R0047-24310 · q-20261015-advance-20261113-r07-finite-flow-network-0035
finite loop-free directed network、逐edge capacity與逐internal-vertex conservation。操作=flow-capacity-audit；資料={"vertices":["s9","a9","t9","u9"],"edges":[["s9","a9"],["a9","s9"],["a9","t9"],["t9","a9"]],"source":"s9","sink":"t9","capacities":[50,50,50,50],"flows":[-10,20,30,10]}。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":["s9","a9","t9","u9"],"edges":[["s9","a9"],["a9","s9"],["a9","t9"],["t9","a9"]],"source":"s9","sink":"t9","capacities":[50,50,50,50],"flows":[-10,20,30,10]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。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。
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。
規則：source_ordered_fields_without_JS · 依賴：G001
操作前
source roles = [[s9, a9, t9, u9], [[s9, a9], [a9, s9], [a9, t9], [t9, a9]], [50, 50, 50, 50], [-10, 20, 30, 10], s9, t9]
操作後
derived = [[a9, s9, t9, u9], [[[a9, s9], 50, 20], [[a9, t9], 50, 30], [[s9, a9], 50, -10], [[t9, a9], 50, 10]], 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。
規則：Flow_zip_source_edges_capacities_flows_BEFORE_order · 依賴：G001, S001
操作前
source roles = [0, [a9, s9], 20, 50]
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。
規則：Flow_EACH_edge_two_sided_capacity_obligation · 依賴：G001, S002
操作前
source roles = [1, [a9, t9], 30, 50]
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。
規則：Flow_EACH_edge_two_sided_capacity_obligation · 依賴：G001, S003
操作前
source roles = [2, [s9, a9], -10, 50]
操作後
derived = [否, 是, 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。
規則：Flow_EACH_edge_two_sided_capacity_obligation · 依賴：G001, S004
操作前
source roles = [3, [t9, a9], 10, 50]
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。
規則：Flow_EACH_edge_two_sided_capacity_obligation · 依賴：G001, S005
操作前
Flow ALL incoming at vertex a9 := sum(-10, 10)
操作後
Flow ALL incoming at vertex a9 = 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。
規則：exact_rational_arithmetic · 依賴：G001, S006
操作前
Flow ALL outgoing at vertex a9 := sum(20, 30)
操作後
Flow ALL outgoing at vertex a9 = 50
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。
規則：exact_rational_arithmetic · 依賴：G001, S007
操作前
source roles = [a9, [2, 3], [0, 1]]
操作後
derived = [0, 50, 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。
規則：Flow_EACH_vertex_complete_directional_incidence · 依賴：G001, S008
操作前
Flow ALL incoming at vertex s9 := sum(20)
操作後
Flow ALL incoming at vertex s9 = 20
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。
規則：exact_rational_arithmetic · 依賴：G001, S009
操作前
Flow ALL outgoing at vertex s9 := sum(-10)
操作後
Flow ALL outgoing at vertex s9 = -10
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。
規則：exact_rational_arithmetic · 依賴：G001, S010
操作前
source roles = [s9, [0], [2]]
操作後
derived = [20, -10, 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。
規則：Flow_EACH_vertex_complete_directional_incidence · 依賴：G001, S011
操作前
Flow ALL incoming at vertex t9 := sum(30)
操作後
Flow ALL incoming at vertex t9 = 30
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。
規則：exact_rational_arithmetic · 依賴：G001, S012
操作前
Flow ALL outgoing at vertex t9 := sum(10)
操作後
Flow ALL outgoing at vertex t9 = 10
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。
規則：exact_rational_arithmetic · 依賴：G001, S013
操作前
source roles = [t9, [1], [3]]
操作後
derived = [30, 10, 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。
規則：Flow_EACH_vertex_complete_directional_incidence · 依賴：G001, S014
操作前
Flow ALL incoming at vertex u9 := sum()
操作後
Flow ALL incoming at vertex u9 = 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。
規則：exact_rational_arithmetic · 依賴：G001, S015
操作前
Flow ALL outgoing at vertex u9 := sum()
操作後
Flow ALL outgoing at vertex u9 = 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。
規則：exact_rational_arithmetic · 依賴：G001, S016
操作前
source roles = [u9, [], []]
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。
規則：Flow_EACH_vertex_complete_directional_incidence · 依賴：G001, S017
操作前
Flow source NET outflow not gross outgoing := sub(-10, 20)
操作後
Flow source NET outflow not gross outgoing = -30
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。
規則：exact_rational_arithmetic · 依賴：G001, S018
操作前
Flow sink NET inflow not gross incoming := sub(30, 10)
操作後
Flow sink NET inflow not gross incoming = 20
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。
規則：exact_rational_arithmetic · 依賴：G001, S019
操作前
source roles = [s9, t9, [[[a9, s9], 20, 50, 是], [[a9, t9], 30, 50, 是], [[s9, a9], -10, 50, 否], [[t9, a9], 10, 50, 是]], [[a9, 0, 50, 否], [u9, 0, 0, 是]], -30, 20]
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。
規則：Flow_value_capacity_conservation_and_maximum_separate · 依賴：G001, S020
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 u9，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 u9，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 t9，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 t9，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 s9，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 s9，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 a9，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 a9，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。(s9, t9, [[[a9, s9], 50, 20], [[a9, t9], 50, 30], [[s9, a9], 50, -10], [[t9, a9], 50, 10]], [[[a9, s9], 20, 50, 是], [[a9, t9], 30, 50, 是], [[s9, a9], -10, 50, 否], [[t9, a9], 10, 50, 是]], 否, [[s9, a9], -10, 50, 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。
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。
input commit: 0f7be61e2c22317a4075561dc822c87d881a50dboriginal question SHA-256: 885603b559f55f2090d5c7a3b8d51afaf9c7712cf705905c3a5eba5fba41d1facandidate record SHA-256: a8c7d55e677fd5a9ce434f6fd3405ba110930885b075843d688cc9426b4598f9semantic review receipt SHA-256: 9b43b9823e9bd2478d6caa5b57ffed161c2c8956af86974e901d5662ad994386question-first witness SHA-256: eb97596068e2d5648033bab2b8f6ca9b653bbe8cb77c0f8e1d16d0c40e46f50bLean: NOT_RUN · human review: NOT_PERFORMED