第 24378 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":["s2","a2","t2","u2"],"edges":[["s2","a2"],["a2","s2"],["a2","t2"],["t2","a2"]],"source":"s2","sink":"t2","capacities":[15,15,15,15],"flows":[12,6,9,3]}。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":["s2","a2","t2","u2"],"edges":[["s2","a2"],["a2","s2"],["a2","t2"],["t2","a2"]],"source":"s2","sink":"t2","capacities":[15,15,15,15],"flows":[12,6,9,3]}。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 = [[s2, a2, t2, u2], [[s2, a2], [a2, s2], [a2, t2], [t2, a2]], [15, 15, 15, 15], [12, 6, 9, 3], s2, t2]

derived = [[a2, s2, t2, u2], [[[a2, s2], 15, 6], [[a2, t2], 15, 9], [[s2, a2], 15, 12], [[t2, a2], 15, 3]], 2, 是]

source roles = [0, [a2, s2], 6, 15]

derived = [是, 是, 是]

source roles = [1, [a2, t2], 9, 15]

derived = [是, 是, 是]

source roles = [2, [s2, a2], 12, 15]

derived = [是, 是, 是]

source roles = [3, [t2, a2], 3, 15]

derived = [是, 是, 是]

Flow ALL incoming at vertex a2 := sum(12, 3)

Flow ALL incoming at vertex a2 = 15

Flow ALL outgoing at vertex a2 := sum(6, 9)

Flow ALL outgoing at vertex a2 = 15

source roles = [a2, [2, 3], [0, 1]]

derived = [15, 15, 是, 是]

Flow ALL incoming at vertex s2 := sum(6)

Flow ALL incoming at vertex s2 = 6

Flow ALL outgoing at vertex s2 := sum(12)

Flow ALL outgoing at vertex s2 = 12

source roles = [s2, [0], [2]]

derived = [6, 12, 否, 否]

Flow ALL incoming at vertex t2 := sum(9)

Flow ALL incoming at vertex t2 = 9

Flow ALL outgoing at vertex t2 := sum(3)

Flow ALL outgoing at vertex t2 = 3

source roles = [t2, [1], [3]]

derived = [9, 3, 否, 否]

Flow ALL incoming at vertex u2 := sum()

Flow ALL incoming at vertex u2 = 0

Flow ALL outgoing at vertex u2 := sum()

Flow ALL outgoing at vertex u2 = 0

source roles = [u2, [], []]

derived = [0, 0, 是, 是]

Flow source NET outflow not gross outgoing := sub(12, 6)

Flow source NET outflow not gross outgoing = 6

Flow sink NET inflow not gross incoming := sub(9, 3)

Flow sink NET inflow not gross incoming = 6

source roles = [s2, t2, [[[a2, s2], 6, 15, 是], [[a2, t2], 9, 15, 是], [[s2, a2], 12, 15, 是], [[t2, a2], 3, 15, 是]], [[a2, 15, 15, 是], [u2, 0, 0, 是]], 6, 6]

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 u2，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 u2，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 t2，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 t2，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 s2，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 s2，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 a2，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 a2，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

(s2, t2, [[[a2, s2], 15, 6], [[a2, t2], 15, 9], [[s2, a2], 15, 12], [[t2, a2], 15, 3]], 6, 6, 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。