第 24316 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-capacity-audit；資料={"vertices":["s15","a15","t15","u15"],"edges":[["s15","a15"],["a15","s15"],["a15","t15"],["t15","a15"]],"source":"s15","sink":"t15","capacities":[80,80,80,80],"flows":[-16,32,48,16]}。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-capacity-audit；資料={"vertices":["s15","a15","t15","u15"],"edges":[["s15","a15"],["a15","s15"],["a15","t15"],["t15","a15"]],"source":"s15","sink":"t15","capacities":[80,80,80,80],"flows":[-16,32,48,16]}。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-capacity-audit]

derived = [[source, sink, edge_capacity_flow_rows, capacity_rows, capacity_feasible, first_failed_edge_or_null, certificate]]

source roles = [[s15, a15, t15, u15], [[s15, a15], [a15, s15], [a15, t15], [t15, a15]], [80, 80, 80, 80], [-16, 32, 48, 16], s15, t15]

derived = [[a15, s15, t15, u15], [[[a15, s15], 80, 32], [[a15, t15], 80, 48], [[s15, a15], 80, -16], [[t15, a15], 80, 16]], 2, 是]

source roles = [0, [a15, s15], 32, 80]

derived = [是, 是, 是]

source roles = [1, [a15, t15], 48, 80]

derived = [是, 是, 是]

source roles = [2, [s15, a15], -16, 80]

derived = [否, 是, 否]

source roles = [3, [t15, a15], 16, 80]

derived = [是, 是, 是]

Flow ALL incoming at vertex a15 := sum(-16, 16)

Flow ALL incoming at vertex a15 = 0

Flow ALL outgoing at vertex a15 := sum(32, 48)

Flow ALL outgoing at vertex a15 = 80

source roles = [a15, [2, 3], [0, 1]]

derived = [0, 80, 否, 是]

Flow ALL incoming at vertex s15 := sum(32)

Flow ALL incoming at vertex s15 = 32

Flow ALL outgoing at vertex s15 := sum(-16)

Flow ALL outgoing at vertex s15 = -16

source roles = [s15, [0], [2]]

derived = [32, -16, 否, 否]

Flow ALL incoming at vertex t15 := sum(48)

Flow ALL incoming at vertex t15 = 48

Flow ALL outgoing at vertex t15 := sum(16)

Flow ALL outgoing at vertex t15 = 16

source roles = [t15, [1], [3]]

derived = [48, 16, 否, 否]

Flow ALL incoming at vertex u15 := sum()

Flow ALL incoming at vertex u15 = 0

Flow ALL outgoing at vertex u15 := sum()

Flow ALL outgoing at vertex u15 = 0

source roles = [u15, [], []]

derived = [0, 0, 是, 是]

Flow source NET outflow not gross outgoing := sub(-16, 32)

Flow source NET outflow not gross outgoing = -48

Flow sink NET inflow not gross incoming := sub(48, 16)

Flow sink NET inflow not gross incoming = 32

source roles = [s15, t15, [[[a15, s15], 32, 80, 是], [[a15, t15], 48, 80, 是], [[s15, a15], -16, 80, 否], [[t15, a15], 16, 80, 是]], [[a15, 0, 80, 否], [u15, 0, 0, 是]], -48, 32]

derived = [否, 否, 否, 否, 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 u15，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 u15，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 t15，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 t15，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 s15，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 s15，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 a15，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 a15，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

(s15, t15, [[[a15, s15], 80, 32], [[a15, t15], 80, 48], [[s15, a15], 80, -16], [[t15, a15], 80, 16]], [[[a15, s15], 32, 80, 是], [[a15, t15], 48, 80, 是], [[s15, a15], -16, 80, 否], [[t15, a15], 16, 80, 是]], 否, [[s15, a15], -16, 80, 否], every_edge_requires_zero_less_equal_flow_less_equal_capacity)

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。