoint/[\"-11/2\",\"4/1\"]"}},"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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r02-real-number-0013","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","language":"zh-Hant","source_mko_ids":["mko-real-number"],"title_zh":"有序有理端點的中點距離","stem_zh":"有理點嵌入實數的有限切片；a=-12/2≤b=13/3。依序求 (中點, 中點−a, b−中點)，用既約正分母分數。","answer":{"kind":"expression","value":"(-5/6, 31/6, 31/6)","explanation_zh":"有序中點與兩側距離=(-5/6, 31/6, 31/6)。精確加總除二，再分別相減核兩側距離。非實數完備性或無理數分類證明，不用近似小數。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["有序有理端點的中點距離"]},"variant":{"family_id":"qf-density14-real-ordered-midpoint","seed":202610142013,"parameters":{"a":[-12,2],"b":[13,3],"domain":"finite_rational_real_slice","operation":"density14_real_ordered_midpoint","semantic_template":"qf-density14-real-ordered-midpoint/[\"-6/1\",\"13/3\"]"}},"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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r02-real-number-0014","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","language":"zh-Hant","source_mko_ids":["mko-real-number"],"title_zh":"有序有理端點的中點距離","stem_zh":"有理點嵌入實數的有限切片；a=-13/2≤b=14/3。依序求 (中點, 中點−a, b−中點)，用既約正分母分數。","answer":{"kind":"expression","value":"(-11/12, 67/12, 67/12)","explanation_zh":"有序中點與兩側距離=(-11/12, 67/12, 67/12)。精確加總除二，再分別相減核兩側距離。非實數完備性或無理數分類證明，不用近似小數。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["有序有理端點的中點距離"]},"variant":{"family_id":"qf-density14-real-ordered-midpoint","seed":202610142014,"parameters":{"a":[-13,2],"b":[14,3],"domain":"finite_rational_real_slice","operation":"density14_real_ordered_midpoint","semantic_template":"qf-density14-real-ordered-midpoint/[\"-13/2\",\"14/3\"]"}},"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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r02-real-number-0015","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","language":"zh-Hant","source_mko_ids":["mko-real-number"],"title_zh":"有序有理端點的中點距離","stem_zh":"有理點嵌入實數的有限切片；a=-14/2≤b=15/3。依序求 (中點, 中點−a, b−中點)，用既約正分母分數。","answer":{"kind":"expression","value":"(-1/1, 6/1, 6/1)","explanation_zh":"有序中點與兩側距離=(-1/1, 6/1, 6/1)。精確加總除二，再分別相減核兩側距離。非實數完備性或無理數分類證明，不用近似小數。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["有序有理端點的中點距離"]},"variant":{"family_id":"qf-density14-real-ordered-midpoint","seed":202610142015,"parameters":{"a":[-14,2],"b":[15,3],"domain":"finite_rational_real_slice","operation":"density14_real_ordered_midpoint","semantic_template":"qf-density14-real-ordered-midpoint/[\"-7/1\",\"5/1\"]"}},"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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r02-real-number-0016","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","language":"zh-Hant","source_mko_ids":["mko-real-number"],"title_zh":"有序有理端點的中點距離","stem_zh":"有理點嵌入實數的有限切片；a=-15/2≤b=16/3。依序求 (中點, 中點−a, b−中點)，用既約正分母分數。","answer":{"kind":"expression","value":"(-13/12, 77/12, 77/12)","explanation_zh":"有序中點與兩側距離=(-13/12, 77/12, 77/12)。精確加總除二，再分別相減核兩側距離。非實數完備性或無理數分類證明，不用近似小數。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["有序有理端點的中點距離"]},"variant":{"family_id":"qf-density14-real-ordered-midpoint","seed":202610142016,"parameters":{"a":[-15,2],"b":[16,3],"domain":"finite_rational_real_slice","operation":"density14_real_ordered_midpoint","semantic_template":"qf-density14-real-ordered-midpoint/[\"-15/2\",\"16/3\"]"}},"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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r02-real-number-0017","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","language":"zh-Hant","source_mko_ids":["mko-real-number"],"title_zh":"有序有理端點的中點距離","stem_zh":"有理點嵌入實數的有限切片；a=-16/2≤b=17/3。依序求 (中點, 中點−a, b−中點)，用既約正分母分數。","answer":{"kind":"expression","value":"(-7/6, 41/6, 41/6)","explanation_zh":"有序中點與兩側距離=(-7/6, 41/6, 41/6)。精確加總除二，再分別相減核兩側距離。非實數完備性或無理數分類證明，不用近似小數。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["有序有理端點的中點距離"]},"variant":{"family_id":"qf-density14-real-ordered-midpoint","seed":202610142017,"parameters":{"a":[-16,2],"b":[17,3],"domain":"finite_rational_real_slice","operation":"density14_real_ordered_midpoint","semantic_template":"qf-density14-real-ordered-midpoint/[\"-8/1\",\"17/3\"]"}},"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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r02-real-number-0018","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","language":"zh-Hant","source_mko_ids":["mko-real-number"],"title_zh":"有序有理端點的中點距離","stem_zh":"有理點嵌入實數的有限切片；a=-17/2≤b=18/3。依序求 (中點, 中點−a, b−中點)，用既約正分母分數。","answer":{"kind":"expression","value":"(-5/4, 29/4, 29/4)","explanation_zh":"有序中點與兩側距離=(-5/4, 29/4, 29/4)。精確加總除二，再分別相減核兩側距離。非實數完備性或無理數分類證明，不用近似小數。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["有序有理端點的中點距離"]},"variant":{"family_id":"qf-density14-real-ordered-midpoint","seed":202610142018,"parameters":{"a":[-17,2],"b":[18,3],"domain":"finite_rational_real_slice","operation":"density14_real_ordered_midpoint","semantic_template":"qf-density14-real-ordered-midpoint/[\"-17/2\",\"6/1\"]"}},"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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r02-real-number-0019","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","language":"zh-Hant","source_mko_ids":["mko-real-number"],"title_zh":"有序有理端點的中點距離","stem_zh":"有理點嵌入實數的有限切片；a=-18/2≤b=19/3。依序求 (中點, 中點−a, b−中點)，用既約正分母分數。","answer":{"kind":"expression","value":"(-4/3, 23/3, 23/3)","explanation_zh":"有序中點與兩側距離=(-4/3, 23/3, 23/3)。精確加總除二，再分別相減核兩側距離。非實數完備性或無理數分類證明，不用近似小數。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["有序有理端點的中點距離"]},"variant":{"family_id":"qf-density14-real-ordered-midpoint","seed":202610142019,"parameters":{"a":[-18,2],"b":[19,3],"domain":"finite_rational_real_slice","operation":"density14_real_ordered_midpoint","semantic_template":"qf-density14-real-ordered-midpoint/[\"-9/1\",\"19/3\"]"}},"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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r02-real-number-0020","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","language":"zh-Hant","source_mko_ids":["mko-real-number"],"title_zh":"有序有理端點的中點距離","stem_zh":"有理點嵌入實數的有限切片；a=-19/2≤b=20/3。依序求 (中點, 中點−a, b−中點)，用既約正分母分數。","answer":{"kind":"expression","value":"(-17/12, 97/12, 97/12)","explanation_zh":"有序中點與兩側距離=(-17/12, 97/12, 97/12)。精確加總除二，再分別相減核兩側距離。非實數完備性或無理數分類證明，不用近似小數。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["有序有理端點的中點距離"]},"variant":{"family_id":"qf-density14-real-ordered-midpoint","seed":202610142020,"parameters":{"a":[-19,2],"b":[20,3],"domain":"finite_rational_real_slice","operation":"density14_real_ordered_midpoint","semantic_template":"qf-density14-real-ordered-midpoint/[\"-19/2\",\"20/3\"]"}},"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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r02-real-number-0021","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","language":"zh-Hant","source_mko_ids":["mko-real-number"],"title_zh":"有序有理端點的中點距離","stem_zh":"有理點嵌入實數的有限切片；a=-20/2≤b=21/3。依序求 (中點, 中點−a, b−中點)，用既約正分母分數。","answer":{"kind":"expression","value":"(-3/2, 17/2, 17/2)","explanation_zh":"有序中點與兩側距離=(-3/2, 17/2, 17/2)。精確加總除二，再分別相減核兩側距離。非實數完備性或無理數分類證明，不用近似小數。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["有序有理端點的中點距離"]},"variant":{"family_id":"qf-density14-real-ordered-midpoint","seed":202610142021,"parameters":{"a":[-20,2],"b":[21,3],"domain":"finite_rational_real_slice","operation":"density14_real_ordered_midpoint","semantic_template":"qf-density14-real-ordered-midpoint/[\"-10/1\",\"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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r02-real-number-0022","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","language":"zh-Hant","source_mko_ids":["mko-real-number"],"title_zh":"有序有理端點的中點距離","stem_zh":"有理點嵌入實數的有限切片；a=-21/2≤b=22/3。依序求 (中點, 中點−a, b−中點)，用既約正分母分數。","answer":{"kind":"expression","value":"(-19/12, 107/12, 107/12)","explanation_zh":"有序中點與兩側距離=(-19/12, 107/12, 107/12)。精確加總除二，再分別相減核兩側距離。非實數完備性或無理數分類證明，不用近似小數。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["有序有理端點的中點距離"]},"variant":{"family_id":"qf-density14-real-ordered-midpoint","seed":202610142022,"parameters":{"a":[-21,2],"b":[22,3],"domain":"finite_rational_real_slice","operation":"density14_real_ordered_midpoint","semantic_template":"qf-density14-real-ordered-midpoint/[\"-21/2\",\"22/3\"]"}},"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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r02-real-number-0023","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","language":"zh-Hant","source_mko_ids":["mko-real-number"],"title_zh":"有序有理端點的中點距離","stem_zh":"有理點嵌入實數的有限切片；a=-22/2≤b=23/3。依序求 (中點, 中點−a, b−中點)，用既約正分母分數。","answer":{"kind":"expression","value":"(-5/3, 28/3, 28/3)","explanation_zh":"有序中點與兩側距離=(-5/3, 28/3, 28/3)。精確加總除二，再分別相減核兩側距離。非實數完備性或無理數分類證明，不用近似小數。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["有序有理端點的中點距離"]},"variant":{"family_id":"qf-density14-real-ordered-midpoint","seed":202610142023,"parameters":{"a":[-22,2],"b":[23,3],"domain":"finite_rational_real_slice","operation":"density14_real_ordered_midpoint","semantic_template":"qf-density14-real-ordered-midpoint/[\"-11/1\",\"23/3\"]"}},"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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r02-real-number-0024","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","language":"zh-Hant","source_mko_ids":["mko-real-number"],"title_zh":"有序有理端點的中點距離","stem_zh":"有理點嵌入實數的有限切片；a=-23/2≤b=24/3。依序求 (中點, 中點−a, b−中點)，用既約正分母分數。","answer":{"kind":"expression","value":"(-7/4, 39/4, 39/4)","explanation_zh":"有序中點與兩側距離=(-7/4, 39/4, 39/4)。精確加總除二，再分別相減核兩側距離。非實數完備性或無理數分類證明，不用近似小數。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["有序有理端點的中點距離"]},"variant":{"family_id":"qf-density14-real-ordered-midpoint","seed":202610142024,"parameters":{"a":[-23,2],"b":[24,3],"domain":"finite_rational_real_slice","operation":"density14_real_ordered_midpoint","semantic_template":"qf-density14-real-ordered-midpoint/[\"-23/2\",\"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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r02-real-number-0025","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","language":"zh-Hant","source_mko_ids":["mko-real-number"],"title_zh":"有序有理端點的中點距離","stem_zh":"有理點嵌入實數的有限切片；a=-24/2≤b=25/3。依序求 (中點, 中點−a, b−中點)，用既約正分母分數。","answer":{"kind":"expression","value":"(-11/6, 61/6, 61/6)","explanation_zh":"有序中點與兩側距離=(-11/6, 61/6, 61/6)。精確加總除二，再分別相減核兩側距離。非實數完備性或無理數分類證明，不用近似小數。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["有序有理端點的中點距離"]},"variant":{"family_id":"qf-density14-real-ordered-midpoint","seed":202610142025,"parameters":{"a":[-24,2],"b":[25,3],"domain":"finite_rational_real_slice","operation":"density14_real_ordered_midpoint","semantic_template":"qf-density14-real-ordered-midpoint/[\"-12/1\",\"25/3\"]"}},"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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r02-real-number-0026","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","language":"zh-Hant","source_mko_ids":["mko-real-number"],"title_zh":"有限點列與最小包含閉區間","stem_zh":"有理點嵌入實數的有限切片；points=[0/1, 0/-2, 0/3]。依序列 (遞增去重點列, 最小端點lo, 最大端點hi, hi−lo)，所求包含區間是[lo,hi]。","answer":{"kind":"expression","value":"([0/1], 0/1, 0/1, 0/1)","explanation_zh":"有限點列與最小閉區間=([0/1], 0/1, 0/1, 0/1)。按有理數實值排序去重，再取兩端；最小包含閉區間非點集拓撲閉包。非實數完備性或無理數分類證明，不用近似小數。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["有限點列與最小包含閉區間"]},"variant":{"family_id":"qf-density14-real-point-closed-hull","seed":202610142026,"parameters":{"points":[[0,1],[0,-2],[0,3]],"domain":"finite_rational_real_slice","operation":"density14_real_point_closed_hull","semantic_template":"qf-density14-real-point-closed-hull/[\"0/1\"]"}},"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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r02-real-number-0027","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","language":"zh-Hant","source_mko_ids":["mko-real-number"],"title_zh":"有限點列與最小包含閉區間","stem_zh":"有理點嵌入實數的有限切片；points=[1/2, -1/2, 1/2]。依序列 (遞增去重點列, 最小端點lo, 最大端點hi, hi−lo)，所求包含區間是[lo,hi]。","answer":{"kind":"expression","value":"([-1/2, 1/2], -1/2, 1/2, 1/1)","explanation_zh":"有限點列與最小閉區間=([-1/2, 1/2], -1/2, 1/2, 1/1)。按有理數實值排序去重，再取兩端；最小包含閉區間非點集拓撲閉包。非實數完備性或無理數分類證明，不用近似小數。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["有限點列與最小包含閉區間"]},"variant":{"family_id":"qf-density14-real-point-closed-hull","seed":202610142027,"parameters":{"points":[[1,2],[-1,2],[1,2]],"domain":"finite_rational_real_slice","operation":"density14_real_point_closed_hull","semantic_template":"qf-density14-real-point-closed-hull/[\"-1/2\",\"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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r02-real-number-0028","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","language":"zh-Hant","source_mko_ids":["mko-real-number"],"title_zh":"有限點列與最小包含閉區間","stem_zh":"有理點嵌入實數的有限切片；points=[-2/2, 0/1, 3/3]。依序列 (遞增去重點列, 最小端點lo, 最大端點hi, hi−lo)，所求包含區間是[lo,hi]。","answer":{"kind":"expression","value":"([-1/1, 0/1, 1/1], -1/1, 1/1, 2/1)","explanation_zh":"有限點列與最小閉區間=([-1/1, 0/1, 1/1], -1/1, 1/1, 2/1)。按有理數實值排序去重，再取兩端；最小包含閉區間非點集拓撲閉包。非實數完備性或無理數分類證明，不用近似小數。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["有限點列與最小包含閉區間"]},"variant":{"family_id":"qf-density14-real-point-closed-hull","seed":202610142028,"parameters":{"points":[[-2,2],[0,1],[3,3]],"domain":"finite_rational_real_slice","operation":"density14_real_point_closed_hull","semantic_template":"qf-density14-real-point-closed-hull/[\"-1/1\",\"0/1\",\"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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r02-real-number-0029","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","language":"zh-Hant","source_mko_ids":["mko-real-number"],"title_zh":"有限點列與最小包含閉區間","stem_zh":"有理點嵌入實數的有限切片；points=[-3/2, 0/1, 4/3]。依序列 (遞增去重點列, 最小端點lo, 最大端點hi, hi−lo)，所求包含區間是[lo,hi]。","answer":{"kind":"expression","value":"([-3/2, 0/1, 4/3], -3/2, 4/3, 17/6)","explanation_zh":"有限點列與最小閉區間=([-3/2, 0/1, 4/3], -3/2, 4/3, 17/6)。按有理數實值排序去重，再取兩端；最小包含閉區間非點集拓撲閉包。非實數完備性或無理數分類證明，不用近似小數。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["有限點列與最小包含閉區間"]},"variant":{"family_id":"qf-density14-real-point-closed-hull","seed":202610142029,"parameters":{"points":[[-3,2],[0,1],[4,3]],"domain":"finite_rational_real_slice","operation":"density14_real_point_closed_hull","semantic_template":"qf-density14-real-point-closed-hull/[\"-3/2\",\"0/1\",\"4/3\"]"}},"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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r02-real-number-0030","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","language":"zh-Hant","source_mko_ids":["mko-real-number"],"title_zh":"有限點列與最小包含閉區間","stem_zh":"有理點嵌入實數的有限切片；points=[-4/2, 0/1, 5/3]。依序列 (遞增去重點列, 最小端點lo, 最大端點hi, hi−lo)，所求包含區間是[lo,hi]。","answer":{"kind":"expression","value":"([-2/1, 0/1, 5/3], -2/1, 5/3, 11/3)","explanation_zh":"有限點列與最小閉區間=([-2/1, 0/1, 5/3], -2/1, 5/3, 11/3)。按有理數實值排序去重，再取兩端；最小包含閉區間非點集拓撲閉包。非實數完備性或無理數分類證明，不用近似小數。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["有限點列與最小包含閉區間"]},"variant":{"family_id":"qf-density14-real-point-closed-hull","seed":202610142030,"parameters":{"points":[[-4,2],[0,1],[5,3]],"domain":"finite_rational_real_slice","operation":"density14_real_point_closed_hull","semantic_template":"qf-density14-real-point-closed-hull/[\"-2/1\",\"0/1\",\"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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r02-real-number-0031","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","language":"zh-Hant","source_mko_ids":["mko-real-number"],"title_zh":"有限點列與最小包含閉區間","stem_zh":"有理點嵌入實數的有限切片；points=[-5/2, 0/1, 6/3]。依序列 (遞增去重點列, 最小端點lo, 最大端點hi, hi−lo)，所求包含區間是[lo,hi]。","answer":{"kind":"expression","value":"([-5/2, 0/1, 2/1], -5/2, 2/1, 9/2)","explanation_zh":"有限點列與最小閉區間=([-5/2, 0/1, 2/1], -5/2, 2/1, 9/2)。按有理數實值排序去重，再取兩端；最小包含閉區間非點集拓撲閉包。非實數完備性或無理數分類證明，不用近似小數。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["有限點列與最小包含閉區間"]},"variant":{"family_id":"qf-density14-real-point-closed-hull","seed":202610142031,"parameters":{"points":[[-5,2],[0,1],[6,3]],"domain":"finite_rational_real_slice","operation":"density14_real_point_closed_hull","semantic_template":"qf-density14-real-point-closed-hull/[\"-5/2\",\"0/1\",\"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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r02-real-number-0032","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","language":"zh-Hant","source_mko_ids":["mko-real-number"],"title_zh":"有限點列與最小包含閉區間","stem_zh":"有理點嵌入實數的有限切片；points=[-6/2, 0/1, 7/3]。依序列 (遞增去重點列, 最小端點lo, 最大端點hi, hi−lo)，所求包含區間是[lo,hi]。","answer":{"kind":"expression","value":"([-3/1, 0/1, 7/3], -3/1, 7/3, 16/3)","explanation_zh":"有限點列與最小閉區間=([-3/1, 0/1, 7/3], -3/1, 7/3, 16/3)。按有理數實值排序去重，再取兩端；最小包含閉區間非點集拓撲閉包。非實數完備性或無理數分類證明，不用近似小數。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["有限點列與最小包含閉區間"]},"variant":{"family_id":"qf-density14-real-point-closed-hull","seed":202610142032,"parameters":{"points":[[-6,2],[0,1],[7,3]],"domain":"finite_rational_real_slice","operation":"density14_real_point_closed_hull","semantic_template":"qf-density14-real-point-closed-hull/[\"-3/1\",\"0/1\",\"7/3\"]"}},"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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r02-real-number-0033","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","language":"zh-Hant","source_mko_ids":["mko-real-number"],"title_zh":"有限點列與最小包含閉區間","stem_zh":"有理點嵌入實數的有限切片；points=[-7/2, 0/1, 8/3]。依序列 (遞增去重點列, 最小端點lo, 最大端點hi, hi−lo)，所求包含區間是[lo,hi]。","answer":{"kind":"expression","value":"([-7/2, 0/1, 8/3], -7/2, 8/3, 37/6)","explanation_zh":"有限點列與最小閉區間=([-7/2, 0/1, 8/3], -7/2, 8/3, 37/6)。按有理數實值排序去重，再取兩端；最小包含閉區間非點集拓撲閉包。非實數完備性或無理數分類證明，不用近似小數。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["有限點列與最小包含閉區間"]},"variant":{"family_id":"qf-density14-real-point-closed-hull","seed":202610142033,"parameters":{"points":[[-7,2],[0,1],[8,3]],"domain":"finite_rational_real_slice","operation":"density14_real_point_closed_hull","semantic_template":"qf-density14-real-point-closed-hull/[\"-7/2\",\"0/1\",\"8/3\"]"}},"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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r02-real-number-0034","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","language":"zh-Hant","source_mko_ids":["mko-real-number"],"title_zh":"有限點列與最小包含閉區間","stem_zh":"有理點嵌入實數的有限切片；points=[-8/2, 0/1, 9/3]。依序列 (遞增去重點列, 最小端點lo, 最大端點hi, hi−lo)，所求包含區間是[lo,hi]。","answer":{"kind":"expression","value":"([-4/1, 0/1, 3/1], -4/1, 3/1, 7/1)","explanation_zh":"有限點列與最小閉區間=([-4/1, 0/1, 3/1], -4/1, 3/1, 7/1)。按有理數實值排序去重，再取兩端；最小包含閉區間非點集拓撲閉包。非實數完備性或無理數分類證明，不用近似小數。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["有限點列與最小包含閉區間"]},"variant":{"family_id":"qf-density14-real-point-closed-hull","seed":202610142034,"parameters":{"points":[[-8,2],[0,1],[9,3]],"domain":"finite_rational_real_slice","operation":"density14_real_point_closed_hull","semantic_template":"qf-density14-real-point-closed-hull/[\"-4/1\",\"0/1\",\"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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r02-real-number-0035","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","language":"zh-Hant","source_mko_ids":["mko-real-number"],"title_zh":"有限點列與最小包含閉區間","stem_zh":"有理點嵌入實數的有限切片；points=[-9/2, 0/1, 10/3]。依序列 (遞增去重點列, 最小端點lo, 最大端點hi, hi−lo)，所求包含區間是[lo,hi]。","answer":{"kind":"expression","value":"([-9/2, 0/1, 10/3], -9/2, 10/3, 47/6)","explanation_zh":"有限點列與最小閉區間=([-9/2, 0/1, 10/3], -9/2, 10/3, 47/6)。按有理數實值排序去重，再取兩端；最小包含閉區間非點集拓撲閉包。非實數完備性或無理數分類證明，不用近似小數。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["有限點列與最小包含閉區間"]},"variant":{"family_id":"qf-density14-real-point-closed-hull","seed":202610142035,"parameters":{"points":[[-9,2],[0,1],[10,3]],"domain":"finite_rational_real_slice","operation":"density14_real_point_closed_hull","semantic_template":"qf-density14-real-point-closed-hull/[\"-9/2\",\"0/1\",\"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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r02-real-number-0036","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","language":"zh-Hant","source_mko_ids":["mko-real-number"],"title_zh":"有限點列與最小包含閉區間","stem_zh":"有理點嵌入實數的有限切片；points=[-10/2, 0/1, 11/3]。依序列 (遞增去重點列, 最小端點lo, 最大端點hi, hi−lo)，所求包含區間是[lo,hi]。","answer":{"kind":"expression","value":"([-5/1, 0/1, 11/3], -5/1, 11/3, 26/3)","explanation_zh":"有限點列與最小閉區間=([-5/1, 0/1, 11/3], -5/1, 11/3, 26/3)。按有理數實值排序去重，再取兩端；最小包含閉區間非點集拓撲閉包。非實數完備性或無理數分類證明，不用近似小數。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["有限點列與最小包含閉區間"]},"variant":{"family_id":"qf-density14-real-point-closed-hull","seed":202610142036,"parameters":{"points":[[-10,2],[0,1],[11,3]],"domain":"finite_rational_real_slice","operation":"density14_real_point_closed_hull","semantic_template":"qf-density14-real-point-closed-hull/[\"-5/1\",\"0/1\",\"11/3\"]"}},"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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r02-real-number-0037","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","language":"zh-Hant","source_mko_ids":["mko-real-number"],"title_zh":"有限點列與最小包含閉區間","stem_zh":"有理點嵌入實數的有限切片；points=[-11/2, 0/1, 12/3]。依序列 (遞增去重點列, 最小端點lo, 最大端點hi, hi−lo)，所求包含區間是[lo,hi]。","answer":{"kind":"expression","value":"([-11/2, 0/1, 4/1], -11/2, 4/1, 19/2)","explanation_zh":"有限點列與最小閉區間=([-11/2, 0/1, 4/1], -11/2, 4/1, 19/2)。按有理數實值排序去重，再取兩端；最小包含閉區間非點集拓撲閉包。非實數完備性或無理數分類證明，不用近似小數。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["有限點列與最小包含閉區間"]},"variant":{"family_id":"qf-density14-real-point-closed-hull","seed":202610142037,"parameters":{"points":[[-11,2],[0,1],[12,3]],"domain":"finite_rational_real_slice","operation":"density14_real_point_closed_hull","semantic_template":"qf-density14-real-point-closed-hull/[\"-11/2\",\"0/1\",\"4/1\"]"}},"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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r02-real-number-0038","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","language":"zh-Hant","source_mko_ids":["mko-real-number"],"title_zh":"有限點列與最小包含閉區間","stem_zh":"有理點嵌入實數的有限切片；points=[-12/2, 0/1, 13/3]。依序列 (遞增去重點列, 最小端點lo, 最大端點hi, hi−lo)，所求包含區間是[lo,hi]。","answer":{"kind":"expression","value":"([-6/1, 0/1, 13/3], -6/1, 13/3, 31/3)","explanation_zh":"有限點列與最小閉區間=([-6/1, 0/1, 13/3], -6/1, 13/3, 31/3)。按有理數實值排序去重，再取兩端；最小包含閉區間非點集拓撲閉包。非實數完備性或無理數分類證明，不用近似小數。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["有限點列與最小包含閉區間"]},"variant":{"family_id":"qf-density14-real-point-closed-hull","seed":202610142038,"parameters":{"points":[[-12,2],[0,1],[13,3]],"domain":"finite_rational_real_slice","operation":"density14_real_point_closed_hull","semantic_template":"qf-density14-real-point-closed-hull/[\"-6/1\",\"0/1\",\"13/3\"]"}},"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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r02-real-number-0039","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","language":"zh-Hant","source_mko_ids":["mko-real-number"],"title_zh":"有限點列與最小包含閉區間","stem_zh":"有理點嵌入實數的有限切片；points=[-13/2, 0/1, 14/3]。依序列 (遞增去重點列, 最小端點lo, 最大端點hi, hi−lo)，所求包含區間是[lo,hi]。","answer":{"kind":"expression","value":"([-13/2, 0/1, 14/3], -13/2, 14/3, 67/6)","explanation_zh":"有限點列與最小閉區間=([-13/2, 0/1, 14/3], -13/2, 14/3, 67/6)。按有理數實值排序去重，再取兩端；最小包含閉區間非點集拓撲閉包。非實數完備性或無理數分類證明，不用近似小數。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["有限點列與最小包含閉區間"]},"variant":{"family_id":"qf-density14-real-point-closed-hull","seed":202610142039,"parameters":{"points":[[-13,2],[0,1],[14,3]],"domain":"finite_rational_real_slice","operation":"density14_real_point_closed_hull","semantic_template":"qf-density14-real-point-closed-hull/[\"-13/2\",\"0/1\",\"14/3\"]"}},"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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r02-real-number-0040","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","language":"zh-Hant","source_mko_ids":["mko-real-number"],"title_zh":"有限點列與最小包含閉區間","stem_zh":"有理點嵌入實數的有限切片；points=[-14/2, 0/1, 15/3]。依序列 (遞增去重點列, 最小端點lo, 最大端點hi, hi−lo)，所求包含區間是[lo,hi]。","answer":{"kind":"expression","value":"([-7/1, 0/1, 5/1], -7/1, 5/1, 12/1)","explanation_zh":"有限點列與最小閉區間=([-7/1, 0/1, 5/1], -7/1, 5/1, 12/1)。按有理數實值排序去重，再取兩端；最小包含閉區間非點集拓撲閉包。非實數完備性或無理數分類證明，不用近似小數。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["有限點列與最小包含閉區間"]},"variant":{"family_id":"qf-density14-real-point-closed-hull","seed":202610142040,"parameters":{"points":[[-14,2],[0,1],[15,3]],"domain":"finite_rational_real_slice","operation":"density14_real_point_closed_hull","semantic_template":"qf-density14-real-point-closed-hull/[\"-7/1\",\"0/1\",\"5/1\"]"}},"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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r02-real-number-0041","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","language":"zh-Hant","source_mko_ids":["mko-real-number"],"title_zh":"有限點列與最小包含閉區間","stem_zh":"有理點嵌入實數的有限切片；points=[-15/2, 0/1, 16/3]。依序列 (遞增去重點列, 最小端點lo, 最大端點hi, hi−lo)，所求包含區間是[lo,hi]。","answer":{"kind":"expression","value":"([-15/2, 0/1, 16/3], -15/2, 16/3, 77/6)","explanation_zh":"有限點列與最小閉區間=([-15/2, 0/1, 16/3], -15/2, 16/3, 77/6)。按有理數實值排序去重，再取兩端；最小包含閉區間非點集拓撲閉包。非實數完備性或無理數分類證明，不用近似小數。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["有限點列與最小包含閉區間"]},"variant":{"family_id":"qf-density14-real-point-closed-hull","seed":202610142041,"parameters":{"points":[[-15,2],[0,1],[16,3]],"domain":"finite_rational_real_slice","operation":"density14_real_point_closed_hull","semantic_template":"qf-density14-real-point-closed-hull/[\"-15/2\",\"0/1\",\"16/3\"]"}},"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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r02-real-number-0042","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","language":"zh-Hant","source_mko_ids":["mko-real-number"],"title_zh":"有限點列與最小包含閉區間","stem_zh":"有理點嵌入實數的有限切片；points=[-16/2, 0/1, 17/3]。依序列 (遞增去重點列, 最小端點lo, 最大端點hi, hi−lo)，所求包含區間是[lo,hi]。","answer":{"kind":"expression","value":"([-8/1, 0/1, 17/3], -8/1, 17/3, 41/3)","explanation_zh":"有限點列與最小閉區間=([-8/1, 0/1, 17/3], -8/1, 17/3, 41/3)。按有理數實值排序去重，再取兩端；最小包含閉區間非點集拓撲閉包。非實數完備性或無理數分類證明，不用近似小數。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["有限點列與最小包含閉區間"]},"variant":{"family_id":"qf-density14-real-point-closed-hull","seed":202610142042,"parameters":{"points":[[-16,2],[0,1],[17,3]],"domain":"finite_rational_real_slice","operation":"density14_real_point_closed_hull","semantic_template":"qf-density14-real-point-closed-hull/[\"-8/1\",\"0/1\",\"17/3\"]"}},"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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r02-real-number-0043","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","language":"zh-Hant","source_mko_ids":["mko-real-number"],"title_zh":"有限點列與最小包含閉區間","stem_zh":"有理點嵌入實數的有限切片；points=[-17/2, 0/1, 18/3]。依序列 (遞增去重點列, 最小端點lo, 最大端點hi, hi−lo)，所求包含區間是[lo,hi]。","answer":{"kind":"expression","value":"([-17/2, 0/1, 6/1], -17/2, 6/1, 29/2)","explanation_zh":"有限點列與最小閉區間=([-17/2, 0/1, 6/1], -17/2, 6/1, 29/2)。按有理數實值排序去重，再取兩端；最小包含閉區間非點集拓撲閉包。非實數完備性或無理數分類證明，不用近似小數。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["有限點列與最小包含閉區間"]},"variant":{"family_id":"qf-density14-real-point-closed-hull","seed":202610142043,"parameters":{"points":[[-17,2],[0,1],[18,3]],"domain":"finite_rational_real_slice","operation":"density14_real_point_closed_hull","semantic_template":"qf-density14-real-point-closed-hull/[\"-17/2\",\"0/1\",\"6/1\"]"}},"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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r02-real-number-0044","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","language":"zh-Hant","source_mko_ids":["mko-real-number"],"title_zh":"有限點列與最小包含閉區間","stem_zh":"有理點嵌入實數的有限切片；points=[-18/2, 0/1, 19/3]。依序列 (遞增去重點列, 最小端點lo, 最大端點hi, hi−lo)，所求包含區間是[lo,hi]。","answer":{"kind":"expression","value":"([-9/1, 0/1, 19/3], -9/1, 19/3, 46/3)","explanation_zh":"有限點列與最小閉區間=([-9/1, 0/1, 19/3], -9/1, 19/3, 46/3)。按有理數實值排序去重，再取兩端；最小包含閉區間非點集拓撲閉包。非實數完備性或無理數分類證明，不用近似小數。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["有限點列與最小包含閉區間"]},"variant":{"family_id":"qf-density14-real-point-closed-hull","seed":202610142044,"parameters":{"points":[[-18,2],[0,1],[19,3]],"domain":"finite_rational_real_slice","operation":"density14_real_point_closed_hull","semantic_template":"qf-density14-real-point-closed-hull/[\"-9/1\",\"0/1\",\"19/3\"]"}},"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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r02-real-number-0045","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","language":"zh-Hant","source_mko_ids":["mko-real-number"],"title_zh":"有限點列與最小包含閉區間","stem_zh":"有理點嵌入實數的有限切片；points=[-19/2, 0/1, 20/3]。依序列 (遞增去重點列, 最小端點lo, 最大端點hi, hi−lo)，所求包含區間是[lo,hi]。","answer":{"kind":"expression","value":"([-19/2, 0/1, 20/3], -19/2, 20/3, 97/6)","explanation_zh":"有限點列與最小閉區間=([-19/2, 0/1, 20/3], -19/2, 20/3, 97/6)。按有理數實值排序去重，再取兩端；最小包含閉區間非點集拓撲閉包。非實數完備性或無理數分類證明，不用近似小數。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["有限點列與最小包含閉區間"]},"variant":{"family_id":"qf-density14-real-point-closed-hull","seed":202610142045,"parameters":{"points":[[-19,2],[0,1],[20,3]],"domain":"finite_rational_real_slice","operation":"density14_real_point_closed_hull","semantic_template":"qf-density14-real-point-closed-hull/[\"-19/2\",\"0/1\",\"20/3\"]"}},"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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r02-real-number-0046","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","language":"zh-Hant","source_mko_ids":["mko-real-number"],"title_zh":"有限點列與最小包含閉區間","stem_zh":"有理點嵌入實數的有限切片；points=[-20/2, 0/1, 21/3]。依序列 (遞增去重點列, 最小端點lo, 最大端點hi, hi−lo)，所求包含區間是[lo,hi]。","answer":{"kind":"expression","value":"([-10/1, 0/1, 7/1], -10/1, 7/1, 17/1)","explanation_zh":"有限點列與最小閉區間=([-10/1, 0/1, 7/1], -10/1, 7/1, 17/1)。按有理數實值排序去重，再取兩端；最小包含閉區間非點集拓撲閉包。非實數完備性或無理數分類證明，不用近似小數。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["有限點列與最小包含閉區間"]},"variant":{"family_id":"qf-density14-real-point-closed-hull","seed":202610142046,"parameters":{"points":[[-20,2],[0,1],[21,3]],"domain":"finite_rational_real_slice","operation":"density14_real_point_closed_hull","semantic_template":"qf-density14-real-point-closed-hull/[\"-10/1\",\"0/1\",\"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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r02-real-number-0047","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","language":"zh-Hant","source_mko_ids":["mko-real-number"],"title_zh":"有限點列與最小包含閉區間","stem_zh":"有理點嵌入實數的有限切片；points=[-21/2, 0/1, 22/3]。依序列 (遞增去重點列, 最小端點lo, 最大端點hi, hi−lo)，所求包含區間是[lo,hi]。","answer":{"kind":"expression","value":"([-21/2, 0/1, 22/3], -21/2, 22/3, 107/6)","explanation_zh":"有限點列與最小閉區間=([-21/2, 0/1, 22/3], -21/2, 22/3, 107/6)。按有理數實值排序去重，再取兩端；最小包含閉區間非點集拓撲閉包。非實數完備性或無理數分類證明，不用近似小數。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["有限點列與最小包含閉區間"]},"variant":{"family_id":"qf-density14-real-point-closed-hull","seed":202610142047,"parameters":{"points":[[-21,2],[0,1],[22,3]],"domain":"finite_rational_real_slice","operation":"density14_real_point_closed_hull","semantic_template":"qf-density14-real-point-closed-hull/[\"-21/2\",\"0/1\",\"22/3\"]"}},"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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r02-real-number-0048","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","language":"zh-Hant","source_mko_ids":["mko-real-number"],"title_zh":"有限點列與最小包含閉區間","stem_zh":"有理點嵌入實數的有限切片；points=[-22/2, 0/1, 23/3]。依序列 (遞增去重點列, 最小端點lo, 最大端點hi, hi−lo)，所求包含區間是[lo,hi]。","answer":{"kind":"expression","value":"([-11/1, 0/1, 23/3], -11/1, 23/3, 56/3)","explanation_zh":"有限點列與最小閉區間=([-11/1, 0/1, 23/3], -11/1, 23/3, 56/3)。按有理數實值排序去重，再取兩端；最小包含閉區間非點集拓撲閉包。非實數完備性或無理數分類證明，不用近似小數。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["有限點列與最小包含閉區間"]},"variant":{"family_id":"qf-density14-real-point-closed-hull","seed":202610142048,"parameters":{"points":[[-22,2],[0,1],[23,3]],"domain":"finite_rational_real_slice","operation":"density14_real_point_closed_hull","semantic_template":"qf-density14-real-point-closed-hull/[\"-11/1\",\"0/1\",\"23/3\"]"}},"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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r02-real-number-0049","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","language":"zh-Hant","source_mko_ids":["mko-real-number"],"title_zh":"有限點列與最小包含閉區間","stem_zh":"有理點嵌入實數的有限切片；points=[-23/2, 0/1, 24/3]。依序列 (遞增去重點列, 最小端點lo, 最大端點hi, hi−lo)，所求包含區間是[lo,hi]。","answer":{"kind":"expression","value":"([-23/2, 0/1, 8/1], -23/2, 8/1, 39/2)","explanation_zh":"有限點列與最小閉區間=([-23/2, 0/1, 8/1], -23/2, 8/1, 39/2)。按有理數實值排序去重，再取兩端；最小包含閉區間非點集拓撲閉包。非實數完備性或無理數分類證明，不用近似小數。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["有限點列與最小包含閉區間"]},"variant":{"family_id":"qf-density14-real-point-closed-hull","seed":202610142049,"parameters":{"points":[[-23,2],[0,1],[24,3]],"domain":"finite_rational_real_slice","operation":"density14_real_point_closed_hull","semantic_template":"qf-density14-real-point-closed-hull/[\"-23/2\",\"0/1\",\"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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r02-real-number-0050","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","language":"zh-Hant","source_mko_ids":["mko-real-number"],"title_zh":"有限點列與最小包含閉區間","stem_zh":"有理點嵌入實數的有限切片；points=[-24/2, 0/1, 25/3]。依序列 (遞增去重點列, 最小端點lo, 最大端點hi, hi−lo)，所求包含區間是[lo,hi]。","answer":{"kind":"expression","value":"([-12/1, 0/1, 25/3], -12/1, 25/3, 61/3)","explanation_zh":"有限點列與最小閉區間=([-12/1, 0/1, 25/3], -12/1, 25/3, 61/3)。按有理數實值排序去重，再取兩端；最小包含閉區間非點集拓撲閉包。非實數完備性或無理數分類證明，不用近似小數。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["有限點列與最小包含閉區間"]},"variant":{"family_id":"qf-density14-real-point-closed-hull","seed":202610142050,"parameters":{"points":[[-24,2],[0,1],[25,3]],"domain":"finite_rational_real_slice","operation":"density14_real_point_closed_hull","semantic_template":"qf-density14-real-point-closed-hull/[\"-12/1\",\"0/1\",\"25/3\"]"}},"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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r02-real-number-0051","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","language":"zh-Hant","source_mko_ids":["mko-real-number"],"title_zh":"兩個閉區間的交與間隔","stem_zh":"有理點嵌入實數的有限切片；閉區間包含端點；I=[0/1, 1/1]、J=[1/1, 2/1]，各端點有序。依序答 (交非空是否, 交的[lo,hi]或[], 分離間隔)。","answer":{"kind":"expression","value":"(是, [1/1, 1/1], 0/1)","explanation_zh":"閉區間交集與分離距離=(是, [1/1, 1/1], 0/1)。取左端最大值、右端最小值，交時間隔0/1；接觸端點不是空集。非實數完備性或無理數分類證明，不用近似小數。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["兩個閉區間的交與間隔"]},"variant":{"family_id":"qf-density14-real-overlap-closed-intervals","seed":202610142051,"parameters":{"I":[[0,1],[1,1]],"J":[[1,1],[2,1]],"domain":"finite_rational_real_slice","operation":"density14_real_overlap_closed_intervals","semantic_template":"qf-density14-real-overlap-closed-intervals/[\"intersection\",\"1/1\",\"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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r02-real-number-0052","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","language":"zh-Hant","source_mko_ids":["mko-real-number"],"title_zh":"兩個閉區間的交與間隔","stem_zh":"有理點嵌入實數的有限切片；閉區間包含端點；I=[0/1, 1/1]、J=[2/1, 3/1]，各端點有序。依序答 (交非空是否, 交的[lo,hi]或[], 分離間隔)。","answer":{"kind":"expression","value":"(否, [], 1/1)","explanation_zh":"閉區間交集與分離距離=(否, [], 1/1)。取左端最大值、右端最小值，交時間隔0/1；接觸端點不是空集。非實數完備性或無理數分類證明，不用近似小數。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["兩個閉區間的交與間隔"]},"variant":{"family_id":"qf-density14-real-overlap-closed-intervals","seed":202610142052,"parameters":{"I":[[0,1],[1,1]],"J":[[2,1],[3,1]],"domain":"finite_rational_real_slice","operation":"density14_real_overlap_closed_intervals","semantic_template":"qf-density14-real-overlap-closed-intervals/[\"disjoint\",\"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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r02-real-number-0053","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","language":"zh-Hant","source_mko_ids":["mko-real-number"],"title_zh":"兩個閉區間的交與間隔","stem_zh":"有理點嵌入實數的有限切片；閉區間包含端點；I=[0/1, 0/1]、J=[0/1, 1/1]，各端點有序。依序答 (交非空是否, 交的[lo,hi]或[], 分離間隔)。","answer":{"kind":"expression","value":"(是, [0/1, 0/1], 0/1)","explanation_zh":"閉區間交集與分離距離=(是, [0/1, 0/1], 0/1)。取左端最大值、右端最小值，交時間隔0/1；接觸端點不是空集。非實數完備性或無理數分類證明，不用近似小數。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["兩個閉區間的交與間隔"]},"variant":{"family_id":"qf-density14-real-overlap-closed-intervals","seed":202610142053,"parameters":{"I":[[0,1],[0,1]],"J":[[0,1],[1,1]],"domain":"finite_rational_real_slice","operation":"density14_real_overlap_closed_intervals","semantic_template":"qf-density14-real-overlap-closed-intervals/[\"intersection\",\"0/1\",\"0/1\"]"}},"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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r02-real-number-0054","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","language":"zh-Hant","source_mko_ids":["mko-real-number"],"title_zh":"兩個閉區間的交與間隔","stem_zh":"有理點嵌入實數的有限切片；閉區間包含端點；I=[-3/2, 3/2]、J=[0/1, 5/2]，各端點有序。依序答 (交非空是否, 交的[lo,hi]或[], 分離間隔)。","answer":{"kind":"expression","value":"(是, [0/1, 3/2], 0/1)","explanation_zh":"閉區間交集與分離距離=(是, [0/1, 3/2], 0/1)。取左端最大值、右端最小值，交時間隔0/1；接觸端點不是空集。非實數完備性或無理數分類證明，不用近似小數。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["兩個閉區間的交與間隔"]},"variant":{"family_id":"qf-density14-real-overlap-closed-intervals","seed":202610142054,"parameters":{"I":[[-3,2],[3,2]],"J":[[0,1],[5,2]],"domain":"finite_rational_real_slice","operation":"density14_real_overlap_closed_intervals","semantic_template":"qf-density14-real-overlap-closed-intervals/[\"intersection\",\"0/1\",\"3/2\"]"}},"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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r02-real-number-0055","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","language":"zh-Hant","source_mko_ids":["mko-real-number"],"title_zh":"兩個閉區間的交與間隔","stem_zh":"有理點嵌入實數的有限切片；閉區間包含端點；I=[-4/2, 4/2]、J=[0/1, 6/2]，各端點有序。依序答 (交非空是否, 交的[lo,hi]或[], 分離間隔)。","answer":{"kind":"expression","value":"(是, [0/1, 2/1], 0/1)","explanation_zh":"閉區間交集與分離距離=(是, [0/1, 2/1], 0/1)。取左端最大值、右端最小值，交時間隔0/1；接觸端點不是空集。非實數完備性或無理數分類證明，不用近似小數。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["兩個閉區間的交與間隔"]},"variant":{"family_id":"qf-density14-real-overlap-closed-intervals","seed":202610142055,"parameters":{"I":[[-4,2],[4,2]],"J":[[0,1],[6,2]],"domain":"finite_rational_real_slice","operation":"density14_real_overlap_closed_intervals","semantic_template":"qf-density14-real-overlap-closed-intervals/[\"intersection\",\"0/1\",\"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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r02-real-number-0056","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","language":"zh-Hant","source_mko_ids":["mko-real-number"],"title_zh":"兩個閉區間的交與間隔","stem_zh":"有理點嵌入實數的有限切片；閉區間包含端點；I=[-5/2, 5/2]、J=[0/1, 7/2]，各端點有序。依序答 (交非空是否, 交的[lo,hi]或[], 分離間隔)。","answer":{"kind":"expression","value":"(是, [0/1, 5/2], 0/1)","explanation_zh":"閉區間交集與分離距離=(是, [0/1, 5/2], 0/1)。取左端最大值、右端最小值，交時間隔0/1；接觸端點不是空集。非實數完備性或無理數分類證明，不用近似小數。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["兩個閉區間的交與間隔"]},"variant":{"family_id":"qf-density14-real-overlap-closed-intervals","seed":202610142056,"parameters":{"I":[[-5,2],[5,2]],"J":[[0,1],[7,2]],"domain":"finite_rational_real_slice","operation":"density14_real_overlap_closed_intervals","semantic_template":"qf-density14-real-overlap-closed-intervals/[\"intersection\",\"0/1\",\"5/2\"]"}},"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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r02-real-number-0057","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","language":"zh-Hant","source_mko_ids":["mko-real-number"],"title_zh":"兩個閉區間的交與間隔","stem_zh":"有理點嵌入實數的有限切片；閉區間包含端點；I=[-6/2, 6/2]、J=[0/1, 8/2]，各端點有序。依序答 (交非空是否, 交的[lo,hi]或[], 分離間隔)。","answer":{"kind":"expression","value":"(是, [0/1, 3/1], 0/1)","explanation_zh":"閉區間交集與分離距離=(是, [0/1, 3/1], 0/1)。取左端最大值、右端最小值，交時間隔0/1；接觸端點不是空集。非實數完備性或無理數分類證明，不用近似小數。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["兩個閉區間的交與間隔"]},"variant":{"family_id":"qf-density14-real-overlap-closed-intervals","seed":202610142057,"parameters":{"I":[[-6,2],[6,2]],"J":[[0,1],[8,2]],"domain":"finite_rational_real_slice","operation":"density14_real_overlap_closed_intervals","semantic_template":"qf-density14-real-overlap-closed-intervals/[\"intersection\",\"0/1\",\"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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r02-real-number-0058","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","language":"zh-Hant","source_mko_ids":["mko-real-number"],"title_zh":"兩個閉區間的交與間隔","stem_zh":"有理點嵌入實數的有限切片；閉區間包含端點；I=[-7/2, 7/2]、J=[0/1, 9/2]，各端點有序。依序答 (交非空是否, 交的[lo,hi]或[], 分離間隔)。","answer":{"kind":"expression","value":"(是, [0/1, 7/2], 0/1)","explanation_zh":"閉區間交集與分離距離=(是, [0/1, 7/2], 0/1)。取左端最大值、右端最小值，交時間隔0/1；接觸端點不是空集。非實數完備性或無理數分類證明，不用近似小數。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["兩個閉區間的交與間隔"]},"variant":{"family_id":"qf-density14-real-overlap-closed-intervals","seed":202610142058,"parameters":{"I":[[-7,2],[7,2]],"J":[[0,1],[9,2]],"domain":"finite_rational_real_slice","operation":"density14_real_overlap_closed_intervals","semantic_template":"qf-density14-real-overlap-closed-intervals/[\"intersection\",\"0/1\",\"7/2\"]"}},"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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r02-real-number-0059","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","language":"zh-Hant","source_mko_ids":["mko-real-number"],"title_zh":"兩個閉區間的交與間隔","stem_zh":"有理點嵌入實數的有限切片；閉區間包含端點；I=[-8/2, 8/2]、J=[0/1, 10/2]，各端點有序。依序答 (交非空是否, 交的[lo,hi]或[], 分離間隔)。","answer":{"kind":"expression","value":"(是, [0/1, 4/1], 0/1)","explanation_zh":"閉區間交集與分離距離=(是, [0/1, 4/1], 0/1)。取左端最大值、右端最小值，交時間隔0/1；接觸端點不是空集。非實數完備性或無理數分類證明，不用近似小數。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["兩個閉區間的交與間隔"]},"variant":{"family_id":"qf-density14-real-overlap-closed-intervals","seed":202610142059,"parameters":{"I":[[-8,2],[8,2]],"J":[[0,1],[10,2]],"domain":"finite_rational_real_slice","operation":"density14_real_overlap_closed_intervals","semantic_template":"qf-density14-real-overlap-closed-intervals/[\"intersection\",\"0/1\",\"4/1\"]"}},"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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r02-real-number-0060","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","language":"zh-Hant","source_mko_ids":["mko-real-number"],"title_zh":"兩個閉區間的交與間隔","stem_zh":"有理點嵌入實數的有限切片；閉區間包含端點；I=[-9/2, 9/2]、J=[0/1, 11/2]，各端點有序。依序答 (交非空是否, 交的[lo,hi]或[], 分離間隔)。","answer":{"kind":"expression","value":"(是, [0/1, 9/2], 0/1)","explanation_zh":"閉區間交集與分離距離=(是, [0/1, 9/2], 0/1)。取左端最大值、右端最小值，交時間隔0/1；接觸端點不是空集。非實數完備性或無理數分類證明，不用近似小數。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["兩個閉區間的交與間隔"]},"variant":{"family_id":"qf-density14-real-overlap-closed-intervals","seed":202610142060,"parameters":{"I":[[-9,2],[9,2]],"J":[[0,1],[11,2]],"domain":"finite_rational_real_slice","operation":"density14_real_overlap_closed_intervals","semantic_template":"qf-density14-real-overlap-closed-intervals/[\"intersection\",\"0/1\",\"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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r02-real-number-0061","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","language":"zh-Hant","source_mko_ids":["mko-real-number"],"title_zh":"兩個閉區間的交與間隔","stem_zh":"有理點嵌入實數的有限切片；閉區間包含端點；I=[-10/2, 10/2]、J=[0/1, 12/2]，各端點有序。依序答 (交非空是否, 交的[lo,hi]或[], 分離間隔)。","answer":{"kind":"expression","value":"(是, [0/1, 5/1], 0/1)","explanation_zh":"閉區間交集與分離距離=(是, [0/1, 5/1], 0/1)。取左端最大值、右端最小值，交時間隔0/1；接觸端點不是空集。非實數完備性或無理數分類證明，不用近似小數。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["兩個閉區間的交與間隔"]},"variant":{"family_id":"qf-density14-real-overlap-closed-intervals","seed":202610142061,"parameters":{"I":[[-10,2],[10,2]],"J":[[0,1],[12,2]],"domain":"finite_rational_real_slice","operation":"density14_real_overlap_closed_intervals","semantic_template":"qf-density14-real-overlap-closed-intervals/[\"intersection\",\"0/1\",\"5/1\"]"}},"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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r02-real-number-0062","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","language":"zh-Hant","source_mko_ids":["mko-real-number"],"title_zh":"兩個閉區間的交與間隔","stem_zh":"有理點嵌入實數的有限切片；閉區間包含端點；I=[-11/2, 11/2]、J=[0/1, 13/2]，各端點有序。依序答 (交非空是否, 交的[lo,hi]或[], 分離間隔)。","answer":{"kind":"expression","value":"(是, [0/1, 11/2], 0/1)","explanation_zh":"閉區間交集與分離距離=(是, [0/1, 11/2], 0/1)。取左端最大值、右端最小值，交時間隔0/1；接觸端點不是空集。非實數完備性或無理數分類證明，不用近似小數。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["兩個閉區間的交與間隔"]},"variant":{"family_id":"qf-density14-real-overlap-closed-intervals","seed":202610142062,"parameters":{"I":[[-11,2],[11,2]],"J":[[0,1],[13,2]],"domain":"finite_rational_real_slice","operation":"density14_real_overlap_closed_intervals","semantic_template":"qf-density14-real-overlap-closed-intervals/[\"intersection\",\"0/1\",\"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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r02-real-number-0063","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","language":"zh-Hant","source_mko_ids":["mko-real-number"],"title_zh":"兩個閉區間的交與間隔","stem_zh":"有理點嵌入實數的有限切片；閉區間包含端點；I=[-12/2, 12/2]、J=[0/1, 14/2]，各端點有序。依序答 (交非空是否, 交的[lo,hi]或[], 分離間隔)。","answer":{"kind":"expression","value":"(是, [0/1, 6/1], 0/1)","explanation_zh":"閉區間交集與分離距離=(是, [0/1, 6/1], 0/1)。取左端最大值、右端最小值，交時間隔0/1；接觸端點不是空集。非實數完備性或無理數分類證明，不用近似小數。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["兩個閉區間的交與間隔"]},"variant":{"family_id":"qf-density14-real-overlap-closed-intervals","seed":202610142063,"parameters":{"I":[[-12,2],[12,2]],"J":[[0,1],[14,2]],"domain":"finite_rational_real_slice","operation":"density14_real_overlap_closed_intervals","semantic_template":"qf-density14-real-overlap-closed-intervals/[\"intersection\",\"0/1\",\"6/1\"]"}},"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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r02-real-number-0064","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","language":"zh-Hant","source_mko_ids":["mko-real-number"],"title_zh":"兩個閉區間的交與間隔","stem_zh":"有理點嵌入實數的有限切片；閉區間包含端點；I=[-13/2, 13/2]、J=[0/1, 15/2]，各端點有序。依序答 (交非空是否, 交的[lo,hi]或[], 分離間隔)。","answer":{"kind":"expression","value":"(是, [0/1, 13/2], 0/1)","explanation_zh":"閉區間交集與分離距離=(是, [0/1, 13/2], 0/1)。取左端最大值、右端最小值，交時間隔0/1；接觸端點不是空集。非實數完備性或無理數分類證明，不用近似小數。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["兩個閉區間的交與間隔"]},"variant":{"family_id":"qf-density14-real-overlap-closed-intervals","seed":202610142064,"parameters":{"I":[[-13,2],[13,2]],"J":[[0,1],[15,2]],"domain":"finite_rational_real_slice","operation":"density14_real_overlap_closed_intervals","semantic_template":"qf-density14-real-overlap-closed-intervals/[\"intersection\",\"0/1\",\"13/2\"]"}},"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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r02-real-number-0065","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","language":"zh-Hant","source_mko_ids":["mko-real-number"],"title_zh":"兩個閉區間的交與間隔","stem_zh":"有理點嵌入實數的有限切片；閉區間包含端點；I=[-14/2, 14/2]、J=[0/1, 16/2]，各端點有序。依序答 (交非空是否, 交的[lo,hi]或[], 分離間隔)。","answer":{"kind":"expression","value":"(是, [0/1, 7/1], 0/1)","explanation_zh":"閉區間交集與分離距離=(是, [0/1, 7/1], 0/1)。取左端最大值、右端最小值，交時間隔0/1；接觸端點不是空集。非實數完備性或無理數分類證明，不用近似小數。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["兩個閉區間的交與間隔"]},"variant":{"family_id":"qf-density14-real-overlap-closed-intervals","seed":202610142065,"parameters":{"I":[[-14,2],[14,2]],"J":[[0,1],[16,2]],"domain":"finite_rational_real_slice","operation":"density14_real_overlap_closed_intervals","semantic_template":"qf-density14-real-overlap-closed-intervals/[\"intersection\",\"0/1\",\"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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r02-real-number-0066","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","language":"zh-Hant","source_mko_ids":["mko-real-number"],"title_zh":"兩個閉區間的交與間隔","stem_zh":"有理點嵌入實數的有限切片；閉區間包含端點；I=[-15/2, 15/2]、J=[0/1, 17/2]，各端點有序。依序答 (交非空是否, 交的[lo,hi]或[], 分離間隔)。","answer":{"kind":"expression","value":"(是, [0/1, 15/2], 0/1)","explanation_zh":"閉區間交集與分離距離=(是, [0/1, 15/2], 0/1)。取左端最大值、右端最小值，交時間隔0/1；接觸端點不是空集。非實數完備性或無理數分類證明，不用近似小數。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["兩個閉區間的交與間隔"]},"variant":{"family_id":"qf-density14-real-overlap-closed-intervals","seed":202610142066,"parameters":{"I":[[-15,2],[15,2]],"J":[[0,1],[17,2]],"domain":"finite_rational_real_slice","operation":"density14_real_overlap_closed_intervals","semantic_template":"qf-density14-real-overlap-closed-intervals/[\"intersection\",\"0/1\",\"15/2\"]"}},"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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r02-real-number-0067","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","language":"zh-Hant","source_mko_ids":["mko-real-number"],"title_zh":"兩個閉區間的交與間隔","stem_zh":"有理點嵌入實數的有限切片；閉區間包含端點；I=[-16/2, 16/2]、J=[0/1, 18/2]，各端點有序。依序答 (交非空是否, 交的[lo,hi]或[], 分離間隔)。","answer":{"kind":"expression","value":"(是, [0/1, 8/1], 0/1)","explanation_zh":"閉區間交集與分離距離=(是, [0/1, 8/1], 0/1)。取左端最大值、右端最小值，交時間隔0/1；接觸端點不是空集。非實數完備性或無理數分類證明，不用近似小數。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["兩個閉區間的交與間隔"]},"variant":{"family_id":"qf-density14-real-overlap-closed-intervals","seed":202610142067,"parameters":{"I":[[-16,2],[16,2]],"J":[[0,1],[18,2]],"domain":"finite_rational_real_slice","operation":"density14_real_overlap_closed_intervals","semantic_template":"qf-density14-real-overlap-closed-intervals/[\"intersection\",\"0/1\",\"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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r02-real-number-0068","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","language":"zh-Hant","source_mko_ids":["mko-real-number"],"title_zh":"兩個閉區間的交與間隔","stem_zh":"有理點嵌入實數的有限切片；閉區間包含端點；I=[-17/2, 17/2]、J=[0/1, 19/2]，各端點有序。依序答 (交非空是否, 交的[lo,hi]或[], 分離間隔)。","answer":{"kind":"expression","value":"(是, [0/1, 17/2], 0/1)","explanation_zh":"閉區間交集與分離距離=(是, [0/1, 17/2], 0/1)。取左端最大值、右端最小值，交時間隔0/1；接觸端點不是空集。非實數完備性或無理數分類證明，不用近似小數。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["兩個閉區間的交與間隔"]},"variant":{"family_id":"qf-density14-real-overlap-closed-intervals","seed":202610142068,"parameters":{"I":[[-17,2],[17,2]],"J":[[0,1],[19,2]],"domain":"finite_rational_real_slice","operation":"density14_real_overlap_closed_intervals","semantic_template":"qf-density14-real-overlap-closed-intervals/[\"intersection\",\"0/1\",\"17/2\"]"}},"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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r02-real-number-0069","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","language":"zh-Hant","source_mko_ids":["mko-real-number"],"title_zh":"兩個閉區間的交與間隔","stem_zh":"有理點嵌入實數的有限切片；閉區間包含端點；I=[-18/2, 18/2]、J=[0/1, 20/2]，各端點有序。依序答 (交非空是否, 交的[lo,hi]或[], 分離間隔)。","answer":{"kind":"expression","value":"(是, [0/1, 9/1], 0/1)","explanation_zh":"閉區間交集與分離距離=(是, [0/1, 9/1], 0/1)。取左端最大值、右端最小值，交時間隔0/1；接觸端點不是空集。非實數完備性或無理數分類證明，不用近似小數。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["兩個閉區間的交與間隔"]},"variant":{"family_id":"qf-density14-real-overlap-closed-intervals","seed":202610142069,"parameters":{"I":[[-18,2],[18,2]],"J":[[0,1],[20,2]],"domain":"finite_rational_real_slice","operation":"density14_real_overlap_closed_intervals","semantic_template":"qf-density14-real-overlap-closed-intervals/[\"intersection\",\"0/1\",\"9/1\"]"}},"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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r02-real-number-0070","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","language":"zh-Hant","source_mko_ids":["mko-real-number"],"title_zh":"兩個閉區間的交與間隔","stem_zh":"有理點嵌入實數的有限切片；閉區間包含端點；I=[-19/2, 19/2]、J=[0/1, 21/2]，各端點有序。依序答 (交非空是否, 交的[lo,hi]或[], 分離間隔)。","answer":{"kind":"expression","value":"(是, [0/1, 19/2], 0/1)","explanation_zh":"閉區間交集與分離距離=(是, [0/1, 19/2], 0/1)。取左端最大值、右端最小值，交時間隔0/1；接觸端點不是空集。非實數完備性或無理數分類證明，不用近似小數。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["兩個閉區間的交與間隔"]},"variant":{"family_id":"qf-density14-real-overlap-closed-intervals","seed":202610142070,"parameters":{"I":[[-19,2],[19,2]],"J":[[0,1],[21,2]],"domain":"finite_rational_real_slice","operation":"density14_real_overlap_closed_intervals","semantic_template":"qf-density14-real-overlap-closed-intervals/[\"intersection\",\"0/1\",\"19/2\"]"}},"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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r02-real-number-0071","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","language":"zh-Hant","source_mko_ids":["mko-real-number"],"title_zh":"兩個閉區間的交與間隔","stem_zh":"有理點嵌入實數的有限切片；閉區間包含端點；I=[-20/2, 20/2]、J=[0/1, 22/2]，各端點有序。依序答 (交非空是否, 交的[lo,hi]或[], 分離間隔)。","answer":{"kind":"expression","value":"(是, [0/1, 10/1], 0/1)","explanation_zh":"閉區間交集與分離距離=(是, [0/1, 10/1], 0/1)。取左端最大值、右端最小值，交時間隔0/1；接觸端點不是空集。非實數完備性或無理數分類證明，不用近似小數。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["兩個閉區間的交與間隔"]},"variant":{"family_id":"qf-density14-real-overlap-closed-intervals","seed":202610142071,"parameters":{"I":[[-20,2],[20,2]],"J":[[0,1],[22,2]],"domain":"finite_rational_real_slice","operation":"density14_real_overlap_closed_intervals","semantic_template":"qf-density14-real-overlap-closed-intervals/[\"intersection\",\"0/1\",\"10/1\"]"}},"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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r02-real-number-0072","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","language":"zh-Hant","source_mko_ids":["mko-real-number"],"title_zh":"兩個閉區間的交與間隔","stem_zh":"有理點嵌入實數的有限切片；閉區間包含端點；I=[-21/2, 21/2]、J=[0/1, 23/2]，各端點有序。依序答 (交非空是否, 交的[lo,hi]或[], 分離間隔)。","answer":{"kind":"expression","value":"(是, [0/1, 21/2], 0/1)","explanation_zh":"閉區間交集與分離距離=(是, [0/1, 21/2], 0/1)。取左端最大值、右端最小值，交時間隔0/1；接觸端點不是空集。非實數完備性或無理數分類證明，不用近似小數。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["兩個閉區間的交與間隔"]},"variant":{"family_id":"qf-density14-real-overlap-closed-intervals","seed":202610142072,"parameters":{"I":[[-21,2],[21,2]],"J":[[0,1],[23,2]],"domain":"finite_rational_real_slice","operation":"density14_real_overlap_closed_intervals","semantic_template":"qf-density14-real-overlap-closed-intervals/[\"intersection\",\"0/1\",\"21/2\"]"}},"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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r02-real-number-0073","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","language":"zh-Hant","source_mko_ids":["mko-real-number"],"title_zh":"兩個閉區間的交與間隔","stem_zh":"有理點嵌入實數的有限切片；閉區間包含端點；I=[-22/2, 22/2]、J=[0/1, 24/2]，各端點有序。依序答 (交非空是否, 交的[lo,hi]或[], 分離間隔)。","answer":{"kind":"expression","value":"(是, [0/1, 11/1], 0/1)","explanation_zh":"閉區間交集與分離距離=(是, [0/1, 11/1], 0/1)。取左端最大值、右端最小值，交時間隔0/1；接觸端點不是空集。非實數完備性或無理數分類證明，不用近似小數。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["兩個閉區間的交與間隔"]},"variant":{"family_id":"qf-density14-real-overlap-closed-intervals","seed":202610142073,"parameters":{"I":[[-22,2],[22,2]],"J":[[0,1],[24,2]],"domain":"finite_rational_real_slice","operation":"density14_real_overlap_closed_intervals","semantic_template":"qf-density14-real-overlap-closed-intervals/[\"intersection\",\"0/1\",\"11/1\"]"}},"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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r02-real-number-0074","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","language":"zh-Hant","source_mko_ids":["mko-real-number"],"title_zh":"兩個閉區間的交與間隔","stem_zh":"有理點嵌入實數的有限切片；閉區間包含端點；I=[-23/2, 23/2]、J=[0/1, 25/2]，各端點有序。依序答 (交非空是否, 交的[lo,hi]或[], 分離間隔)。","answer":{"kind":"expression","value":"(是, [0/1, 23/2], 0/1)","explanation_zh":"閉區間交集與分離距離=(是, [0/1, 23/2], 0/1)。取左端最大值、右端最小值，交時間隔0/1；接觸端點不是空集。非實數完備性或無理數分類證明，不用近似小數。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["兩個閉區間的交與間隔"]},"variant":{"family_id":"qf-density14-real-overlap-closed-intervals","seed":202610142074,"parameters":{"I":[[-23,2],[23,2]],"J":[[0,1],[25,2]],"domain":"finite_rational_real_slice","operation":"density14_real_overlap_closed_intervals","semantic_template":"qf-density14-real-overlap-closed-intervals/[\"intersection\",\"0/1\",\"23/2\"]"}},"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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r02-real-number-0075","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","language":"zh-Hant","source_mko_ids":["mko-real-number"],"title_zh":"兩個閉區間的交與間隔","stem_zh":"有理點嵌入實數的有限切片；閉區間包含端點；I=[-24/2, 24/2]、J=[0/1, 26/2]，各端點有序。依序答 (交非空是否, 交的[lo,hi]或[], 分離間隔)。","answer":{"kind":"expression","value":"(是, [0/1, 12/1], 0/1)","explanation_zh":"閉區間交集與分離距離=(是, [0/1, 12/1], 0/1)。取左端最大值、右端最小值，交時間隔0/1；接觸端點不是空集。非實數完備性或無理數分類證明，不用近似小數。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["兩個閉區間的交與間隔"]},"variant":{"family_id":"qf-density14-real-overlap-closed-intervals","seed":202610142075,"parameters":{"I":[[-24,2],[24,2]],"J":[[0,1],[26,2]],"domain":"finite_rational_real_slice","operation":"density14_real_overlap_closed_intervals","semantic_template":"qf-density14-real-overlap-closed-intervals/[\"intersection\",\"0/1\",\"12/1\"]"}},"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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r02-real-number-0076","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","language":"zh-Hant","source_mko_ids":["mko-real-number"],"title_zh":"有理點的有向偏移與距離","stem_zh":"有理點嵌入實數的有限切片；origin=-12/2，有序points=[-10/1, -13/2, -6/1, -16/3]。依序列 (各point−origin, 各偏移絕對值)，保留四欄次序。","answer":{"kind":"expression","value":"([-4/1, -1/2, 0/1, 2/3], [4/1, 1/2, 0/1, 2/3])","explanation_zh":"有理點有向偏移與距離=([-4/1, -1/2, 0/1, 2/3], [4/1, 1/2, 0/1, 2/3])。先精確相減保留方向，再取非負絕對值。非實數完備性或無理數分類證明，不用近似小數。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["有理點的有向偏移與距離"]},"variant":{"family_id":"qf-density14-real-point-offset-distances","seed":202610142076,"parameters":{"origin":[-12,2],"points":[[-10,1],[-13,2],[-6,1],[-16,3]],"domain":"finite_rational_real_slice","operation":"density14_real_point_offset_distances","semantic_template":"qf-density14-real-point-offset-distances/[\"-4/1\",\"-1/2\",\"0/1\",\"2/3\"]"}},"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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r02-real-number-0077","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","language":"zh-Hant","source_mko_ids":["mko-real-number"],"title_zh":"有理點的有向偏移與距離","stem_zh":"有理點嵌入實數的有限切片；origin=-11/2，有序points=[-55/6, -6/1, -11/2, -29/6]。依序列 (各point−origin, 各偏移絕對值)，保留四欄次序。","answer":{"kind":"expression","value":"([-11/3, -1/2, 0/1, 2/3], [11/3, 1/2, 0/1, 2/3])","explanation_zh":"有理點有向偏移與距離=([-11/3, -1/2, 0/1, 2/3], [11/3, 1/2, 0/1, 2/3])。先精確相減保留方向，再取非負絕對值。非實數完備性或無理數分類證明，不用近似小數。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["有理點的有向偏移與距離"]},"variant":{"family_id":"qf-density14-real-point-offset-distances","seed":202610142077,"parameters":{"origin":[-11,2],"points":[[-55,6],[-6,1],[-11,2],[-29,6]],"domain":"finite_rational_real_slice","operation":"density14_real_point_offset_distances","semantic_template":"qf-density14-real-point-offset-distances/[\"-11/3\",\"-1/2\",\"0/1\",\"2/3\"]"}},"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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r02-real-number-0078","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","language":"zh-Hant","source_mko_ids":["mko-real-number"],"title_zh":"有理點的有向偏移與距離","stem_zh":"有理點嵌入實數的有限切片；origin=-10/2，有序points=[-25/3, -11/2, -5/1, -13/3]。依序列 (各point−origin, 各偏移絕對值)，保留四欄次序。","answer":{"kind":"expression","value":"([-10/3, -1/2, 0/1, 2/3], [10/3, 1/2, 0/1, 2/3])","explanation_zh":"有理點有向偏移與距離=([-10/3, -1/2, 0/1, 2/3], [10/3, 1/2, 0/1, 2/3])。先精確相減保留方向，再取非負絕對值。非實數完備性或無理數分類證明，不用近似小數。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["有理點的有向偏移與距離"]},"variant":{"family_id":"qf-density14-real-point-offset-distances","seed":202610142078,"parameters":{"origin":[-10,2],"points":[[-25,3],[-11,2],[-5,1],[-13,3]],"domain":"finite_rational_real_slice","operation":"density14_real_point_offset_distances","semantic_template":"qf-density14-real-point-offset-distances/[\"-10/3\",\"-1/2\",\"0/1\",\"2/3\"]"}},"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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r02-real-number-0079","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","language":"zh-Hant","source_mko_ids":["mko-real-number"],"title_zh":"有理點的有向偏移與距離","stem_zh":"有理點嵌入實數的有限切片；origin=-9/2，有序points=[-15/2, -5/1, -9/2, -23/6]。依序列 (各point−origin, 各偏移絕對值)，保留四欄次序。","answer":{"kind":"expression","value":"([-3/1, -1/2, 0/1, 2/3], [3/1, 1/2, 0/1, 2/3])","explanation_zh":"有理點有向偏移與距離=([-3/1, -1/2, 0/1, 2/3], [3/1, 1/2, 0/1, 2/3])。先精確相減保留方向，再取非負絕對值。非實數完備性或無理數分類證明，不用近似小數。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["有理點的有向偏移與距離"]},"variant":{"family_id":"qf-density14-real-point-offset-distances","seed":202610142079,"parameters":{"origin":[-9,2],"points":[[-15,2],[-5,1],[-9,2],[-23,6]],"domain":"finite_rational_real_slice","operation":"density14_real_point_offset_distances","semantic_template":"qf-density14-real-point-offset-distances/[\"-3/1\",\"-1/2\",\"0/1\",\"2/3\"]"}},"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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r02-real-number-0080","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","language":"zh-Hant","source_mko_ids":["mko-real-number"],"title_zh":"有理點的有向偏移與距離","stem_zh":"有理點嵌入實數的有限切片；origin=-8/2，有序points=[-20/3, -9/2, -4/1, -10/3]。依序列 (各point−origin, 各偏移絕對值)，保留四欄次序。","answer":{"kind":"expression","value":"([-8/3, -1/2, 0/1, 2/3], [8/3, 1/2, 0/1, 2/3])","explanation_zh":"有理點有向偏移與距離=([-8/3, -1/2, 0/1, 2/3], [8/3, 1/2, 0/1, 2/3])。先精確相減保留方向，再取非負絕對值。非實數完備性或無理數分類證明，不用近似小數。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["有理點的有向偏移與距離"]},"variant":{"family_id":"qf-density14-real-point-offset-distances","seed":202610142080,"parameters":{"origin":[-8,2],"points":[[-20,3],[-9,2],[-4,1],[-10,3]],"domain":"finite_rational_real_slice","operation":"density14_real_point_offset_distances","semantic_template":"qf-density14-real-point-offset-distances/[\"-8/3\",\"-1/2\",\"0/1\",\"2/3\"]"}},"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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r02-real-number-0081","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","language":"zh-Hant","source_mko_ids":["mko-real-number"],"title_zh":"有理點的有向偏移與距離","stem_zh":"有理點嵌入實數的有限切片；origin=-7/2，有序points=[-35/6, -4/1, -7/2, -17/6]。依序列 (各point−origin, 各偏移絕對值)，保留四欄次序。","answer":{"kind":"expression","value":"([-7/3, -1/2, 0/1, 2/3], [7/3, 1/2, 0/1, 2/3])","explanation_zh":"有理點有向偏移與距離=([-7/3, -1/2, 0/1, 2/3], [7/3, 1/2, 0/1, 2/3])。先精確相減保留方向，再取非負絕對值。非實數完備性或無理數分類證明，不用近似小數。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["有理點的有向偏移與距離"]},"variant":{"family_id":"qf-density14-real-point-offset-distances","seed":202610142081,"parameters":{"origin":[-7,2],"points":[[-35,6],[-4,1],[-7,2],[-17,6]],"domain":"finite_rational_real_slice","operation":"density14_real_point_offset_distances","semantic_template":"qf-density14-real-point-offset-distances/[\"-7/3\",\"-1/2\",\"0/1\",\"2/3\"]"}},"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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r02-real-number-0082","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","language":"zh-Hant","source_mko_ids":["mko-real-number"],"title_zh":"有理點的有向偏移與距離","stem_zh":"有理點嵌入實數的有限切片；origin=-6/2，有序points=[-5/1, -7/2, -3/1, -7/3]。依序列 (各point−origin, 各偏移絕對值)，保留四欄次序。","answer":{"kind":"expression","value":"([-2/1, -1/2, 0/1, 2/3], [2/1, 1/2, 0/1, 2/3])","explanation_zh":"有理點有向偏移與距離=([-2/1, -1/2, 0/1, 2/3], [2/1, 1/2, 0/1, 2/3])。先精確相減保留方向，再取非負絕對值。非實數完備性或無理數分類證明，不用近似小數。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["有理點的有向偏移與距離"]},"variant":{"family_id":"qf-density14-real-point-offset-distances","seed":202610142082,"parameters":{"origin":[-6,2],"points":[[-5,1],[-7,2],[-3,1],[-7,3]],"domain":"finite_rational_real_slice","operation":"density14_real_point_offset_distances","semantic_template":"qf-density14-real-point-offset-distances/[\"-2/1\",\"-1/2\",\"0/1\",\"2/3\"]"}},"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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r02-real-number-0083","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","language":"zh-Hant","source_mko_ids":["mko-real-number"],"title_zh":"有理點的有向偏移與距離","stem_zh":"有理點嵌入實數的有限切片；origin=-5/2，有序points=[-25/6, -3/1, -5/2, -11/6]。依序列 (各point−origin, 各偏移絕對值)，保留四欄次序。","answer":{"kind":"expression","value":"([-5/3, -1/2, 0/1, 2/3], [5/3, 1/2, 0/1, 2/3])","explanation_zh":"有理點有向偏移與距離=([-5/3, -1/2, 0/1, 2/3], [5/3, 1/2, 0/1, 2/3])。先精確相減保留方向，再取非負絕對值。非實數完備性或無理數分類證明，不用近似小數。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["有理點的有向偏移與距離"]},"variant":{"family_id":"qf-density14-real-point-offset-distances","seed":202610142083,"parameters":{"origin":[-5,2],"points":[[-25,6],[-3,1],[-5,2],[-11,6]],"domain":"finite_rational_real_slice","operation":"density14_real_point_offset_distances","semantic_template":"qf-density14-real-point-offset-distances/[\"-5/3\",\"-1/2\",\"0/1\",\"2/3\"]"}},"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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r02-real-number-0084","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","language":"zh-Hant","source_mko_ids":["mko-real-number"],"title_zh":"有理點的有向偏移與距離","stem_zh":"有理點嵌入實數的有限切片；origin=-4/2，有序points=[-10/3, -5/2, -2/1, -4/3]。依序列 (各point−origin, 各偏移絕對值)，保留四欄次序。","answer":{"kind":"expression","value":"([-4/3, -1/2, 0/1, 2/3], [4/3, 1/2, 0/1, 2/3])","explanation_zh":"有理點有向偏移與距離=([-4/3, -1/2, 0/1, 2/3], [4/3, 1/2, 0/1, 2/3])。先精確相減保留方向，再取非負絕對值。非實數完備性或無理數分類證明，不用近似小數。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["有理點的有向偏移與距離"]},"variant":{"family_id":"qf-density14-real-point-offset-distances","seed":202610142084,"parameters":{"origin":[-4,2],"points":[[-10,3],[-5,2],[-2,1],[-4,3]],"domain":"finite_rational_real_slice","operation":"density14_real_point_offset_distances","semantic_template":"qf-density14-real-point-offset-distances/[\"-4/3\",\"-1/2\",\"0/1\",\"2/3\"]"}},"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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r02-real-number-0085","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","language":"zh-Hant","source_mko_ids":["mko-real-number"],"title_zh":"有理點的有向偏移與距離","stem_zh":"有理點嵌入實數的有限切片；origin=-3/2，有序points=[-5/2, -2/1, -3/2, -5/6]。依序列 (各point−origin, 各偏移絕對值)，保留四欄次序。","answer":{"kind":"expression","value":"([-1/1, -1/2, 0/1, 2/3], [1/1, 1/2, 0/1, 2/3])","explanation_zh":"有理點有向偏移與距離=([-1/1, -1/2, 0/1, 2/3], [1/1, 1/2, 0/1, 2/3])。先精確相減保留方向，再取非負絕對值。非實數完備性或無理數分類證明，不用近似小數。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["有理點的有向偏移與距離"]},"variant":{"family_id":"qf-density14-real-point-offset-distances","seed":202610142085,"parameters":{"origin":[-3,2],"points":[[-5,2],[-2,1],[-3,2],[-5,6]],"domain":"finite_rational_real_slice","operation":"density14_real_point_offset_distances","semantic_template":"qf-density14-real-point-offset-distances/[\"-1/1\",\"-1/2\",\"0/1\",\"2/3\"]"}},"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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r02-real-number-0086","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","language":"zh-Hant","source_mko_ids":["mko-real-number"],"title_zh":"有理點的有向偏移與距離","stem_zh":"有理點嵌入實數的有限切片；origin=-2/2，有序points=[-5/3, -3/2, -1/1, -1/3]。依序列 (各point−origin, 各偏移絕對值)，保留四欄次序。","answer":{"kind":"expression","value":"([-2/3, -1/2, 0/1, 2/3], [2/3, 1/2, 0/1, 2/3])","explanation_zh":"有理點有向偏移與距離=([-2/3, -1/2, 0/1, 2/3], [2/3, 1/2, 0/1, 2/3])。先精確相減保留方向，再取非負絕對值。非實數完備性或無理數分類證明，不用近似小數。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["有理點的有向偏移與距離"]},"variant":{"family_id":"qf-density14-real-point-offset-distances","seed":202610142086,"parameters":{"origin":[-2,2],"points":[[-5,3],[-3,2],[-1,1],[-1,3]],"domain":"finite_rational_real_slice","operation":"density14_real_point_offset_distances","semantic_template":"qf-density14-real-point-offset-distances/[\"-2/3\",\"-1/2\",\"0/1\",\"2/3\"]"}},"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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r02-real-number-0087","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","language":"zh-Hant","source_mko_ids":["mko-real-number"],"title_zh":"有理點的有向偏移與距離","stem_zh":"有理點嵌入實數的有限切片；origin=-1/2，有序points=[-5/6, -1/1, -1/2, 1/6]。依序列 (各point−origin, 各偏移絕對值)，保留四欄次序。","answer":{"kind":"expression","value":"([-1/3, -1/2, 0/1, 2/3], [1/3, 1/2, 0/1, 2/3])","explanation_zh":"有理點有向偏移與距離=([-1/3, -1/2, 0/1, 2/3], [1/3, 1/2, 0/1, 2/3])。先精確相減保留方向，再取非負絕對值。非實數完備性或無理數分類證明，不用近似小數。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["有理點的有向偏移與距離"]},"variant":{"family_id":"qf-density14-real-point-offset-distances","seed":202610142087,"parameters":{"origin":[-1,2],"points":[[-5,6],[-1,1],[-1,2],[1,6]],"domain":"finite_rational_real_slice","operation":"density14_real_point_offset_distances","semantic_template":"qf-density14-real-point-offset-distances/[\"-1/3\",\"-1/2\",\"0/1\",\"2/3\"]"}},"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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r02-real-number-0088","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","language":"zh-Hant","source_mko_ids":["mko-real-number"],"title_zh":"有理點的有向偏移與距離","stem_zh":"有理點嵌入實數的有限切片；origin=0/2，有序points=[0/1, -1/2, 0/1, 2/3]。依序列 (各point−origin, 各偏移絕對值)，保留四欄次序。","answer":{"kind":"expression","value":"([0/1, -1/2, 0/1, 2/3], [0/1, 1/2, 0/1, 2/3])","explanation_zh":"有理點有向偏移與距離=([0/1, -1/2, 0/1, 2/3], [0/1, 1/2, 0/1, 2/3])。先精確相減保留方向，再取非負絕對值。非實數完備性或無理數分類證明，不用近似小數。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["有理點的有向偏移與距離"]},"variant":{"family_id":"qf-density14-real-point-offset-distances","seed":202610142088,"parameters":{"origin":[0,2],"points":[[0,1],[-1,2],[0,1],[2,3]],"domain":"finite_rational_real_slice","operation":"density14_real_point_offset_distances","semantic_template":"qf-density14-real-point-offset-distances/[\"0/1\",\"-1/2\",\"0/1\",\"2/3\"]"}},"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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r02-real-number-0089","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","language":"zh-Hant","source_mko_ids":["mko-real-number"],"title_zh":"有理點的有向偏移與距離","stem_zh":"有理點嵌入實數的有限切片；origin=1/2，有序points=[5/6, 0/1, 1/2, 7/6]。依序列 (各point−origin, 各偏移絕對值)，保留四欄次序。","answer":{"kind":"expression","value":"([1/3, -1/2, 0/1, 2/3], [1/3, 1/2, 0/1, 2/3])","explanation_zh":"有理點有向偏移與距離=([1/3, -1/2, 0/1, 2/3], [1/3, 1/2, 0/1, 2/3])。先精確相減保留方向，再取非負絕對值。非實數完備性或無理數分類證明，不用近似小數。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["有理點的有向偏移與距離"]},"variant":{"family_id":"qf-density14-real-point-offset-distances","seed":202610142089,"parameters":{"origin":[1,2],"points":[[5,6],[0,1],[1,2],[7,6]],"domain":"finite_rational_real_slice","operation":"density14_real_point_offset_distances","semantic_template":"qf-density14-real-point-offset-distances/[\"1/3\",\"-1/2\",\"0/1\",\"2/3\"]"}},"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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r02-real-number-0090","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","language":"zh-Hant","source_mko_ids":["mko-real-number"],"title_zh":"有理點的有向偏移與距離","stem_zh":"有理點嵌入實數的有限切片；origin=2/2，有序points=[5/3, 1/2, 1/1, 5/3]。依序列 (各point−origin, 各偏移絕對值)，保留四欄次序。","answer":{"kind":"expression","value":"([2/3, -1/2, 0/1, 2/3], [2/3, 1/2, 0/1, 2/3])","explanation_zh":"有理點有向偏移與距離=([2/3, -1/2, 0/1, 2/3], [2/3, 1/2, 0/1, 2/3])。先精確相減保留方向，再取非負絕對值。非實數完備性或無理數分類證明，不用近似小數。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["有理點的有向偏移與距離"]},"variant":{"family_id":"qf-density14-real-point-offset-distances","seed":202610142090,"parameters":{"origin":[2,2],"points":[[5,3],[1,2],[1,1],[5,3]],"domain":"finite_rational_real_slice","operation":"density14_real_point_offset_distances","semantic_template":"qf-density14-real-point-offset-distances/[\"2/3\",\"-1/2\",\"0/1\",\"2/3\"]"}},"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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r02-real-number-0091","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","language":"zh-Hant","source_mko_ids":["mko-real-number"],"title_zh":"有理點的有向偏移與距離","stem_zh":"有理點嵌入實數的有限切片；origin=3/2，有序points=[5/2, 1/1, 3/2, 13/6]。依序列 (各point−origin, 各偏移絕對值)，保留四欄次序。","answer":{"kind":"expression","value":"([1/1, -1/2, 0/1, 2/3], [1/1, 1/2, 0/1, 2/3])","explanation_zh":"有理點有向偏移與距離=([1/1, -1/2, 0/1, 2/3], [1/1, 1/2, 0/1, 2/3])。先精確相減保留方向，再取非負絕對值。非實數完備性或無理數分類證明，不用近似小數。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["有理點的有向偏移與距離"]},"variant":{"family_id":"qf-density14-real-point-offset-distances","seed":202610142091,"parameters":{"origin":[3,2],"points":[[5,2],[1,1],[3,2],[13,6]],"domain":"finite_rational_real_slice","operation":"density14_real_point_offset_distances","semantic_template":"qf-density14-real-point-offset-distances/[\"1/1\",\"-1/2\",\"0/1\",\"2/3\"]"}},"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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r02-real-number-0092","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","language":"zh-Hant","source_mko_ids":["mko-real-number"],"title_zh":"有理點的有向偏移與距離","stem_zh":"有理點嵌入實數的有限切片；origin=4/2，有序points=[10/3, 3/2, 2/1, 8/3]。依序列 (各point−origin, 各偏移絕對值)，保留四欄次序。","answer":{"kind":"expression","value":"([4/3, -1/2, 0/1, 2/3], [4/3, 1/2, 0/1, 2/3])","explanation_zh":"有理點有向偏移與距離=([4/3, -1/2, 0/1, 2/3], [4/3, 1/2, 0/1, 2/3])。先精確相減保留方向，再取非負絕對值。非實數完備性或無理數分類證明，不用近似小數。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["有理點的有向偏移與距離"]},"variant":{"family_id":"qf-density14-real-point-offset-distances","seed":202610142092,"parameters":{"origin":[4,2],"points":[[10,3],[3,2],[2,1],[8,3]],"domain":"finite_rational_real_slice","operation":"density14_real_point_offset_distances","semantic_template":"qf-density14-real-point-offset-distances/[\"4/3\",\"-1/2\",\"0/1\",\"2/3\"]"}},"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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r02-real-number-0093","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","language":"zh-Hant","source_mko_ids":["mko-real-number"],"title_zh":"有理點的有向偏移與距離","stem_zh":"有理點嵌入實數的有限切片；origin=5/2，有序points=[25/6, 2/1, 5/2, 19/6]。依序列 (各point−origin, 各偏移絕對值)，保留四欄次序。","answer":{"kind":"expression","value":"([5/3, -1/2, 0/1, 2/3], [5/3, 1/2, 0/1, 2/3])","explanation_zh":"有理點有向偏移與距離=([5/3, -1/2, 0/1, 2/3], [5/3, 1/2, 0/1, 2/3])。先精確相減保留方向，再取非負絕對值。非實數完備性或無理數分類證明，不用近似小數。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["有理點的有向偏移與距離"]},"variant":{"family_id":"qf-density14-real-point-offset-distances","seed":202610142093,"parameters":{"origin":[5,2],"points":[[25,6],[2,1],[5,2],[19,6]],"domain":"finite_rational_real_slice","operation":"density14_real_point_offset_distances","semantic_template":"qf-density14-real-point-offset-distances/[\"5/3\",\"-1/2\",\"0/1\",\"2/3\"]"}},"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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r02-real-number-0094","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","language":"zh-Hant","source_mko_ids":["mko-real-number"],"title_zh":"有理點的有向偏移與距離","stem_zh":"有理點嵌入實數的有限切片；origin=6/2，有序points=[5/1, 5/2, 3/1, 11/3]。依序列 (各point−origin, 各偏移絕對值)，保留四欄次序。","answer":{"kind":"expression","value":"([2/1, -1/2, 0/1, 2/3], [2/1, 1/2, 0/1, 2/3])","explanation_zh":"有理點有向偏移與距離=([2/1, -1/2, 0/1, 2/3], [2/1, 1/2, 0/1, 2/3])。先精確相減保留方向，再取非負絕對值。非實數完備性或無理數分類證明，不用近似小數。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["有理點的有向偏移與距離"]},"variant":{"family_id":"qf-density14-real-point-offset-distances","seed":202610142094,"parameters":{"origin":[6,2],"points":[[5,1],[5,2],[3,1],[11,3]],"domain":"finite_rational_real_slice","operation":"density14_real_point_offset_distances","semantic_template":"qf-density14-real-point-offset-distances/[\"2/1\",\"-1/2\",\"0/1\",\"2/3\"]"}},"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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r02-real-number-0095","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","language":"zh-Hant","source_mko_ids":["mko-real-number"],"title_zh":"有理點的有向偏移與距離","stem_zh":"有理點嵌入實數的有限切片；origin=7/2，有序points=[35/6, 3/1, 7/2, 25/6]。依序列 (各point−origin, 各偏移絕對值)，保留四欄次序。","answer":{"kind":"expression","value":"([7/3, -1/2, 0/1, 2/3], [7/3, 1/2, 0/1, 2/3])","explanation_zh":"有理點有向偏移與距離=([7/3, -1/2, 0/1, 2/3], [7/3, 1/2, 0/1, 2/3])。先精確相減保留方向，再取非負絕對值。非實數完備性或無理數分類證明，不用近似小數。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["有理點的有向偏移與距離"]},"variant":{"family_id":"qf-density14-real-point-offset-distances","seed":202610142095,"parameters":{"origin":[7,2],"points":[[35,6],[3,1],[7,2],[25,6]],"domain":"finite_rational_real_slice","operation":"density14_real_point_offset_distances","semantic_template":"qf-density14-real-point-offset-distances/[\"7/3\",\"-1/2\",\"0/1\",\"2/3\"]"}},"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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r02-real-number-0096","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","language":"zh-Hant","source_mko_ids":["mko-real-number"],"title_zh":"有理點的有向偏移與距離","stem_zh":"有理點嵌入實數的有限切片；origin=8/2，有序points=[20/3, 7/2, 4/1, 14/3]。依序列 (各point−origin, 各偏移絕對值)，保留四欄次序。","answer":{"kind":"expression","value":"([8/3, -1/2, 0/1, 2/3], [8/3, 1/2, 0/1, 2/3])","explanation_zh":"有理點有向偏移與距離=([8/3, -1/2, 0/1, 2/3], [8/3, 1/2, 0/1, 2/3])。先精確相減保留方向，再取非負絕對值。非實數完備性或無理數分類證明，不用近似小數。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["有理點的有向偏移與距離"]},"variant":{"family_id":"qf-density14-real-point-offset-distances","seed":202610142096,"parameters":{"origin":[8,2],"points":[[20,3],[7,2],[4,1],[14,3]],"domain":"finite_rational_real_slice","operation":"density14_real_point_offset_distances","semantic_template":"qf-density14-real-point-offset-distances/[\"8/3\",\"-1/2\",\"0/1\",\"2/3\"]"}},"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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r02-real-number-0097","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","language":"zh-Hant","source_mko_ids":["mko-real-number"],"title_zh":"有理點的有向偏移與距離","stem_zh":"有理點嵌入實數的有限切片；origin=9/2，有序points=[15/2, 4/1, 9/2, 31/6]。依序列 (各point−origin, 各偏移絕對值)，保留四欄次序。","answer":{"kind":"expression","value":"([3/1, -1/2, 0/1, 2/3], [3/1, 1/2, 0/1, 2/3])","explanation_zh":"有理點有向偏移與距離=([3/1, -1/2, 0/1, 2/3], [3/1, 1/2, 0/1, 2/3])。先精確相減保留方向，再取非負絕對值。非實數完備性或無理數分類證明，不用近似小數。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["有理點的有向偏移與距離"]},"variant":{"family_id":"qf-density14-real-point-offset-distances","seed":202610142097,"parameters":{"origin":[9,2],"points":[[15,2],[4,1],[9,2],[31,6]],"domain":"finite_rational_real_slice","operation":"density14_real_point_offset_distances","semantic_template":"qf-density14-real-point-offset-distances/[\"3/1\",\"-1/2\",\"0/1\",\"2/3\"]"}},"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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r02-real-number-0098","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","language":"zh-Hant","source_mko_ids":["mko-real-number"],"title_zh":"有理點的有向偏移與距離","stem_zh":"有理點嵌入實數的有限切片；origin=10/2，有序points=[25/3, 9/2, 5/1, 17/3]。依序列 (各point−origin, 各偏移絕對值)，保留四欄次序。","answer":{"kind":"expression","value":"([10/3, -1/2, 0/1, 2/3], [10/3, 1/2, 0/1, 2/3])","explanation_zh":"有理點有向偏移與距離=([10/3, -1/2, 0/1, 2/3], [10/3, 1/2, 0/1, 2/3])。先精確相減保留方向，再取非負絕對值。非實數完備性或無理數分類證明，不用近似小數。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["有理點的有向偏移與距離"]},"variant":{"family_id":"qf-density14-real-point-offset-distances","seed":202610142098,"parameters":{"origin":[10,2],"points":[[25,3],[9,2],[5,1],[17,3]],"domain":"finite_rational_real_slice","operation":"density14_real_point_offset_distances","semantic_template":"qf-density14-real-point-offset-distances/[\"10/3\",\"-1/2\",\"0/1\",\"2/3\"]"}},"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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r02-real-number-0099","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","language":"zh-Hant","source_mko_ids":["mko-real-number"],"title_zh":"有理點的有向偏移與距離","stem_zh":"有理點嵌入實數的有限切片；origin=11/2，有序points=[55/6, 5/1, 11/2, 37/6]。依序列 (各point−origin, 各偏移絕對值)，保留四欄次序。","answer":{"kind":"expression","value":"([11/3, -1/2, 0/1, 2/3], [11/3, 1/2, 0/1, 2/3])","explanation_zh":"有理點有向偏移與距離=([11/3, -1/2, 0/1, 2/3], [11/3, 1/2, 0/1, 2/3])。先精確相減保留方向，再取非負絕對值。非實數完備性或無理數分類證明，不用近似小數。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["有理點的有向偏移與距離"]},"variant":{"family_id":"qf-density14-real-point-offset-distances","seed":202610142099,"parameters":{"origin":[11,2],"points":[[55,6],[5,1],[11,2],[37,6]],"domain":"finite_rational_real_slice","operation":"density14_real_point_offset_distances","semantic_template":"qf-density14-real-point-offset-distances/[\"11/3\",\"-1/2\",\"0/1\",\"2/3\"]"}},"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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r02-real-number-0100","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","language":"zh-Hant","source_mko_ids":["mko-real-number"],"title_zh":"有理點的有向偏移與距離","stem_zh":"有理點嵌入實數的有限切片；origin=12/2，有序points=[10/1, 11/2, 6/1, 20/3]。依序列 (各point−origin, 各偏移絕對值)，保留四欄次序。","answer":{"kind":"expression","value":"([4/1, -1/2, 0/1, 2/3], [4/1, 1/2, 0/1, 2/3])","explanation_zh":"有理點有向偏移與距離=([4/1, -1/2, 0/1, 2/3], [4/1, 1/2, 0/1, 2/3])。先精確相減保留方向，再取非負絕對值。非實數完備性或無理數分類證明，不用近似小數。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["有理點的有向偏移與距離"]},"variant":{"family_id":"qf-density14-real-point-offset-distances","seed":202610142100,"parameters":{"origin":[12,2],"points":[[10,1],[11,2],[6,1],[20,3]],"domain":"finite_rational_real_slice","operation":"density14_real_point_offset_distances","semantic_template":"qf-density14-real-point-offset-distances/[\"4/1\",\"-1/2\",\"0/1\",\"2/3\"]"}},"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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}