第 24382 題：有限流網路

原題與已知條件

finite loop-free directed network、逐edge capacity與逐internal-vertex conservation。操作=flow-value-cross-check；資料={"vertices":["s6","a6","t6","u6"],"edges":[["s6","a6"],["a6","s6"],["a6","t6"],["t6","a6"]],"source":"s6","sink":"t6","capacities":[35,35,35,35],"flows":[28,14,21,7]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。

• G001 — finite loop-free directed network、逐edge capacity與逐internal-vertex conservation。操作=flow-value-cross-check；資料={"vertices":["s6","a6","t6","u6"],"edges":[["s6","a6"],["a6","s6"],["a6","t6"],["t6","a6"]],"source":"s6","sink":"t6","capacities":[35,35,35,35],"flows":[28,14,21,7]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。

定義與適用範圍

沿用原題明示的論域、假設與來源 MKO 定義。本題已完成第二對話數理與中英語義覆核，並由原始參數逐題獨立重算模型、答案與每個算術步驟；原題仍為 candidate。

• mko-finite-flow-network

逐步推導

source roles = [flow-value-cross-check]

derived = [[source, sink, edge_capacity_flow_rows, source_net_outflow, sink_net_inflow, net_values_equal, capacity_feasible, conservation_feasible, feasible, maximum_or_min_cut_certificate, boundary]]

source roles = [[s6, a6, t6, u6], [[s6, a6], [a6, s6], [a6, t6], [t6, a6]], [35, 35, 35, 35], [28, 14, 21, 7], s6, t6]

derived = [[a6, s6, t6, u6], [[[a6, s6], 35, 14], [[a6, t6], 35, 21], [[s6, a6], 35, 28], [[t6, a6], 35, 7]], 2, 是]

source roles = [0, [a6, s6], 14, 35]

derived = [是, 是, 是]

source roles = [1, [a6, t6], 21, 35]

derived = [是, 是, 是]

source roles = [2, [s6, a6], 28, 35]

derived = [是, 是, 是]

source roles = [3, [t6, a6], 7, 35]

derived = [是, 是, 是]

Flow ALL incoming at vertex a6 := sum(28, 7)

Flow ALL incoming at vertex a6 = 35

Flow ALL outgoing at vertex a6 := sum(14, 21)

Flow ALL outgoing at vertex a6 = 35

source roles = [a6, [2, 3], [0, 1]]

derived = [35, 35, 是, 是]

Flow ALL incoming at vertex s6 := sum(14)

Flow ALL incoming at vertex s6 = 14

Flow ALL outgoing at vertex s6 := sum(28)

Flow ALL outgoing at vertex s6 = 28

source roles = [s6, [0], [2]]

derived = [14, 28, 否, 否]

Flow ALL incoming at vertex t6 := sum(21)

Flow ALL incoming at vertex t6 = 21

Flow ALL outgoing at vertex t6 := sum(7)

Flow ALL outgoing at vertex t6 = 7

source roles = [t6, [1], [3]]

derived = [21, 7, 否, 否]

Flow ALL incoming at vertex u6 := sum()

Flow ALL incoming at vertex u6 = 0

Flow ALL outgoing at vertex u6 := sum()

Flow ALL outgoing at vertex u6 = 0

source roles = [u6, [], []]

derived = [0, 0, 是, 是]

Flow source NET outflow not gross outgoing := sub(28, 14)

Flow source NET outflow not gross outgoing = 14

Flow sink NET inflow not gross incoming := sub(21, 7)

Flow sink NET inflow not gross incoming = 14

source roles = [s6, t6, [[[a6, s6], 14, 35, 是], [[a6, t6], 21, 35, 是], [[s6, a6], 28, 35, 是], [[t6, a6], 7, 35, 是]], [[a6, 35, 35, 是], [u6, 0, 0, 是]], 14, 14]

derived = [是, 是, 是, 是, 否]

逐步反向核對

• S021 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S020 · premise_and_result_audit — 以精確有理數重算 Flow sink NET inflow not gross incoming，不以小數近似替代。
• S019 · premise_and_result_audit — 以精確有理數重算 Flow source NET outflow not gross outgoing，不以小數近似替代。
• S018 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S017 · premise_and_result_audit — 以精確有理數重算 Flow ALL outgoing at vertex u6，不以小數近似替代。
• S016 · premise_and_result_audit — 以精確有理數重算 Flow ALL incoming at vertex u6，不以小數近似替代。
• S015 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S014 · premise_and_result_audit — 以精確有理數重算 Flow ALL outgoing at vertex t6，不以小數近似替代。
• S013 · premise_and_result_audit — 以精確有理數重算 Flow ALL incoming at vertex t6，不以小數近似替代。
• S012 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S011 · premise_and_result_audit — 以精確有理數重算 Flow ALL outgoing at vertex s6，不以小數近似替代。
• S010 · premise_and_result_audit — 以精確有理數重算 Flow ALL incoming at vertex s6，不以小數近似替代。
• S009 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S008 · premise_and_result_audit — 以精確有理數重算 Flow ALL outgoing at vertex a6，不以小數近似替代。
• S007 · premise_and_result_audit — 以精確有理數重算 Flow ALL incoming at vertex a6，不以小數近似替代。
• S006 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S005 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S004 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S003 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S002 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S001 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。

結論

(s6, t6, [[[a6, s6], 35, 14], [[a6, t6], 35, 21], [[s6, a6], 35, 28], [[t6, a6], 35, 7]], 14, 14, 是, 是, 是, 是, 否, feasible_flow_value_cross_check_does_not_claim_maximality_or_min_cut_equality)

依上列完整代值及條件核對得到原題答案；原題編號、格式與狀態保持不變。

本題覆核與來源綁定

覆核日期：2026-10-26

原題、答案、步驟表達式、數值 AST 與既有歷史保持原樣；原來的候選狀態屬於來源歷史，不覆寫為形式化證明。