# 第 24323 題：有限流網路 [返回所在分頁 122](https://ocme.evemisslab.com/zh-Hant/questions/page/122/) [← 逐題詳解目錄](https://ocme.evemisslab.com/zh-Hant/questions/) · [原題批次](https://ocme.evemisslab.com/zh-Hant/practice/batch-20261015-advance-20261113-r07-finite-flow-network-foundations/#q-20261015-advance-20261113-r07-finite-flow-network-0048) 本頁是已完成第二對話 AI 數理與中英語義覆核的教學詳解，只支持本題明示的模型與前提。沒有人工審查或新 Lean 證明，也不提升原題或 MKO 的 Canonical 真值狀態。 R0047-24323 · q-20261015-advance-20261113-r07-finite-flow-network-0048 ## 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":["s22","a22","t22","u22"],"edges":[["s22","a22"],["a22","s22"],["a22","t22"],["t22","a22"]],"source":"s22","sink":"t22","capacities":[115,115,115,115],"flows":[138,46,69,23]}。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":["s22","a22","t22","u22"],"edges":[["s22","a22"],["a22","s22"],["a22","t22"],["t22","a22"]],"source":"s22","sink":"t22","capacities":[115,115,115,115],"flows":[138,46,69,23]}。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](https://ocme.evemisslab.com/zh-Hant/mko/mko-finite-flow-network/) ## 逐步推導 S001 · implies操作前 source roles = [flow-capacity-audit] 操作後 derived = [[source, sink, edge_capacity_flow_rows, capacity_rows, capacity_feasible, first_failed_edge_or_null, certificate]] 條件：無 本頁列完整來源欄序，不需JavaScript；gradient boundary依原mode選6／8／7欄，不補欄也不由答案猜mode。 步驟識別與依賴規則：source_ordered_fields_without_JS · 依賴：G001 S002 · implies操作前 source roles = [[s22, a22, t22, u22], [[s22, a22], [a22, s22], [a22, t22], [t22, a22]], [115, 115, 115, 115], [138, 46, 69, 23], s22, t22] 操作後 derived = [[a22, s22, t22, u22], [[[a22, s22], 115, 46], [[a22, t22], 115, 69], [[s22, a22], 115, 138], [[t22, a22], 115, 23]], 2, 是] 條件：無 先依原index配好edge／capacity／flow，再按來源English canonical(edge)序排；vertices用UTF16。反向邊是不同edge，zero容量與孤立點合法，負flow保留供容量判失敗，不修改。 步驟識別與依賴規則：Flow_zip_source_edges_capacities_flows_BEFORE_order · 依賴：G001, S001 S003 · implies操作前 source roles = [0, [a22, s22], 46, 115] 操作後 derived = [是, 是, 是] 條件：無 每條有向邊都要0<=flow<=capacity；網路signature合格並不保證候選flow可行。 步驟識別與依賴規則：Flow_EACH_edge_two_sided_capacity_obligation · 依賴：G001, S002 S004 · implies操作前 source roles = [1, [a22, t22], 69, 115] 操作後 derived = [是, 是, 是] 條件：無 每條有向邊都要0<=flow<=capacity；網路signature合格並不保證候選flow可行。 步驟識別與依賴規則：Flow_EACH_edge_two_sided_capacity_obligation · 依賴：G001, S003 S005 · implies操作前 source roles = [2, [s22, a22], 138, 115] 操作後 derived = [是, 否, 否] 條件：無 每條有向邊都要0<=flow<=capacity；網路signature合格並不保證候選flow可行。 步驟識別與依賴規則：Flow_EACH_edge_two_sided_capacity_obligation · 依賴：G001, S004 S006 · implies操作前 source roles = [3, [t22, a22], 23, 115] 操作後 derived = [是, 是, 是] 條件：無 每條有向邊都要0<=flow<=capacity；網路signature合格並不保證候選flow可行。 步驟識別與依賴規則：Flow_EACH_edge_two_sided_capacity_obligation · 依賴：G001, S005 S007 · calculation操作前 Flow ALL incoming at vertex a22 := sum(138, 23) 操作後 Flow ALL incoming at vertex a22 = 161 條件：無 依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。 步驟識別與依賴規則：exact_rational_arithmetic · 依賴：G001, S006 S008 · calculation操作前 Flow ALL outgoing at vertex a22 := sum(46, 69) 操作後 Flow ALL outgoing at vertex a22 = 115 條件：無 依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。 步驟識別與依賴規則：exact_rational_arithmetic · 依賴：G001, S007 S009 · implies操作前 source roles = [a22, [2, 3], [0, 1]] 操作後 derived = [161, 115, 否, 是] 條件：無 入邊／出邊各依完整清單加總；孤立點是0=0。守恆只要求internal vertices，源匯仍計完整balance以求net值。 步驟識別與依賴規則：Flow_EACH_vertex_complete_directional_incidence · 依賴：G001, S008 S010 · calculation操作前 Flow ALL incoming at vertex s22 := sum(46) 操作後 Flow ALL incoming at vertex s22 = 46 條件：無 依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。 步驟識別與依賴規則：exact_rational_arithmetic · 依賴：G001, S009 S011 · calculation操作前 Flow ALL outgoing at vertex s22 := sum(138) 操作後 Flow ALL outgoing at vertex s22 = 138 條件：無 依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。 步驟識別與依賴規則：exact_rational_arithmetic · 依賴：G001, S010 S012 · implies操作前 source roles = [s22, [0], [2]] 操作後 derived = [46, 138, 否, 否] 條件：無 入邊／出邊各依完整清單加總；孤立點是0=0。守恆只要求internal vertices，源匯仍計完整balance以求net值。 步驟識別與依賴規則：Flow_EACH_vertex_complete_directional_incidence · 依賴：G001, S011 S013 · calculation操作前 Flow ALL incoming at vertex t22 := sum(69) 操作後 Flow ALL incoming at vertex t22 = 69 條件：無 依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。 步驟識別與依賴規則：exact_rational_arithmetic · 依賴：G001, S012 S014 · calculation操作前 Flow ALL outgoing at vertex t22 := sum(23) 操作後 Flow ALL outgoing at vertex t22 = 23 條件：無 依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。 步驟識別與依賴規則：exact_rational_arithmetic · 依賴：G001, S013 S015 · implies操作前 source roles = [t22, [1], [3]] 操作後 derived = [69, 23, 否, 否] 條件：無 入邊／出邊各依完整清單加總；孤立點是0=0。守恆只要求internal vertices，源匯仍計完整balance以求net值。 步驟識別與依賴規則：Flow_EACH_vertex_complete_directional_incidence · 依賴：G001, S014 S016 · calculation操作前 Flow ALL incoming at vertex u22 := sum() 操作後 Flow ALL incoming at vertex u22 = 0 條件：無 依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。 步驟識別與依賴規則：exact_rational_arithmetic · 依賴：G001, S015 S017 · calculation操作前 Flow ALL outgoing at vertex u22 := sum() 操作後 Flow ALL outgoing at vertex u22 = 0 條件：無 依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。 步驟識別與依賴規則：exact_rational_arithmetic · 依賴：G001, S016 S018 · implies操作前 source roles = [u22, [], []] 操作後 derived = [0, 0, 是, 是] 條件：無 入邊／出邊各依完整清單加總；孤立點是0=0。守恆只要求internal vertices，源匯仍計完整balance以求net值。 步驟識別與依賴規則：Flow_EACH_vertex_complete_directional_incidence · 依賴：G001, S017 S019 · calculation操作前 Flow source NET outflow not gross outgoing := sub(138, 46) 操作後 Flow source NET outflow not gross outgoing = 92 條件：無 依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。 步驟識別與依賴規則：exact_rational_arithmetic · 依賴：G001, S018 S020 · calculation操作前 Flow sink NET inflow not gross incoming := sub(69, 23) 操作後 Flow sink NET inflow not gross incoming = 46 條件：無 依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。 步驟識別與依賴規則：exact_rational_arithmetic · 依賴：G001, S019 S021 · implies操作前 source roles = [s22, t22, [[[a22, s22], 46, 115, 是], [[a22, t22], 69, 115, 是], [[s22, a22], 138, 115, 否], [[t22, a22], 23, 115, 是]], [[a22, 161, 115, 否], [u22, 0, 0, 是]], 92, 46] 操作後 derived = [否, 否, 否, 否, 否] 條件：無 全vertex總in−out中每edge相消，故全部internal守恆必得兩net相等；反向不成立，internal imbalance可互抵。容量、逐點守恆、兩net值分別核，feasible不推出maximum/min-cut。最後No是不提供最優證書，不否定特例恰好最優。 步驟識別與依賴規則：Flow_value_capacity_conservation_and_maximum_separate · 依賴：G001, S020 ## 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 u22，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 u22，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 t22，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 t22，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 s22，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 s22，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 a22，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 a22，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 (s22, t22, [[[a22, s22], 115, 46], [[a22, t22], 115, 69], [[s22, a22], 115, 138], [[t22, a22], 115, 23]], [[[a22, s22], 46, 115, 是], [[a22, t22], 69, 115, 是], [[s22, a22], 138, 115, 否], [[t22, a22], 23, 115, 是]], 否, [[s22, a22], 138, 115, 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。 來源雜湊與覆核憑證 - input commit: 0f7be61e2c22317a4075561dc822c87d881a50db - original question SHA-256: 38b9b619c1e12ab945f212629014eacb4a54bce7856994d4599de1262392048a - candidate record SHA-256: 8a81feda95d6eb53faa2e724245a1d53f3391cfdfa5f54f774c7e25a2645aec1 - semantic review receipt SHA-256: 9b43b9823e9bd2478d6caa5b57ffed161c2c8956af86974e901d5662ad994386 - question-first witness SHA-256: eb97596068e2d5648033bab2b8f6ca9b653bbe8cb77c0f8e1d16d0c40e46f50b - Lean: NOT_RUN · human review: NOT_PERFORMED [同源完整 JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261015-advance-20261113-r07-finite-flow-network-0048.json)