第 24333 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-conservation-audit；資料={"vertices":["s7","a7","t7","u7"],"edges":[["s7","a7"],["a7","s7"],["a7","t7"],["t7","a7"]],"source":"s7","sink":"t7","capacities":[40,40,40,40],"flows":[32,16,16,8]}。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-conservation-audit；資料={"vertices":["s7","a7","t7","u7"],"edges":[["s7","a7"],["a7","s7"],["a7","t7"],["t7","a7"]],"source":"s7","sink":"t7","capacities":[40,40,40,40],"flows":[32,16,16,8]}。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-conservation-audit]

derived = [[source, sink, edge_capacity_flow_rows, internal_balance_rows, conservation_feasible, first_failed_vertex_or_null, certificate]]

source roles = [[s7, a7, t7, u7], [[s7, a7], [a7, s7], [a7, t7], [t7, a7]], [40, 40, 40, 40], [32, 16, 16, 8], s7, t7]

derived = [[a7, s7, t7, u7], [[[a7, s7], 40, 16], [[a7, t7], 40, 16], [[s7, a7], 40, 32], [[t7, a7], 40, 8]], 2, 是]

source roles = [0, [a7, s7], 16, 40]

derived = [是, 是, 是]

source roles = [1, [a7, t7], 16, 40]

derived = [是, 是, 是]

source roles = [2, [s7, a7], 32, 40]

derived = [是, 是, 是]

source roles = [3, [t7, a7], 8, 40]

derived = [是, 是, 是]

Flow ALL incoming at vertex a7 := sum(32, 8)

Flow ALL incoming at vertex a7 = 40

Flow ALL outgoing at vertex a7 := sum(16, 16)

Flow ALL outgoing at vertex a7 = 32

source roles = [a7, [2, 3], [0, 1]]

derived = [40, 32, 否, 是]

Flow ALL incoming at vertex s7 := sum(16)

Flow ALL incoming at vertex s7 = 16

Flow ALL outgoing at vertex s7 := sum(32)

Flow ALL outgoing at vertex s7 = 32

source roles = [s7, [0], [2]]

derived = [16, 32, 否, 否]

Flow ALL incoming at vertex t7 := sum(16)

Flow ALL incoming at vertex t7 = 16

Flow ALL outgoing at vertex t7 := sum(8)

Flow ALL outgoing at vertex t7 = 8

source roles = [t7, [1], [3]]

derived = [16, 8, 否, 否]

Flow ALL incoming at vertex u7 := sum()

Flow ALL incoming at vertex u7 = 0

Flow ALL outgoing at vertex u7 := sum()

Flow ALL outgoing at vertex u7 = 0

source roles = [u7, [], []]

derived = [0, 0, 是, 是]

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

Flow source NET outflow not gross outgoing = 16

Flow sink NET inflow not gross incoming := sub(16, 8)

Flow sink NET inflow not gross incoming = 8

source roles = [s7, t7, [[[a7, s7], 16, 40, 是], [[a7, t7], 16, 40, 是], [[s7, a7], 32, 40, 是], [[t7, a7], 8, 40, 是]], [[a7, 40, 32, 否], [u7, 0, 0, 是]], 16, 8]

derived = [否, 是, 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 u7，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 u7，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 t7，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 t7，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 s7，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 s7，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 a7，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 a7，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

(s7, t7, [[[a7, s7], 40, 16], [[a7, t7], 40, 16], [[s7, a7], 40, 32], [[t7, a7], 40, 8]], [[a7, 40, 32, 否], [u7, 0, 0, 是]], 否, [a7, 40, 32, 否], every_internal_vertex_requires_complete_inflow_equals_outflow)

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。