R0047-24399 · q-20261015-advance-20261113-r07-finite-flow-network-0099
finite loop-free directed network、逐edge capacity與逐internal-vertex conservation。操作=flow-value-cross-check；資料={"vertices":["s23","a23","t23","u23"],"edges":[["s23","a23"],["a23","s23"],["a23","t23"],["t23","a23"]],"source":"s23","sink":"t23","capacities":[120,120,120,120],"flows":[96,48,72,24]}。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-value-cross-check；資料={"vertices":["s23","a23","t23","u23"],"edges":[["s23","a23"],["a23","s23"],["a23","t23"],["t23","a23"]],"source":"s23","sink":"t23","capacities":[120,120,120,120],"flows":[96,48,72,24]}。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-value-cross-check]
操作後
derived = [[source, sink, edge_capacity_flow_rows, source_net_outflow, sink_net_inflow, net_values_equal, capacity_feasible, conservation_feasible, feasible, maximum_or_min_cut_certificate, boundary]]
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 = [[s23, a23, t23, u23], [[s23, a23], [a23, s23], [a23, t23], [t23, a23]], [120, 120, 120, 120], [96, 48, 72, 24], s23, t23]
操作後
derived = [[a23, s23, t23, u23], [[[a23, s23], 120, 48], [[a23, t23], 120, 72], [[s23, a23], 120, 96], [[t23, a23], 120, 24]], 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, [a23, s23], 48, 120]
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, [a23, t23], 72, 120]
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, [s23, a23], 96, 120]
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, S004
操作前
source roles = [3, [t23, a23], 24, 120]
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 a23 := sum(96, 24)
操作後
Flow ALL incoming at vertex a23 = 120
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 a23 := sum(48, 72)
操作後
Flow ALL outgoing at vertex a23 = 120
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 = [a23, [2, 3], [0, 1]]
操作後
derived = [120, 120, 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, S008
操作前
Flow ALL incoming at vertex s23 := sum(48)
操作後
Flow ALL incoming at vertex s23 = 48
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 s23 := sum(96)
操作後
Flow ALL outgoing at vertex s23 = 96
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 = [s23, [0], [2]]
操作後
derived = [48, 96, 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 t23 := sum(72)
操作後
Flow ALL incoming at vertex t23 = 72
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 t23 := sum(24)
操作後
Flow ALL outgoing at vertex t23 = 24
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 = [t23, [1], [3]]
操作後
derived = [72, 24, 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 u23 := sum()
操作後
Flow ALL incoming at vertex u23 = 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 u23 := sum()
操作後
Flow ALL outgoing at vertex u23 = 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 = [u23, [], []]
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(96, 48)
操作後
Flow source NET outflow not gross outgoing = 48
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(72, 24)
操作後
Flow sink NET inflow not gross incoming = 48
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 = [s23, t23, [[[a23, s23], 48, 120, 是], [[a23, t23], 72, 120, 是], [[s23, a23], 96, 120, 是], [[t23, a23], 24, 120, 是]], [[a23, 120, 120, 是], [u23, 0, 0, 是]], 48, 48]
操作後
derived = [是, 是, 是, 是, 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 u23，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 u23，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 t23，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 t23，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 s23，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 s23，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 a23，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 a23，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。(s23, t23, [[[a23, s23], 120, 48], [[a23, t23], 120, 72], [[s23, a23], 120, 96], [[t23, a23], 120, 24]], 48, 48, 是, 是, 是, 是, 否, feasible_flow_value_cross_check_does_not_claim_maximality_or_min_cut_equality)
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。
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