第 20857 題：有限有向圖

原題與已知條件

完整有限ordered-pair edge set與loop-free simple digraph convention。操作=digraph-loop-boundary；資料={"n":5,"edges":[[0,0],[0,1],[1,2],[1,3],[2,3],[2,4],[3,4],[4,0]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。

• G001 — 完整有限ordered-pair edge set與loop-free simple digraph convention。操作=digraph-loop-boundary；資料={"n":5,"edges":[[0,0],[0,1],[1,2],[1,3],[2,3],[2,4],[3,4],[4,0]]}。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-directed-graph

逐步推導

source operation=digraph-loop-boundary;original snapshot and mathematics retained

explicit ordered source output fields=[loops, declared_model_qualified, out_degrees, in_degrees, boundary]

ORIGINAL complete directed carrier Fin(5);ORIGINAL lexicographically canonical ORDERED arcs=[[0, 0], [0, 1], [1, 2], [1, 3], [2, 3], [2, 4], [3, 4], [4, 0]]

all endpoints in range and same-direction arcs unique;actual loops=[[0, 0]];temporary loop-boundary extended data,NOT main loop-free qualification

digraph vertex0 exact outgoing count := sum(1, 1)

digraph vertex0 exact outgoing count = 2

digraph vertex0 exact incoming count := sum(1, 1)

digraph vertex0 exact incoming count = 2

ORIGINAL vertex=0;ALL outgoing original-arc indices=[0, 1];ALL incoming indices=[0, 7]

outdegree=2;indegree=2;a loop at this vertex contributes ONCE to EACH separate count

digraph vertex1 exact outgoing count := sum(1, 1)

digraph vertex1 exact outgoing count = 2

digraph vertex1 exact incoming count := sum(1)

digraph vertex1 exact incoming count = 1

ORIGINAL vertex=1;ALL outgoing original-arc indices=[2, 3];ALL incoming indices=[1]

outdegree=2;indegree=1;a loop at this vertex contributes ONCE to EACH separate count

digraph vertex2 exact outgoing count := sum(1, 1)

digraph vertex2 exact outgoing count = 2

digraph vertex2 exact incoming count := sum(1)

digraph vertex2 exact incoming count = 1

ORIGINAL vertex=2;ALL outgoing original-arc indices=[4, 5];ALL incoming indices=[2]

outdegree=2;indegree=1;a loop at this vertex contributes ONCE to EACH separate count

digraph vertex3 exact outgoing count := sum(1)

digraph vertex3 exact outgoing count = 1

digraph vertex3 exact incoming count := sum(1, 1)

digraph vertex3 exact incoming count = 2

ORIGINAL vertex=3;ALL outgoing original-arc indices=[6];ALL incoming indices=[3, 4]

outdegree=1;indegree=2;a loop at this vertex contributes ONCE to EACH separate count

digraph vertex4 exact outgoing count := sum(1)

digraph vertex4 exact outgoing count = 1

digraph vertex4 exact incoming count := sum(1, 1)

digraph vertex4 exact incoming count = 2

ORIGINAL vertex=4;ALL outgoing original-arc indices=[7];ALL incoming indices=[5, 6]

outdegree=1;indegree=2;a loop at this vertex contributes ONCE to EACH separate count

digraph complete outgoing-degree sum := sum(2, 2, 2, 1, 1)

digraph complete outgoing-degree sum = 8

digraph complete incoming-degree sum := sum(2, 1, 1, 2, 2)

digraph complete incoming-degree sum = 8

ALL vertex-order outdegrees=[2, 2, 2, 1, 1];indegrees=[2, 1, 1, 2, 2];complete directed arc count=8

each arc counted once at its source and once at its target;SUM(out)=8=|E| and SUM(in)=8=|E|;the SEPARATE counting rule is |E|,not an imported undirected2|E| rule (numbers may coincide when E=0)

ACTUAL loops in supplied ordered arc set=[[0, 0]];extended-data counts out=[2, 2, 2, 1, 1],in=[2, 1, 1, 2, 2]

declared MAIN loop-free model qualification=否;loops may exist under a DIFFERENT declared convention;not a denial of all digraph definitions

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 — 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 digraph complete incoming-degree sum，不以小數近似替代。
• S018 · premise_and_result_audit — 以精確有理數重算 digraph complete outgoing-degree sum，不以小數近似替代。
• S017 · 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。
• S016 · premise_and_result_audit — 以精確有理數重算 digraph vertex4 exact incoming count，不以小數近似替代。
• S015 · premise_and_result_audit — 以精確有理數重算 digraph vertex4 exact outgoing count，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 digraph vertex3 exact incoming count，不以小數近似替代。
• S012 · premise_and_result_audit — 以精確有理數重算 digraph vertex3 exact outgoing count，不以小數近似替代。
• S011 · 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。
• S010 · premise_and_result_audit — 以精確有理數重算 digraph vertex2 exact incoming count，不以小數近似替代。
• S009 · premise_and_result_audit — 以精確有理數重算 digraph vertex2 exact outgoing count，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 digraph vertex1 exact incoming count，不以小數近似替代。
• S006 · premise_and_result_audit — 以精確有理數重算 digraph vertex1 exact outgoing count，不以小數近似替代。
• 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 — 以精確有理數重算 digraph vertex0 exact incoming count，不以小數近似替代。
• S003 · premise_and_result_audit — 以精確有理數重算 digraph vertex0 exact outgoing count，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

([[0, 0]], 否, [2, 2, 2, 1, 1], [2, 1, 1, 2, 2], loops_rejected_by_declared_loop_free_simple_digraph_convention)

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。