第 19527 題：Clique與獨立集

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原題與已知條件
定義與適用範圍
逐步推導
逐步反向核對
結論
本題覆核與來源綁定

R0038-19527 · q-20261014-advance-20261106-r02-clique-and-independent-set-0052

原題與已知條件

完整有限簡單圖與全部候選頂點對。操作=clique-exact-extrema；資料={"n":6,"edges":[[0,1],[0,2],[0,3],[1,2],[3,4]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。

G001 — 完整有限簡單圖與全部候選頂點對。操作=clique-exact-extrema；資料={"n":6,"edges":[[0,1],[0,2],[0,3],[1,2],[3,4]]}。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-clique-and-independent-set

逐步推導

S001 · implies

操作前

source operation=clique-exact-extrema;original snapshot and mathematics retained

操作後

explicit ordered source output fields=[maximum_clique, clique_number, maximum_independent_set, independence_number, all_subsets_checked, certificate]

條件：無

原題快照保留歷史介面標記；本頁直接列來源欄位順序，不需JavaScript或外部表。欄位標籤不自動成為數學證據，No依各題聲稱範圍解讀。

步驟識別與依賴

規則：source_ordered_interface_explicit_without_javascript · 依賴：G001

S002 · implies

操作前

complete ORIGINAL finite carrier Fin(6);ALL canonical simple undirected original edges=[[0, 1], [0, 2], [0, 3], [1, 2], [3, 4]]

操作後

all vertices and edge indices retained;extrema slice n<=9 includes empty carrier;complement reverses ONLY distinct-vertex adjacency

條件：無

空圖與孤立點都保留；complement只在不同頂點間反轉有無邊，不新增自環。這是bounded完整圖證書，不從某個局部子圖推原圖極值。

步驟識別與依賴

規則：clique_complete_graph_carrier_not_sample_or_loop_complement · 依賴：G001, S001

S003 · implies

操作前

next ORIGINAL vertex bitmask=0;decoded increasing vertices=[];ALL pair rows=[];previous best bitmasks=[0, 0]

操作後

clique=是;independent=是;size=0;strict-larger ONLY update gives best clique=[] at mask0,best independent=[] at mask0

條件：無

每個mask恰表示一個原carrier子集，逐所有pair判兩種資格。只在嚴格更大時更新，因此同大小保最先數字bitmask，不重新按頂點列表lex排序；本列不可用抽樣替代。

步驟識別與依賴

規則：clique_every_bitmask_exact_extrema_strict_tie_order · 依賴：G001, S002

S004 · implies

操作前

next ORIGINAL vertex bitmask=1;decoded increasing vertices=[0];ALL pair rows=[];previous best bitmasks=[0, 0]

操作後

clique=是;independent=是;size=1;strict-larger ONLY update gives best clique=[0] at mask1,best independent=[0] at mask1

條件：無

步驟識別與依賴

規則：clique_every_bitmask_exact_extrema_strict_tie_order · 依賴：G001, S003

S005 · implies

操作前

next ORIGINAL vertex bitmask=2;decoded increasing vertices=[1];ALL pair rows=[];previous best bitmasks=[1, 1]

操作後

clique=是;independent=是;size=1;strict-larger ONLY update gives best clique=[0] at mask1,best independent=[0] at mask1

條件：無

步驟識別與依賴

規則：clique_every_bitmask_exact_extrema_strict_tie_order · 依賴：G001, S004

S006 · implies

操作前

next ORIGINAL vertex bitmask=3;decoded increasing vertices=[0, 1];ALL pair rows=[[0, 1, 是]];previous best bitmasks=[1, 1]

操作後

clique=是;independent=否;size=2;strict-larger ONLY update gives best clique=[0, 1] at mask3,best independent=[0] at mask1

條件：無

步驟識別與依賴

規則：clique_every_bitmask_exact_extrema_strict_tie_order · 依賴：G001, S005

S007 · implies

操作前

next ORIGINAL vertex bitmask=4;decoded increasing vertices=[2];ALL pair rows=[];previous best bitmasks=[3, 1]

操作後

clique=是;independent=是;size=1;strict-larger ONLY update gives best clique=[0, 1] at mask3,best independent=[0] at mask1

條件：無

步驟識別與依賴

規則：clique_every_bitmask_exact_extrema_strict_tie_order · 依賴：G001, S006

S008 · implies

操作前

next ORIGINAL vertex bitmask=5;decoded increasing vertices=[0, 2];ALL pair rows=[[0, 2, 是]];previous best bitmasks=[3, 1]

操作後

clique=是;independent=否;size=2;strict-larger ONLY update gives best clique=[0, 1] at mask3,best independent=[0] at mask1

條件：無

步驟識別與依賴

規則：clique_every_bitmask_exact_extrema_strict_tie_order · 依賴：G001, S007

S009 · implies

操作前

next ORIGINAL vertex bitmask=6;decoded increasing vertices=[1, 2];ALL pair rows=[[1, 2, 是]];previous best bitmasks=[3, 1]

操作後

clique=是;independent=否;size=2;strict-larger ONLY update gives best clique=[0, 1] at mask3,best independent=[0] at mask1

條件：無

步驟識別與依賴

規則：clique_every_bitmask_exact_extrema_strict_tie_order · 依賴：G001, S008

S010 · implies

操作前

next ORIGINAL vertex bitmask=7;decoded increasing vertices=[0, 1, 2];ALL pair rows=[[0, 1, 是], [0, 2, 是], [1, 2, 是]];previous best bitmasks=[3, 1]

操作後

clique=是;independent=否;size=3;strict-larger ONLY update gives best clique=[0, 1, 2] at mask7,best independent=[0] at mask1

條件：無

步驟識別與依賴

規則：clique_every_bitmask_exact_extrema_strict_tie_order · 依賴：G001, S009

S011 · implies

操作前

next ORIGINAL vertex bitmask=8;decoded increasing vertices=[3];ALL pair rows=[];previous best bitmasks=[7, 1]

操作後

clique=是;independent=是;size=1;strict-larger ONLY update gives best clique=[0, 1, 2] at mask7,best independent=[0] at mask1

條件：無

步驟識別與依賴

規則：clique_every_bitmask_exact_extrema_strict_tie_order · 依賴：G001, S010

S012 · implies

操作前

next ORIGINAL vertex bitmask=9;decoded increasing vertices=[0, 3];ALL pair rows=[[0, 3, 是]];previous best bitmasks=[7, 1]

操作後

clique=是;independent=否;size=2;strict-larger ONLY update gives best clique=[0, 1, 2] at mask7,best independent=[0] at mask1

條件：無

步驟識別與依賴

規則：clique_every_bitmask_exact_extrema_strict_tie_order · 依賴：G001, S011

S013 · implies

操作前

next ORIGINAL vertex bitmask=10;decoded increasing vertices=[1, 3];ALL pair rows=[[1, 3, 否]];previous best bitmasks=[7, 1]

操作後

clique=否;independent=是;size=2;strict-larger ONLY update gives best clique=[0, 1, 2] at mask7,best independent=[1, 3] at mask10

條件：無

步驟識別與依賴

規則：clique_every_bitmask_exact_extrema_strict_tie_order · 依賴：G001, S012

S014 · implies

操作前

next ORIGINAL vertex bitmask=11;decoded increasing vertices=[0, 1, 3];ALL pair rows=[[0, 1, 是], [0, 3, 是], [1, 3, 否]];previous best bitmasks=[7, 10]

操作後

clique=否;independent=否;size=3;strict-larger ONLY update gives best clique=[0, 1, 2] at mask7,best independent=[1, 3] at mask10

條件：無

步驟識別與依賴

規則：clique_every_bitmask_exact_extrema_strict_tie_order · 依賴：G001, S013

S015 · implies

操作前

next ORIGINAL vertex bitmask=12;decoded increasing vertices=[2, 3];ALL pair rows=[[2, 3, 否]];previous best bitmasks=[7, 10]

操作後

clique=否;independent=是;size=2;strict-larger ONLY update gives best clique=[0, 1, 2] at mask7,best independent=[1, 3] at mask10

條件：無

步驟識別與依賴

規則：clique_every_bitmask_exact_extrema_strict_tie_order · 依賴：G001, S014

S016 · implies

操作前

next ORIGINAL vertex bitmask=13;decoded increasing vertices=[0, 2, 3];ALL pair rows=[[0, 2, 是], [0, 3, 是], [2, 3, 否]];previous best bitmasks=[7, 10]

操作後

clique=否;independent=否;size=3;strict-larger ONLY update gives best clique=[0, 1, 2] at mask7,best independent=[1, 3] at mask10

條件：無

步驟識別與依賴

規則：clique_every_bitmask_exact_extrema_strict_tie_order · 依賴：G001, S015

S017 · implies

操作前

next ORIGINAL vertex bitmask=14;decoded increasing vertices=[1, 2, 3];ALL pair rows=[[1, 2, 是], [1, 3, 否], [2, 3, 否]];previous best bitmasks=[7, 10]

操作後

clique=否;independent=否;size=3;strict-larger ONLY update gives best clique=[0, 1, 2] at mask7,best independent=[1, 3] at mask10

條件：無

步驟識別與依賴

規則：clique_every_bitmask_exact_extrema_strict_tie_order · 依賴：G001, S016

S018 · implies

操作前

next ORIGINAL vertex bitmask=15;decoded increasing vertices=[0, 1, 2, 3];ALL pair rows=[[0, 1, 是], [0, 2, 是], [0, 3, 是], [1, 2, 是], [1, 3, 否], [2, 3, 否]];previous best bitmasks=[7, 10]

操作後

clique=否;independent=否;size=4;strict-larger ONLY update gives best clique=[0, 1, 2] at mask7,best independent=[1, 3] at mask10

條件：無

步驟識別與依賴

規則：clique_every_bitmask_exact_extrema_strict_tie_order · 依賴：G001, S017

S019 · implies

操作前

next ORIGINAL vertex bitmask=16;decoded increasing vertices=[4];ALL pair rows=[];previous best bitmasks=[7, 10]

操作後

clique=是;independent=是;size=1;strict-larger ONLY update gives best clique=[0, 1, 2] at mask7,best independent=[1, 3] at mask10

條件：無

步驟識別與依賴

規則：clique_every_bitmask_exact_extrema_strict_tie_order · 依賴：G001, S018

S020 · implies

操作前

next ORIGINAL vertex bitmask=17;decoded increasing vertices=[0, 4];ALL pair rows=[[0, 4, 否]];previous best bitmasks=[7, 10]

操作後

clique=否;independent=是;size=2;strict-larger ONLY update gives best clique=[0, 1, 2] at mask7,best independent=[1, 3] at mask10

條件：無

步驟識別與依賴

規則：clique_every_bitmask_exact_extrema_strict_tie_order · 依賴：G001, S019

S021 · implies

操作前

next ORIGINAL vertex bitmask=18;decoded increasing vertices=[1, 4];ALL pair rows=[[1, 4, 否]];previous best bitmasks=[7, 10]

操作後

clique=否;independent=是;size=2;strict-larger ONLY update gives best clique=[0, 1, 2] at mask7,best independent=[1, 3] at mask10

條件：無

步驟識別與依賴

規則：clique_every_bitmask_exact_extrema_strict_tie_order · 依賴：G001, S020

S022 · implies

操作前

next ORIGINAL vertex bitmask=19;decoded increasing vertices=[0, 1, 4];ALL pair rows=[[0, 1, 是], [0, 4, 否], [1, 4, 否]];previous best bitmasks=[7, 10]

操作後

clique=否;independent=否;size=3;strict-larger ONLY update gives best clique=[0, 1, 2] at mask7,best independent=[1, 3] at mask10

條件：無

步驟識別與依賴

規則：clique_every_bitmask_exact_extrema_strict_tie_order · 依賴：G001, S021

S023 · implies

操作前

next ORIGINAL vertex bitmask=20;decoded increasing vertices=[2, 4];ALL pair rows=[[2, 4, 否]];previous best bitmasks=[7, 10]

操作後

clique=否;independent=是;size=2;strict-larger ONLY update gives best clique=[0, 1, 2] at mask7,best independent=[1, 3] at mask10

條件：無

步驟識別與依賴

規則：clique_every_bitmask_exact_extrema_strict_tie_order · 依賴：G001, S022

S024 · implies

操作前

next ORIGINAL vertex bitmask=21;decoded increasing vertices=[0, 2, 4];ALL pair rows=[[0, 2, 是], [0, 4, 否], [2, 4, 否]];previous best bitmasks=[7, 10]

操作後

clique=否;independent=否;size=3;strict-larger ONLY update gives best clique=[0, 1, 2] at mask7,best independent=[1, 3] at mask10

條件：無

步驟識別與依賴

規則：clique_every_bitmask_exact_extrema_strict_tie_order · 依賴：G001, S023

S025 · implies

操作前

next ORIGINAL vertex bitmask=22;decoded increasing vertices=[1, 2, 4];ALL pair rows=[[1, 2, 是], [1, 4, 否], [2, 4, 否]];previous best bitmasks=[7, 10]

操作後

clique=否;independent=否;size=3;strict-larger ONLY update gives best clique=[0, 1, 2] at mask7,best independent=[1, 3] at mask10

條件：無

步驟識別與依賴

規則：clique_every_bitmask_exact_extrema_strict_tie_order · 依賴：G001, S024

S026 · implies

操作前

next ORIGINAL vertex bitmask=23;decoded increasing vertices=[0, 1, 2, 4];ALL pair rows=[[0, 1, 是], [0, 2, 是], [0, 4, 否], [1, 2, 是], [1, 4, 否], [2, 4, 否]];previous best bitmasks=[7, 10]

操作後

clique=否;independent=否;size=4;strict-larger ONLY update gives best clique=[0, 1, 2] at mask7,best independent=[1, 3] at mask10

條件：無

步驟識別與依賴

規則：clique_every_bitmask_exact_extrema_strict_tie_order · 依賴：G001, S025

S027 · implies

操作前

next ORIGINAL vertex bitmask=24;decoded increasing vertices=[3, 4];ALL pair rows=[[3, 4, 是]];previous best bitmasks=[7, 10]

操作後

clique=是;independent=否;size=2;strict-larger ONLY update gives best clique=[0, 1, 2] at mask7,best independent=[1, 3] at mask10

條件：無

步驟識別與依賴

規則：clique_every_bitmask_exact_extrema_strict_tie_order · 依賴：G001, S026

S028 · implies

操作前

next ORIGINAL vertex bitmask=25;decoded increasing vertices=[0, 3, 4];ALL pair rows=[[0, 3, 是], [0, 4, 否], [3, 4, 是]];previous best bitmasks=[7, 10]

操作後

clique=否;independent=否;size=3;strict-larger ONLY update gives best clique=[0, 1, 2] at mask7,best independent=[1, 3] at mask10

條件：無

步驟識別與依賴

規則：clique_every_bitmask_exact_extrema_strict_tie_order · 依賴：G001, S027

S029 · implies

操作前

next ORIGINAL vertex bitmask=26;decoded increasing vertices=[1, 3, 4];ALL pair rows=[[1, 3, 否], [1, 4, 否], [3, 4, 是]];previous best bitmasks=[7, 10]

操作後

clique=否;independent=否;size=3;strict-larger ONLY update gives best clique=[0, 1, 2] at mask7,best independent=[1, 3] at mask10

條件：無

步驟識別與依賴

規則：clique_every_bitmask_exact_extrema_strict_tie_order · 依賴：G001, S028

S030 · implies

操作前

next ORIGINAL vertex bitmask=27;decoded increasing vertices=[0, 1, 3, 4];ALL pair rows=[[0, 1, 是], [0, 3, 是], [0, 4, 否], [1, 3, 否], [1, 4, 否], [3, 4, 是]];previous best bitmasks=[7, 10]

操作後

clique=否;independent=否;size=4;strict-larger ONLY update gives best clique=[0, 1, 2] at mask7,best independent=[1, 3] at mask10

條件：無

步驟識別與依賴

規則：clique_every_bitmask_exact_extrema_strict_tie_order · 依賴：G001, S029

S031 · implies

操作前

next ORIGINAL vertex bitmask=28;decoded increasing vertices=[2, 3, 4];ALL pair rows=[[2, 3, 否], [2, 4, 否], [3, 4, 是]];previous best bitmasks=[7, 10]

操作後

clique=否;independent=否;size=3;strict-larger ONLY update gives best clique=[0, 1, 2] at mask7,best independent=[1, 3] at mask10

條件：無

步驟識別與依賴

規則：clique_every_bitmask_exact_extrema_strict_tie_order · 依賴：G001, S030

S032 · implies

操作前

next ORIGINAL vertex bitmask=29;decoded increasing vertices=[0, 2, 3, 4];ALL pair rows=[[0, 2, 是], [0, 3, 是], [0, 4, 否], [2, 3, 否], [2, 4, 否], [3, 4, 是]];previous best bitmasks=[7, 10]

操作後

clique=否;independent=否;size=4;strict-larger ONLY update gives best clique=[0, 1, 2] at mask7,best independent=[1, 3] at mask10

條件：無

步驟識別與依賴

規則：clique_every_bitmask_exact_extrema_strict_tie_order · 依賴：G001, S031

S033 · implies

操作前

next ORIGINAL vertex bitmask=30;decoded increasing vertices=[1, 2, 3, 4];ALL pair rows=[[1, 2, 是], [1, 3, 否], [1, 4, 否], [2, 3, 否], [2, 4, 否], [3, 4, 是]];previous best bitmasks=[7, 10]

操作後

clique=否;independent=否;size=4;strict-larger ONLY update gives best clique=[0, 1, 2] at mask7,best independent=[1, 3] at mask10

條件：無

步驟識別與依賴

規則：clique_every_bitmask_exact_extrema_strict_tie_order · 依賴：G001, S032

S034 · implies

操作前

next ORIGINAL vertex bitmask=31;decoded increasing vertices=[0, 1, 2, 3, 4];ALL pair rows=[[0, 1, 是], [0, 2, 是], [0, 3, 是], [0, 4, 否], [1, 2, 是], [1, 3, 否], [1, 4, 否], [2, 3, 否], [2, 4, 否], [3, 4, 是]];previous best bitmasks=[7, 10]

操作後

clique=否;independent=否;size=5;strict-larger ONLY update gives best clique=[0, 1, 2] at mask7,best independent=[1, 3] at mask10

條件：無

步驟識別與依賴

規則：clique_every_bitmask_exact_extrema_strict_tie_order · 依賴：G001, S033

S035 · implies

操作前

next ORIGINAL vertex bitmask=32;decoded increasing vertices=[5];ALL pair rows=[];previous best bitmasks=[7, 10]

操作後

clique=是;independent=是;size=1;strict-larger ONLY update gives best clique=[0, 1, 2] at mask7,best independent=[1, 3] at mask10

條件：無

步驟識別與依賴

規則：clique_every_bitmask_exact_extrema_strict_tie_order · 依賴：G001, S034

S036 · implies

操作前

next ORIGINAL vertex bitmask=33;decoded increasing vertices=[0, 5];ALL pair rows=[[0, 5, 否]];previous best bitmasks=[7, 10]

操作後

clique=否;independent=是;size=2;strict-larger ONLY update gives best clique=[0, 1, 2] at mask7,best independent=[1, 3] at mask10

條件：無

步驟識別與依賴

規則：clique_every_bitmask_exact_extrema_strict_tie_order · 依賴：G001, S035

S037 · implies

操作前

next ORIGINAL vertex bitmask=34;decoded increasing vertices=[1, 5];ALL pair rows=[[1, 5, 否]];previous best bitmasks=[7, 10]

操作後

clique=否;independent=是;size=2;strict-larger ONLY update gives best clique=[0, 1, 2] at mask7,best independent=[1, 3] at mask10

條件：無

步驟識別與依賴

規則：clique_every_bitmask_exact_extrema_strict_tie_order · 依賴：G001, S036

S038 · implies

操作前

next ORIGINAL vertex bitmask=35;decoded increasing vertices=[0, 1, 5];ALL pair rows=[[0, 1, 是], [0, 5, 否], [1, 5, 否]];previous best bitmasks=[7, 10]

操作後

clique=否;independent=否;size=3;strict-larger ONLY update gives best clique=[0, 1, 2] at mask7,best independent=[1, 3] at mask10

條件：無

步驟識別與依賴

規則：clique_every_bitmask_exact_extrema_strict_tie_order · 依賴：G001, S037

S039 · implies

操作前

next ORIGINAL vertex bitmask=36;decoded increasing vertices=[2, 5];ALL pair rows=[[2, 5, 否]];previous best bitmasks=[7, 10]

操作後

clique=否;independent=是;size=2;strict-larger ONLY update gives best clique=[0, 1, 2] at mask7,best independent=[1, 3] at mask10

條件：無

步驟識別與依賴

規則：clique_every_bitmask_exact_extrema_strict_tie_order · 依賴：G001, S038

S040 · implies

操作前

next ORIGINAL vertex bitmask=37;decoded increasing vertices=[0, 2, 5];ALL pair rows=[[0, 2, 是], [0, 5, 否], [2, 5, 否]];previous best bitmasks=[7, 10]

操作後

clique=否;independent=否;size=3;strict-larger ONLY update gives best clique=[0, 1, 2] at mask7,best independent=[1, 3] at mask10

條件：無

步驟識別與依賴

規則：clique_every_bitmask_exact_extrema_strict_tie_order · 依賴：G001, S039

S041 · implies

操作前

next ORIGINAL vertex bitmask=38;decoded increasing vertices=[1, 2, 5];ALL pair rows=[[1, 2, 是], [1, 5, 否], [2, 5, 否]];previous best bitmasks=[7, 10]

操作後

clique=否;independent=否;size=3;strict-larger ONLY update gives best clique=[0, 1, 2] at mask7,best independent=[1, 3] at mask10

條件：無

步驟識別與依賴

規則：clique_every_bitmask_exact_extrema_strict_tie_order · 依賴：G001, S040

S042 · implies

操作前

next ORIGINAL vertex bitmask=39;decoded increasing vertices=[0, 1, 2, 5];ALL pair rows=[[0, 1, 是], [0, 2, 是], [0, 5, 否], [1, 2, 是], [1, 5, 否], [2, 5, 否]];previous best bitmasks=[7, 10]

操作後

clique=否;independent=否;size=4;strict-larger ONLY update gives best clique=[0, 1, 2] at mask7,best independent=[1, 3] at mask10

條件：無

步驟識別與依賴

規則：clique_every_bitmask_exact_extrema_strict_tie_order · 依賴：G001, S041

S043 · implies

操作前

next ORIGINAL vertex bitmask=40;decoded increasing vertices=[3, 5];ALL pair rows=[[3, 5, 否]];previous best bitmasks=[7, 10]

操作後

clique=否;independent=是;size=2;strict-larger ONLY update gives best clique=[0, 1, 2] at mask7,best independent=[1, 3] at mask10

條件：無

步驟識別與依賴

規則：clique_every_bitmask_exact_extrema_strict_tie_order · 依賴：G001, S042

S044 · implies

操作前

next ORIGINAL vertex bitmask=41;decoded increasing vertices=[0, 3, 5];ALL pair rows=[[0, 3, 是], [0, 5, 否], [3, 5, 否]];previous best bitmasks=[7, 10]

操作後

clique=否;independent=否;size=3;strict-larger ONLY update gives best clique=[0, 1, 2] at mask7,best independent=[1, 3] at mask10

條件：無

步驟識別與依賴

規則：clique_every_bitmask_exact_extrema_strict_tie_order · 依賴：G001, S043

S045 · implies

操作前

next ORIGINAL vertex bitmask=42;decoded increasing vertices=[1, 3, 5];ALL pair rows=[[1, 3, 否], [1, 5, 否], [3, 5, 否]];previous best bitmasks=[7, 10]

操作後

clique=否;independent=是;size=3;strict-larger ONLY update gives best clique=[0, 1, 2] at mask7,best independent=[1, 3, 5] at mask42

條件：無

步驟識別與依賴

規則：clique_every_bitmask_exact_extrema_strict_tie_order · 依賴：G001, S044

S046 · implies

操作前

next ORIGINAL vertex bitmask=43;decoded increasing vertices=[0, 1, 3, 5];ALL pair rows=[[0, 1, 是], [0, 3, 是], [0, 5, 否], [1, 3, 否], [1, 5, 否], [3, 5, 否]];previous best bitmasks=[7, 42]

操作後

clique=否;independent=否;size=4;strict-larger ONLY update gives best clique=[0, 1, 2] at mask7,best independent=[1, 3, 5] at mask42

條件：無

步驟識別與依賴

規則：clique_every_bitmask_exact_extrema_strict_tie_order · 依賴：G001, S045

S047 · implies

操作前

next ORIGINAL vertex bitmask=44;decoded increasing vertices=[2, 3, 5];ALL pair rows=[[2, 3, 否], [2, 5, 否], [3, 5, 否]];previous best bitmasks=[7, 42]

操作後

clique=否;independent=是;size=3;strict-larger ONLY update gives best clique=[0, 1, 2] at mask7,best independent=[1, 3, 5] at mask42

條件：無

步驟識別與依賴

規則：clique_every_bitmask_exact_extrema_strict_tie_order · 依賴：G001, S046

S048 · implies

操作前

next ORIGINAL vertex bitmask=45;decoded increasing vertices=[0, 2, 3, 5];ALL pair rows=[[0, 2, 是], [0, 3, 是], [0, 5, 否], [2, 3, 否], [2, 5, 否], [3, 5, 否]];previous best bitmasks=[7, 42]

操作後

clique=否;independent=否;size=4;strict-larger ONLY update gives best clique=[0, 1, 2] at mask7,best independent=[1, 3, 5] at mask42

條件：無

步驟識別與依賴

規則：clique_every_bitmask_exact_extrema_strict_tie_order · 依賴：G001, S047

S049 · implies

操作前

next ORIGINAL vertex bitmask=46;decoded increasing vertices=[1, 2, 3, 5];ALL pair rows=[[1, 2, 是], [1, 3, 否], [1, 5, 否], [2, 3, 否], [2, 5, 否], [3, 5, 否]];previous best bitmasks=[7, 42]

操作後

clique=否;independent=否;size=4;strict-larger ONLY update gives best clique=[0, 1, 2] at mask7,best independent=[1, 3, 5] at mask42

條件：無

步驟識別與依賴

規則：clique_every_bitmask_exact_extrema_strict_tie_order · 依賴：G001, S048

S050 · implies

操作前

next ORIGINAL vertex bitmask=47;decoded increasing vertices=[0, 1, 2, 3, 5];ALL pair rows=[[0, 1, 是], [0, 2, 是], [0, 3, 是], [0, 5, 否], [1, 2, 是], [1, 3, 否], [1, 5, 否], [2, 3, 否], [2, 5, 否], [3, 5, 否]];previous best bitmasks=[7, 42]

操作後

clique=否;independent=否;size=5;strict-larger ONLY update gives best clique=[0, 1, 2] at mask7,best independent=[1, 3, 5] at mask42

條件：無

步驟識別與依賴

規則：clique_every_bitmask_exact_extrema_strict_tie_order · 依賴：G001, S049

S051 · implies

操作前

next ORIGINAL vertex bitmask=48;decoded increasing vertices=[4, 5];ALL pair rows=[[4, 5, 否]];previous best bitmasks=[7, 42]

操作後

clique=否;independent=是;size=2;strict-larger ONLY update gives best clique=[0, 1, 2] at mask7,best independent=[1, 3, 5] at mask42

條件：無

步驟識別與依賴

規則：clique_every_bitmask_exact_extrema_strict_tie_order · 依賴：G001, S050

S052 · implies

操作前

next ORIGINAL vertex bitmask=49;decoded increasing vertices=[0, 4, 5];ALL pair rows=[[0, 4, 否], [0, 5, 否], [4, 5, 否]];previous best bitmasks=[7, 42]

操作後

clique=否;independent=是;size=3;strict-larger ONLY update gives best clique=[0, 1, 2] at mask7,best independent=[1, 3, 5] at mask42

條件：無

步驟識別與依賴

規則：clique_every_bitmask_exact_extrema_strict_tie_order · 依賴：G001, S051

S053 · implies

操作前

next ORIGINAL vertex bitmask=50;decoded increasing vertices=[1, 4, 5];ALL pair rows=[[1, 4, 否], [1, 5, 否], [4, 5, 否]];previous best bitmasks=[7, 42]

操作後

clique=否;independent=是;size=3;strict-larger ONLY update gives best clique=[0, 1, 2] at mask7,best independent=[1, 3, 5] at mask42

條件：無

步驟識別與依賴

規則：clique_every_bitmask_exact_extrema_strict_tie_order · 依賴：G001, S052

S054 · implies

操作前

next ORIGINAL vertex bitmask=51;decoded increasing vertices=[0, 1, 4, 5];ALL pair rows=[[0, 1, 是], [0, 4, 否], [0, 5, 否], [1, 4, 否], [1, 5, 否], [4, 5, 否]];previous best bitmasks=[7, 42]

操作後

clique=否;independent=否;size=4;strict-larger ONLY update gives best clique=[0, 1, 2] at mask7,best independent=[1, 3, 5] at mask42

條件：無

步驟識別與依賴

規則：clique_every_bitmask_exact_extrema_strict_tie_order · 依賴：G001, S053

S055 · implies

操作前

next ORIGINAL vertex bitmask=52;decoded increasing vertices=[2, 4, 5];ALL pair rows=[[2, 4, 否], [2, 5, 否], [4, 5, 否]];previous best bitmasks=[7, 42]

操作後

clique=否;independent=是;size=3;strict-larger ONLY update gives best clique=[0, 1, 2] at mask7,best independent=[1, 3, 5] at mask42

條件：無

步驟識別與依賴

規則：clique_every_bitmask_exact_extrema_strict_tie_order · 依賴：G001, S054

S056 · implies

操作前

next ORIGINAL vertex bitmask=53;decoded increasing vertices=[0, 2, 4, 5];ALL pair rows=[[0, 2, 是], [0, 4, 否], [0, 5, 否], [2, 4, 否], [2, 5, 否], [4, 5, 否]];previous best bitmasks=[7, 42]

操作後

clique=否;independent=否;size=4;strict-larger ONLY update gives best clique=[0, 1, 2] at mask7,best independent=[1, 3, 5] at mask42

條件：無

步驟識別與依賴

規則：clique_every_bitmask_exact_extrema_strict_tie_order · 依賴：G001, S055

S057 · implies

操作前

next ORIGINAL vertex bitmask=54;decoded increasing vertices=[1, 2, 4, 5];ALL pair rows=[[1, 2, 是], [1, 4, 否], [1, 5, 否], [2, 4, 否], [2, 5, 否], [4, 5, 否]];previous best bitmasks=[7, 42]

操作後

clique=否;independent=否;size=4;strict-larger ONLY update gives best clique=[0, 1, 2] at mask7,best independent=[1, 3, 5] at mask42

條件：無

步驟識別與依賴

規則：clique_every_bitmask_exact_extrema_strict_tie_order · 依賴：G001, S056

S058 · implies

操作前

next ORIGINAL vertex bitmask=55;decoded increasing vertices=[0, 1, 2, 4, 5];ALL pair rows=[[0, 1, 是], [0, 2, 是], [0, 4, 否], [0, 5, 否], [1, 2, 是], [1, 4, 否], [1, 5, 否], [2, 4, 否], [2, 5, 否], [4, 5, 否]];previous best bitmasks=[7, 42]

操作後

clique=否;independent=否;size=5;strict-larger ONLY update gives best clique=[0, 1, 2] at mask7,best independent=[1, 3, 5] at mask42

條件：無

步驟識別與依賴

規則：clique_every_bitmask_exact_extrema_strict_tie_order · 依賴：G001, S057

S059 · implies

操作前

next ORIGINAL vertex bitmask=56;decoded increasing vertices=[3, 4, 5];ALL pair rows=[[3, 4, 是], [3, 5, 否], [4, 5, 否]];previous best bitmasks=[7, 42]

操作後

clique=否;independent=否;size=3;strict-larger ONLY update gives best clique=[0, 1, 2] at mask7,best independent=[1, 3, 5] at mask42

條件：無

步驟識別與依賴

規則：clique_every_bitmask_exact_extrema_strict_tie_order · 依賴：G001, S058

S060 · implies

操作前

next ORIGINAL vertex bitmask=57;decoded increasing vertices=[0, 3, 4, 5];ALL pair rows=[[0, 3, 是], [0, 4, 否], [0, 5, 否], [3, 4, 是], [3, 5, 否], [4, 5, 否]];previous best bitmasks=[7, 42]

操作後

clique=否;independent=否;size=4;strict-larger ONLY update gives best clique=[0, 1, 2] at mask7,best independent=[1, 3, 5] at mask42

條件：無

步驟識別與依賴

規則：clique_every_bitmask_exact_extrema_strict_tie_order · 依賴：G001, S059

S061 · implies

操作前

next ORIGINAL vertex bitmask=58;decoded increasing vertices=[1, 3, 4, 5];ALL pair rows=[[1, 3, 否], [1, 4, 否], [1, 5, 否], [3, 4, 是], [3, 5, 否], [4, 5, 否]];previous best bitmasks=[7, 42]

操作後

clique=否;independent=否;size=4;strict-larger ONLY update gives best clique=[0, 1, 2] at mask7,best independent=[1, 3, 5] at mask42

條件：無

步驟識別與依賴

規則：clique_every_bitmask_exact_extrema_strict_tie_order · 依賴：G001, S060

S062 · implies

操作前

next ORIGINAL vertex bitmask=59;decoded increasing vertices=[0, 1, 3, 4, 5];ALL pair rows=[[0, 1, 是], [0, 3, 是], [0, 4, 否], [0, 5, 否], [1, 3, 否], [1, 4, 否], [1, 5, 否], [3, 4, 是], [3, 5, 否], [4, 5, 否]];previous best bitmasks=[7, 42]

操作後

clique=否;independent=否;size=5;strict-larger ONLY update gives best clique=[0, 1, 2] at mask7,best independent=[1, 3, 5] at mask42

條件：無

步驟識別與依賴

規則：clique_every_bitmask_exact_extrema_strict_tie_order · 依賴：G001, S061

S063 · implies

操作前

next ORIGINAL vertex bitmask=60;decoded increasing vertices=[2, 3, 4, 5];ALL pair rows=[[2, 3, 否], [2, 4, 否], [2, 5, 否], [3, 4, 是], [3, 5, 否], [4, 5, 否]];previous best bitmasks=[7, 42]

操作後

clique=否;independent=否;size=4;strict-larger ONLY update gives best clique=[0, 1, 2] at mask7,best independent=[1, 3, 5] at mask42

條件：無

步驟識別與依賴

規則：clique_every_bitmask_exact_extrema_strict_tie_order · 依賴：G001, S062

S064 · implies

操作前

next ORIGINAL vertex bitmask=61;decoded increasing vertices=[0, 2, 3, 4, 5];ALL pair rows=[[0, 2, 是], [0, 3, 是], [0, 4, 否], [0, 5, 否], [2, 3, 否], [2, 4, 否], [2, 5, 否], [3, 4, 是], [3, 5, 否], [4, 5, 否]];previous best bitmasks=[7, 42]

操作後

clique=否;independent=否;size=5;strict-larger ONLY update gives best clique=[0, 1, 2] at mask7,best independent=[1, 3, 5] at mask42

條件：無

步驟識別與依賴

規則：clique_every_bitmask_exact_extrema_strict_tie_order · 依賴：G001, S063

S065 · implies

操作前

next ORIGINAL vertex bitmask=62;decoded increasing vertices=[1, 2, 3, 4, 5];ALL pair rows=[[1, 2, 是], [1, 3, 否], [1, 4, 否], [1, 5, 否], [2, 3, 否], [2, 4, 否], [2, 5, 否], [3, 4, 是], [3, 5, 否], [4, 5, 否]];previous best bitmasks=[7, 42]

操作後

clique=否;independent=否;size=5;strict-larger ONLY update gives best clique=[0, 1, 2] at mask7,best independent=[1, 3, 5] at mask42

條件：無

步驟識別與依賴

規則：clique_every_bitmask_exact_extrema_strict_tie_order · 依賴：G001, S064

S066 · implies

操作前

next ORIGINAL vertex bitmask=63;decoded increasing vertices=[0, 1, 2, 3, 4, 5];ALL pair rows=[[0, 1, 是], [0, 2, 是], [0, 3, 是], [0, 4, 否], [0, 5, 否], [1, 2, 是], [1, 3, 否], [1, 4, 否], [1, 5, 否], [2, 3, 否], [2, 4, 否], [2, 5, 否], [3, 4, 是], [3, 5, 否], [4, 5, 否]];previous best bitmasks=[7, 42]

操作後

clique=否;independent=否;size=6;strict-larger ONLY update gives best clique=[0, 1, 2] at mask7,best independent=[1, 3, 5] at mask42

條件：無

步驟識別與依賴

規則：clique_every_bitmask_exact_extrema_strict_tie_order · 依賴：G001, S065

S067 · implies

操作前

ALL masks0..63 processed once;best clique=[0, 1, 2] at mask7;best independent=[1, 3, 5] at mask42

操作後

every possible subset covered;both actual witnesses valid;no valid subset larger;omega=3,alpha=3;FIRST numeric-mask ties retained

條件：無

完整枚舉覆蓋全部候選給全域上界，實際合格witness達到此界給下界。二者相等才是maximum；不能拿無法加一個頂點的maximal代替。空carrier的兩個極值都是0。

步驟識別與依賴

規則：clique_global_upper_and_attained_lower_not_maximal_only · 依賴：G001, S066

逐步反向核對

S067 · 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。
S066 · 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。
S065 · 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。
S064 · 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。
S063 · 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。
S062 · 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。
S061 · 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。
S060 · 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。
S059 · 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。
S058 · 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。
S057 · 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。
S056 · 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。
S055 · 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。
S054 · 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。
S053 · 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。
S052 · 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。
S051 · 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。
S050 · 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。
S049 · 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。
S048 · 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。
S047 · 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。
S046 · 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。
S045 · 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。
S044 · 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。
S043 · 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。
S042 · 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。
S041 · 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。
S040 · 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。
S039 · 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。
S038 · 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。
S037 · 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。
S036 · 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。
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 — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
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 — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
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 — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
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 — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
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 — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
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 — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
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 — 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 — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
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 — 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 — 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 — 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 — 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 — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
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 — 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 — 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 — 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 — 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 — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
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

([0, 1, 2], 3, [1, 3, 5], 3, 是, exhaustive_all_vertex_subsets)

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: 9dd803d78e772a921fc3d13e5cc07e21e161a09d56dd658bcb597e08d44aa4d4
candidate record SHA-256: ab4ae763271d042ba4d5a860150b49b4c6ca98743b30a46383f3e57a909a127e
semantic review receipt SHA-256: d165cad30849a3aba67390a59c572c0d965269d4b405e43df36da19e28d85612
question-first witness SHA-256: c222ca47a07939c019865a1e4d9ce633205cdee68db92bf96a96f5683a62ebc7
Lean: NOT_RUN · human review: NOT_PERFORMED

同源完整 JSON