{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r06-path-connected-space-0001","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","language":"zh-Hant","source_mko_ids":["mko-path-connected-space"],"title_zh":"道路連通空間","stem_zh":"unit-interval path的continuity、兩端點與全range containment責任。操作=path-interval-linear；資料={\"left\":-2,\"right\":3,\"start\":0,\"end\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-2, 3, 0, 1, gamma(t)=0+(1)t, gamma(0)=start, gamma(1)=end, 是, convex_combination_stays_in_interval)","explanation_zh":"Wave6第十一層核對=(-2, 3, 0, 1, gamma(t)=0+(1)t, gamma(0)=start, gamma(1)=end, 是, convex_combination_stays_in_interval)；依完整明示資料獨立重算；linear／box／concatenated paths逐段exact核對；有限samples、graph walk或單一端點對不足；connected一般不推出path connected。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["道路連通空間"]},"variant":{"family_id":"qf-mod48-path-interval-linear","seed":202611176001,"parameters":{"left":-2,"right":3,"start":0,"end":1,"domain":"Exact_explicit_path_connectedness","operation":"path-interval-linear","semantic_template":"path-interval-linear|{end:[1,1],left:[-2,1],right:[3,1],start:[0,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r06-path-connected-space-0002","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","language":"zh-Hant","source_mko_ids":["mko-path-connected-space"],"title_zh":"道路連通空間","stem_zh":"unit-interval path的continuity、兩端點與全range containment責任。操作=path-interval-linear；資料={\"left\":-3,\"right\":4,\"start\":-1,\"end\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-3, 4, -1, 2, gamma(t)=-1+(3)t, gamma(0)=start, gamma(1)=end, 是, convex_combination_stays_in_interval)","explanation_zh":"Wave6第十一層核對=(-3, 4, -1, 2, gamma(t)=-1+(3)t, gamma(0)=start, gamma(1)=end, 是, convex_combination_stays_in_interval)；依完整明示資料獨立重算；linear／box／concatenated paths逐段exact核對；有限samples、graph walk或單一端點對不足；connected一般不推出path connected。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["道路連通空間"]},"variant":{"family_id":"qf-mod48-path-interval-linear","seed":202611176002,"parameters":{"left":-3,"right":4,"start":-1,"end":2,"domain":"Exact_explicit_path_connectedness","operation":"path-interval-linear","semantic_template":"path-interval-linear|{end:[2,1],left:[-3,1],right:[4,1],start:[-1,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r06-path-connected-space-0003","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","language":"zh-Hant","source_mko_ids":["mko-path-connected-space"],"title_zh":"道路連通空間","stem_zh":"unit-interval path的continuity、兩端點與全range containment責任。操作=path-interval-linear；資料={\"left\":-4,\"right\":5,\"start\":-2,\"end\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-4, 5, -2, 3, gamma(t)=-2+(5)t, gamma(0)=start, gamma(1)=end, 是, convex_combination_stays_in_interval)","explanation_zh":"Wave6第十一層核對=(-4, 5, -2, 3, gamma(t)=-2+(5)t, gamma(0)=start, gamma(1)=end, 是, convex_combination_stays_in_interval)；依完整明示資料獨立重算；linear／box／concatenated paths逐段exact核對；有限samples、graph walk或單一端點對不足；connected一般不推出path connected。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["道路連通空間"]},"variant":{"family_id":"qf-mod48-path-interval-linear","seed":202611176003,"parameters":{"left":-4,"right":5,"start":-2,"end":3,"domain":"Exact_explicit_path_connectedness","operation":"path-interval-linear","semantic_template":"path-interval-linear|{end:[3,1],left:[-4,1],right:[5,1],start:[-2,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r06-path-connected-space-0004","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","language":"zh-Hant","source_mko_ids":["mko-path-connected-space"],"title_zh":"道路連通空間","stem_zh":"unit-interval path的continuity、兩端點與全range containment責任。操作=path-interval-linear；資料={\"left\":-5,\"right\":6,\"start\":-3,\"end\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-5, 6, -3, 4, gamma(t)=-3+(7)t, gamma(0)=start, gamma(1)=end, 是, convex_combination_stays_in_interval)","explanation_zh":"Wave6第十一層核對=(-5, 6, -3, 4, gamma(t)=-3+(7)t, gamma(0)=start, gamma(1)=end, 是, convex_combination_stays_in_interval)；依完整明示資料獨立重算；linear／box／concatenated paths逐段exact核對；有限samples、graph walk或單一端點對不足；connected一般不推出path connected。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["道路連通空間"]},"variant":{"family_id":"qf-mod48-path-interval-linear","seed":202611176004,"parameters":{"left":-5,"right":6,"start":-3,"end":4,"domain":"Exact_explicit_path_connectedness","operation":"path-interval-linear","semantic_template":"path-interval-linear|{end:[4,1],left:[-5,1],right:[6,1],start:[-3,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r06-path-connected-space-0005","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","language":"zh-Hant","source_mko_ids":["mko-path-connected-space"],"title_zh":"道路連通空間","stem_zh":"unit-interval path的continuity、兩端點與全range containment責任。操作=path-interval-linear；資料={\"left\":-6,\"right\":7,\"start\":-4,\"end\":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-6, 7, -4, 5, gamma(t)=-4+(9)t, gamma(0)=start, gamma(1)=end, 是, convex_combination_stays_in_interval)","explanation_zh":"Wave6第十一層核對=(-6, 7, -4, 5, gamma(t)=-4+(9)t, gamma(0)=start, gamma(1)=end, 是, convex_combination_stays_in_interval)；依完整明示資料獨立重算；linear／box／concatenated paths逐段exact核對；有限samples、graph walk或單一端點對不足；connected一般不推出path connected。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["道路連通空間"]},"variant":{"family_id":"qf-mod48-path-interval-linear","seed":202611176005,"parameters":{"left":-6,"right":7,"start":-4,"end":5,"domain":"Exact_explicit_path_connectedness","operation":"path-interval-linear","semantic_template":"path-interval-linear|{end:[5,1],left:[-6,1],right:[7,1],start:[-4,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r06-path-connected-space-0006","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","language":"zh-Hant","source_mko_ids":["mko-path-connected-space"],"title_zh":"道路連通空間","stem_zh":"unit-interval path的continuity、兩端點與全range containment責任。操作=path-interval-linear；資料={\"left\":-7,\"right\":8,\"start\":-5,\"end\":6}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-7, 8, -5, 6, gamma(t)=-5+(11)t, gamma(0)=start, gamma(1)=end, 是, convex_combination_stays_in_interval)","explanation_zh":"Wave6第十一層核對=(-7, 8, -5, 6, gamma(t)=-5+(11)t, gamma(0)=start, gamma(1)=end, 是, convex_combination_stays_in_interval)；依完整明示資料獨立重算；linear／box／concatenated paths逐段exact核對；有限samples、graph walk或單一端點對不足；connected一般不推出path connected。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["道路連通空間"]},"variant":{"family_id":"qf-mod48-path-interval-linear","seed":202611176006,"parameters":{"left":-7,"right":8,"start":-5,"end":6,"domain":"Exact_explicit_path_connectedness","operation":"path-interval-linear","semantic_template":"path-interval-linear|{end:[6,1],left:[-7,1],right:[8,1],start:[-5,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r06-path-connected-space-0007","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","language":"zh-Hant","source_mko_ids":["mko-path-connected-space"],"title_zh":"道路連通空間","stem_zh":"unit-interval path的continuity、兩端點與全range containment責任。操作=path-interval-linear；資料={\"left\":-8,\"right\":9,\"start\":-6,\"end\":7}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-8, 9, -6, 7, gamma(t)=-6+(13)t, gamma(0)=start, gamma(1)=end, 是, convex_combination_stays_in_interval)","explanation_zh":"Wave6第十一層核對=(-8, 9, -6, 7, gamma(t)=-6+(13)t, gamma(0)=start, gamma(1)=end, 是, convex_combination_stays_in_interval)；依完整明示資料獨立重算；linear／box／concatenated paths逐段exact核對；有限samples、graph walk或單一端點對不足；connected一般不推出path connected。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["道路連通空間"]},"variant":{"family_id":"qf-mod48-path-interval-linear","seed":202611176007,"parameters":{"left":-8,"right":9,"start":-6,"end":7,"domain":"Exact_explicit_path_connectedness","operation":"path-interval-linear","semantic_template":"path-interval-linear|{end:[7,1],left:[-8,1],right:[9,1],start:[-6,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r06-path-connected-space-0008","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","language":"zh-Hant","source_mko_ids":["mko-path-connected-space"],"title_zh":"道路連通空間","stem_zh":"unit-interval path的continuity、兩端點與全range containment責任。操作=path-interval-linear；資料={\"left\":-9,\"right\":10,\"start\":-7,\"end\":8}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-9, 10, -7, 8, gamma(t)=-7+(15)t, gamma(0)=start, gamma(1)=end, 是, convex_combination_stays_in_interval)","explanation_zh":"Wave6第十一層核對=(-9, 10, -7, 8, gamma(t)=-7+(15)t, gamma(0)=start, gamma(1)=end, 是, convex_combination_stays_in_interval)；依完整明示資料獨立重算；linear／box／concatenated paths逐段exact核對；有限samples、graph walk或單一端點對不足；connected一般不推出path connected。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["道路連通空間"]},"variant":{"family_id":"qf-mod48-path-interval-linear","seed":202611176008,"parameters":{"left":-9,"right":10,"start":-7,"end":8,"domain":"Exact_explicit_path_connectedness","operation":"path-interval-linear","semantic_template":"path-interval-linear|{end:[8,1],left:[-9,1],right:[10,1],start:[-7,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r06-path-connected-space-0009","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","language":"zh-Hant","source_mko_ids":["mko-path-connected-space"],"title_zh":"道路連通空間","stem_zh":"unit-interval path的continuity、兩端點與全range containment責任。操作=path-interval-linear；資料={\"left\":-10,\"right\":11,\"start\":-8,\"end\":9}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-10, 11, -8, 9, gamma(t)=-8+(17)t, gamma(0)=start, gamma(1)=end, 是, convex_combination_stays_in_interval)","explanation_zh":"Wave6第十一層核對=(-10, 11, -8, 9, gamma(t)=-8+(17)t, gamma(0)=start, gamma(1)=end, 是, convex_combination_stays_in_interval)；依完整明示資料獨立重算；linear／box／concatenated paths逐段exact核對；有限samples、graph walk或單一端點對不足；connected一般不推出path connected。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["道路連通空間"]},"variant":{"family_id":"qf-mod48-path-interval-linear","seed":202611176009,"parameters":{"left":-10,"right":11,"start":-8,"end":9,"domain":"Exact_explicit_path_connectedness","operation":"path-interval-linear","semantic_template":"path-interval-linear|{end:[9,1],left:[-10,1],right:[11,1],start:[-8,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r06-path-connected-space-0010","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","language":"zh-Hant","source_mko_ids":["mko-path-connected-space"],"title_zh":"道路連通空間","stem_zh":"unit-interval path的continuity、兩端點與全range containment責任。操作=path-interval-linear；資料={\"left\":-11,\"right\":12,\"start\":-9,\"end\":10}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-11, 12, -9, 10, gamma(t)=-9+(19)t, gamma(0)=start, gamma(1)=end, 是, convex_combination_stays_in_interval)","explanation_zh":"Wave6第十一層核對=(-11, 12, -9, 10, gamma(t)=-9+(19)t, gamma(0)=start, gamma(1)=end, 是, convex_combination_stays_in_interval)；依完整明示資料獨立重算；linear／box／concatenated paths逐段exact核對；有限samples、graph walk或單一端點對不足；connected一般不推出path connected。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["道路連通空間"]},"variant":{"family_id":"qf-mod48-path-interval-linear","seed":202611176010,"parameters":{"left":-11,"right":12,"start":-9,"end":10,"domain":"Exact_explicit_path_connectedness","operation":"path-interval-linear","semantic_template":"path-interval-linear|{end:[10,1],left:[-11,1],right:[12,1],start:[-9,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r06-path-connected-space-0011","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","language":"zh-Hant","source_mko_ids":["mko-path-connected-space"],"title_zh":"道路連通空間","stem_zh":"unit-interval path的continuity、兩端點與全range containment責任。操作=path-interval-linear；資料={\"left\":-12,\"right\":13,\"start\":-10,\"end\":11}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-12, 13, -10, 11, gamma(t)=-10+(21)t, gamma(0)=start, gamma(1)=end, 是, convex_combination_stays_in_interval)","explanation_zh":"Wave6第十一層核對=(-12, 13, -10, 11, gamma(t)=-10+(21)t, gamma(0)=start, gamma(1)=end, 是, convex_combination_stays_in_interval)；依完整明示資料獨立重算；linear／box／concatenated paths逐段exact核對；有限samples、graph walk或單一端點對不足；connected一般不推出path connected。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["道路連通空間"]},"variant":{"family_id":"qf-mod48-path-interval-linear","seed":202611176011,"parameters":{"left":-12,"right":13,"start":-10,"end":11,"domain":"Exact_explicit_path_connectedness","operation":"path-interval-linear","semantic_template":"path-interval-linear|{end:[11,1],left:[-12,1],right:[13,1],start:[-10,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r06-path-connected-space-0012","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","language":"zh-Hant","source_mko_ids":["mko-path-connected-space"],"title_zh":"道路連通空間","stem_zh":"unit-interval path的continuity、兩端點與全range containment責任。操作=path-interval-linear；資料={\"left\":-13,\"right\":14,\"start\":-11,\"end\":12}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-13, 14, -11, 12, gamma(t)=-11+(23)t, gamma(0)=start, gamma(1)=end, 是, convex_combination_stays_in_interval)","explanation_zh":"Wave6第十一層核對=(-13, 14, -11, 12, gamma(t)=-11+(23)t, gamma(0)=start, gamma(1)=end, 是, convex_combination_stays_in_interval)；依完整明示資料獨立重算；linear／box／concatenated paths逐段exact核對；有限samples、graph walk或單一端點對不足；connected一般不推出path connected。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["道路連通空間"]},"variant":{"family_id":"qf-mod48-path-interval-linear","seed":202611176012,"parameters":{"left":-13,"right":14,"start":-11,"end":12,"domain":"Exact_explicit_path_connectedness","operation":"path-interval-linear","semantic_template":"path-interval-linear|{end:[12,1],left:[-13,1],right:[14,1],start:[-11,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r06-path-connected-space-0013","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","language":"zh-Hant","source_mko_ids":["mko-path-connected-space"],"title_zh":"道路連通空間","stem_zh":"unit-interval path的continuity、兩端點與全range containment責任。操作=path-interval-linear；資料={\"left\":-14,\"right\":15,\"start\":-12,\"end\":13}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-14, 15, -12, 13, gamma(t)=-12+(25)t, gamma(0)=start, gamma(1)=end, 是, convex_combination_stays_in_interval)","explanation_zh":"Wave6第十一層核對=(-14, 15, -12, 13, gamma(t)=-12+(25)t, gamma(0)=start, gamma(1)=end, 是, convex_combination_stays_in_interval)；依完整明示資料獨立重算；linear／box／concatenated paths逐段exact核對；有限samples、graph walk或單一端點對不足；connected一般不推出path connected。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["道路連通空間"]},"variant":{"family_id":"qf-mod48-path-interval-linear","seed":202611176013,"parameters":{"left":-14,"right":15,"start":-12,"end":13,"domain":"Exact_explicit_path_connectedness","operation":"path-interval-linear","semantic_template":"path-interval-linear|{end:[13,1],left:[-14,1],right:[15,1],start:[-12,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r06-path-connected-space-0014","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","language":"zh-Hant","source_mko_ids":["mko-path-connected-space"],"title_zh":"道路連通空間","stem_zh":"unit-interval path的continuity、兩端點與全range containment責任。操作=path-interval-linear；資料={\"left\":-15,\"right\":16,\"start\":-13,\"end\":14}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-15, 16, -13, 14, gamma(t)=-13+(27)t, gamma(0)=start, gamma(1)=end, 是, convex_combination_stays_in_interval)","explanation_zh":"Wave6第十一層核對=(-15, 16, -13, 14, gamma(t)=-13+(27)t, gamma(0)=start, gamma(1)=end, 是, convex_combination_stays_in_interval)；依完整明示資料獨立重算；linear／box／concatenated paths逐段exact核對；有限samples、graph walk或單一端點對不足；connected一般不推出path connected。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["道路連通空間"]},"variant":{"family_id":"qf-mod48-path-interval-linear","seed":202611176014,"parameters":{"left":-15,"right":16,"start":-13,"end":14,"domain":"Exact_explicit_path_connectedness","operation":"path-interval-linear","semantic_template":"path-interval-linear|{end:[14,1],left:[-15,1],right:[16,1],start:[-13,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r06-path-connected-space-0015","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","language":"zh-Hant","source_mko_ids":["mko-path-connected-space"],"title_zh":"道路連通空間","stem_zh":"unit-interval path的continuity、兩端點與全range containment責任。操作=path-interval-linear；資料={\"left\":-16,\"right\":17,\"start\":-14,\"end\":15}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-16, 17, -14, 15, gamma(t)=-14+(29)t, gamma(0)=start, gamma(1)=end, 是, convex_combination_stays_in_interval)","explanation_zh":"Wave6第十一層核對=(-16, 17, -14, 15, gamma(t)=-14+(29)t, gamma(0)=start, gamma(1)=end, 是, convex_combination_stays_in_interval)；依完整明示資料獨立重算；linear／box／concatenated paths逐段exact核對；有限samples、graph walk或單一端點對不足；connected一般不推出path connected。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["道路連通空間"]},"variant":{"family_id":"qf-mod48-path-interval-linear","seed":202611176015,"parameters":{"left":-16,"right":17,"start":-14,"end":15,"domain":"Exact_explicit_path_connectedness","operation":"path-interval-linear","semantic_template":"path-interval-linear|{end:[15,1],left:[-16,1],right:[17,1],start:[-14,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r06-path-connected-space-0016","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","language":"zh-Hant","source_mko_ids":["mko-path-connected-space"],"title_zh":"道路連通空間","stem_zh":"unit-interval path的continuity、兩端點與全range containment責任。操作=path-interval-linear；資料={\"left\":-17,\"right\":18,\"start\":-15,\"end\":16}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-17, 18, -15, 16, gamma(t)=-15+(31)t, gamma(0)=start, gamma(1)=end, 是, convex_combination_stays_in_interval)","explanation_zh":"Wave6第十一層核對=(-17, 18, -15, 16, gamma(t)=-15+(31)t, gamma(0)=start, gamma(1)=end, 是, convex_combination_stays_in_interval)；依完整明示資料獨立重算；linear／box／concatenated paths逐段exact核對；有限samples、graph walk或單一端點對不足；connected一般不推出path connected。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["道路連通空間"]},"variant":{"family_id":"qf-mod48-path-interval-linear","seed":202611176016,"parameters":{"left":-17,"right":18,"start":-15,"end":16,"domain":"Exact_explicit_path_connectedness","operation":"path-interval-linear","semantic_template":"path-interval-linear|{end:[16,1],left:[-17,1],right:[18,1],start:[-15,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r06-path-connected-space-0017","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","language":"zh-Hant","source_mko_ids":["mko-path-connected-space"],"title_zh":"道路連通空間","stem_zh":"unit-interval path的continuity、兩端點與全range containment責任。操作=path-interval-linear；資料={\"left\":-18,\"right\":19,\"start\":-16,\"end\":17}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-18, 19, -16, 17, gamma(t)=-16+(33)t, gamma(0)=start, gamma(1)=end, 是, convex_combination_stays_in_interval)","explanation_zh":"Wave6第十一層核對=(-18, 19, -16, 17, gamma(t)=-16+(33)t, gamma(0)=start, gamma(1)=end, 是, convex_combination_stays_in_interval)；依完整明示資料獨立重算；linear／box／concatenated paths逐段exact核對；有限samples、graph walk或單一端點對不足；connected一般不推出path connected。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["道路連通空間"]},"variant":{"family_id":"qf-mod48-path-interval-linear","seed":202611176017,"parameters":{"left":-18,"right":19,"start":-16,"end":17,"domain":"Exact_explicit_path_connectedness","operation":"path-interval-linear","semantic_template":"path-interval-linear|{end:[17,1],left:[-18,1],right:[19,1],start:[-16,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r06-path-connected-space-0018","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","language":"zh-Hant","source_mko_ids":["mko-path-connected-space"],"title_zh":"道路連通空間","stem_zh":"unit-interval path的continuity、兩端點與全range containment責任。操作=path-interval-linear；資料={\"left\":-19,\"right\":20,\"start\":-17,\"end\":18}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-19, 20, -17, 18, gamma(t)=-17+(35)t, gamma(0)=start, gamma(1)=end, 是, convex_combination_stays_in_interval)","explanation_zh":"Wave6第十一層核對=(-19, 20, -17, 18, gamma(t)=-17+(35)t, gamma(0)=start, gamma(1)=end, 是, convex_combination_stays_in_interval)；依完整明示資料獨立重算；linear／box／concatenated paths逐段exact核對；有限samples、graph walk或單一端點對不足；connected一般不推出path connected。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["道路連通空間"]},"variant":{"family_id":"qf-mod48-path-interval-linear","seed":202611176018,"parameters":{"left":-19,"right":20,"start":-17,"end":18,"domain":"Exact_explicit_path_connectedness","operation":"path-interval-linear","semantic_template":"path-interval-linear|{end:[18,1],left:[-19,1],right:[20,1],start:[-17,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r06-path-connected-space-0019","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","language":"zh-Hant","source_mko_ids":["mko-path-connected-space"],"title_zh":"道路連通空間","stem_zh":"unit-interval path的continuity、兩端點與全range containment責任。操作=path-interval-linear；資料={\"left\":-20,\"right\":21,\"start\":-18,\"end\":19}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-20, 21, -18, 19, gamma(t)=-18+(37)t, gamma(0)=start, gamma(1)=end, 是, convex_combination_stays_in_interval)","explanation_zh":"Wave6第十一層核對=(-20, 21, -18, 19, gamma(t)=-18+(37)t, gamma(0)=start, gamma(1)=end, 是, convex_combination_stays_in_interval)；依完整明示資料獨立重算；linear／box／concatenated paths逐段exact核對；有限samples、graph walk或單一端點對不足；connected一般不推出path connected。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["道路連通空間"]},"variant":{"family_id":"qf-mod48-path-interval-linear","seed":202611176019,"parameters":{"left":-20,"right":21,"start":-18,"end":19,"domain":"Exact_explicit_path_connectedness","operation":"path-interval-linear","semantic_template":"path-interval-linear|{end:[19,1],left:[-20,1],right:[21,1],start:[-18,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r06-path-connected-space-0020","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","language":"zh-Hant","source_mko_ids":["mko-path-connected-space"],"title_zh":"道路連通空間","stem_zh":"unit-interval path的continuity、兩端點與全range containment責任。操作=path-interval-linear；資料={\"left\":-21,\"right\":22,\"start\":-19,\"end\":20}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-21, 22, -19, 20, gamma(t)=-19+(39)t, gamma(0)=start, gamma(1)=end, 是, convex_combination_stays_in_interval)","explanation_zh":"Wave6第十一層核對=(-21, 22, -19, 20, gamma(t)=-19+(39)t, gamma(0)=start, gamma(1)=end, 是, convex_combination_stays_in_interval)；依完整明示資料獨立重算；linear／box／concatenated paths逐段exact核對；有限samples、graph walk或單一端點對不足；connected一般不推出path connected。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["道路連通空間"]},"variant":{"family_id":"qf-mod48-path-interval-linear","seed":202611176020,"parameters":{"left":-21,"right":22,"start":-19,"end":20,"domain":"Exact_explicit_path_connectedness","operation":"path-interval-linear","semantic_template":"path-interval-linear|{end:[20,1],left:[-21,1],right:[22,1],start:[-19,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r06-path-connected-space-0021","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","language":"zh-Hant","source_mko_ids":["mko-path-connected-space"],"title_zh":"道路連通空間","stem_zh":"unit-interval path的continuity、兩端點與全range containment責任。操作=path-interval-linear；資料={\"left\":-22,\"right\":23,\"start\":-20,\"end\":21}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-22, 23, -20, 21, gamma(t)=-20+(41)t, gamma(0)=start, gamma(1)=end, 是, convex_combination_stays_in_interval)","explanation_zh":"Wave6第十一層核對=(-22, 23, -20, 21, gamma(t)=-20+(41)t, gamma(0)=start, gamma(1)=end, 是, convex_combination_stays_in_interval)；依完整明示資料獨立重算；linear／box／concatenated paths逐段exact核對；有限samples、graph walk或單一端點對不足；connected一般不推出path connected。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["道路連通空間"]},"variant":{"family_id":"qf-mod48-path-interval-linear","seed":202611176021,"parameters":{"left":-22,"right":23,"start":-20,"end":21,"domain":"Exact_explicit_path_connectedness","operation":"path-interval-linear","semantic_template":"path-interval-linear|{end:[21,1],left:[-22,1],right:[23,1],start:[-20,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r06-path-connected-space-0022","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","language":"zh-Hant","source_mko_ids":["mko-path-connected-space"],"title_zh":"道路連通空間","stem_zh":"unit-interval path的continuity、兩端點與全range containment責任。操作=path-interval-linear；資料={\"left\":-23,\"right\":24,\"start\":-21,\"end\":22}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-23, 24, -21, 22, gamma(t)=-21+(43)t, gamma(0)=start, gamma(1)=end, 是, convex_combination_stays_in_interval)","explanation_zh":"Wave6第十一層核對=(-23, 24, -21, 22, gamma(t)=-21+(43)t, gamma(0)=start, gamma(1)=end, 是, convex_combination_stays_in_interval)；依完整明示資料獨立重算；linear／box／concatenated paths逐段exact核對；有限samples、graph walk或單一端點對不足；connected一般不推出path connected。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["道路連通空間"]},"variant":{"family_id":"qf-mod48-path-interval-linear","seed":202611176022,"parameters":{"left":-23,"right":24,"start":-21,"end":22,"domain":"Exact_explicit_path_connectedness","operation":"path-interval-linear","semantic_template":"path-interval-linear|{end:[22,1],left:[-23,1],right:[24,1],start:[-21,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r06-path-connected-space-0023","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","language":"zh-Hant","source_mko_ids":["mko-path-connected-space"],"title_zh":"道路連通空間","stem_zh":"unit-interval path的continuity、兩端點與全range containment責任。操作=path-interval-linear；資料={\"left\":-24,\"right\":25,\"start\":-22,\"end\":23}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-24, 25, -22, 23, gamma(t)=-22+(45)t, gamma(0)=start, gamma(1)=end, 是, convex_combination_stays_in_interval)","explanation_zh":"Wave6第十一層核對=(-24, 25, -22, 23, gamma(t)=-22+(45)t, gamma(0)=start, gamma(1)=end, 是, convex_combination_stays_in_interval)；依完整明示資料獨立重算；linear／box／concatenated paths逐段exact核對；有限samples、graph walk或單一端點對不足；connected一般不推出path connected。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["道路連通空間"]},"variant":{"family_id":"qf-mod48-path-interval-linear","seed":202611176023,"parameters":{"left":-24,"right":25,"start":-22,"end":23,"domain":"Exact_explicit_path_connectedness","operation":"path-interval-linear","semantic_template":"path-interval-linear|{end:[23,1],left:[-24,1],right:[25,1],start:[-22,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r06-path-connected-space-0024","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","language":"zh-Hant","source_mko_ids":["mko-path-connected-space"],"title_zh":"道路連通空間","stem_zh":"unit-interval path的continuity、兩端點與全range containment責任。操作=path-interval-linear；資料={\"left\":-25,\"right\":26,\"start\":-23,\"end\":24}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-25, 26, -23, 24, gamma(t)=-23+(47)t, gamma(0)=start, gamma(1)=end, 是, convex_combination_stays_in_interval)","explanation_zh":"Wave6第十一層核對=(-25, 26, -23, 24, gamma(t)=-23+(47)t, gamma(0)=start, gamma(1)=end, 是, convex_combination_stays_in_interval)；依完整明示資料獨立重算；linear／box／concatenated paths逐段exact核對；有限samples、graph walk或單一端點對不足；connected一般不推出path connected。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["道路連通空間"]},"variant":{"family_id":"qf-mod48-path-interval-linear","seed":202611176024,"parameters":{"left":-25,"right":26,"start":-23,"end":24,"domain":"Exact_explicit_path_connectedness","operation":"path-interval-linear","semantic_template":"path-interval-linear|{end:[24,1],left:[-25,1],right:[26,1],start:[-23,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r06-path-connected-space-0025","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","language":"zh-Hant","source_mko_ids":["mko-path-connected-space"],"title_zh":"道路連通空間","stem_zh":"unit-interval path的continuity、兩端點與全range containment責任。操作=path-interval-linear；資料={\"left\":-26,\"right\":27,\"start\":-24,\"end\":25}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-26, 27, -24, 25, gamma(t)=-24+(49)t, gamma(0)=start, gamma(1)=end, 是, convex_combination_stays_in_interval)","explanation_zh":"Wave6第十一層核對=(-26, 27, -24, 25, gamma(t)=-24+(49)t, gamma(0)=start, gamma(1)=end, 是, convex_combination_stays_in_interval)；依完整明示資料獨立重算；linear／box／concatenated paths逐段exact核對；有限samples、graph walk或單一端點對不足；connected一般不推出path connected。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["道路連通空間"]},"variant":{"family_id":"qf-mod48-path-interval-linear","seed":202611176025,"parameters":{"left":-26,"right":27,"start":-24,"end":25,"domain":"Exact_explicit_path_connectedness","operation":"path-interval-linear","semantic_template":"path-interval-linear|{end:[25,1],left:[-26,1],right:[27,1],start:[-24,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r06-path-connected-space-0026","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","language":"zh-Hant","source_mko_ids":["mko-path-connected-space"],"title_zh":"道路連通空間","stem_zh":"unit-interval path的continuity、兩端點與全range containment責任。操作=path-box-linear；資料={\"lower\":[-2,-3],\"upper\":[3,4],\"start\":[-1,-2],\"end\":[2,3]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([-2, -3], [3, 4], [-1, -2], [2, 3], [-1+(3)t, -2+(5)t], 是, coordinatewise_convex_combination_stays_in_box)","explanation_zh":"Wave6第十一層核對=([-2, -3], [3, 4], [-1, -2], [2, 3], [-1+(3)t, -2+(5)t], 是, coordinatewise_convex_combination_stays_in_box)；依完整明示資料獨立重算；linear／box／concatenated paths逐段exact核對；有限samples、graph walk或單一端點對不足；connected一般不推出path connected。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["道路連通空間"]},"variant":{"family_id":"qf-mod48-path-box-linear","seed":202611176026,"parameters":{"lower":[-2,-3],"upper":[3,4],"start":[-1,-2],"end":[2,3],"domain":"Exact_explicit_path_connectedness","operation":"path-box-linear","semantic_template":"path-box-linear|{end:[[2,1],[3,1]],lower:[[-2,1],[-3,1]],start:[[-1,1],[-2,1]],upper:[[3,1],[4,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r06-path-connected-space-0027","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","language":"zh-Hant","source_mko_ids":["mko-path-connected-space"],"title_zh":"道路連通空間","stem_zh":"unit-interval path的continuity、兩端點與全range containment責任。操作=path-box-linear；資料={\"lower\":[-3,-4,-5],\"upper\":[4,5,6],\"start\":[-2,-3,-4],\"end\":[3,4,5]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([-3, -4, -5], [4, 5, 6], [-2, -3, -4], [3, 4, 5], [-2+(5)t, -3+(7)t, -4+(9)t], 是, coordinatewise_convex_combination_stays_in_box)","explanation_zh":"Wave6第十一層核對=([-3, -4, -5], [4, 5, 6], [-2, -3, -4], [3, 4, 5], [-2+(5)t, -3+(7)t, -4+(9)t], 是, coordinatewise_convex_combination_stays_in_box)；依完整明示資料獨立重算；linear／box／concatenated paths逐段exact核對；有限samples、graph walk或單一端點對不足；connected一般不推出path connected。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["道路連通空間"]},"variant":{"family_id":"qf-mod48-path-box-linear","seed":202611176027,"parameters":{"lower":[-3,-4,-5],"upper":[4,5,6],"start":[-2,-3,-4],"end":[3,4,5],"domain":"Exact_explicit_path_connectedness","operation":"path-box-linear","semantic_template":"path-box-linear|{end:[[3,1],[4,1],[5,1]],lower:[[-3,1],[-4,1],[-5,1]],start:[[-2,1],[-3,1],[-4,1]],upper:[[4,1],[5,1],[6,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r06-path-connected-space-0028","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","language":"zh-Hant","source_mko_ids":["mko-path-connected-space"],"title_zh":"道路連通空間","stem_zh":"unit-interval path的continuity、兩端點與全range containment責任。操作=path-box-linear；資料={\"lower\":[-4,-5,-6,-7],\"upper\":[5,6,7,8],\"start\":[-3,-4,-5,-6],\"end\":[4,5,6,7]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([-4, -5, -6, -7], [5, 6, 7, 8], [-3, -4, -5, -6], [4, 5, 6, 7], [-3+(7)t, -4+(9)t, -5+(11)t, -6+(13)t], 是, coordinatewise_convex_combination_stays_in_box)","explanation_zh":"Wave6第十一層核對=([-4, -5, -6, -7], [5, 6, 7, 8], [-3, -4, -5, -6], [4, 5, 6, 7], [-3+(7)t, -4+(9)t, -5+(11)t, -6+(13)t], 是, coordinatewise_convex_combination_stays_in_box)；依完整明示資料獨立重算；linear／box／concatenated paths逐段exact核對；有限samples、graph walk或單一端點對不足；connected一般不推出path connected。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["道路連通空間"]},"variant":{"family_id":"qf-mod48-path-box-linear","seed":202611176028,"parameters":{"lower":[-4,-5,-6,-7],"upper":[5,6,7,8],"start":[-3,-4,-5,-6],"end":[4,5,6,7],"domain":"Exact_explicit_path_connectedness","operation":"path-box-linear","semantic_template":"path-box-linear|{end:[[4,1],[5,1],[6,1],[7,1]],lower:[[-4,1],[-5,1],[-6,1],[-7,1]],start:[[-3,1],[-4,1],[-5,1],[-6,1]],upper:[[5,1],[6,1],[7,1],[8,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r06-path-connected-space-0029","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","language":"zh-Hant","source_mko_ids":["mko-path-connected-space"],"title_zh":"道路連通空間","stem_zh":"unit-interval path的continuity、兩端點與全range containment責任。操作=path-box-linear；資料={\"lower\":[-5,-6],\"upper\":[6,7],\"start\":[-4,-5],\"end\":[5,6]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([-5, -6], [6, 7], [-4, -5], [5, 6], [-4+(9)t, -5+(11)t], 是, coordinatewise_convex_combination_stays_in_box)","explanation_zh":"Wave6第十一層核對=([-5, -6], [6, 7], [-4, -5], [5, 6], [-4+(9)t, -5+(11)t], 是, coordinatewise_convex_combination_stays_in_box)；依完整明示資料獨立重算；linear／box／concatenated paths逐段exact核對；有限samples、graph walk或單一端點對不足；connected一般不推出path connected。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["道路連通空間"]},"variant":{"family_id":"qf-mod48-path-box-linear","seed":202611176029,"parameters":{"lower":[-5,-6],"upper":[6,7],"start":[-4,-5],"end":[5,6],"domain":"Exact_explicit_path_connectedness","operation":"path-box-linear","semantic_template":"path-box-linear|{end:[[5,1],[6,1]],lower:[[-5,1],[-6,1]],start:[[-4,1],[-5,1]],upper:[[6,1],[7,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r06-path-connected-space-0030","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","language":"zh-Hant","source_mko_ids":["mko-path-connected-space"],"title_zh":"道路連通空間","stem_zh":"unit-interval path的continuity、兩端點與全range containment責任。操作=path-box-linear；資料={\"lower\":[-6,-7,-8],\"upper\":[7,8,9],\"start\":[-5,-6,-7],\"end\":[6,7,8]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([-6, -7, -8], [7, 8, 9], [-5, -6, -7], [6, 7, 8], [-5+(11)t, -6+(13)t, -7+(15)t], 是, coordinatewise_convex_combination_stays_in_box)","explanation_zh":"Wave6第十一層核對=([-6, -7, -8], [7, 8, 9], [-5, -6, -7], [6, 7, 8], [-5+(11)t, -6+(13)t, -7+(15)t], 是, coordinatewise_convex_combination_stays_in_box)；依完整明示資料獨立重算；linear／box／concatenated paths逐段exact核對；有限samples、graph walk或單一端點對不足；connected一般不推出path connected。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["道路連通空間"]},"variant":{"family_id":"qf-mod48-path-box-linear","seed":202611176030,"parameters":{"lower":[-6,-7,-8],"upper":[7,8,9],"start":[-5,-6,-7],"end":[6,7,8],"domain":"Exact_explicit_path_connectedness","operation":"path-box-linear","semantic_template":"path-box-linear|{end:[[6,1],[7,1],[8,1]],lower:[[-6,1],[-7,1],[-8,1]],start:[[-5,1],[-6,1],[-7,1]],upper:[[7,1],[8,1],[9,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r06-path-connected-space-0031","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","language":"zh-Hant","source_mko_ids":["mko-path-connected-space"],"title_zh":"道路連通空間","stem_zh":"unit-interval path的continuity、兩端點與全range containment責任。操作=path-box-linear；資料={\"lower\":[-7,-8,-9,-10],\"upper\":[8,9,10,11],\"start\":[-6,-7,-8,-9],\"end\":[7,8,9,10]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([-7, -8, -9, -10], [8, 9, 10, 11], [-6, -7, -8, -9], [7, 8, 9, 10], [-6+(13)t, -7+(15)t, -8+(17)t, -9+(19)t], 是, coordinatewise_convex_combination_stays_in_box)","explanation_zh":"Wave6第十一層核對=([-7, -8, -9, -10], [8, 9, 10, 11], [-6, -7, -8, -9], [7, 8, 9, 10], [-6+(13)t, -7+(15)t, -8+(17)t, -9+(19)t], 是, coordinatewise_convex_combination_stays_in_box)；依完整明示資料獨立重算；linear／box／concatenated paths逐段exact核對；有限samples、graph walk或單一端點對不足；connected一般不推出path connected。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["道路連通空間"]},"variant":{"family_id":"qf-mod48-path-box-linear","seed":202611176031,"parameters":{"lower":[-7,-8,-9,-10],"upper":[8,9,10,11],"start":[-6,-7,-8,-9],"end":[7,8,9,10],"domain":"Exact_explicit_path_connectedness","operation":"path-box-linear","semantic_template":"path-box-linear|{end:[[7,1],[8,1],[9,1],[10,1]],lower:[[-7,1],[-8,1],[-9,1],[-10,1]],start:[[-6,1],[-7,1],[-8,1],[-9,1]],upper:[[8,1],[9,1],[10,1],[11,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r06-path-connected-space-0032","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","language":"zh-Hant","source_mko_ids":["mko-path-connected-space"],"title_zh":"道路連通空間","stem_zh":"unit-interval path的continuity、兩端點與全range containment責任。操作=path-box-linear；資料={\"lower\":[-8,-9],\"upper\":[9,10],\"start\":[-7,-8],\"end\":[8,9]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([-8, -9], [9, 10], [-7, -8], [8, 9], [-7+(15)t, -8+(17)t], 是, coordinatewise_convex_combination_stays_in_box)","explanation_zh":"Wave6第十一層核對=([-8, -9], [9, 10], [-7, -8], [8, 9], [-7+(15)t, -8+(17)t], 是, coordinatewise_convex_combination_stays_in_box)；依完整明示資料獨立重算；linear／box／concatenated paths逐段exact核對；有限samples、graph walk或單一端點對不足；connected一般不推出path connected。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["道路連通空間"]},"variant":{"family_id":"qf-mod48-path-box-linear","seed":202611176032,"parameters":{"lower":[-8,-9],"upper":[9,10],"start":[-7,-8],"end":[8,9],"domain":"Exact_explicit_path_connectedness","operation":"path-box-linear","semantic_template":"path-box-linear|{end:[[8,1],[9,1]],lower:[[-8,1],[-9,1]],start:[[-7,1],[-8,1]],upper:[[9,1],[10,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r06-path-connected-space-0033","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","language":"zh-Hant","source_mko_ids":["mko-path-connected-space"],"title_zh":"道路連通空間","stem_zh":"unit-interval path的continuity、兩端點與全range containment責任。操作=path-box-linear；資料={\"lower\":[-9,-10,-11],\"upper\":[10,11,12],\"start\":[-8,-9,-10],\"end\":[9,10,11]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([-9, -10, -11], [10, 11, 12], [-8, -9, -10], [9, 10, 11], [-8+(17)t, -9+(19)t, -10+(21)t], 是, coordinatewise_convex_combination_stays_in_box)","explanation_zh":"Wave6第十一層核對=([-9, -10, -11], [10, 11, 12], [-8, -9, -10], [9, 10, 11], [-8+(17)t, -9+(19)t, -10+(21)t], 是, coordinatewise_convex_combination_stays_in_box)；依完整明示資料獨立重算；linear／box／concatenated paths逐段exact核對；有限samples、graph walk或單一端點對不足；connected一般不推出path connected。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["道路連通空間"]},"variant":{"family_id":"qf-mod48-path-box-linear","seed":202611176033,"parameters":{"lower":[-9,-10,-11],"upper":[10,11,12],"start":[-8,-9,-10],"end":[9,10,11],"domain":"Exact_explicit_path_connectedness","operation":"path-box-linear","semantic_template":"path-box-linear|{end:[[9,1],[10,1],[11,1]],lower:[[-9,1],[-10,1],[-11,1]],start:[[-8,1],[-9,1],[-10,1]],upper:[[10,1],[11,1],[12,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r06-path-connected-space-0034","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","language":"zh-Hant","source_mko_ids":["mko-path-connected-space"],"title_zh":"道路連通空間","stem_zh":"unit-interval path的continuity、兩端點與全range containment責任。操作=path-box-linear；資料={\"lower\":[-10,-11,-12,-13],\"upper\":[11,12,13,14],\"start\":[-9,-10,-11,-12],\"end\":[10,11,12,13]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([-10, -11, -12, -13], [11, 12, 13, 14], [-9, -10, -11, -12], [10, 11, 12, 13], [-9+(19)t, -10+(21)t, -11+(23)t, -12+(25)t], 是, coordinatewise_convex_combination_stays_in_box)","explanation_zh":"Wave6第十一層核對=([-10, -11, -12, -13], [11, 12, 13, 14], [-9, -10, -11, -12], [10, 11, 12, 13], [-9+(19)t, -10+(21)t, -11+(23)t, -12+(25)t], 是, coordinatewise_convex_combination_stays_in_box)；依完整明示資料獨立重算；linear／box／concatenated paths逐段exact核對；有限samples、graph walk或單一端點對不足；connected一般不推出path connected。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["道路連通空間"]},"variant":{"family_id":"qf-mod48-path-box-linear","seed":202611176034,"parameters":{"lower":[-10,-11,-12,-13],"upper":[11,12,13,14],"start":[-9,-10,-11,-12],"end":[10,11,12,13],"domain":"Exact_explicit_path_connectedness","operation":"path-box-linear","semantic_template":"path-box-linear|{end:[[10,1],[11,1],[12,1],[13,1]],lower:[[-10,1],[-11,1],[-12,1],[-13,1]],start:[[-9,1],[-10,1],[-11,1],[-12,1]],upper:[[11,1],[12,1],[13,1],[14,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r06-path-connected-space-0035","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","language":"zh-Hant","source_mko_ids":["mko-path-connected-space"],"title_zh":"道路連通空間","stem_zh":"unit-interval path的continuity、兩端點與全range containment責任。操作=path-box-linear；資料={\"lower\":[-11,-12],\"upper\":[12,13],\"start\":[-10,-11],\"end\":[11,12]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([-11, -12], [12, 13], [-10, -11], [11, 12], [-10+(21)t, -11+(23)t], 是, coordinatewise_convex_combination_stays_in_box)","explanation_zh":"Wave6第十一層核對=([-11, -12], [12, 13], [-10, -11], [11, 12], [-10+(21)t, -11+(23)t], 是, coordinatewise_convex_combination_stays_in_box)；依完整明示資料獨立重算；linear／box／concatenated paths逐段exact核對；有限samples、graph walk或單一端點對不足；connected一般不推出path connected。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["道路連通空間"]},"variant":{"family_id":"qf-mod48-path-box-linear","seed":202611176035,"parameters":{"lower":[-11,-12],"upper":[12,13],"start":[-10,-11],"end":[11,12],"domain":"Exact_explicit_path_connectedness","operation":"path-box-linear","semantic_template":"path-box-linear|{end:[[11,1],[12,1]],lower:[[-11,1],[-12,1]],start:[[-10,1],[-11,1]],upper:[[12,1],[13,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r06-path-connected-space-0036","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","language":"zh-Hant","source_mko_ids":["mko-path-connected-space"],"title_zh":"道路連通空間","stem_zh":"unit-interval path的continuity、兩端點與全range containment責任。操作=path-box-linear；資料={\"lower\":[-12,-13,-14],\"upper\":[13,14,15],\"start\":[-11,-12,-13],\"end\":[12,13,14]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([-12, -13, -14], [13, 14, 15], [-11, -12, -13], [12, 13, 14], [-11+(23)t, -12+(25)t, -13+(27)t], 是, coordinatewise_convex_combination_stays_in_box)","explanation_zh":"Wave6第十一層核對=([-12, -13, -14], [13, 14, 15], [-11, -12, -13], [12, 13, 14], [-11+(23)t, -12+(25)t, -13+(27)t], 是, coordinatewise_convex_combination_stays_in_box)；依完整明示資料獨立重算；linear／box／concatenated paths逐段exact核對；有限samples、graph walk或單一端點對不足；connected一般不推出path connected。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["道路連通空間"]},"variant":{"family_id":"qf-mod48-path-box-linear","seed":202611176036,"parameters":{"lower":[-12,-13,-14],"upper":[13,14,15],"start":[-11,-12,-13],"end":[12,13,14],"domain":"Exact_explicit_path_connectedness","operation":"path-box-linear","semantic_template":"path-box-linear|{end:[[12,1],[13,1],[14,1]],lower:[[-12,1],[-13,1],[-14,1]],start:[[-11,1],[-12,1],[-13,1]],upper:[[13,1],[14,1],[15,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r06-path-connected-space-0037","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","language":"zh-Hant","source_mko_ids":["mko-path-connected-space"],"title_zh":"道路連通空間","stem_zh":"unit-interval path的continuity、兩端點與全range containment責任。操作=path-box-linear；資料={\"lower\":[-13,-14,-15,-16],\"upper\":[14,15,16,17],\"start\":[-12,-13,-14,-15],\"end\":[13,14,15,16]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([-13, -14, -15, -16], [14, 15, 16, 17], [-12, -13, -14, -15], [13, 14, 15, 16], [-12+(25)t, -13+(27)t, -14+(29)t, -15+(31)t], 是, coordinatewise_convex_combination_stays_in_box)","explanation_zh":"Wave6第十一層核對=([-13, -14, -15, -16], [14, 15, 16, 17], [-12, -13, -14, -15], [13, 14, 15, 16], [-12+(25)t, -13+(27)t, -14+(29)t, -15+(31)t], 是, coordinatewise_convex_combination_stays_in_box)；依完整明示資料獨立重算；linear／box／concatenated paths逐段exact核對；有限samples、graph walk或單一端點對不足；connected一般不推出path connected。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["道路連通空間"]},"variant":{"family_id":"qf-mod48-path-box-linear","seed":202611176037,"parameters":{"lower":[-13,-14,-15,-16],"upper":[14,15,16,17],"start":[-12,-13,-14,-15],"end":[13,14,15,16],"domain":"Exact_explicit_path_connectedness","operation":"path-box-linear","semantic_template":"path-box-linear|{end:[[13,1],[14,1],[15,1],[16,1]],lower:[[-13,1],[-14,1],[-15,1],[-16,1]],start:[[-12,1],[-13,1],[-14,1],[-15,1]],upper:[[14,1],[15,1],[16,1],[17,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r06-path-connected-space-0038","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","language":"zh-Hant","source_mko_ids":["mko-path-connected-space"],"title_zh":"道路連通空間","stem_zh":"unit-interval path的continuity、兩端點與全range containment責任。操作=path-box-linear；資料={\"lower\":[-14,-15],\"upper\":[15,16],\"start\":[-13,-14],\"end\":[14,15]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([-14, -15], [15, 16], [-13, -14], [14, 15], [-13+(27)t, -14+(29)t], 是, coordinatewise_convex_combination_stays_in_box)","explanation_zh":"Wave6第十一層核對=([-14, -15], [15, 16], [-13, -14], [14, 15], [-13+(27)t, -14+(29)t], 是, coordinatewise_convex_combination_stays_in_box)；依完整明示資料獨立重算；linear／box／concatenated paths逐段exact核對；有限samples、graph walk或單一端點對不足；connected一般不推出path connected。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["道路連通空間"]},"variant":{"family_id":"qf-mod48-path-box-linear","seed":202611176038,"parameters":{"lower":[-14,-15],"upper":[15,16],"start":[-13,-14],"end":[14,15],"domain":"Exact_explicit_path_connectedness","operation":"path-box-linear","semantic_template":"path-box-linear|{end:[[14,1],[15,1]],lower:[[-14,1],[-15,1]],start:[[-13,1],[-14,1]],upper:[[15,1],[16,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r06-path-connected-space-0039","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","language":"zh-Hant","source_mko_ids":["mko-path-connected-space"],"title_zh":"道路連通空間","stem_zh":"unit-interval path的continuity、兩端點與全range containment責任。操作=path-box-linear；資料={\"lower\":[-15,-16,-17],\"upper\":[16,17,18],\"start\":[-14,-15,-16],\"end\":[15,16,17]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([-15, -16, -17], [16, 17, 18], [-14, -15, -16], [15, 16, 17], [-14+(29)t, -15+(31)t, -16+(33)t], 是, coordinatewise_convex_combination_stays_in_box)","explanation_zh":"Wave6第十一層核對=([-15, -16, -17], [16, 17, 18], [-14, -15, -16], [15, 16, 17], [-14+(29)t, -15+(31)t, -16+(33)t], 是, coordinatewise_convex_combination_stays_in_box)；依完整明示資料獨立重算；linear／box／concatenated paths逐段exact核對；有限samples、graph walk或單一端點對不足；connected一般不推出path connected。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["道路連通空間"]},"variant":{"family_id":"qf-mod48-path-box-linear","seed":202611176039,"parameters":{"lower":[-15,-16,-17],"upper":[16,17,18],"start":[-14,-15,-16],"end":[15,16,17],"domain":"Exact_explicit_path_connectedness","operation":"path-box-linear","semantic_template":"path-box-linear|{end:[[15,1],[16,1],[17,1]],lower:[[-15,1],[-16,1],[-17,1]],start:[[-14,1],[-15,1],[-16,1]],upper:[[16,1],[17,1],[18,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r06-path-connected-space-0040","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","language":"zh-Hant","source_mko_ids":["mko-path-connected-space"],"title_zh":"道路連通空間","stem_zh":"unit-interval path的continuity、兩端點與全range containment責任。操作=path-box-linear；資料={\"lower\":[-16,-17,-18,-19],\"upper\":[17,18,19,20],\"start\":[-15,-16,-17,-18],\"end\":[16,17,18,19]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([-16, -17, -18, -19], [17, 18, 19, 20], [-15, -16, -17, -18], [16, 17, 18, 19], [-15+(31)t, -16+(33)t, -17+(35)t, -18+(37)t], 是, coordinatewise_convex_combination_stays_in_box)","explanation_zh":"Wave6第十一層核對=([-16, -17, -18, -19], [17, 18, 19, 20], [-15, -16, -17, -18], [16, 17, 18, 19], [-15+(31)t, -16+(33)t, -17+(35)t, -18+(37)t], 是, coordinatewise_convex_combination_stays_in_box)；依完整明示資料獨立重算；linear／box／concatenated paths逐段exact核對；有限samples、graph walk或單一端點對不足；connected一般不推出path connected。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["道路連通空間"]},"variant":{"family_id":"qf-mod48-path-box-linear","seed":202611176040,"parameters":{"lower":[-16,-17,-18,-19],"upper":[17,18,19,20],"start":[-15,-16,-17,-18],"end":[16,17,18,19],"domain":"Exact_explicit_path_connectedness","operation":"path-box-linear","semantic_template":"path-box-linear|{end:[[16,1],[17,1],[18,1],[19,1]],lower:[[-16,1],[-17,1],[-18,1],[-19,1]],start:[[-15,1],[-16,1],[-17,1],[-18,1]],upper:[[17,1],[18,1],[19,1],[20,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r06-path-connected-space-0041","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","language":"zh-Hant","source_mko_ids":["mko-path-connected-space"],"title_zh":"道路連通空間","stem_zh":"unit-interval path的continuity、兩端點與全range containment責任。操作=path-box-linear；資料={\"lower\":[-17,-18],\"upper\":[18,19],\"start\":[-16,-17],\"end\":[17,18]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([-17, -18], [18, 19], [-16, -17], [17, 18], [-16+(33)t, -17+(35)t], 是, coordinatewise_convex_combination_stays_in_box)","explanation_zh":"Wave6第十一層核對=([-17, -18], [18, 19], [-16, -17], [17, 18], [-16+(33)t, -17+(35)t], 是, coordinatewise_convex_combination_stays_in_box)；依完整明示資料獨立重算；linear／box／concatenated paths逐段exact核對；有限samples、graph walk或單一端點對不足；connected一般不推出path connected。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["道路連通空間"]},"variant":{"family_id":"qf-mod48-path-box-linear","seed":202611176041,"parameters":{"lower":[-17,-18],"upper":[18,19],"start":[-16,-17],"end":[17,18],"domain":"Exact_explicit_path_connectedness","operation":"path-box-linear","semantic_template":"path-box-linear|{end:[[17,1],[18,1]],lower:[[-17,1],[-18,1]],start:[[-16,1],[-17,1]],upper:[[18,1],[19,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r06-path-connected-space-0042","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","language":"zh-Hant","source_mko_ids":["mko-path-connected-space"],"title_zh":"道路連通空間","stem_zh":"unit-interval path的continuity、兩端點與全range containment責任。操作=path-box-linear；資料={\"lower\":[-18,-19,-20],\"upper\":[19,20,21],\"start\":[-17,-18,-19],\"end\":[18,19,20]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([-18, -19, -20], [19, 20, 21], [-17, -18, -19], [18, 19, 20], [-17+(35)t, -18+(37)t, -19+(39)t], 是, coordinatewise_convex_combination_stays_in_box)","explanation_zh":"Wave6第十一層核對=([-18, -19, -20], [19, 20, 21], [-17, -18, -19], [18, 19, 20], [-17+(35)t, -18+(37)t, -19+(39)t], 是, coordinatewise_convex_combination_stays_in_box)；依完整明示資料獨立重算；linear／box／concatenated paths逐段exact核對；有限samples、graph walk或單一端點對不足；connected一般不推出path connected。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["道路連通空間"]},"variant":{"family_id":"qf-mod48-path-box-linear","seed":202611176042,"parameters":{"lower":[-18,-19,-20],"upper":[19,20,21],"start":[-17,-18,-19],"end":[18,19,20],"domain":"Exact_explicit_path_connectedness","operation":"path-box-linear","semantic_template":"path-box-linear|{end:[[18,1],[19,1],[20,1]],lower:[[-18,1],[-19,1],[-20,1]],start:[[-17,1],[-18,1],[-19,1]],upper:[[19,1],[20,1],[21,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r06-path-connected-space-0043","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","language":"zh-Hant","source_mko_ids":["mko-path-connected-space"],"title_zh":"道路連通空間","stem_zh":"unit-interval path的continuity、兩端點與全range containment責任。操作=path-box-linear；資料={\"lower\":[-19,-20,-21,-22],\"upper\":[20,21,22,23],\"start\":[-18,-19,-20,-21],\"end\":[19,20,21,22]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([-19, -20, -21, -22], [20, 21, 22, 23], [-18, -19, -20, -21], [19, 20, 21, 22], [-18+(37)t, -19+(39)t, -20+(41)t, -21+(43)t], 是, coordinatewise_convex_combination_stays_in_box)","explanation_zh":"Wave6第十一層核對=([-19, -20, -21, -22], [20, 21, 22, 23], [-18, -19, -20, -21], [19, 20, 21, 22], [-18+(37)t, -19+(39)t, -20+(41)t, -21+(43)t], 是, coordinatewise_convex_combination_stays_in_box)；依完整明示資料獨立重算；linear／box／concatenated paths逐段exact核對；有限samples、graph walk或單一端點對不足；connected一般不推出path connected。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["道路連通空間"]},"variant":{"family_id":"qf-mod48-path-box-linear","seed":202611176043,"parameters":{"lower":[-19,-20,-21,-22],"upper":[20,21,22,23],"start":[-18,-19,-20,-21],"end":[19,20,21,22],"domain":"Exact_explicit_path_connectedness","operation":"path-box-linear","semantic_template":"path-box-linear|{end:[[19,1],[20,1],[21,1],[22,1]],lower:[[-19,1],[-20,1],[-21,1],[-22,1]],start:[[-18,1],[-19,1],[-20,1],[-21,1]],upper:[[20,1],[21,1],[22,1],[23,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r06-path-connected-space-0044","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","language":"zh-Hant","source_mko_ids":["mko-path-connected-space"],"title_zh":"道路連通空間","stem_zh":"unit-interval path的continuity、兩端點與全range containment責任。操作=path-box-linear；資料={\"lower\":[-20,-21],\"upper\":[21,22],\"start\":[-19,-20],\"end\":[20,21]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([-20, -21], [21, 22], [-19, -20], [20, 21], [-19+(39)t, -20+(41)t], 是, coordinatewise_convex_combination_stays_in_box)","explanation_zh":"Wave6第十一層核對=([-20, -21], [21, 22], [-19, -20], [20, 21], [-19+(39)t, -20+(41)t], 是, coordinatewise_convex_combination_stays_in_box)；依完整明示資料獨立重算；linear／box／concatenated paths逐段exact核對；有限samples、graph walk或單一端點對不足；connected一般不推出path connected。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["道路連通空間"]},"variant":{"family_id":"qf-mod48-path-box-linear","seed":202611176044,"parameters":{"lower":[-20,-21],"upper":[21,22],"start":[-19,-20],"end":[20,21],"domain":"Exact_explicit_path_connectedness","operation":"path-box-linear","semantic_template":"path-box-linear|{end:[[20,1],[21,1]],lower:[[-20,1],[-21,1]],start:[[-19,1],[-20,1]],upper:[[21,1],[22,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r06-path-connected-space-0045","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","language":"zh-Hant","source_mko_ids":["mko-path-connected-space"],"title_zh":"道路連通空間","stem_zh":"unit-interval path的continuity、兩端點與全range containment責任。操作=path-box-linear；資料={\"lower\":[-21,-22,-23],\"upper\":[22,23,24],\"start\":[-20,-21,-22],\"end\":[21,22,23]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([-21, -22, -23], [22, 23, 24], [-20, -21, -22], [21, 22, 23], [-20+(41)t, -21+(43)t, -22+(45)t], 是, coordinatewise_convex_combination_stays_in_box)","explanation_zh":"Wave6第十一層核對=([-21, -22, -23], [22, 23, 24], [-20, -21, -22], [21, 22, 23], [-20+(41)t, -21+(43)t, -22+(45)t], 是, coordinatewise_convex_combination_stays_in_box)；依完整明示資料獨立重算；linear／box／concatenated paths逐段exact核對；有限samples、graph walk或單一端點對不足；connected一般不推出path connected。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["道路連通空間"]},"variant":{"family_id":"qf-mod48-path-box-linear","seed":202611176045,"parameters":{"lower":[-21,-22,-23],"upper":[22,23,24],"start":[-20,-21,-22],"end":[21,22,23],"domain":"Exact_explicit_path_connectedness","operation":"path-box-linear","semantic_template":"path-box-linear|{end:[[21,1],[22,1],[23,1]],lower:[[-21,1],[-22,1],[-23,1]],start:[[-20,1],[-21,1],[-22,1]],upper:[[22,1],[23,1],[24,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r06-path-connected-space-0046","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","language":"zh-Hant","source_mko_ids":["mko-path-connected-space"],"title_zh":"道路連通空間","stem_zh":"unit-interval path的continuity、兩端點與全range containment責任。操作=path-box-linear；資料={\"lower\":[-22,-23,-24,-25],\"upper\":[23,24,25,26],\"start\":[-21,-22,-23,-24],\"end\":[22,23,24,25]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([-22, -23, -24, -25], [23, 24, 25, 26], [-21, -22, -23, -24], [22, 23, 24, 25], [-21+(43)t, -22+(45)t, -23+(47)t, -24+(49)t], 是, coordinatewise_convex_combination_stays_in_box)","explanation_zh":"Wave6第十一層核對=([-22, -23, -24, -25], [23, 24, 25, 26], [-21, -22, -23, -24], [22, 23, 24, 25], [-21+(43)t, -22+(45)t, -23+(47)t, -24+(49)t], 是, coordinatewise_convex_combination_stays_in_box)；依完整明示資料獨立重算；linear／box／concatenated paths逐段exact核對；有限samples、graph walk或單一端點對不足；connected一般不推出path connected。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["道路連通空間"]},"variant":{"family_id":"qf-mod48-path-box-linear","seed":202611176046,"parameters":{"lower":[-22,-23,-24,-25],"upper":[23,24,25,26],"start":[-21,-22,-23,-24],"end":[22,23,24,25],"domain":"Exact_explicit_path_connectedness","operation":"path-box-linear","semantic_template":"path-box-linear|{end:[[22,1],[23,1],[24,1],[25,1]],lower:[[-22,1],[-23,1],[-24,1],[-25,1]],start:[[-21,1],[-22,1],[-23,1],[-24,1]],upper:[[23,1],[24,1],[25,1],[26,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r06-path-connected-space-0047","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","language":"zh-Hant","source_mko_ids":["mko-path-connected-space"],"title_zh":"道路連通空間","stem_zh":"unit-interval path的continuity、兩端點與全range containment責任。操作=path-box-linear；資料={\"lower\":[-23,-24],\"upper\":[24,25],\"start\":[-22,-23],\"end\":[23,24]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([-23, -24], [24, 25], [-22, -23], [23, 24], [-22+(45)t, -23+(47)t], 是, coordinatewise_convex_combination_stays_in_box)","explanation_zh":"Wave6第十一層核對=([-23, -24], [24, 25], [-22, -23], [23, 24], [-22+(45)t, -23+(47)t], 是, coordinatewise_convex_combination_stays_in_box)；依完整明示資料獨立重算；linear／box／concatenated paths逐段exact核對；有限samples、graph walk或單一端點對不足；connected一般不推出path connected。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["道路連通空間"]},"variant":{"family_id":"qf-mod48-path-box-linear","seed":202611176047,"parameters":{"lower":[-23,-24],"upper":[24,25],"start":[-22,-23],"end":[23,24],"domain":"Exact_explicit_path_connectedness","operation":"path-box-linear","semantic_template":"path-box-linear|{end:[[23,1],[24,1]],lower:[[-23,1],[-24,1]],start:[[-22,1],[-23,1]],upper:[[24,1],[25,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r06-path-connected-space-0048","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","language":"zh-Hant","source_mko_ids":["mko-path-connected-space"],"title_zh":"道路連通空間","stem_zh":"unit-interval path的continuity、兩端點與全range containment責任。操作=path-box-linear；資料={\"lower\":[-24,-25,-26],\"upper\":[25,26,27],\"start\":[-23,-24,-25],\"end\":[24,25,26]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([-24, -25, -26], [25, 26, 27], [-23, -24, -25], [24, 25, 26], [-23+(47)t, -24+(49)t, -25+(51)t], 是, coordinatewise_convex_combination_stays_in_box)","explanation_zh":"Wave6第十一層核對=([-24, -25, -26], [25, 26, 27], [-23, -24, -25], [24, 25, 26], [-23+(47)t, -24+(49)t, -25+(51)t], 是, coordinatewise_convex_combination_stays_in_box)；依完整明示資料獨立重算；linear／box／concatenated paths逐段exact核對；有限samples、graph walk或單一端點對不足；connected一般不推出path connected。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["道路連通空間"]},"variant":{"family_id":"qf-mod48-path-box-linear","seed":202611176048,"parameters":{"lower":[-24,-25,-26],"upper":[25,26,27],"start":[-23,-24,-25],"end":[24,25,26],"domain":"Exact_explicit_path_connectedness","operation":"path-box-linear","semantic_template":"path-box-linear|{end:[[24,1],[25,1],[26,1]],lower:[[-24,1],[-25,1],[-26,1]],start:[[-23,1],[-24,1],[-25,1]],upper:[[25,1],[26,1],[27,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r06-path-connected-space-0049","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","language":"zh-Hant","source_mko_ids":["mko-path-connected-space"],"title_zh":"道路連通空間","stem_zh":"unit-interval path的continuity、兩端點與全range containment責任。操作=path-box-linear；資料={\"lower\":[-25,-26,-27,-28],\"upper\":[26,27,28,29],\"start\":[-24,-25,-26,-27],\"end\":[25,26,27,28]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([-25, -26, -27, -28], [26, 27, 28, 29], [-24, -25, -26, -27], [25, 26, 27, 28], [-24+(49)t, -25+(51)t, -26+(53)t, -27+(55)t], 是, coordinatewise_convex_combination_stays_in_box)","explanation_zh":"Wave6第十一層核對=([-25, -26, -27, -28], [26, 27, 28, 29], [-24, -25, -26, -27], [25, 26, 27, 28], [-24+(49)t, -25+(51)t, -26+(53)t, -27+(55)t], 是, coordinatewise_convex_combination_stays_in_box)；依完整明示資料獨立重算；linear／box／concatenated paths逐段exact核對；有限samples、graph walk或單一端點對不足；connected一般不推出path connected。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["道路連通空間"]},"variant":{"family_id":"qf-mod48-path-box-linear","seed":202611176049,"parameters":{"lower":[-25,-26,-27,-28],"upper":[26,27,28,29],"start":[-24,-25,-26,-27],"end":[25,26,27,28],"domain":"Exact_explicit_path_connectedness","operation":"path-box-linear","semantic_template":"path-box-linear|{end:[[25,1],[26,1],[27,1],[28,1]],lower:[[-25,1],[-26,1],[-27,1],[-28,1]],start:[[-24,1],[-25,1],[-26,1],[-27,1]],upper:[[26,1],[27,1],[28,1],[29,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r06-path-connected-space-0050","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","language":"zh-Hant","source_mko_ids":["mko-path-connected-space"],"title_zh":"道路連通空間","stem_zh":"unit-interval path的continuity、兩端點與全range containment責任。操作=path-box-linear；資料={\"lower\":[-26,-27],\"upper\":[27,28],\"start\":[-25,-26],\"end\":[26,27]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([-26, -27], [27, 28], [-25, -26], [26, 27], [-25+(51)t, -26+(53)t], 是, coordinatewise_convex_combination_stays_in_box)","explanation_zh":"Wave6第十一層核對=([-26, -27], [27, 28], [-25, -26], [26, 27], [-25+(51)t, -26+(53)t], 是, coordinatewise_convex_combination_stays_in_box)；依完整明示資料獨立重算；linear／box／concatenated paths逐段exact核對；有限samples、graph walk或單一端點對不足；connected一般不推出path connected。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["道路連通空間"]},"variant":{"family_id":"qf-mod48-path-box-linear","seed":202611176050,"parameters":{"lower":[-26,-27],"upper":[27,28],"start":[-25,-26],"end":[26,27],"domain":"Exact_explicit_path_connectedness","operation":"path-box-linear","semantic_template":"path-box-linear|{end:[[26,1],[27,1]],lower:[[-26,1],[-27,1]],start:[[-25,1],[-26,1]],upper:[[27,1],[28,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r06-path-connected-space-0051","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","language":"zh-Hant","source_mko_ids":["mko-path-connected-space"],"title_zh":"道路連通空間","stem_zh":"unit-interval path的continuity、兩端點與全range containment責任。操作=path-concatenation；資料={\"left\":-3,\"right\":4,\"start\":0,\"middle\":2,\"end\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, 2, 0, 0<=t<=1/2:0+2t(2), 1/2<=t<=1:2+(2t-1)(-2), both_branches_meet_at_middle, 是)","explanation_zh":"Wave6第十一層核對=(0, 2, 0, 0<=t<=1/2:0+2t(2), 1/2<=t<=1:2+(2t-1)(-2), both_branches_meet_at_middle, 是)；依完整明示資料獨立重算；linear／box／concatenated paths逐段exact核對；有限samples、graph walk或單一端點對不足；connected一般不推出path connected。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["道路連通空間"]},"variant":{"family_id":"qf-mod48-path-concatenation","seed":202611176051,"parameters":{"left":-3,"right":4,"start":0,"middle":2,"end":0,"domain":"Exact_explicit_path_connectedness","operation":"path-concatenation","semantic_template":"path-concatenation|{end:[0,1],left:[-3,1],middle:[2,1],right:[4,1],start:[0,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r06-path-connected-space-0052","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","language":"zh-Hant","source_mko_ids":["mko-path-connected-space"],"title_zh":"道路連通空間","stem_zh":"unit-interval path的continuity、兩端點與全range containment責任。操作=path-concatenation；資料={\"left\":-4,\"right\":5,\"start\":-1,\"middle\":3,\"end\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-1, 3, 1, 0<=t<=1/2:-1+2t(4), 1/2<=t<=1:3+(2t-1)(-2), both_branches_meet_at_middle, 是)","explanation_zh":"Wave6第十一層核對=(-1, 3, 1, 0<=t<=1/2:-1+2t(4), 1/2<=t<=1:3+(2t-1)(-2), both_branches_meet_at_middle, 是)；依完整明示資料獨立重算；linear／box／concatenated paths逐段exact核對；有限samples、graph walk或單一端點對不足；connected一般不推出path connected。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["道路連通空間"]},"variant":{"family_id":"qf-mod48-path-concatenation","seed":202611176052,"parameters":{"left":-4,"right":5,"start":-1,"middle":3,"end":1,"domain":"Exact_explicit_path_connectedness","operation":"path-concatenation","semantic_template":"path-concatenation|{end:[1,1],left:[-4,1],middle:[3,1],right:[5,1],start:[-1,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r06-path-connected-space-0053","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","language":"zh-Hant","source_mko_ids":["mko-path-connected-space"],"title_zh":"道路連通空間","stem_zh":"unit-interval path的continuity、兩端點與全range containment責任。操作=path-concatenation；資料={\"left\":-5,\"right\":6,\"start\":-2,\"middle\":4,\"end\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-2, 4, 2, 0<=t<=1/2:-2+2t(6), 1/2<=t<=1:4+(2t-1)(-2), both_branches_meet_at_middle, 是)","explanation_zh":"Wave6第十一層核對=(-2, 4, 2, 0<=t<=1/2:-2+2t(6), 1/2<=t<=1:4+(2t-1)(-2), both_branches_meet_at_middle, 是)；依完整明示資料獨立重算；linear／box／concatenated paths逐段exact核對；有限samples、graph walk或單一端點對不足；connected一般不推出path connected。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["道路連通空間"]},"variant":{"family_id":"qf-mod48-path-concatenation","seed":202611176053,"parameters":{"left":-5,"right":6,"start":-2,"middle":4,"end":2,"domain":"Exact_explicit_path_connectedness","operation":"path-concatenation","semantic_template":"path-concatenation|{end:[2,1],left:[-5,1],middle:[4,1],right:[6,1],start:[-2,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r06-path-connected-space-0054","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","language":"zh-Hant","source_mko_ids":["mko-path-connected-space"],"title_zh":"道路連通空間","stem_zh":"unit-interval path的continuity、兩端點與全range containment責任。操作=path-concatenation；資料={\"left\":-6,\"right\":7,\"start\":-3,\"middle\":5,\"end\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-3, 5, 3, 0<=t<=1/2:-3+2t(8), 1/2<=t<=1:5+(2t-1)(-2), both_branches_meet_at_middle, 是)","explanation_zh":"Wave6第十一層核對=(-3, 5, 3, 0<=t<=1/2:-3+2t(8), 1/2<=t<=1:5+(2t-1)(-2), both_branches_meet_at_middle, 是)；依完整明示資料獨立重算；linear／box／concatenated paths逐段exact核對；有限samples、graph walk或單一端點對不足；connected一般不推出path connected。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["道路連通空間"]},"variant":{"family_id":"qf-mod48-path-concatenation","seed":202611176054,"parameters":{"left":-6,"right":7,"start":-3,"middle":5,"end":3,"domain":"Exact_explicit_path_connectedness","operation":"path-concatenation","semantic_template":"path-concatenation|{end:[3,1],left:[-6,1],middle:[5,1],right:[7,1],start:[-3,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r06-path-connected-space-0055","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","language":"zh-Hant","source_mko_ids":["mko-path-connected-space"],"title_zh":"道路連通空間","stem_zh":"unit-interval path的continuity、兩端點與全range containment責任。操作=path-concatenation；資料={\"left\":-7,\"right\":8,\"start\":-4,\"middle\":6,\"end\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-4, 6, 4, 0<=t<=1/2:-4+2t(10), 1/2<=t<=1:6+(2t-1)(-2), both_branches_meet_at_middle, 是)","explanation_zh":"Wave6第十一層核對=(-4, 6, 4, 0<=t<=1/2:-4+2t(10), 1/2<=t<=1:6+(2t-1)(-2), both_branches_meet_at_middle, 是)；依完整明示資料獨立重算；linear／box／concatenated paths逐段exact核對；有限samples、graph walk或單一端點對不足；connected一般不推出path connected。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["道路連通空間"]},"variant":{"family_id":"qf-mod48-path-concatenation","seed":202611176055,"parameters":{"left":-7,"right":8,"start":-4,"middle":6,"end":4,"domain":"Exact_explicit_path_connectedness","operation":"path-concatenation","semantic_template":"path-concatenation|{end:[4,1],left:[-7,1],middle:[6,1],right:[8,1],start:[-4,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r06-path-connected-space-0056","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","language":"zh-Hant","source_mko_ids":["mko-path-connected-space"],"title_zh":"道路連通空間","stem_zh":"unit-interval path的continuity、兩端點與全range containment責任。操作=path-concatenation；資料={\"left\":-8,\"right\":9,\"start\":-5,\"middle\":7,\"end\":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-5, 7, 5, 0<=t<=1/2:-5+2t(12), 1/2<=t<=1:7+(2t-1)(-2), both_branches_meet_at_middle, 是)","explanation_zh":"Wave6第十一層核對=(-5, 7, 5, 0<=t<=1/2:-5+2t(12), 1/2<=t<=1:7+(2t-1)(-2), both_branches_meet_at_middle, 是)；依完整明示資料獨立重算；linear／box／concatenated paths逐段exact核對；有限samples、graph walk或單一端點對不足；connected一般不推出path connected。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["道路連通空間"]},"variant":{"family_id":"qf-mod48-path-concatenation","seed":202611176056,"parameters":{"left":-8,"right":9,"start":-5,"middle":7,"end":5,"domain":"Exact_explicit_path_connectedness","operation":"path-concatenation","semantic_template":"path-concatenation|{end:[5,1],left:[-8,1],middle:[7,1],right:[9,1],start:[-5,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r06-path-connected-space-0057","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","language":"zh-Hant","source_mko_ids":["mko-path-connected-space"],"title_zh":"道路連通空間","stem_zh":"unit-interval path的continuity、兩端點與全range containment責任。操作=path-concatenation；資料={\"left\":-9,\"right\":10,\"start\":-6,\"middle\":8,\"end\":6}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-6, 8, 6, 0<=t<=1/2:-6+2t(14), 1/2<=t<=1:8+(2t-1)(-2), both_branches_meet_at_middle, 是)","explanation_zh":"Wave6第十一層核對=(-6, 8, 6, 0<=t<=1/2:-6+2t(14), 1/2<=t<=1:8+(2t-1)(-2), both_branches_meet_at_middle, 是)；依完整明示資料獨立重算；linear／box／concatenated paths逐段exact核對；有限samples、graph walk或單一端點對不足；connected一般不推出path connected。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["道路連通空間"]},"variant":{"family_id":"qf-mod48-path-concatenation","seed":202611176057,"parameters":{"left":-9,"right":10,"start":-6,"middle":8,"end":6,"domain":"Exact_explicit_path_connectedness","operation":"path-concatenation","semantic_template":"path-concatenation|{end:[6,1],left:[-9,1],middle:[8,1],right:[10,1],start:[-6,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r06-path-connected-space-0058","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","language":"zh-Hant","source_mko_ids":["mko-path-connected-space"],"title_zh":"道路連通空間","stem_zh":"unit-interval path的continuity、兩端點與全range containment責任。操作=path-concatenation；資料={\"left\":-10,\"right\":11,\"start\":-7,\"middle\":9,\"end\":7}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-7, 9, 7, 0<=t<=1/2:-7+2t(16), 1/2<=t<=1:9+(2t-1)(-2), both_branches_meet_at_middle, 是)","explanation_zh":"Wave6第十一層核對=(-7, 9, 7, 0<=t<=1/2:-7+2t(16), 1/2<=t<=1:9+(2t-1)(-2), both_branches_meet_at_middle, 是)；依完整明示資料獨立重算；linear／box／concatenated paths逐段exact核對；有限samples、graph walk或單一端點對不足；connected一般不推出path connected。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["道路連通空間"]},"variant":{"family_id":"qf-mod48-path-concatenation","seed":202611176058,"parameters":{"left":-10,"right":11,"start":-7,"middle":9,"end":7,"domain":"Exact_explicit_path_connectedness","operation":"path-concatenation","semantic_template":"path-concatenation|{end:[7,1],left:[-10,1],middle:[9,1],right:[11,1],start:[-7,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r06-path-connected-space-0059","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","language":"zh-Hant","source_mko_ids":["mko-path-connected-space"],"title_zh":"道路連通空間","stem_zh":"unit-interval path的continuity、兩端點與全range containment責任。操作=path-concatenation；資料={\"left\":-11,\"right\":12,\"start\":-8,\"middle\":10,\"end\":8}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-8, 10, 8, 0<=t<=1/2:-8+2t(18), 1/2<=t<=1:10+(2t-1)(-2), both_branches_meet_at_middle, 是)","explanation_zh":"Wave6第十一層核對=(-8, 10, 8, 0<=t<=1/2:-8+2t(18), 1/2<=t<=1:10+(2t-1)(-2), both_branches_meet_at_middle, 是)；依完整明示資料獨立重算；linear／box／concatenated paths逐段exact核對；有限samples、graph walk或單一端點對不足；connected一般不推出path connected。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["道路連通空間"]},"variant":{"family_id":"qf-mod48-path-concatenation","seed":202611176059,"parameters":{"left":-11,"right":12,"start":-8,"middle":10,"end":8,"domain":"Exact_explicit_path_connectedness","operation":"path-concatenation","semantic_template":"path-concatenation|{end:[8,1],left:[-11,1],middle:[10,1],right:[12,1],start:[-8,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r06-path-connected-space-0060","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","language":"zh-Hant","source_mko_ids":["mko-path-connected-space"],"title_zh":"道路連通空間","stem_zh":"unit-interval path的continuity、兩端點與全range containment責任。操作=path-concatenation；資料={\"left\":-12,\"right\":13,\"start\":-9,\"middle\":11,\"end\":9}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-9, 11, 9, 0<=t<=1/2:-9+2t(20), 1/2<=t<=1:11+(2t-1)(-2), both_branches_meet_at_middle, 是)","explanation_zh":"Wave6第十一層核對=(-9, 11, 9, 0<=t<=1/2:-9+2t(20), 1/2<=t<=1:11+(2t-1)(-2), both_branches_meet_at_middle, 是)；依完整明示資料獨立重算；linear／box／concatenated paths逐段exact核對；有限samples、graph walk或單一端點對不足；connected一般不推出path connected。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["道路連通空間"]},"variant":{"family_id":"qf-mod48-path-concatenation","seed":202611176060,"parameters":{"left":-12,"right":13,"start":-9,"middle":11,"end":9,"domain":"Exact_explicit_path_connectedness","operation":"path-concatenation","semantic_template":"path-concatenation|{end:[9,1],left:[-12,1],middle:[11,1],right:[13,1],start:[-9,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r06-path-connected-space-0061","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","language":"zh-Hant","source_mko_ids":["mko-path-connected-space"],"title_zh":"道路連通空間","stem_zh":"unit-interval path的continuity、兩端點與全range containment責任。操作=path-concatenation；資料={\"left\":-13,\"right\":14,\"start\":-10,\"middle\":12,\"end\":10}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-10, 12, 10, 0<=t<=1/2:-10+2t(22), 1/2<=t<=1:12+(2t-1)(-2), both_branches_meet_at_middle, 是)","explanation_zh":"Wave6第十一層核對=(-10, 12, 10, 0<=t<=1/2:-10+2t(22), 1/2<=t<=1:12+(2t-1)(-2), both_branches_meet_at_middle, 是)；依完整明示資料獨立重算；linear／box／concatenated paths逐段exact核對；有限samples、graph walk或單一端點對不足；connected一般不推出path connected。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["道路連通空間"]},"variant":{"family_id":"qf-mod48-path-concatenation","seed":202611176061,"parameters":{"left":-13,"right":14,"start":-10,"middle":12,"end":10,"domain":"Exact_explicit_path_connectedness","operation":"path-concatenation","semantic_template":"path-concatenation|{end:[10,1],left:[-13,1],middle:[12,1],right:[14,1],start:[-10,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r06-path-connected-space-0062","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","language":"zh-Hant","source_mko_ids":["mko-path-connected-space"],"title_zh":"道路連通空間","stem_zh":"unit-interval path的continuity、兩端點與全range containment責任。操作=path-concatenation；資料={\"left\":-14,\"right\":15,\"start\":-11,\"middle\":13,\"end\":11}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-11, 13, 11, 0<=t<=1/2:-11+2t(24), 1/2<=t<=1:13+(2t-1)(-2), both_branches_meet_at_middle, 是)","explanation_zh":"Wave6第十一層核對=(-11, 13, 11, 0<=t<=1/2:-11+2t(24), 1/2<=t<=1:13+(2t-1)(-2), both_branches_meet_at_middle, 是)；依完整明示資料獨立重算；linear／box／concatenated paths逐段exact核對；有限samples、graph walk或單一端點對不足；connected一般不推出path connected。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["道路連通空間"]},"variant":{"family_id":"qf-mod48-path-concatenation","seed":202611176062,"parameters":{"left":-14,"right":15,"start":-11,"middle":13,"end":11,"domain":"Exact_explicit_path_connectedness","operation":"path-concatenation","semantic_template":"path-concatenation|{end:[11,1],left:[-14,1],middle:[13,1],right:[15,1],start:[-11,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r06-path-connected-space-0063","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","language":"zh-Hant","source_mko_ids":["mko-path-connected-space"],"title_zh":"道路連通空間","stem_zh":"unit-interval path的continuity、兩端點與全range containment責任。操作=path-concatenation；資料={\"left\":-15,\"right\":16,\"start\":-12,\"middle\":14,\"end\":12}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-12, 14, 12, 0<=t<=1/2:-12+2t(26), 1/2<=t<=1:14+(2t-1)(-2), both_branches_meet_at_middle, 是)","explanation_zh":"Wave6第十一層核對=(-12, 14, 12, 0<=t<=1/2:-12+2t(26), 1/2<=t<=1:14+(2t-1)(-2), both_branches_meet_at_middle, 是)；依完整明示資料獨立重算；linear／box／concatenated paths逐段exact核對；有限samples、graph walk或單一端點對不足；connected一般不推出path connected。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["道路連通空間"]},"variant":{"family_id":"qf-mod48-path-concatenation","seed":202611176063,"parameters":{"left":-15,"right":16,"start":-12,"middle":14,"end":12,"domain":"Exact_explicit_path_connectedness","operation":"path-concatenation","semantic_template":"path-concatenation|{end:[12,1],left:[-15,1],middle:[14,1],right:[16,1],start:[-12,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r06-path-connected-space-0064","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","language":"zh-Hant","source_mko_ids":["mko-path-connected-space"],"title_zh":"道路連通空間","stem_zh":"unit-interval path的continuity、兩端點與全range containment責任。操作=path-concatenation；資料={\"left\":-16,\"right\":17,\"start\":-13,\"middle\":15,\"end\":13}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-13, 15, 13, 0<=t<=1/2:-13+2t(28), 1/2<=t<=1:15+(2t-1)(-2), both_branches_meet_at_middle, 是)","explanation_zh":"Wave6第十一層核對=(-13, 15, 13, 0<=t<=1/2:-13+2t(28), 1/2<=t<=1:15+(2t-1)(-2), both_branches_meet_at_middle, 是)；依完整明示資料獨立重算；linear／box／concatenated paths逐段exact核對；有限samples、graph walk或單一端點對不足；connected一般不推出path connected。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["道路連通空間"]},"variant":{"family_id":"qf-mod48-path-concatenation","seed":202611176064,"parameters":{"left":-16,"right":17,"start":-13,"middle":15,"end":13,"domain":"Exact_explicit_path_connectedness","operation":"path-concatenation","semantic_template":"path-concatenation|{end:[13,1],left:[-16,1],middle:[15,1],right:[17,1],start:[-13,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r06-path-connected-space-0065","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","language":"zh-Hant","source_mko_ids":["mko-path-connected-space"],"title_zh":"道路連通空間","stem_zh":"unit-interval path的continuity、兩端點與全range containment責任。操作=path-concatenation；資料={\"left\":-17,\"right\":18,\"start\":-14,\"middle\":16,\"end\":14}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-14, 16, 14, 0<=t<=1/2:-14+2t(30), 1/2<=t<=1:16+(2t-1)(-2), both_branches_meet_at_middle, 是)","explanation_zh":"Wave6第十一層核對=(-14, 16, 14, 0<=t<=1/2:-14+2t(30), 1/2<=t<=1:16+(2t-1)(-2), both_branches_meet_at_middle, 是)；依完整明示資料獨立重算；linear／box／concatenated paths逐段exact核對；有限samples、graph walk或單一端點對不足；connected一般不推出path connected。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["道路連通空間"]},"variant":{"family_id":"qf-mod48-path-concatenation","seed":202611176065,"parameters":{"left":-17,"right":18,"start":-14,"middle":16,"end":14,"domain":"Exact_explicit_path_connectedness","operation":"path-concatenation","semantic_template":"path-concatenation|{end:[14,1],left:[-17,1],middle:[16,1],right:[18,1],start:[-14,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r06-path-connected-space-0066","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","language":"zh-Hant","source_mko_ids":["mko-path-connected-space"],"title_zh":"道路連通空間","stem_zh":"unit-interval path的continuity、兩端點與全range containment責任。操作=path-concatenation；資料={\"left\":-18,\"right\":19,\"start\":-15,\"middle\":17,\"end\":15}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-15, 17, 15, 0<=t<=1/2:-15+2t(32), 1/2<=t<=1:17+(2t-1)(-2), both_branches_meet_at_middle, 是)","explanation_zh":"Wave6第十一層核對=(-15, 17, 15, 0<=t<=1/2:-15+2t(32), 1/2<=t<=1:17+(2t-1)(-2), both_branches_meet_at_middle, 是)；依完整明示資料獨立重算；linear／box／concatenated paths逐段exact核對；有限samples、graph walk或單一端點對不足；connected一般不推出path connected。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["道路連通空間"]},"variant":{"family_id":"qf-mod48-path-concatenation","seed":202611176066,"parameters":{"left":-18,"right":19,"start":-15,"middle":17,"end":15,"domain":"Exact_explicit_path_connectedness","operation":"path-concatenation","semantic_template":"path-concatenation|{end:[15,1],left:[-18,1],middle:[17,1],right:[19,1],start:[-15,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r06-path-connected-space-0067","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","language":"zh-Hant","source_mko_ids":["mko-path-connected-space"],"title_zh":"道路連通空間","stem_zh":"unit-interval path的continuity、兩端點與全range containment責任。操作=path-concatenation；資料={\"left\":-19,\"right\":20,\"start\":-16,\"middle\":18,\"end\":16}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-16, 18, 16, 0<=t<=1/2:-16+2t(34), 1/2<=t<=1:18+(2t-1)(-2), both_branches_meet_at_middle, 是)","explanation_zh":"Wave6第十一層核對=(-16, 18, 16, 0<=t<=1/2:-16+2t(34), 1/2<=t<=1:18+(2t-1)(-2), both_branches_meet_at_middle, 是)；依完整明示資料獨立重算；linear／box／concatenated paths逐段exact核對；有限samples、graph walk或單一端點對不足；connected一般不推出path connected。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["道路連通空間"]},"variant":{"family_id":"qf-mod48-path-concatenation","seed":202611176067,"parameters":{"left":-19,"right":20,"start":-16,"middle":18,"end":16,"domain":"Exact_explicit_path_connectedness","operation":"path-concatenation","semantic_template":"path-concatenation|{end:[16,1],left:[-19,1],middle:[18,1],right:[20,1],start:[-16,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r06-path-connected-space-0068","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","language":"zh-Hant","source_mko_ids":["mko-path-connected-space"],"title_zh":"道路連通空間","stem_zh":"unit-interval path的continuity、兩端點與全range containment責任。操作=path-concatenation；資料={\"left\":-20,\"right\":21,\"start\":-17,\"middle\":19,\"end\":17}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-17, 19, 17, 0<=t<=1/2:-17+2t(36), 1/2<=t<=1:19+(2t-1)(-2), both_branches_meet_at_middle, 是)","explanation_zh":"Wave6第十一層核對=(-17, 19, 17, 0<=t<=1/2:-17+2t(36), 1/2<=t<=1:19+(2t-1)(-2), both_branches_meet_at_middle, 是)；依完整明示資料獨立重算；linear／box／concatenated paths逐段exact核對；有限samples、graph walk或單一端點對不足；connected一般不推出path connected。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["道路連通空間"]},"variant":{"family_id":"qf-mod48-path-concatenation","seed":202611176068,"parameters":{"left":-20,"right":21,"start":-17,"middle":19,"end":17,"domain":"Exact_explicit_path_connectedness","operation":"path-concatenation","semantic_template":"path-concatenation|{end:[17,1],left:[-20,1],middle:[19,1],right:[21,1],start:[-17,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r06-path-connected-space-0069","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","language":"zh-Hant","source_mko_ids":["mko-path-connected-space"],"title_zh":"道路連通空間","stem_zh":"unit-interval path的continuity、兩端點與全range containment責任。操作=path-concatenation；資料={\"left\":-21,\"right\":22,\"start\":-18,\"middle\":20,\"end\":18}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-18, 20, 18, 0<=t<=1/2:-18+2t(38), 1/2<=t<=1:20+(2t-1)(-2), both_branches_meet_at_middle, 是)","explanation_zh":"Wave6第十一層核對=(-18, 20, 18, 0<=t<=1/2:-18+2t(38), 1/2<=t<=1:20+(2t-1)(-2), both_branches_meet_at_middle, 是)；依完整明示資料獨立重算；linear／box／concatenated paths逐段exact核對；有限samples、graph walk或單一端點對不足；connected一般不推出path connected。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["道路連通空間"]},"variant":{"family_id":"qf-mod48-path-concatenation","seed":202611176069,"parameters":{"left":-21,"right":22,"start":-18,"middle":20,"end":18,"domain":"Exact_explicit_path_connectedness","operation":"path-concatenation","semantic_template":"path-concatenation|{end:[18,1],left:[-21,1],middle:[20,1],right:[22,1],start:[-18,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r06-path-connected-space-0070","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","language":"zh-Hant","source_mko_ids":["mko-path-connected-space"],"title_zh":"道路連通空間","stem_zh":"unit-interval path的continuity、兩端點與全range containment責任。操作=path-concatenation；資料={\"left\":-22,\"right\":23,\"start\":-19,\"middle\":21,\"end\":19}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-19, 21, 19, 0<=t<=1/2:-19+2t(40), 1/2<=t<=1:21+(2t-1)(-2), both_branches_meet_at_middle, 是)","explanation_zh":"Wave6第十一層核對=(-19, 21, 19, 0<=t<=1/2:-19+2t(40), 1/2<=t<=1:21+(2t-1)(-2), both_branches_meet_at_middle, 是)；依完整明示資料獨立重算；linear／box／concatenated paths逐段exact核對；有限samples、graph walk或單一端點對不足；connected一般不推出path connected。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["道路連通空間"]},"variant":{"family_id":"qf-mod48-path-concatenation","seed":202611176070,"parameters":{"left":-22,"right":23,"start":-19,"middle":21,"end":19,"domain":"Exact_explicit_path_connectedness","operation":"path-concatenation","semantic_template":"path-concatenation|{end:[19,1],left:[-22,1],middle:[21,1],right:[23,1],start:[-19,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r06-path-connected-space-0071","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","language":"zh-Hant","source_mko_ids":["mko-path-connected-space"],"title_zh":"道路連通空間","stem_zh":"unit-interval path的continuity、兩端點與全range containment責任。操作=path-concatenation；資料={\"left\":-23,\"right\":24,\"start\":-20,\"middle\":22,\"end\":20}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-20, 22, 20, 0<=t<=1/2:-20+2t(42), 1/2<=t<=1:22+(2t-1)(-2), both_branches_meet_at_middle, 是)","explanation_zh":"Wave6第十一層核對=(-20, 22, 20, 0<=t<=1/2:-20+2t(42), 1/2<=t<=1:22+(2t-1)(-2), both_branches_meet_at_middle, 是)；依完整明示資料獨立重算；linear／box／concatenated paths逐段exact核對；有限samples、graph walk或單一端點對不足；connected一般不推出path connected。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["道路連通空間"]},"variant":{"family_id":"qf-mod48-path-concatenation","seed":202611176071,"parameters":{"left":-23,"right":24,"start":-20,"middle":22,"end":20,"domain":"Exact_explicit_path_connectedness","operation":"path-concatenation","semantic_template":"path-concatenation|{end:[20,1],left:[-23,1],middle:[22,1],right:[24,1],start:[-20,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r06-path-connected-space-0072","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","language":"zh-Hant","source_mko_ids":["mko-path-connected-space"],"title_zh":"道路連通空間","stem_zh":"unit-interval path的continuity、兩端點與全range containment責任。操作=path-concatenation；資料={\"left\":-24,\"right\":25,\"start\":-21,\"middle\":23,\"end\":21}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-21, 23, 21, 0<=t<=1/2:-21+2t(44), 1/2<=t<=1:23+(2t-1)(-2), both_branches_meet_at_middle, 是)","explanation_zh":"Wave6第十一層核對=(-21, 23, 21, 0<=t<=1/2:-21+2t(44), 1/2<=t<=1:23+(2t-1)(-2), both_branches_meet_at_middle, 是)；依完整明示資料獨立重算；linear／box／concatenated paths逐段exact核對；有限samples、graph walk或單一端點對不足；connected一般不推出path connected。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["道路連通空間"]},"variant":{"family_id":"qf-mod48-path-concatenation","seed":202611176072,"parameters":{"left":-24,"right":25,"start":-21,"middle":23,"end":21,"domain":"Exact_explicit_path_connectedness","operation":"path-concatenation","semantic_template":"path-concatenation|{end:[21,1],left:[-24,1],middle:[23,1],right:[25,1],start:[-21,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r06-path-connected-space-0073","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","language":"zh-Hant","source_mko_ids":["mko-path-connected-space"],"title_zh":"道路連通空間","stem_zh":"unit-interval path的continuity、兩端點與全range containment責任。操作=path-concatenation；資料={\"left\":-25,\"right\":26,\"start\":-22,\"middle\":24,\"end\":22}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-22, 24, 22, 0<=t<=1/2:-22+2t(46), 1/2<=t<=1:24+(2t-1)(-2), both_branches_meet_at_middle, 是)","explanation_zh":"Wave6第十一層核對=(-22, 24, 22, 0<=t<=1/2:-22+2t(46), 1/2<=t<=1:24+(2t-1)(-2), both_branches_meet_at_middle, 是)；依完整明示資料獨立重算；linear／box／concatenated paths逐段exact核對；有限samples、graph walk或單一端點對不足；connected一般不推出path connected。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["道路連通空間"]},"variant":{"family_id":"qf-mod48-path-concatenation","seed":202611176073,"parameters":{"left":-25,"right":26,"start":-22,"middle":24,"end":22,"domain":"Exact_explicit_path_connectedness","operation":"path-concatenation","semantic_template":"path-concatenation|{end:[22,1],left:[-25,1],middle:[24,1],right:[26,1],start:[-22,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r06-path-connected-space-0074","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","language":"zh-Hant","source_mko_ids":["mko-path-connected-space"],"title_zh":"道路連通空間","stem_zh":"unit-interval path的continuity、兩端點與全range containment責任。操作=path-concatenation；資料={\"left\":-26,\"right\":27,\"start\":-23,\"middle\":25,\"end\":23}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-23, 25, 23, 0<=t<=1/2:-23+2t(48), 1/2<=t<=1:25+(2t-1)(-2), both_branches_meet_at_middle, 是)","explanation_zh":"Wave6第十一層核對=(-23, 25, 23, 0<=t<=1/2:-23+2t(48), 1/2<=t<=1:25+(2t-1)(-2), both_branches_meet_at_middle, 是)；依完整明示資料獨立重算；linear／box／concatenated paths逐段exact核對；有限samples、graph walk或單一端點對不足；connected一般不推出path connected。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["道路連通空間"]},"variant":{"family_id":"qf-mod48-path-concatenation","seed":202611176074,"parameters":{"left":-26,"right":27,"start":-23,"middle":25,"end":23,"domain":"Exact_explicit_path_connectedness","operation":"path-concatenation","semantic_template":"path-concatenation|{end:[23,1],left:[-26,1],middle:[25,1],right:[27,1],start:[-23,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r06-path-connected-space-0075","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","language":"zh-Hant","source_mko_ids":["mko-path-connected-space"],"title_zh":"道路連通空間","stem_zh":"unit-interval path的continuity、兩端點與全range containment責任。操作=path-concatenation；資料={\"left\":-27,\"right\":28,\"start\":-24,\"middle\":26,\"end\":24}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-24, 26, 24, 0<=t<=1/2:-24+2t(50), 1/2<=t<=1:26+(2t-1)(-2), both_branches_meet_at_middle, 是)","explanation_zh":"Wave6第十一層核對=(-24, 26, 24, 0<=t<=1/2:-24+2t(50), 1/2<=t<=1:26+(2t-1)(-2), both_branches_meet_at_middle, 是)；依完整明示資料獨立重算；linear／box／concatenated paths逐段exact核對；有限samples、graph walk或單一端點對不足；connected一般不推出path connected。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["道路連通空間"]},"variant":{"family_id":"qf-mod48-path-concatenation","seed":202611176075,"parameters":{"left":-27,"right":28,"start":-24,"middle":26,"end":24,"domain":"Exact_explicit_path_connectedness","operation":"path-concatenation","semantic_template":"path-concatenation|{end:[24,1],left:[-27,1],middle:[26,1],right:[28,1],start:[-24,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r06-path-connected-space-0076","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","language":"zh-Hant","source_mko_ids":["mko-path-connected-space"],"title_zh":"道路連通空間","stem_zh":"unit-interval path的continuity、兩端點與全range containment責任。操作=path-connected-boundary；資料={\"mode\":\"disconnected_gap\",\"scale\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(disconnected_gap, 1, X=(-infinity,0] union [1,infinity), IVT_blocks_a_path_between_components, 否)","explanation_zh":"Wave6第十一層核對=(disconnected_gap, 1, X=(-infinity,0] union [1,infinity), IVT_blocks_a_path_between_components, 否)；依完整明示資料獨立重算；linear／box／concatenated paths逐段exact核對；有限samples、graph walk或單一端點對不足；connected一般不推出path connected。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["道路連通空間"]},"variant":{"family_id":"qf-mod48-path-connected-boundary","seed":202611176076,"parameters":{"mode":"disconnected_gap","scale":1,"domain":"Exact_explicit_path_connectedness","operation":"path-connected-boundary","semantic_template":"path-connected-boundary|{mode:\"disconnected_gap\",scale:[1,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r06-path-connected-space-0077","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","language":"zh-Hant","source_mko_ids":["mko-path-connected-space"],"title_zh":"道路連通空間","stem_zh":"unit-interval path的continuity、兩端點與全range containment責任。操作=path-connected-boundary；資料={\"mode\":\"discrete_two_point\",\"scale\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(discrete_two_point, 2, X={0,2}_with_discrete_topology, continuous_image_of_connected_[0,1]_cannot_meet_both_points, 否)","explanation_zh":"Wave6第十一層核對=(discrete_two_point, 2, X={0,2}_with_discrete_topology, continuous_image_of_connected_[0,1]_cannot_meet_both_points, 否)；依完整明示資料獨立重算；linear／box／concatenated paths逐段exact核對；有限samples、graph walk或單一端點對不足；connected一般不推出path connected。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["道路連通空間"]},"variant":{"family_id":"qf-mod48-path-connected-boundary","seed":202611176077,"parameters":{"mode":"discrete_two_point","scale":2,"domain":"Exact_explicit_path_connectedness","operation":"path-connected-boundary","semantic_template":"path-connected-boundary|{mode:\"discrete_two_point\",scale:[2,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r06-path-connected-space-0078","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","language":"zh-Hant","source_mko_ids":["mko-path-connected-space"],"title_zh":"道路連通空間","stem_zh":"unit-interval path的continuity、兩端點與全range containment責任。操作=path-connected-boundary；資料={\"mode\":\"endpoint_only\",\"scale\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(endpoint_only, 3, candidate_samples_gamma(0)=0_gamma(1/2)=3_gamma(1)=0, three_samples_do_not_define_a_continuous_map_or_control_its_full_range, 否)","explanation_zh":"Wave6第十一層核對=(endpoint_only, 3, candidate_samples_gamma(0)=0_gamma(1/2)=3_gamma(1)=0, three_samples_do_not_define_a_continuous_map_or_control_its_full_range, 否)；依完整明示資料獨立重算；linear／box／concatenated paths逐段exact核對；有限samples、graph walk或單一端點對不足；connected一般不推出path connected。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["道路連通空間"]},"variant":{"family_id":"qf-mod48-path-connected-boundary","seed":202611176078,"parameters":{"mode":"endpoint_only","scale":3,"domain":"Exact_explicit_path_connectedness","operation":"path-connected-boundary","semantic_template":"path-connected-boundary|{mode:\"endpoint_only\",scale:[3,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r06-path-connected-space-0079","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","language":"zh-Hant","source_mko_ids":["mko-path-connected-space"],"title_zh":"道路連通空間","stem_zh":"unit-interval path的continuity、兩端點與全range containment責任。操作=path-connected-boundary；資料={\"mode\":\"connected_not_path\",\"scale\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(connected_not_path, 4, topologists_sine_curve_scaled, connected_but_not_path_connected, 否)","explanation_zh":"Wave6第十一層核對=(connected_not_path, 4, topologists_sine_curve_scaled, connected_but_not_path_connected, 否)；依完整明示資料獨立重算；linear／box／concatenated paths逐段exact核對；有限samples、graph walk或單一端點對不足；connected一般不推出path connected。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["道路連通空間"]},"variant":{"family_id":"qf-mod48-path-connected-boundary","seed":202611176079,"parameters":{"mode":"connected_not_path","scale":4,"domain":"Exact_explicit_path_connectedness","operation":"path-connected-boundary","semantic_template":"path-connected-boundary|{mode:\"connected_not_path\",scale:[4,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r06-path-connected-space-0080","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","language":"zh-Hant","source_mko_ids":["mko-path-connected-space"],"title_zh":"道路連通空間","stem_zh":"unit-interval path的continuity、兩端點與全range containment責任。操作=path-connected-boundary；資料={\"mode\":\"disconnected_gap\",\"scale\":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(disconnected_gap, 5, X=(-infinity,0] union [5,infinity), IVT_blocks_a_path_between_components, 否)","explanation_zh":"Wave6第十一層核對=(disconnected_gap, 5, X=(-infinity,0] union [5,infinity), IVT_blocks_a_path_between_components, 否)；依完整明示資料獨立重算；linear／box／concatenated paths逐段exact核對；有限samples、graph walk或單一端點對不足；connected一般不推出path connected。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["道路連通空間"]},"variant":{"family_id":"qf-mod48-path-connected-boundary","seed":202611176080,"parameters":{"mode":"disconnected_gap","scale":5,"domain":"Exact_explicit_path_connectedness","operation":"path-connected-boundary","semantic_template":"path-connected-boundary|{mode:\"disconnected_gap\",scale:[5,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r06-path-connected-space-0081","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","language":"zh-Hant","source_mko_ids":["mko-path-connected-space"],"title_zh":"道路連通空間","stem_zh":"unit-interval path的continuity、兩端點與全range containment責任。操作=path-connected-boundary；資料={\"mode\":\"discrete_two_point\",\"scale\":6}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(discrete_two_point, 6, X={0,6}_with_discrete_topology, continuous_image_of_connected_[0,1]_cannot_meet_both_points, 否)","explanation_zh":"Wave6第十一層核對=(discrete_two_point, 6, X={0,6}_with_discrete_topology, continuous_image_of_connected_[0,1]_cannot_meet_both_points, 否)；依完整明示資料獨立重算；linear／box／concatenated paths逐段exact核對；有限samples、graph walk或單一端點對不足；connected一般不推出path connected。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["道路連通空間"]},"variant":{"family_id":"qf-mod48-path-connected-boundary","seed":202611176081,"parameters":{"mode":"discrete_two_point","scale":6,"domain":"Exact_explicit_path_connectedness","operation":"path-connected-boundary","semantic_template":"path-connected-boundary|{mode:\"discrete_two_point\",scale:[6,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r06-path-connected-space-0082","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","language":"zh-Hant","source_mko_ids":["mko-path-connected-space"],"title_zh":"道路連通空間","stem_zh":"unit-interval path的continuity、兩端點與全range containment責任。操作=path-connected-boundary；資料={\"mode\":\"endpoint_only\",\"scale\":7}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(endpoint_only, 7, candidate_samples_gamma(0)=0_gamma(1/2)=7_gamma(1)=0, three_samples_do_not_define_a_continuous_map_or_control_its_full_range, 否)","explanation_zh":"Wave6第十一層核對=(endpoint_only, 7, candidate_samples_gamma(0)=0_gamma(1/2)=7_gamma(1)=0, three_samples_do_not_define_a_continuous_map_or_control_its_full_range, 否)；依完整明示資料獨立重算；linear／box／concatenated paths逐段exact核對；有限samples、graph walk或單一端點對不足；connected一般不推出path connected。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["道路連通空間"]},"variant":{"family_id":"qf-mod48-path-connected-boundary","seed":202611176082,"parameters":{"mode":"endpoint_only","scale":7,"domain":"Exact_explicit_path_connectedness","operation":"path-connected-boundary","semantic_template":"path-connected-boundary|{mode:\"endpoint_only\",scale:[7,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r06-path-connected-space-0083","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","language":"zh-Hant","source_mko_ids":["mko-path-connected-space"],"title_zh":"道路連通空間","stem_zh":"unit-interval path的continuity、兩端點與全range containment責任。操作=path-connected-boundary；資料={\"mode\":\"connected_not_path\",\"scale\":8}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(connected_not_path, 8, topologists_sine_curve_scaled, connected_but_not_path_connected, 否)","explanation_zh":"Wave6第十一層核對=(connected_not_path, 8, topologists_sine_curve_scaled, connected_but_not_path_connected, 否)；依完整明示資料獨立重算；linear／box／concatenated paths逐段exact核對；有限samples、graph walk或單一端點對不足；connected一般不推出path connected。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["道路連通空間"]},"variant":{"family_id":"qf-mod48-path-connected-boundary","seed":202611176083,"parameters":{"mode":"connected_not_path","scale":8,"domain":"Exact_explicit_path_connectedness","operation":"path-connected-boundary","semantic_template":"path-connected-boundary|{mode:\"connected_not_path\",scale:[8,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r06-path-connected-space-0084","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","language":"zh-Hant","source_mko_ids":["mko-path-connected-space"],"title_zh":"道路連通空間","stem_zh":"unit-interval path的continuity、兩端點與全range containment責任。操作=path-connected-boundary；資料={\"mode\":\"disconnected_gap\",\"scale\":9}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(disconnected_gap, 9, X=(-infinity,0] union [9,infinity), IVT_blocks_a_path_between_components, 否)","explanation_zh":"Wave6第十一層核對=(disconnected_gap, 9, X=(-infinity,0] union [9,infinity), IVT_blocks_a_path_between_components, 否)；依完整明示資料獨立重算；linear／box／concatenated paths逐段exact核對；有限samples、graph walk或單一端點對不足；connected一般不推出path connected。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["道路連通空間"]},"variant":{"family_id":"qf-mod48-path-connected-boundary","seed":202611176084,"parameters":{"mode":"disconnected_gap","scale":9,"domain":"Exact_explicit_path_connectedness","operation":"path-connected-boundary","semantic_template":"path-connected-boundary|{mode:\"disconnected_gap\",scale:[9,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r06-path-connected-space-0085","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","language":"zh-Hant","source_mko_ids":["mko-path-connected-space"],"title_zh":"道路連通空間","stem_zh":"unit-interval path的continuity、兩端點與全range containment責任。操作=path-connected-boundary；資料={\"mode\":\"discrete_two_point\",\"scale\":10}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(discrete_two_point, 10, X={0,10}_with_discrete_topology, continuous_image_of_connected_[0,1]_cannot_meet_both_points, 否)","explanation_zh":"Wave6第十一層核對=(discrete_two_point, 10, X={0,10}_with_discrete_topology, continuous_image_of_connected_[0,1]_cannot_meet_both_points, 否)；依完整明示資料獨立重算；linear／box／concatenated paths逐段exact核對；有限samples、graph walk或單一端點對不足；connected一般不推出path connected。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["道路連通空間"]},"variant":{"family_id":"qf-mod48-path-connected-boundary","seed":202611176085,"parameters":{"mode":"discrete_two_point","scale":10,"domain":"Exact_explicit_path_connectedness","operation":"path-connected-boundary","semantic_template":"path-connected-boundary|{mode:\"discrete_two_point\",scale:[10,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r06-path-connected-space-0086","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","language":"zh-Hant","source_mko_ids":["mko-path-connected-space"],"title_zh":"道路連通空間","stem_zh":"unit-interval path的continuity、兩端點與全range containment責任。操作=path-connected-boundary；資料={\"mode\":\"endpoint_only\",\"scale\":11}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(endpoint_only, 11, candidate_samples_gamma(0)=0_gamma(1/2)=11_gamma(1)=0, three_samples_do_not_define_a_continuous_map_or_control_its_full_range, 否)","explanation_zh":"Wave6第十一層核對=(endpoint_only, 11, candidate_samples_gamma(0)=0_gamma(1/2)=11_gamma(1)=0, three_samples_do_not_define_a_continuous_map_or_control_its_full_range, 否)；依完整明示資料獨立重算；linear／box／concatenated paths逐段exact核對；有限samples、graph walk或單一端點對不足；connected一般不推出path connected。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["道路連通空間"]},"variant":{"family_id":"qf-mod48-path-connected-boundary","seed":202611176086,"parameters":{"mode":"endpoint_only","scale":11,"domain":"Exact_explicit_path_connectedness","operation":"path-connected-boundary","semantic_template":"path-connected-boundary|{mode:\"endpoint_only\",scale:[11,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r06-path-connected-space-0087","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","language":"zh-Hant","source_mko_ids":["mko-path-connected-space"],"title_zh":"道路連通空間","stem_zh":"unit-interval path的continuity、兩端點與全range containment責任。操作=path-connected-boundary；資料={\"mode\":\"connected_not_path\",\"scale\":12}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(connected_not_path, 12, topologists_sine_curve_scaled, connected_but_not_path_connected, 否)","explanation_zh":"Wave6第十一層核對=(connected_not_path, 12, topologists_sine_curve_scaled, connected_but_not_path_connected, 否)；依完整明示資料獨立重算；linear／box／concatenated paths逐段exact核對；有限samples、graph walk或單一端點對不足；connected一般不推出path connected。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["道路連通空間"]},"variant":{"family_id":"qf-mod48-path-connected-boundary","seed":202611176087,"parameters":{"mode":"connected_not_path","scale":12,"domain":"Exact_explicit_path_connectedness","operation":"path-connected-boundary","semantic_template":"path-connected-boundary|{mode:\"connected_not_path\",scale:[12,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r06-path-connected-space-0088","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","language":"zh-Hant","source_mko_ids":["mko-path-connected-space"],"title_zh":"道路連通空間","stem_zh":"unit-interval path的continuity、兩端點與全range containment責任。操作=path-connected-boundary；資料={\"mode\":\"disconnected_gap\",\"scale\":13}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(disconnected_gap, 13, X=(-infinity,0] union [13,infinity), IVT_blocks_a_path_between_components, 否)","explanation_zh":"Wave6第十一層核對=(disconnected_gap, 13, X=(-infinity,0] union [13,infinity), IVT_blocks_a_path_between_components, 否)；依完整明示資料獨立重算；linear／box／concatenated paths逐段exact核對；有限samples、graph walk或單一端點對不足；connected一般不推出path connected。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["道路連通空間"]},"variant":{"family_id":"qf-mod48-path-connected-boundary","seed":202611176088,"parameters":{"mode":"disconnected_gap","scale":13,"domain":"Exact_explicit_path_connectedness","operation":"path-connected-boundary","semantic_template":"path-connected-boundary|{mode:\"disconnected_gap\",scale:[13,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r06-path-connected-space-0089","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","language":"zh-Hant","source_mko_ids":["mko-path-connected-space"],"title_zh":"道路連通空間","stem_zh":"unit-interval path的continuity、兩端點與全range containment責任。操作=path-connected-boundary；資料={\"mode\":\"discrete_two_point\",\"scale\":14}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(discrete_two_point, 14, X={0,14}_with_discrete_topology, continuous_image_of_connected_[0,1]_cannot_meet_both_points, 否)","explanation_zh":"Wave6第十一層核對=(discrete_two_point, 14, X={0,14}_with_discrete_topology, continuous_image_of_connected_[0,1]_cannot_meet_both_points, 否)；依完整明示資料獨立重算；linear／box／concatenated paths逐段exact核對；有限samples、graph walk或單一端點對不足；connected一般不推出path connected。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["道路連通空間"]},"variant":{"family_id":"qf-mod48-path-connected-boundary","seed":202611176089,"parameters":{"mode":"discrete_two_point","scale":14,"domain":"Exact_explicit_path_connectedness","operation":"path-connected-boundary","semantic_template":"path-connected-boundary|{mode:\"discrete_two_point\",scale:[14,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r06-path-connected-space-0090","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","language":"zh-Hant","source_mko_ids":["mko-path-connected-space"],"title_zh":"道路連通空間","stem_zh":"unit-interval path的continuity、兩端點與全range containment責任。操作=path-connected-boundary；資料={\"mode\":\"endpoint_only\",\"scale\":15}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(endpoint_only, 15, candidate_samples_gamma(0)=0_gamma(1/2)=15_gamma(1)=0, three_samples_do_not_define_a_continuous_map_or_control_its_full_range, 否)","explanation_zh":"Wave6第十一層核對=(endpoint_only, 15, candidate_samples_gamma(0)=0_gamma(1/2)=15_gamma(1)=0, three_samples_do_not_define_a_continuous_map_or_control_its_full_range, 否)；依完整明示資料獨立重算；linear／box／concatenated paths逐段exact核對；有限samples、graph walk或單一端點對不足；connected一般不推出path connected。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["道路連通空間"]},"variant":{"family_id":"qf-mod48-path-connected-boundary","seed":202611176090,"parameters":{"mode":"endpoint_only","scale":15,"domain":"Exact_explicit_path_connectedness","operation":"path-connected-boundary","semantic_template":"path-connected-boundary|{mode:\"endpoint_only\",scale:[15,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r06-path-connected-space-0091","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","language":"zh-Hant","source_mko_ids":["mko-path-connected-space"],"title_zh":"道路連通空間","stem_zh":"unit-interval path的continuity、兩端點與全range containment責任。操作=path-connected-boundary；資料={\"mode\":\"connected_not_path\",\"scale\":16}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(connected_not_path, 16, topologists_sine_curve_scaled, connected_but_not_path_connected, 否)","explanation_zh":"Wave6第十一層核對=(connected_not_path, 16, topologists_sine_curve_scaled, connected_but_not_path_connected, 否)；依完整明示資料獨立重算；linear／box／concatenated paths逐段exact核對；有限samples、graph walk或單一端點對不足；connected一般不推出path connected。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["道路連通空間"]},"variant":{"family_id":"qf-mod48-path-connected-boundary","seed":202611176091,"parameters":{"mode":"connected_not_path","scale":16,"domain":"Exact_explicit_path_connectedness","operation":"path-connected-boundary","semantic_template":"path-connected-boundary|{mode:\"connected_not_path\",scale:[16,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r06-path-connected-space-0092","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","language":"zh-Hant","source_mko_ids":["mko-path-connected-space"],"title_zh":"道路連通空間","stem_zh":"unit-interval path的continuity、兩端點與全range containment責任。操作=path-connected-boundary；資料={\"mode\":\"disconnected_gap\",\"scale\":17}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(disconnected_gap, 17, X=(-infinity,0] union [17,infinity), IVT_blocks_a_path_between_components, 否)","explanation_zh":"Wave6第十一層核對=(disconnected_gap, 17, X=(-infinity,0] union [17,infinity), IVT_blocks_a_path_between_components, 否)；依完整明示資料獨立重算；linear／box／concatenated paths逐段exact核對；有限samples、graph walk或單一端點對不足；connected一般不推出path connected。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["道路連通空間"]},"variant":{"family_id":"qf-mod48-path-connected-boundary","seed":202611176092,"parameters":{"mode":"disconnected_gap","scale":17,"domain":"Exact_explicit_path_connectedness","operation":"path-connected-boundary","semantic_template":"path-connected-boundary|{mode:\"disconnected_gap\",scale:[17,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r06-path-connected-space-0093","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","language":"zh-Hant","source_mko_ids":["mko-path-connected-space"],"title_zh":"道路連通空間","stem_zh":"unit-interval path的continuity、兩端點與全range containment責任。操作=path-connected-boundary；資料={\"mode\":\"discrete_two_point\",\"scale\":18}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(discrete_two_point, 18, X={0,18}_with_discrete_topology, continuous_image_of_connected_[0,1]_cannot_meet_both_points, 否)","explanation_zh":"Wave6第十一層核對=(discrete_two_point, 18, X={0,18}_with_discrete_topology, continuous_image_of_connected_[0,1]_cannot_meet_both_points, 否)；依完整明示資料獨立重算；linear／box／concatenated paths逐段exact核對；有限samples、graph walk或單一端點對不足；connected一般不推出path connected。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["道路連通空間"]},"variant":{"family_id":"qf-mod48-path-connected-boundary","seed":202611176093,"parameters":{"mode":"discrete_two_point","scale":18,"domain":"Exact_explicit_path_connectedness","operation":"path-connected-boundary","semantic_template":"path-connected-boundary|{mode:\"discrete_two_point\",scale:[18,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r06-path-connected-space-0094","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","language":"zh-Hant","source_mko_ids":["mko-path-connected-space"],"title_zh":"道路連通空間","stem_zh":"unit-interval path的continuity、兩端點與全range containment責任。操作=path-connected-boundary；資料={\"mode\":\"endpoint_only\",\"scale\":19}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(endpoint_only, 19, candidate_samples_gamma(0)=0_gamma(1/2)=19_gamma(1)=0, three_samples_do_not_define_a_continuous_map_or_control_its_full_range, 否)","explanation_zh":"Wave6第十一層核對=(endpoint_only, 19, candidate_samples_gamma(0)=0_gamma(1/2)=19_gamma(1)=0, three_samples_do_not_define_a_continuous_map_or_control_its_full_range, 否)；依完整明示資料獨立重算；linear／box／concatenated paths逐段exact核對；有限samples、graph walk或單一端點對不足；connected一般不推出path connected。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["道路連通空間"]},"variant":{"family_id":"qf-mod48-path-connected-boundary","seed":202611176094,"parameters":{"mode":"endpoint_only","scale":19,"domain":"Exact_explicit_path_connectedness","operation":"path-connected-boundary","semantic_template":"path-connected-boundary|{mode:\"endpoint_only\",scale:[19,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r06-path-connected-space-0095","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","language":"zh-Hant","source_mko_ids":["mko-path-connected-space"],"title_zh":"道路連通空間","stem_zh":"unit-interval path的continuity、兩端點與全range containment責任。操作=path-connected-boundary；資料={\"mode\":\"connected_not_path\",\"scale\":20}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(connected_not_path, 20, topologists_sine_curve_scaled, connected_but_not_path_connected, 否)","explanation_zh":"Wave6第十一層核對=(connected_not_path, 20, topologists_sine_curve_scaled, connected_but_not_path_connected, 否)；依完整明示資料獨立重算；linear／box／concatenated paths逐段exact核對；有限samples、graph walk或單一端點對不足；connected一般不推出path connected。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["道路連通空間"]},"variant":{"family_id":"qf-mod48-path-connected-boundary","seed":202611176095,"parameters":{"mode":"connected_not_path","scale":20,"domain":"Exact_explicit_path_connectedness","operation":"path-connected-boundary","semantic_template":"path-connected-boundary|{mode:\"connected_not_path\",scale:[20,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r06-path-connected-space-0096","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","language":"zh-Hant","source_mko_ids":["mko-path-connected-space"],"title_zh":"道路連通空間","stem_zh":"unit-interval path的continuity、兩端點與全range containment責任。操作=path-connected-boundary；資料={\"mode\":\"disconnected_gap\",\"scale\":21}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(disconnected_gap, 21, X=(-infinity,0] union [21,infinity), IVT_blocks_a_path_between_components, 否)","explanation_zh":"Wave6第十一層核對=(disconnected_gap, 21, X=(-infinity,0] union [21,infinity), IVT_blocks_a_path_between_components, 否)；依完整明示資料獨立重算；linear／box／concatenated paths逐段exact核對；有限samples、graph walk或單一端點對不足；connected一般不推出path connected。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["道路連通空間"]},"variant":{"family_id":"qf-mod48-path-connected-boundary","seed":202611176096,"parameters":{"mode":"disconnected_gap","scale":21,"domain":"Exact_explicit_path_connectedness","operation":"path-connected-boundary","semantic_template":"path-connected-boundary|{mode:\"disconnected_gap\",scale:[21,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r06-path-connected-space-0097","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","language":"zh-Hant","source_mko_ids":["mko-path-connected-space"],"title_zh":"道路連通空間","stem_zh":"unit-interval path的continuity、兩端點與全range containment責任。操作=path-connected-boundary；資料={\"mode\":\"discrete_two_point\",\"scale\":22}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(discrete_two_point, 22, X={0,22}_with_discrete_topology, continuous_image_of_connected_[0,1]_cannot_meet_both_points, 否)","explanation_zh":"Wave6第十一層核對=(discrete_two_point, 22, X={0,22}_with_discrete_topology, continuous_image_of_connected_[0,1]_cannot_meet_both_points, 否)；依完整明示資料獨立重算；linear／box／concatenated paths逐段exact核對；有限samples、graph walk或單一端點對不足；connected一般不推出path connected。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["道路連通空間"]},"variant":{"family_id":"qf-mod48-path-connected-boundary","seed":202611176097,"parameters":{"mode":"discrete_two_point","scale":22,"domain":"Exact_explicit_path_connectedness","operation":"path-connected-boundary","semantic_template":"path-connected-boundary|{mode:\"discrete_two_point\",scale:[22,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r06-path-connected-space-0098","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","language":"zh-Hant","source_mko_ids":["mko-path-connected-space"],"title_zh":"道路連通空間","stem_zh":"unit-interval path的continuity、兩端點與全range containment責任。操作=path-connected-boundary；資料={\"mode\":\"endpoint_only\",\"scale\":23}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(endpoint_only, 23, candidate_samples_gamma(0)=0_gamma(1/2)=23_gamma(1)=0, three_samples_do_not_define_a_continuous_map_or_control_its_full_range, 否)","explanation_zh":"Wave6第十一層核對=(endpoint_only, 23, candidate_samples_gamma(0)=0_gamma(1/2)=23_gamma(1)=0, three_samples_do_not_define_a_continuous_map_or_control_its_full_range, 否)；依完整明示資料獨立重算；linear／box／concatenated paths逐段exact核對；有限samples、graph walk或單一端點對不足；connected一般不推出path connected。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["道路連通空間"]},"variant":{"family_id":"qf-mod48-path-connected-boundary","seed":202611176098,"parameters":{"mode":"endpoint_only","scale":23,"domain":"Exact_explicit_path_connectedness","operation":"path-connected-boundary","semantic_template":"path-connected-boundary|{mode:\"endpoint_only\",scale:[23,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r06-path-connected-space-0099","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","language":"zh-Hant","source_mko_ids":["mko-path-connected-space"],"title_zh":"道路連通空間","stem_zh":"unit-interval path的continuity、兩端點與全range containment責任。操作=path-connected-boundary；資料={\"mode\":\"connected_not_path\",\"scale\":24}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(connected_not_path, 24, topologists_sine_curve_scaled, connected_but_not_path_connected, 否)","explanation_zh":"Wave6第十一層核對=(connected_not_path, 24, topologists_sine_curve_scaled, connected_but_not_path_connected, 否)；依完整明示資料獨立重算；linear／box／concatenated paths逐段exact核對；有限samples、graph walk或單一端點對不足；connected一般不推出path connected。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["道路連通空間"]},"variant":{"family_id":"qf-mod48-path-connected-boundary","seed":202611176099,"parameters":{"mode":"connected_not_path","scale":24,"domain":"Exact_explicit_path_connectedness","operation":"path-connected-boundary","semantic_template":"path-connected-boundary|{mode:\"connected_not_path\",scale:[24,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r06-path-connected-space-0100","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","language":"zh-Hant","source_mko_ids":["mko-path-connected-space"],"title_zh":"道路連通空間","stem_zh":"unit-interval path的continuity、兩端點與全range containment責任。操作=path-connected-boundary；資料={\"mode\":\"disconnected_gap\",\"scale\":25}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(disconnected_gap, 25, X=(-infinity,0] union [25,infinity), IVT_blocks_a_path_between_components, 否)","explanation_zh":"Wave6第十一層核對=(disconnected_gap, 25, X=(-infinity,0] union [25,infinity), IVT_blocks_a_path_between_components, 否)；依完整明示資料獨立重算；linear／box／concatenated paths逐段exact核對；有限samples、graph walk或單一端點對不足；connected一般不推出path connected。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["道路連通空間"]},"variant":{"family_id":"qf-mod48-path-connected-boundary","seed":202611176100,"parameters":{"mode":"disconnected_gap","scale":25,"domain":"Exact_explicit_path_connectedness","operation":"path-connected-boundary","semantic_template":"path-connected-boundary|{mode:\"disconnected_gap\",scale:[25,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}