第 25982 題：最大流最小割定理

原題與已知條件

完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-minimum-cut-audit；資料={"vertices":["s","a0","a1","a2","a3","t"],"edges":[["s","a0"],["a0","t"],["s","a1"],["a1","t"],["s","a2"],["a2","t"],["s","a3"],["a3","t"],["a0","s"]],"source":"s","sink":"t","capacities":[8,8,9,9,10,10,11,11,13]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。

• G001 — 完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-minimum-cut-audit；資料={"vertices":["s","a0","a1","a2","a3","t"],"edges":[["s","a0"],["a0","t"],["s","a1"],["a1","t"],["s","a2"],["a2","t"],["s","a3"],["a3","t"],["a0","s"]],"source":"s","sink":"t","capacities":[8,8,9,9,10,10,11,11,13]}。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-max-flow-min-cut-theorem

逐步推導

source roles = [mfmc-minimum-cut-audit]

derived = [[cut_rows, minimum_capacity, minimum_cuts, certificate]]

source roles = [[s, a0, a1, a2, a3, t], [[s, a0], [a0, t], [s, a1], [a1, t], [s, a2], [a2, t], [s, a3], [a3, t], [a0, s]], [8, 8, 9, 9, 10, 10, 11, 11, 13], null, s, t]

derived = [[a0, a1, a2, a3, s, t], [[a0, s], [a0, t], [a1, t], [a2, t], [a3, t], [s, a0], [s, a1], [s, a2], [s, a3]], [13, 8, 9, 10, 11, 8, 9, 10, 11], null]

MFMC cut mask0 OUTWARD capacity := sum(8, 9, 10, 11)

MFMC cut mask0 OUTWARD capacity = 38

source roles = [0, [a0, a1, a2, a3], [s]]

derived = [[5, 6, 7, 8], [0], [[[s, a0], 8], [[s, a1], 9], [[s, a2], 10], [[s, a3], 11]], 38]

MFMC cut mask1 OUTWARD capacity := sum(8, 9, 10, 11)

MFMC cut mask1 OUTWARD capacity = 38

source roles = [1, [a0, a1, a2, a3], [a0, s]]

derived = [[1, 6, 7, 8], [], [[[a0, t], 8], [[s, a1], 9], [[s, a2], 10], [[s, a3], 11]], 38]

MFMC cut mask2 OUTWARD capacity := sum(9, 8, 10, 11)

MFMC cut mask2 OUTWARD capacity = 38

source roles = [2, [a0, a1, a2, a3], [a1, s]]

derived = [[2, 5, 7, 8], [0], [[[a1, t], 9], [[s, a0], 8], [[s, a2], 10], [[s, a3], 11]], 38]

MFMC cut mask3 OUTWARD capacity := sum(8, 9, 10, 11)

MFMC cut mask3 OUTWARD capacity = 38

source roles = [3, [a0, a1, a2, a3], [a0, a1, s]]

derived = [[1, 2, 7, 8], [], [[[a0, t], 8], [[a1, t], 9], [[s, a2], 10], [[s, a3], 11]], 38]

MFMC cut mask4 OUTWARD capacity := sum(10, 8, 9, 11)

MFMC cut mask4 OUTWARD capacity = 38

source roles = [4, [a0, a1, a2, a3], [a2, s]]

derived = [[3, 5, 6, 8], [0], [[[a2, t], 10], [[s, a0], 8], [[s, a1], 9], [[s, a3], 11]], 38]

MFMC cut mask5 OUTWARD capacity := sum(8, 10, 9, 11)

MFMC cut mask5 OUTWARD capacity = 38

source roles = [5, [a0, a1, a2, a3], [a0, a2, s]]

derived = [[1, 3, 6, 8], [], [[[a0, t], 8], [[a2, t], 10], [[s, a1], 9], [[s, a3], 11]], 38]

MFMC cut mask6 OUTWARD capacity := sum(9, 10, 8, 11)

MFMC cut mask6 OUTWARD capacity = 38

source roles = [6, [a0, a1, a2, a3], [a1, a2, s]]

derived = [[2, 3, 5, 8], [0], [[[a1, t], 9], [[a2, t], 10], [[s, a0], 8], [[s, a3], 11]], 38]

MFMC cut mask7 OUTWARD capacity := sum(8, 9, 10, 11)

MFMC cut mask7 OUTWARD capacity = 38

source roles = [7, [a0, a1, a2, a3], [a0, a1, a2, s]]

derived = [[1, 2, 3, 8], [], [[[a0, t], 8], [[a1, t], 9], [[a2, t], 10], [[s, a3], 11]], 38]

MFMC cut mask8 OUTWARD capacity := sum(11, 8, 9, 10)

MFMC cut mask8 OUTWARD capacity = 38

source roles = [8, [a0, a1, a2, a3], [a3, s]]

derived = [[4, 5, 6, 7], [0], [[[a3, t], 11], [[s, a0], 8], [[s, a1], 9], [[s, a2], 10]], 38]

MFMC cut mask9 OUTWARD capacity := sum(8, 11, 9, 10)

MFMC cut mask9 OUTWARD capacity = 38

source roles = [9, [a0, a1, a2, a3], [a0, a3, s]]

derived = [[1, 4, 6, 7], [], [[[a0, t], 8], [[a3, t], 11], [[s, a1], 9], [[s, a2], 10]], 38]

MFMC cut mask10 OUTWARD capacity := sum(9, 11, 8, 10)

MFMC cut mask10 OUTWARD capacity = 38

source roles = [10, [a0, a1, a2, a3], [a1, a3, s]]

derived = [[2, 4, 5, 7], [0], [[[a1, t], 9], [[a3, t], 11], [[s, a0], 8], [[s, a2], 10]], 38]

MFMC cut mask11 OUTWARD capacity := sum(8, 9, 11, 10)

MFMC cut mask11 OUTWARD capacity = 38

source roles = [11, [a0, a1, a2, a3], [a0, a1, a3, s]]

derived = [[1, 2, 4, 7], [], [[[a0, t], 8], [[a1, t], 9], [[a3, t], 11], [[s, a2], 10]], 38]

MFMC cut mask12 OUTWARD capacity := sum(10, 11, 8, 9)

MFMC cut mask12 OUTWARD capacity = 38

source roles = [12, [a0, a1, a2, a3], [a2, a3, s]]

derived = [[3, 4, 5, 6], [0], [[[a2, t], 10], [[a3, t], 11], [[s, a0], 8], [[s, a1], 9]], 38]

MFMC cut mask13 OUTWARD capacity := sum(8, 10, 11, 9)

MFMC cut mask13 OUTWARD capacity = 38

source roles = [13, [a0, a1, a2, a3], [a0, a2, a3, s]]

derived = [[1, 3, 4, 6], [], [[[a0, t], 8], [[a2, t], 10], [[a3, t], 11], [[s, a1], 9]], 38]

MFMC cut mask14 OUTWARD capacity := sum(9, 10, 11, 8)

MFMC cut mask14 OUTWARD capacity = 38

source roles = [14, [a0, a1, a2, a3], [a1, a2, a3, s]]

derived = [[2, 3, 4, 5], [0], [[[a1, t], 9], [[a2, t], 10], [[a3, t], 11], [[s, a0], 8]], 38]

MFMC cut mask15 OUTWARD capacity := sum(8, 9, 10, 11)

MFMC cut mask15 OUTWARD capacity = 38

source roles = [15, [a0, a1, a2, a3], [a0, a1, a2, a3, s]]

derived = [[1, 2, 3, 4], [], [[[a0, t], 8], [[a1, t], 9], [[a2, t], 10], [[a3, t], 11]], 38]

source roles = [[a0, a1, a2, a3, s, t], [a0, a1, a2, a3]]

derived = [16, 16, 38, [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s], [a3, s], [a0, a3, s], [a1, a3, s], [a0, a1, a3, s], [a2, a3, s], [a0, a2, a3, s], [a1, a2, a3, s], [a0, a1, a2, a3, s]]]

逐步反向核對

• S035 · 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。
• S034 · 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。
• S033 · premise_and_result_audit — 以精確有理數重算 MFMC cut mask15 OUTWARD capacity，不以小數近似替代。
• S032 · 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。
• S031 · premise_and_result_audit — 以精確有理數重算 MFMC cut mask14 OUTWARD capacity，不以小數近似替代。
• S030 · 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。
• S029 · premise_and_result_audit — 以精確有理數重算 MFMC cut mask13 OUTWARD capacity，不以小數近似替代。
• S028 · 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。
• S027 · premise_and_result_audit — 以精確有理數重算 MFMC cut mask12 OUTWARD capacity，不以小數近似替代。
• S026 · 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。
• S025 · premise_and_result_audit — 以精確有理數重算 MFMC cut mask11 OUTWARD capacity，不以小數近似替代。
• S024 · 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。
• S023 · premise_and_result_audit — 以精確有理數重算 MFMC cut mask10 OUTWARD capacity，不以小數近似替代。
• S022 · 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。
• S021 · premise_and_result_audit — 以精確有理數重算 MFMC cut mask9 OUTWARD capacity，不以小數近似替代。
• S020 · 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。
• S019 · premise_and_result_audit — 以精確有理數重算 MFMC cut mask8 OUTWARD capacity，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 MFMC cut mask7 OUTWARD capacity，不以小數近似替代。
• S016 · 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。
• S015 · premise_and_result_audit — 以精確有理數重算 MFMC cut mask6 OUTWARD capacity，不以小數近似替代。
• S014 · 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。
• S013 · premise_and_result_audit — 以精確有理數重算 MFMC cut mask5 OUTWARD capacity，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 MFMC cut mask4 OUTWARD capacity，不以小數近似替代。
• S010 · 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。
• S009 · premise_and_result_audit — 以精確有理數重算 MFMC cut mask3 OUTWARD capacity，不以小數近似替代。
• S008 · 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。
• S007 · premise_and_result_audit — 以精確有理數重算 MFMC cut mask2 OUTWARD capacity，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 — 以精確有理數重算 MFMC cut mask1 OUTWARD capacity，不以小數近似替代。
• 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 — 以精確有理數重算 MFMC cut mask0 OUTWARD capacity，不以小數近似替代。
• 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

([[[s], [[[s, a0], 8], [[s, a1], 9], [[s, a2], 10], [[s, a3], 11]], 38], [[a0, s], [[[a0, t], 8], [[s, a1], 9], [[s, a2], 10], [[s, a3], 11]], 38], [[a1, s], [[[a1, t], 9], [[s, a0], 8], [[s, a2], 10], [[s, a3], 11]], 38], [[a0, a1, s], [[[a0, t], 8], [[a1, t], 9], [[s, a2], 10], [[s, a3], 11]], 38], [[a2, s], [[[a2, t], 10], [[s, a0], 8], [[s, a1], 9], [[s, a3], 11]], 38], [[a0, a2, s], [[[a0, t], 8], [[a2, t], 10], [[s, a1], 9], [[s, a3], 11]], 38], [[a1, a2, s], [[[a1, t], 9], [[a2, t], 10], [[s, a0], 8], [[s, a3], 11]], 38], [[a0, a1, a2, s], [[[a0, t], 8], [[a1, t], 9], [[a2, t], 10], [[s, a3], 11]], 38], [[a3, s], [[[a3, t], 11], [[s, a0], 8], [[s, a1], 9], [[s, a2], 10]], 38], [[a0, a3, s], [[[a0, t], 8], [[a3, t], 11], [[s, a1], 9], [[s, a2], 10]], 38], [[a1, a3, s], [[[a1, t], 9], [[a3, t], 11], [[s, a0], 8], [[s, a2], 10]], 38], [[a0, a1, a3, s], [[[a0, t], 8], [[a1, t], 9], [[a3, t], 11], [[s, a2], 10]], 38], [[a2, a3, s], [[[a2, t], 10], [[a3, t], 11], [[s, a0], 8], [[s, a1], 9]], 38], [[a0, a2, a3, s], [[[a0, t], 8], [[a2, t], 10], [[a3, t], 11], [[s, a1], 9]], 38], [[a1, a2, a3, s], [[[a1, t], 9], [[a2, t], 10], [[a3, t], 11], [[s, a0], 8]], 38], [[a0, a1, a2, a3, s], [[[a0, t], 8], [[a1, t], 9], [[a2, t], 10], [[a3, t], 11]], 38]], 38, [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s], [a3, s], [a0, a3, s], [a1, a3, s], [a0, a1, a3, s], [a2, a3, s], [a0, a2, a3, s], [a1, a2, a3, s], [a0, a1, a2, a3, s]], minimum_cut_is_selected_from_the_complete_cut_family)

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。