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道路連通空間基礎 · 脈衝 R06
本批已通過 Schema、去重、來源快照與獨立答案重算；仍未標記為人工審定。
本題庫已有對應的中英逐步詳解。原題的 candidate 與英文 ai_candidate 狀態保留；AI 覆核、有限計算與 Lean 形式證明分開標示。
batch-20261015-advance-20261117-r06-path-connected-space-foundations · 2026-10-15 · 100 題
欄序為來源綁定的呈現補充；原始 JSONL 與數學答案不變。
欄序與來源綁定資料
1. 道路連通空間
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 [left, right, start, end, path_formula, start_statement, end_statement, range_certificate, certificate] 欄序回答。
查看答案與解釋
(-2, 3, 0, 1, gamma(t)=0+(1)t, gamma(0)=start, gamma(1)=end, 是, convex_combination_stays_in_interval)
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。
本題逐步詳解（AI 二次覆核）
道路連通空間 · qf-mod48-path-interval-linear · candidate
2. 道路連通空間
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 [left, right, start, end, path_formula, start_statement, end_statement, range_certificate, certificate] 欄序回答。
查看答案與解釋
(-3, 4, -1, 2, gamma(t)=-1+(3)t, gamma(0)=start, gamma(1)=end, 是, convex_combination_stays_in_interval)
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。
本題逐步詳解（AI 二次覆核）
道路連通空間 · qf-mod48-path-interval-linear · candidate
3. 道路連通空間
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 [left, right, start, end, path_formula, start_statement, end_statement, range_certificate, certificate] 欄序回答。
查看答案與解釋
(-4, 5, -2, 3, gamma(t)=-2+(5)t, gamma(0)=start, gamma(1)=end, 是, convex_combination_stays_in_interval)
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。
本題逐步詳解（AI 二次覆核）
道路連通空間 · qf-mod48-path-interval-linear · candidate
4. 道路連通空間
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 [left, right, start, end, path_formula, start_statement, end_statement, range_certificate, certificate] 欄序回答。
查看答案與解釋
(-5, 6, -3, 4, gamma(t)=-3+(7)t, gamma(0)=start, gamma(1)=end, 是, convex_combination_stays_in_interval)
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。
本題逐步詳解（AI 二次覆核）
道路連通空間 · qf-mod48-path-interval-linear · candidate
5. 道路連通空間
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 [left, right, start, end, path_formula, start_statement, end_statement, range_certificate, certificate] 欄序回答。
查看答案與解釋
(-6, 7, -4, 5, gamma(t)=-4+(9)t, gamma(0)=start, gamma(1)=end, 是, convex_combination_stays_in_interval)
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。
本題逐步詳解（AI 二次覆核）
道路連通空間 · qf-mod48-path-interval-linear · candidate
6. 道路連通空間
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 [left, right, start, end, path_formula, start_statement, end_statement, range_certificate, certificate] 欄序回答。
查看答案與解釋
(-7, 8, -5, 6, gamma(t)=-5+(11)t, gamma(0)=start, gamma(1)=end, 是, convex_combination_stays_in_interval)
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。
本題逐步詳解（AI 二次覆核）
道路連通空間 · qf-mod48-path-interval-linear · candidate
7. 道路連通空間
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 [left, right, start, end, path_formula, start_statement, end_statement, range_certificate, certificate] 欄序回答。
查看答案與解釋
(-8, 9, -6, 7, gamma(t)=-6+(13)t, gamma(0)=start, gamma(1)=end, 是, convex_combination_stays_in_interval)
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。
本題逐步詳解（AI 二次覆核）
道路連通空間 · qf-mod48-path-interval-linear · candidate
8. 道路連通空間
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 [left, right, start, end, path_formula, start_statement, end_statement, range_certificate, certificate] 欄序回答。
查看答案與解釋
(-9, 10, -7, 8, gamma(t)=-7+(15)t, gamma(0)=start, gamma(1)=end, 是, convex_combination_stays_in_interval)
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。
本題逐步詳解（AI 二次覆核）
道路連通空間 · qf-mod48-path-interval-linear · candidate
9. 道路連通空間
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 [left, right, start, end, path_formula, start_statement, end_statement, range_certificate, certificate] 欄序回答。
查看答案與解釋
(-10, 11, -8, 9, gamma(t)=-8+(17)t, gamma(0)=start, gamma(1)=end, 是, convex_combination_stays_in_interval)
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。
本題逐步詳解（AI 二次覆核）
道路連通空間 · qf-mod48-path-interval-linear · candidate
10. 道路連通空間
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 [left, right, start, end, path_formula, start_statement, end_statement, range_certificate, certificate] 欄序回答。
查看答案與解釋
(-11, 12, -9, 10, gamma(t)=-9+(19)t, gamma(0)=start, gamma(1)=end, 是, convex_combination_stays_in_interval)
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。
本題逐步詳解（AI 二次覆核）
道路連通空間 · qf-mod48-path-interval-linear · candidate
11. 道路連通空間
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 [left, right, start, end, path_formula, start_statement, end_statement, range_certificate, certificate] 欄序回答。
查看答案與解釋
(-12, 13, -10, 11, gamma(t)=-10+(21)t, gamma(0)=start, gamma(1)=end, 是, convex_combination_stays_in_interval)
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。
本題逐步詳解（AI 二次覆核）
道路連通空間 · qf-mod48-path-interval-linear · candidate
12. 道路連通空間
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 [left, right, start, end, path_formula, start_statement, end_statement, range_certificate, certificate] 欄序回答。
查看答案與解釋
(-13, 14, -11, 12, gamma(t)=-11+(23)t, gamma(0)=start, gamma(1)=end, 是, convex_combination_stays_in_interval)
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。
本題逐步詳解（AI 二次覆核）
道路連通空間 · qf-mod48-path-interval-linear · candidate
13. 道路連通空間
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 [left, right, start, end, path_formula, start_statement, end_statement, range_certificate, certificate] 欄序回答。
查看答案與解釋
(-14, 15, -12, 13, gamma(t)=-12+(25)t, gamma(0)=start, gamma(1)=end, 是, convex_combination_stays_in_interval)
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。
本題逐步詳解（AI 二次覆核）
道路連通空間 · qf-mod48-path-interval-linear · candidate
14. 道路連通空間
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 [left, right, start, end, path_formula, start_statement, end_statement, range_certificate, certificate] 欄序回答。
查看答案與解釋
(-15, 16, -13, 14, gamma(t)=-13+(27)t, gamma(0)=start, gamma(1)=end, 是, convex_combination_stays_in_interval)
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。
本題逐步詳解（AI 二次覆核）
道路連通空間 · qf-mod48-path-interval-linear · candidate
15. 道路連通空間
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 [left, right, start, end, path_formula, start_statement, end_statement, range_certificate, certificate] 欄序回答。
查看答案與解釋
(-16, 17, -14, 15, gamma(t)=-14+(29)t, gamma(0)=start, gamma(1)=end, 是, convex_combination_stays_in_interval)
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。
本題逐步詳解（AI 二次覆核）
道路連通空間 · qf-mod48-path-interval-linear · candidate
16. 道路連通空間
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 [left, right, start, end, path_formula, start_statement, end_statement, range_certificate, certificate] 欄序回答。
查看答案與解釋
(-17, 18, -15, 16, gamma(t)=-15+(31)t, gamma(0)=start, gamma(1)=end, 是, convex_combination_stays_in_interval)
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。
本題逐步詳解（AI 二次覆核）
道路連通空間 · qf-mod48-path-interval-linear · candidate
17. 道路連通空間
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 [left, right, start, end, path_formula, start_statement, end_statement, range_certificate, certificate] 欄序回答。
查看答案與解釋
(-18, 19, -16, 17, gamma(t)=-16+(33)t, gamma(0)=start, gamma(1)=end, 是, convex_combination_stays_in_interval)
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。
本題逐步詳解（AI 二次覆核）
道路連通空間 · qf-mod48-path-interval-linear · candidate
18. 道路連通空間
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 [left, right, start, end, path_formula, start_statement, end_statement, range_certificate, certificate] 欄序回答。
查看答案與解釋
(-19, 20, -17, 18, gamma(t)=-17+(35)t, gamma(0)=start, gamma(1)=end, 是, convex_combination_stays_in_interval)
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。
本題逐步詳解（AI 二次覆核）
道路連通空間 · qf-mod48-path-interval-linear · candidate
19. 道路連通空間
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 [left, right, start, end, path_formula, start_statement, end_statement, range_certificate, certificate] 欄序回答。
查看答案與解釋
(-20, 21, -18, 19, gamma(t)=-18+(37)t, gamma(0)=start, gamma(1)=end, 是, convex_combination_stays_in_interval)
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。
本題逐步詳解（AI 二次覆核）
道路連通空間 · qf-mod48-path-interval-linear · candidate
20. 道路連通空間
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 [left, right, start, end, path_formula, start_statement, end_statement, range_certificate, certificate] 欄序回答。
查看答案與解釋
(-21, 22, -19, 20, gamma(t)=-19+(39)t, gamma(0)=start, gamma(1)=end, 是, convex_combination_stays_in_interval)
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。
本題逐步詳解（AI 二次覆核）
道路連通空間 · qf-mod48-path-interval-linear · candidate
21. 道路連通空間
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 [left, right, start, end, path_formula, start_statement, end_statement, range_certificate, certificate] 欄序回答。
查看答案與解釋
(-22, 23, -20, 21, gamma(t)=-20+(41)t, gamma(0)=start, gamma(1)=end, 是, convex_combination_stays_in_interval)
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。
本題逐步詳解（AI 二次覆核）
道路連通空間 · qf-mod48-path-interval-linear · candidate
22. 道路連通空間
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 [left, right, start, end, path_formula, start_statement, end_statement, range_certificate, certificate] 欄序回答。
查看答案與解釋
(-23, 24, -21, 22, gamma(t)=-21+(43)t, gamma(0)=start, gamma(1)=end, 是, convex_combination_stays_in_interval)
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。
本題逐步詳解（AI 二次覆核）
道路連通空間 · qf-mod48-path-interval-linear · candidate
23. 道路連通空間
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 [left, right, start, end, path_formula, start_statement, end_statement, range_certificate, certificate] 欄序回答。
查看答案與解釋
(-24, 25, -22, 23, gamma(t)=-22+(45)t, gamma(0)=start, gamma(1)=end, 是, convex_combination_stays_in_interval)
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。
本題逐步詳解（AI 二次覆核）
道路連通空間 · qf-mod48-path-interval-linear · candidate
24. 道路連通空間
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 [left, right, start, end, path_formula, start_statement, end_statement, range_certificate, certificate] 欄序回答。
查看答案與解釋
(-25, 26, -23, 24, gamma(t)=-23+(47)t, gamma(0)=start, gamma(1)=end, 是, convex_combination_stays_in_interval)
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。
本題逐步詳解（AI 二次覆核）
道路連通空間 · qf-mod48-path-interval-linear · candidate
25. 道路連通空間
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 [left, right, start, end, path_formula, start_statement, end_statement, range_certificate, certificate] 欄序回答。
查看答案與解釋
(-26, 27, -24, 25, gamma(t)=-24+(49)t, gamma(0)=start, gamma(1)=end, 是, convex_combination_stays_in_interval)
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。
本題逐步詳解（AI 二次覆核）
道路連通空間 · qf-mod48-path-interval-linear · candidate
26. 道路連通空間
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 [lower, upper, start, end, coordinate_formulas, range_certificate, certificate] 欄序回答。
查看答案與解釋
([-2, -3], [3, 4], [-1, -2], [2, 3], [-1+(3)t, -2+(5)t], 是, coordinatewise_convex_combination_stays_in_box)
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。
本題逐步詳解（AI 二次覆核）
道路連通空間 · qf-mod48-path-box-linear · candidate
27. 道路連通空間
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 [lower, upper, start, end, coordinate_formulas, range_certificate, certificate] 欄序回答。
查看答案與解釋
([-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)
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。
本題逐步詳解（AI 二次覆核）
道路連通空間 · qf-mod48-path-box-linear · candidate
28. 道路連通空間
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 [lower, upper, start, end, coordinate_formulas, range_certificate, certificate] 欄序回答。
查看答案與解釋
([-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)
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。
本題逐步詳解（AI 二次覆核）
道路連通空間 · qf-mod48-path-box-linear · candidate
29. 道路連通空間
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 [lower, upper, start, end, coordinate_formulas, range_certificate, certificate] 欄序回答。
查看答案與解釋
([-5, -6], [6, 7], [-4, -5], [5, 6], [-4+(9)t, -5+(11)t], 是, coordinatewise_convex_combination_stays_in_box)
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。
本題逐步詳解（AI 二次覆核）
道路連通空間 · qf-mod48-path-box-linear · candidate
30. 道路連通空間
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 [lower, upper, start, end, coordinate_formulas, range_certificate, certificate] 欄序回答。
查看答案與解釋
([-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)
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。
本題逐步詳解（AI 二次覆核）
道路連通空間 · qf-mod48-path-box-linear · candidate
31. 道路連通空間
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 [lower, upper, start, end, coordinate_formulas, range_certificate, certificate] 欄序回答。
查看答案與解釋
([-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)
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。
本題逐步詳解（AI 二次覆核）
道路連通空間 · qf-mod48-path-box-linear · candidate
32. 道路連通空間
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 [lower, upper, start, end, coordinate_formulas, range_certificate, certificate] 欄序回答。
查看答案與解釋
([-8, -9], [9, 10], [-7, -8], [8, 9], [-7+(15)t, -8+(17)t], 是, coordinatewise_convex_combination_stays_in_box)
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。
本題逐步詳解（AI 二次覆核）
道路連通空間 · qf-mod48-path-box-linear · candidate
33. 道路連通空間
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 [lower, upper, start, end, coordinate_formulas, range_certificate, certificate] 欄序回答。
查看答案與解釋
([-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)
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。
本題逐步詳解（AI 二次覆核）
道路連通空間 · qf-mod48-path-box-linear · candidate
34. 道路連通空間
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 [lower, upper, start, end, coordinate_formulas, range_certificate, certificate] 欄序回答。
查看答案與解釋
([-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)
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。
本題逐步詳解（AI 二次覆核）
道路連通空間 · qf-mod48-path-box-linear · candidate
35. 道路連通空間
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 [lower, upper, start, end, coordinate_formulas, range_certificate, certificate] 欄序回答。
查看答案與解釋
([-11, -12], [12, 13], [-10, -11], [11, 12], [-10+(21)t, -11+(23)t], 是, coordinatewise_convex_combination_stays_in_box)
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。
本題逐步詳解（AI 二次覆核）
道路連通空間 · qf-mod48-path-box-linear · candidate
36. 道路連通空間
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 [lower, upper, start, end, coordinate_formulas, range_certificate, certificate] 欄序回答。
查看答案與解釋
([-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)
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。
本題逐步詳解（AI 二次覆核）
道路連通空間 · qf-mod48-path-box-linear · candidate
37. 道路連通空間
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 [lower, upper, start, end, coordinate_formulas, range_certificate, certificate] 欄序回答。
查看答案與解釋
([-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)
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。
本題逐步詳解（AI 二次覆核）
道路連通空間 · qf-mod48-path-box-linear · candidate
38. 道路連通空間
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 [lower, upper, start, end, coordinate_formulas, range_certificate, certificate] 欄序回答。
查看答案與解釋
([-14, -15], [15, 16], [-13, -14], [14, 15], [-13+(27)t, -14+(29)t], 是, coordinatewise_convex_combination_stays_in_box)
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。
本題逐步詳解（AI 二次覆核）
道路連通空間 · qf-mod48-path-box-linear · candidate
39. 道路連通空間
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 [lower, upper, start, end, coordinate_formulas, range_certificate, certificate] 欄序回答。
查看答案與解釋
([-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)
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。
本題逐步詳解（AI 二次覆核）
道路連通空間 · qf-mod48-path-box-linear · candidate
40. 道路連通空間
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 [lower, upper, start, end, coordinate_formulas, range_certificate, certificate] 欄序回答。
查看答案與解釋
([-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)
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。
本題逐步詳解（AI 二次覆核）
道路連通空間 · qf-mod48-path-box-linear · candidate
41. 道路連通空間
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 [lower, upper, start, end, coordinate_formulas, range_certificate, certificate] 欄序回答。
查看答案與解釋
([-17, -18], [18, 19], [-16, -17], [17, 18], [-16+(33)t, -17+(35)t], 是, coordinatewise_convex_combination_stays_in_box)
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。
本題逐步詳解（AI 二次覆核）
道路連通空間 · qf-mod48-path-box-linear · candidate
42. 道路連通空間
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 [lower, upper, start, end, coordinate_formulas, range_certificate, certificate] 欄序回答。
查看答案與解釋
([-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)
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。
本題逐步詳解（AI 二次覆核）
道路連通空間 · qf-mod48-path-box-linear · candidate
43. 道路連通空間
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 [lower, upper, start, end, coordinate_formulas, range_certificate, certificate] 欄序回答。
查看答案與解釋
([-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)
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。
本題逐步詳解（AI 二次覆核）
道路連通空間 · qf-mod48-path-box-linear · candidate
44. 道路連通空間
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 [lower, upper, start, end, coordinate_formulas, range_certificate, certificate] 欄序回答。
查看答案與解釋
([-20, -21], [21, 22], [-19, -20], [20, 21], [-19+(39)t, -20+(41)t], 是, coordinatewise_convex_combination_stays_in_box)
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。
本題逐步詳解（AI 二次覆核）
道路連通空間 · qf-mod48-path-box-linear · candidate
45. 道路連通空間
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 [lower, upper, start, end, coordinate_formulas, range_certificate, certificate] 欄序回答。
查看答案與解釋
([-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)
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。
本題逐步詳解（AI 二次覆核）
道路連通空間 · qf-mod48-path-box-linear · candidate
46. 道路連通空間
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 [lower, upper, start, end, coordinate_formulas, range_certificate, certificate] 欄序回答。
查看答案與解釋
([-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)
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。
本題逐步詳解（AI 二次覆核）
道路連通空間 · qf-mod48-path-box-linear · candidate
47. 道路連通空間
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 [lower, upper, start, end, coordinate_formulas, range_certificate, certificate] 欄序回答。
查看答案與解釋
([-23, -24], [24, 25], [-22, -23], [23, 24], [-22+(45)t, -23+(47)t], 是, coordinatewise_convex_combination_stays_in_box)
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。
本題逐步詳解（AI 二次覆核）
道路連通空間 · qf-mod48-path-box-linear · candidate
48. 道路連通空間
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 [lower, upper, start, end, coordinate_formulas, range_certificate, certificate] 欄序回答。
查看答案與解釋
([-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)
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。
本題逐步詳解（AI 二次覆核）
道路連通空間 · qf-mod48-path-box-linear · candidate
49. 道路連通空間
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 [lower, upper, start, end, coordinate_formulas, range_certificate, certificate] 欄序回答。
查看答案與解釋
([-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)
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。
本題逐步詳解（AI 二次覆核）
道路連通空間 · qf-mod48-path-box-linear · candidate
50. 道路連通空間
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 [lower, upper, start, end, coordinate_formulas, range_certificate, certificate] 欄序回答。
查看答案與解釋
([-26, -27], [27, 28], [-25, -26], [26, 27], [-25+(51)t, -26+(53)t], 是, coordinatewise_convex_combination_stays_in_box)
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。
本題逐步詳解（AI 二次覆核）
道路連通空間 · qf-mod48-path-box-linear · candidate
51. 道路連通空間
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 [start, middle, end, first_branch, second_branch, join_statement, path_certificate] 欄序回答。
查看答案與解釋
(0, 2, 0, 0<=t<=1/2:0+2t(2), 1/2<=t<=1:2+(2t-1)(-2), both_branches_meet_at_middle, 是)
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。
本題逐步詳解（AI 二次覆核）
道路連通空間 · qf-mod48-path-concatenation · candidate
52. 道路連通空間
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 [start, middle, end, first_branch, second_branch, join_statement, path_certificate] 欄序回答。
查看答案與解釋
(-1, 3, 1, 0<=t<=1/2:-1+2t(4), 1/2<=t<=1:3+(2t-1)(-2), both_branches_meet_at_middle, 是)
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。
本題逐步詳解（AI 二次覆核）
道路連通空間 · qf-mod48-path-concatenation · candidate
53. 道路連通空間
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 [start, middle, end, first_branch, second_branch, join_statement, path_certificate] 欄序回答。
查看答案與解釋
(-2, 4, 2, 0<=t<=1/2:-2+2t(6), 1/2<=t<=1:4+(2t-1)(-2), both_branches_meet_at_middle, 是)
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。
本題逐步詳解（AI 二次覆核）
道路連通空間 · qf-mod48-path-concatenation · candidate
54. 道路連通空間
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 [start, middle, end, first_branch, second_branch, join_statement, path_certificate] 欄序回答。
查看答案與解釋
(-3, 5, 3, 0<=t<=1/2:-3+2t(8), 1/2<=t<=1:5+(2t-1)(-2), both_branches_meet_at_middle, 是)
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。
本題逐步詳解（AI 二次覆核）
道路連通空間 · qf-mod48-path-concatenation · candidate
55. 道路連通空間
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 [start, middle, end, first_branch, second_branch, join_statement, path_certificate] 欄序回答。
查看答案與解釋
(-4, 6, 4, 0<=t<=1/2:-4+2t(10), 1/2<=t<=1:6+(2t-1)(-2), both_branches_meet_at_middle, 是)
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。
本題逐步詳解（AI 二次覆核）
道路連通空間 · qf-mod48-path-concatenation · candidate
56. 道路連通空間
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 [start, middle, end, first_branch, second_branch, join_statement, path_certificate] 欄序回答。
查看答案與解釋
(-5, 7, 5, 0<=t<=1/2:-5+2t(12), 1/2<=t<=1:7+(2t-1)(-2), both_branches_meet_at_middle, 是)
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。
本題逐步詳解（AI 二次覆核）
道路連通空間 · qf-mod48-path-concatenation · candidate
57. 道路連通空間
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 [start, middle, end, first_branch, second_branch, join_statement, path_certificate] 欄序回答。
查看答案與解釋
(-6, 8, 6, 0<=t<=1/2:-6+2t(14), 1/2<=t<=1:8+(2t-1)(-2), both_branches_meet_at_middle, 是)
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。
本題逐步詳解（AI 二次覆核）
道路連通空間 · qf-mod48-path-concatenation · candidate
58. 道路連通空間
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 [start, middle, end, first_branch, second_branch, join_statement, path_certificate] 欄序回答。
查看答案與解釋
(-7, 9, 7, 0<=t<=1/2:-7+2t(16), 1/2<=t<=1:9+(2t-1)(-2), both_branches_meet_at_middle, 是)
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。
本題逐步詳解（AI 二次覆核）
道路連通空間 · qf-mod48-path-concatenation · candidate
59. 道路連通空間
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 [start, middle, end, first_branch, second_branch, join_statement, path_certificate] 欄序回答。
查看答案與解釋
(-8, 10, 8, 0<=t<=1/2:-8+2t(18), 1/2<=t<=1:10+(2t-1)(-2), both_branches_meet_at_middle, 是)
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。
本題逐步詳解（AI 二次覆核）
道路連通空間 · qf-mod48-path-concatenation · candidate
60. 道路連通空間
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 [start, middle, end, first_branch, second_branch, join_statement, path_certificate] 欄序回答。
查看答案與解釋
(-9, 11, 9, 0<=t<=1/2:-9+2t(20), 1/2<=t<=1:11+(2t-1)(-2), both_branches_meet_at_middle, 是)
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。
本題逐步詳解（AI 二次覆核）
道路連通空間 · qf-mod48-path-concatenation · candidate
61. 道路連通空間
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 [start, middle, end, first_branch, second_branch, join_statement, path_certificate] 欄序回答。
查看答案與解釋
(-10, 12, 10, 0<=t<=1/2:-10+2t(22), 1/2<=t<=1:12+(2t-1)(-2), both_branches_meet_at_middle, 是)
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。
本題逐步詳解（AI 二次覆核）
道路連通空間 · qf-mod48-path-concatenation · candidate
62. 道路連通空間
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 [start, middle, end, first_branch, second_branch, join_statement, path_certificate] 欄序回答。
查看答案與解釋
(-11, 13, 11, 0<=t<=1/2:-11+2t(24), 1/2<=t<=1:13+(2t-1)(-2), both_branches_meet_at_middle, 是)
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。
本題逐步詳解（AI 二次覆核）
道路連通空間 · qf-mod48-path-concatenation · candidate
63. 道路連通空間
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 [start, middle, end, first_branch, second_branch, join_statement, path_certificate] 欄序回答。
查看答案與解釋
(-12, 14, 12, 0<=t<=1/2:-12+2t(26), 1/2<=t<=1:14+(2t-1)(-2), both_branches_meet_at_middle, 是)
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。
本題逐步詳解（AI 二次覆核）
道路連通空間 · qf-mod48-path-concatenation · candidate
64. 道路連通空間
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 [start, middle, end, first_branch, second_branch, join_statement, path_certificate] 欄序回答。
查看答案與解釋
(-13, 15, 13, 0<=t<=1/2:-13+2t(28), 1/2<=t<=1:15+(2t-1)(-2), both_branches_meet_at_middle, 是)
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。
本題逐步詳解（AI 二次覆核）
道路連通空間 · qf-mod48-path-concatenation · candidate
65. 道路連通空間
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 [start, middle, end, first_branch, second_branch, join_statement, path_certificate] 欄序回答。
查看答案與解釋
(-14, 16, 14, 0<=t<=1/2:-14+2t(30), 1/2<=t<=1:16+(2t-1)(-2), both_branches_meet_at_middle, 是)
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。
本題逐步詳解（AI 二次覆核）
道路連通空間 · qf-mod48-path-concatenation · candidate
66. 道路連通空間
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 [start, middle, end, first_branch, second_branch, join_statement, path_certificate] 欄序回答。
查看答案與解釋
(-15, 17, 15, 0<=t<=1/2:-15+2t(32), 1/2<=t<=1:17+(2t-1)(-2), both_branches_meet_at_middle, 是)
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。
本題逐步詳解（AI 二次覆核）
道路連通空間 · qf-mod48-path-concatenation · candidate
67. 道路連通空間
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 [start, middle, end, first_branch, second_branch, join_statement, path_certificate] 欄序回答。
查看答案與解釋
(-16, 18, 16, 0<=t<=1/2:-16+2t(34), 1/2<=t<=1:18+(2t-1)(-2), both_branches_meet_at_middle, 是)
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。
本題逐步詳解（AI 二次覆核）
道路連通空間 · qf-mod48-path-concatenation · candidate
68. 道路連通空間
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 [start, middle, end, first_branch, second_branch, join_statement, path_certificate] 欄序回答。
查看答案與解釋
(-17, 19, 17, 0<=t<=1/2:-17+2t(36), 1/2<=t<=1:19+(2t-1)(-2), both_branches_meet_at_middle, 是)
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。
本題逐步詳解（AI 二次覆核）
道路連通空間 · qf-mod48-path-concatenation · candidate
69. 道路連通空間
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 [start, middle, end, first_branch, second_branch, join_statement, path_certificate] 欄序回答。
查看答案與解釋
(-18, 20, 18, 0<=t<=1/2:-18+2t(38), 1/2<=t<=1:20+(2t-1)(-2), both_branches_meet_at_middle, 是)
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。
本題逐步詳解（AI 二次覆核）
道路連通空間 · qf-mod48-path-concatenation · candidate
70. 道路連通空間
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 [start, middle, end, first_branch, second_branch, join_statement, path_certificate] 欄序回答。
查看答案與解釋
(-19, 21, 19, 0<=t<=1/2:-19+2t(40), 1/2<=t<=1:21+(2t-1)(-2), both_branches_meet_at_middle, 是)
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。
本題逐步詳解（AI 二次覆核）
道路連通空間 · qf-mod48-path-concatenation · candidate
71. 道路連通空間
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 [start, middle, end, first_branch, second_branch, join_statement, path_certificate] 欄序回答。
查看答案與解釋
(-20, 22, 20, 0<=t<=1/2:-20+2t(42), 1/2<=t<=1:22+(2t-1)(-2), both_branches_meet_at_middle, 是)
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。
本題逐步詳解（AI 二次覆核）
道路連通空間 · qf-mod48-path-concatenation · candidate
72. 道路連通空間
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 [start, middle, end, first_branch, second_branch, join_statement, path_certificate] 欄序回答。
查看答案與解釋
(-21, 23, 21, 0<=t<=1/2:-21+2t(44), 1/2<=t<=1:23+(2t-1)(-2), both_branches_meet_at_middle, 是)
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。
本題逐步詳解（AI 二次覆核）
道路連通空間 · qf-mod48-path-concatenation · candidate
73. 道路連通空間
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 [start, middle, end, first_branch, second_branch, join_statement, path_certificate] 欄序回答。
查看答案與解釋
(-22, 24, 22, 0<=t<=1/2:-22+2t(46), 1/2<=t<=1:24+(2t-1)(-2), both_branches_meet_at_middle, 是)
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。
本題逐步詳解（AI 二次覆核）
道路連通空間 · qf-mod48-path-concatenation · candidate
74. 道路連通空間
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 [start, middle, end, first_branch, second_branch, join_statement, path_certificate] 欄序回答。
查看答案與解釋
(-23, 25, 23, 0<=t<=1/2:-23+2t(48), 1/2<=t<=1:25+(2t-1)(-2), both_branches_meet_at_middle, 是)
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。
本題逐步詳解（AI 二次覆核）
道路連通空間 · qf-mod48-path-concatenation · candidate
75. 道路連通空間
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 [start, middle, end, first_branch, second_branch, join_statement, path_certificate] 欄序回答。
查看答案與解釋
(-24, 26, 24, 0<=t<=1/2:-24+2t(50), 1/2<=t<=1:26+(2t-1)(-2), both_branches_meet_at_middle, 是)
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。
本題逐步詳解（AI 二次覆核）
道路連通空間 · qf-mod48-path-concatenation · candidate
76. 道路連通空間
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 [mode, scale, example_description, scope_or_obstruction, path_connectedness_inference_claim] 欄序回答。
endpoint_only只給samples，不宣稱其有完整continuous path。connected_not_path是來源命名的sine-curve反例範圍，不是本題實作完整curve或一般反向定理證明。
查看答案與解釋
(disconnected_gap, 1, X=(-infinity,0] union [1,infinity), IVT_blocks_a_path_between_components, 否)
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。
本題逐步詳解（AI 二次覆核）
道路連通空間 · qf-mod48-path-connected-boundary · candidate
77. 道路連通空間
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 [mode, scale, example_description, scope_or_obstruction, path_connectedness_inference_claim] 欄序回答。
endpoint_only只給samples，不宣稱其有完整continuous path。connected_not_path是來源命名的sine-curve反例範圍，不是本題實作完整curve或一般反向定理證明。
查看答案與解釋
(discrete_two_point, 2, X={0,2}_with_discrete_topology, continuous_image_of_connected_[0,1]_cannot_meet_both_points, 否)
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。
本題逐步詳解（AI 二次覆核）
道路連通空間 · qf-mod48-path-connected-boundary · candidate
78. 道路連通空間
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 [mode, scale, example_description, scope_or_obstruction, path_connectedness_inference_claim] 欄序回答。
endpoint_only只給samples，不宣稱其有完整continuous path。connected_not_path是來源命名的sine-curve反例範圍，不是本題實作完整curve或一般反向定理證明。
查看答案與解釋
(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)
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。
本題逐步詳解（AI 二次覆核）
道路連通空間 · qf-mod48-path-connected-boundary · candidate
79. 道路連通空間
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 [mode, scale, example_description, scope_or_obstruction, path_connectedness_inference_claim] 欄序回答。
endpoint_only只給samples，不宣稱其有完整continuous path。connected_not_path是來源命名的sine-curve反例範圍，不是本題實作完整curve或一般反向定理證明。
查看答案與解釋
(connected_not_path, 4, topologists_sine_curve_scaled, connected_but_not_path_connected, 否)
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。
本題逐步詳解（AI 二次覆核）
道路連通空間 · qf-mod48-path-connected-boundary · candidate
80. 道路連通空間
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 [mode, scale, example_description, scope_or_obstruction, path_connectedness_inference_claim] 欄序回答。
endpoint_only只給samples，不宣稱其有完整continuous path。connected_not_path是來源命名的sine-curve反例範圍，不是本題實作完整curve或一般反向定理證明。
查看答案與解釋
(disconnected_gap, 5, X=(-infinity,0] union [5,infinity), IVT_blocks_a_path_between_components, 否)
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。
本題逐步詳解（AI 二次覆核）
道路連通空間 · qf-mod48-path-connected-boundary · candidate
81. 道路連通空間
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 [mode, scale, example_description, scope_or_obstruction, path_connectedness_inference_claim] 欄序回答。
endpoint_only只給samples，不宣稱其有完整continuous path。connected_not_path是來源命名的sine-curve反例範圍，不是本題實作完整curve或一般反向定理證明。
查看答案與解釋
(discrete_two_point, 6, X={0,6}_with_discrete_topology, continuous_image_of_connected_[0,1]_cannot_meet_both_points, 否)
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。
本題逐步詳解（AI 二次覆核）
道路連通空間 · qf-mod48-path-connected-boundary · candidate
82. 道路連通空間
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 [mode, scale, example_description, scope_or_obstruction, path_connectedness_inference_claim] 欄序回答。
endpoint_only只給samples，不宣稱其有完整continuous path。connected_not_path是來源命名的sine-curve反例範圍，不是本題實作完整curve或一般反向定理證明。
查看答案與解釋
(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)
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。
本題逐步詳解（AI 二次覆核）
道路連通空間 · qf-mod48-path-connected-boundary · candidate
83. 道路連通空間
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 [mode, scale, example_description, scope_or_obstruction, path_connectedness_inference_claim] 欄序回答。
endpoint_only只給samples，不宣稱其有完整continuous path。connected_not_path是來源命名的sine-curve反例範圍，不是本題實作完整curve或一般反向定理證明。
查看答案與解釋
(connected_not_path, 8, topologists_sine_curve_scaled, connected_but_not_path_connected, 否)
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。
本題逐步詳解（AI 二次覆核）
道路連通空間 · qf-mod48-path-connected-boundary · candidate
84. 道路連通空間
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 [mode, scale, example_description, scope_or_obstruction, path_connectedness_inference_claim] 欄序回答。
endpoint_only只給samples，不宣稱其有完整continuous path。connected_not_path是來源命名的sine-curve反例範圍，不是本題實作完整curve或一般反向定理證明。
查看答案與解釋
(disconnected_gap, 9, X=(-infinity,0] union [9,infinity), IVT_blocks_a_path_between_components, 否)
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。
本題逐步詳解（AI 二次覆核）
道路連通空間 · qf-mod48-path-connected-boundary · candidate
85. 道路連通空間
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 [mode, scale, example_description, scope_or_obstruction, path_connectedness_inference_claim] 欄序回答。
endpoint_only只給samples，不宣稱其有完整continuous path。connected_not_path是來源命名的sine-curve反例範圍，不是本題實作完整curve或一般反向定理證明。
查看答案與解釋
(discrete_two_point, 10, X={0,10}_with_discrete_topology, continuous_image_of_connected_[0,1]_cannot_meet_both_points, 否)
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。
本題逐步詳解（AI 二次覆核）
道路連通空間 · qf-mod48-path-connected-boundary · candidate
86. 道路連通空間
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 [mode, scale, example_description, scope_or_obstruction, path_connectedness_inference_claim] 欄序回答。
endpoint_only只給samples，不宣稱其有完整continuous path。connected_not_path是來源命名的sine-curve反例範圍，不是本題實作完整curve或一般反向定理證明。
查看答案與解釋
(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)
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。
本題逐步詳解（AI 二次覆核）
道路連通空間 · qf-mod48-path-connected-boundary · candidate
87. 道路連通空間
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 [mode, scale, example_description, scope_or_obstruction, path_connectedness_inference_claim] 欄序回答。
endpoint_only只給samples，不宣稱其有完整continuous path。connected_not_path是來源命名的sine-curve反例範圍，不是本題實作完整curve或一般反向定理證明。
查看答案與解釋
(connected_not_path, 12, topologists_sine_curve_scaled, connected_but_not_path_connected, 否)
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。
本題逐步詳解（AI 二次覆核）
道路連通空間 · qf-mod48-path-connected-boundary · candidate
88. 道路連通空間
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 [mode, scale, example_description, scope_or_obstruction, path_connectedness_inference_claim] 欄序回答。
endpoint_only只給samples，不宣稱其有完整continuous path。connected_not_path是來源命名的sine-curve反例範圍，不是本題實作完整curve或一般反向定理證明。
查看答案與解釋
(disconnected_gap, 13, X=(-infinity,0] union [13,infinity), IVT_blocks_a_path_between_components, 否)
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。
本題逐步詳解（AI 二次覆核）
道路連通空間 · qf-mod48-path-connected-boundary · candidate
89. 道路連通空間
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 [mode, scale, example_description, scope_or_obstruction, path_connectedness_inference_claim] 欄序回答。
endpoint_only只給samples，不宣稱其有完整continuous path。connected_not_path是來源命名的sine-curve反例範圍，不是本題實作完整curve或一般反向定理證明。
查看答案與解釋
(discrete_two_point, 14, X={0,14}_with_discrete_topology, continuous_image_of_connected_[0,1]_cannot_meet_both_points, 否)
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。
本題逐步詳解（AI 二次覆核）
道路連通空間 · qf-mod48-path-connected-boundary · candidate
90. 道路連通空間
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 [mode, scale, example_description, scope_or_obstruction, path_connectedness_inference_claim] 欄序回答。
endpoint_only只給samples，不宣稱其有完整continuous path。connected_not_path是來源命名的sine-curve反例範圍，不是本題實作完整curve或一般反向定理證明。
查看答案與解釋
(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)
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。
本題逐步詳解（AI 二次覆核）
道路連通空間 · qf-mod48-path-connected-boundary · candidate
91. 道路連通空間
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 [mode, scale, example_description, scope_or_obstruction, path_connectedness_inference_claim] 欄序回答。
endpoint_only只給samples，不宣稱其有完整continuous path。connected_not_path是來源命名的sine-curve反例範圍，不是本題實作完整curve或一般反向定理證明。
查看答案與解釋
(connected_not_path, 16, topologists_sine_curve_scaled, connected_but_not_path_connected, 否)
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。
本題逐步詳解（AI 二次覆核）
道路連通空間 · qf-mod48-path-connected-boundary · candidate
92. 道路連通空間
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 [mode, scale, example_description, scope_or_obstruction, path_connectedness_inference_claim] 欄序回答。
endpoint_only只給samples，不宣稱其有完整continuous path。connected_not_path是來源命名的sine-curve反例範圍，不是本題實作完整curve或一般反向定理證明。
查看答案與解釋
(disconnected_gap, 17, X=(-infinity,0] union [17,infinity), IVT_blocks_a_path_between_components, 否)
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。
本題逐步詳解（AI 二次覆核）
道路連通空間 · qf-mod48-path-connected-boundary · candidate
93. 道路連通空間
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 [mode, scale, example_description, scope_or_obstruction, path_connectedness_inference_claim] 欄序回答。
endpoint_only只給samples，不宣稱其有完整continuous path。connected_not_path是來源命名的sine-curve反例範圍，不是本題實作完整curve或一般反向定理證明。
查看答案與解釋
(discrete_two_point, 18, X={0,18}_with_discrete_topology, continuous_image_of_connected_[0,1]_cannot_meet_both_points, 否)
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。
本題逐步詳解（AI 二次覆核）
道路連通空間 · qf-mod48-path-connected-boundary · candidate
94. 道路連通空間
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 [mode, scale, example_description, scope_or_obstruction, path_connectedness_inference_claim] 欄序回答。
endpoint_only只給samples，不宣稱其有完整continuous path。connected_not_path是來源命名的sine-curve反例範圍，不是本題實作完整curve或一般反向定理證明。
查看答案與解釋
(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)
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。
本題逐步詳解（AI 二次覆核）
道路連通空間 · qf-mod48-path-connected-boundary · candidate
95. 道路連通空間
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 [mode, scale, example_description, scope_or_obstruction, path_connectedness_inference_claim] 欄序回答。
endpoint_only只給samples，不宣稱其有完整continuous path。connected_not_path是來源命名的sine-curve反例範圍，不是本題實作完整curve或一般反向定理證明。
查看答案與解釋
(connected_not_path, 20, topologists_sine_curve_scaled, connected_but_not_path_connected, 否)
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。
本題逐步詳解（AI 二次覆核）
道路連通空間 · qf-mod48-path-connected-boundary · candidate
96. 道路連通空間
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 [mode, scale, example_description, scope_or_obstruction, path_connectedness_inference_claim] 欄序回答。
endpoint_only只給samples，不宣稱其有完整continuous path。connected_not_path是來源命名的sine-curve反例範圍，不是本題實作完整curve或一般反向定理證明。
查看答案與解釋
(disconnected_gap, 21, X=(-infinity,0] union [21,infinity), IVT_blocks_a_path_between_components, 否)
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。
本題逐步詳解（AI 二次覆核）
道路連通空間 · qf-mod48-path-connected-boundary · candidate
97. 道路連通空間
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 [mode, scale, example_description, scope_or_obstruction, path_connectedness_inference_claim] 欄序回答。
endpoint_only只給samples，不宣稱其有完整continuous path。connected_not_path是來源命名的sine-curve反例範圍，不是本題實作完整curve或一般反向定理證明。
查看答案與解釋
(discrete_two_point, 22, X={0,22}_with_discrete_topology, continuous_image_of_connected_[0,1]_cannot_meet_both_points, 否)
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。
本題逐步詳解（AI 二次覆核）
道路連通空間 · qf-mod48-path-connected-boundary · candidate
98. 道路連通空間
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 [mode, scale, example_description, scope_or_obstruction, path_connectedness_inference_claim] 欄序回答。
endpoint_only只給samples，不宣稱其有完整continuous path。connected_not_path是來源命名的sine-curve反例範圍，不是本題實作完整curve或一般反向定理證明。
查看答案與解釋
(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)
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。
本題逐步詳解（AI 二次覆核）
道路連通空間 · qf-mod48-path-connected-boundary · candidate
99. 道路連通空間
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 [mode, scale, example_description, scope_or_obstruction, path_connectedness_inference_claim] 欄序回答。
endpoint_only只給samples，不宣稱其有完整continuous path。connected_not_path是來源命名的sine-curve反例範圍，不是本題實作完整curve或一般反向定理證明。
查看答案與解釋
(connected_not_path, 24, topologists_sine_curve_scaled, connected_but_not_path_connected, 否)
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。
本題逐步詳解（AI 二次覆核）
道路連通空間 · qf-mod48-path-connected-boundary · candidate
100. 道路連通空間
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 [mode, scale, example_description, scope_or_obstruction, path_connectedness_inference_claim] 欄序回答。
endpoint_only只給samples，不宣稱其有完整continuous path。connected_not_path是來源命名的sine-curve反例範圍，不是本題實作完整curve或一般反向定理證明。
查看答案與解釋
(disconnected_gap, 25, X=(-infinity,0] union [25,infinity), IVT_blocks_a_path_between_components, 否)
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。
本題逐步詳解（AI 二次覆核）
道路連通空間 · qf-mod48-path-connected-boundary · candidate
原始題目 JSONL · 驗收報告