# Question 19530: Exact Extremal Sizes Need the Complete Vertex-Subset Search [Return to catalogue page 98](https://ocme.evemisslab.com/en/questions/page/98/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations/#q-20261014-advance-20261106-r02-clique-and-independent-set-0055) This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth. R0038-19530 · q-20261014-advance-20261106-r02-clique-and-independent-set-0055 ## Original question and givens Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [1, 2], [2, 4], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Give (maximum_clique, clique_number, maximum_independent_set, independence_number, all_subsets_checked, certificate). - G001 — Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [1, 2], [2, 4], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Give (maximum_clique, clique_number, maximum_independent_set, independence_number, all_subsets_checked, certificate). ## Definition and scope Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate. - [mko-clique-and-independent-set](https://ocme.evemisslab.com/en/mko/mko-clique-and-independent-set/) ## Step-by-step derivation S001 · impliesBefore source operation=clique-exact-extrema;original snapshot and mathematics retained After explicit ordered source output fields=[maximum_clique, clique_number, maximum_independent_set, independence_number, all_subsets_checked, certificate] Conditions: none Keep the historical interface marker in the original snapshot while listing ordered source fields here,without JavaScript or external lookup.Field labels are not mathematical evidence,and No retains the operation's claim scope. Step identifiers and dependenciesRule: source_ordered_interface_explicit_without_javascript · Uses: G001 S002 · impliesBefore complete ORIGINAL finite carrier Fin(6);ALL canonical simple undirected original edges=[[0, 1], [0, 2], [1, 2], [2, 4], [3, 4]] After all vertices and edge indices retained;extrema slice n<=9 includes empty carrier;complement reverses ONLY distinct-vertex adjacency Conditions: none Keep empty carriers and isolates.Complementation reverses adjacency only between distinct vertices,without adding loops.This is a bounded complete-graph certificate,not an extremal inference from a sampled subgraph. Step identifiers and dependenciesRule: clique_complete_graph_carrier_not_sample_or_loop_complement · Uses: G001, S001 S003 · impliesBefore next ORIGINAL vertex bitmask=0;decoded increasing vertices=[];ALL pair rows=[];previous best bitmasks=[0, 0] After clique=Yes;independent=Yes;size=0;strict-larger ONLY update gives best clique=[] at mask0,best independent=[] at mask0 Conditions: none Each mask denotes exactly one subset of the original carrier.Audit all pairs and update only for a strictly larger valid size.Ties retain the first numeric bitmask rather than lexicographically sorting vertex lists;sampling cannot replace a row. Step identifiers and dependenciesRule: clique_every_bitmask_exact_extrema_strict_tie_order · Uses: G001, S002 S004 · impliesBefore next ORIGINAL vertex bitmask=1;decoded increasing vertices=[0];ALL pair rows=[];previous best bitmasks=[0, 0] After clique=Yes;independent=Yes;size=1;strict-larger ONLY update gives best clique=[0] at mask1,best independent=[0] at mask1 Conditions: none Each mask denotes exactly one subset of the original carrier.Audit all pairs and update only for a strictly larger valid size.Ties retain the first numeric bitmask rather than lexicographically sorting vertex lists;sampling cannot replace a row. Step identifiers and dependenciesRule: clique_every_bitmask_exact_extrema_strict_tie_order · Uses: G001, S003 S005 · impliesBefore next ORIGINAL vertex bitmask=2;decoded increasing vertices=[1];ALL pair rows=[];previous best bitmasks=[1, 1] After clique=Yes;independent=Yes;size=1;strict-larger ONLY update gives best clique=[0] at mask1,best independent=[0] at mask1 Conditions: none Each mask denotes exactly one subset of the original carrier.Audit all pairs and update only for a strictly larger valid size.Ties retain the first numeric bitmask rather than lexicographically sorting vertex lists;sampling cannot replace a row. Step identifiers and dependenciesRule: clique_every_bitmask_exact_extrema_strict_tie_order · Uses: G001, S004 S006 · impliesBefore next ORIGINAL vertex bitmask=3;decoded increasing vertices=[0, 1];ALL pair rows=[[0, 1, Yes]];previous best bitmasks=[1, 1] After clique=Yes;independent=No;size=2;strict-larger ONLY update gives best clique=[0, 1] at mask3,best independent=[0] at mask1 Conditions: none Each mask denotes exactly one subset of the original carrier.Audit all pairs and update only for a strictly larger valid size.Ties retain the first numeric bitmask rather than lexicographically sorting vertex lists;sampling cannot replace a row. Step identifiers and dependenciesRule: clique_every_bitmask_exact_extrema_strict_tie_order · Uses: G001, S005 S007 · impliesBefore next ORIGINAL vertex bitmask=4;decoded increasing vertices=[2];ALL pair rows=[];previous best bitmasks=[3, 1] After clique=Yes;independent=Yes;size=1;strict-larger ONLY update gives best clique=[0, 1] at mask3,best independent=[0] at mask1 Conditions: none Each mask denotes exactly one subset of the original carrier.Audit all pairs and update only for a strictly larger valid size.Ties retain the first numeric bitmask rather than lexicographically sorting vertex lists;sampling cannot replace a row. Step identifiers and dependenciesRule: clique_every_bitmask_exact_extrema_strict_tie_order · Uses: G001, S006 S008 · impliesBefore next ORIGINAL vertex bitmask=5;decoded increasing vertices=[0, 2];ALL pair rows=[[0, 2, Yes]];previous best bitmasks=[3, 1] After clique=Yes;independent=No;size=2;strict-larger ONLY update gives best clique=[0, 1] at mask3,best independent=[0] at mask1 Conditions: none Each mask denotes exactly one subset of the original carrier.Audit all pairs and update only for a strictly larger valid size.Ties retain the first numeric bitmask rather than lexicographically sorting vertex lists;sampling cannot replace a row. Step identifiers and dependenciesRule: clique_every_bitmask_exact_extrema_strict_tie_order · Uses: G001, S007 S009 · impliesBefore next ORIGINAL vertex bitmask=6;decoded increasing vertices=[1, 2];ALL pair rows=[[1, 2, Yes]];previous best bitmasks=[3, 1] After clique=Yes;independent=No;size=2;strict-larger ONLY update gives best clique=[0, 1] at mask3,best independent=[0] at mask1 Conditions: none Each mask denotes exactly one subset of the original carrier.Audit all pairs and update only for a strictly larger valid size.Ties retain the first numeric bitmask rather than lexicographically sorting vertex lists;sampling cannot replace a row. Step identifiers and dependenciesRule: clique_every_bitmask_exact_extrema_strict_tie_order · Uses: G001, S008 S010 · impliesBefore next ORIGINAL vertex bitmask=7;decoded increasing vertices=[0, 1, 2];ALL pair rows=[[0, 1, Yes], [0, 2, Yes], [1, 2, Yes]];previous best bitmasks=[3, 1] After clique=Yes;independent=No;size=3;strict-larger ONLY update gives best clique=[0, 1, 2] at mask7,best independent=[0] at mask1 Conditions: none Each mask denotes exactly one subset of the original carrier.Audit all pairs and update only for a strictly larger valid size.Ties retain the first numeric bitmask rather than lexicographically sorting vertex lists;sampling cannot replace a row. Step identifiers and dependenciesRule: clique_every_bitmask_exact_extrema_strict_tie_order · Uses: G001, S009 S011 · impliesBefore next ORIGINAL vertex bitmask=8;decoded increasing vertices=[3];ALL pair rows=[];previous best bitmasks=[7, 1] After clique=Yes;independent=Yes;size=1;strict-larger ONLY update gives best clique=[0, 1, 2] at mask7,best independent=[0] at mask1 Conditions: none Each mask denotes exactly one subset of the original carrier.Audit all pairs and update only for a strictly larger valid size.Ties retain the first numeric bitmask rather than lexicographically sorting vertex lists;sampling cannot replace a row. Step identifiers and dependenciesRule: clique_every_bitmask_exact_extrema_strict_tie_order · Uses: G001, S010 S012 · impliesBefore next ORIGINAL vertex bitmask=9;decoded increasing vertices=[0, 3];ALL pair rows=[[0, 3, No]];previous best bitmasks=[7, 1] After clique=No;independent=Yes;size=2;strict-larger ONLY update gives best clique=[0, 1, 2] at mask7,best independent=[0, 3] at mask9 Conditions: none Each mask denotes exactly one subset of the original carrier.Audit all pairs and update only for a strictly larger valid size.Ties retain the first numeric bitmask rather than lexicographically sorting vertex lists;sampling cannot replace a row. Step identifiers and dependenciesRule: clique_every_bitmask_exact_extrema_strict_tie_order · Uses: G001, S011 S013 · impliesBefore next ORIGINAL vertex bitmask=10;decoded increasing vertices=[1, 3];ALL pair rows=[[1, 3, No]];previous best bitmasks=[7, 9] After clique=No;independent=Yes;size=2;strict-larger ONLY update gives best clique=[0, 1, 2] at mask7,best independent=[0, 3] at mask9 Conditions: none Each mask denotes exactly one subset of the original carrier.Audit all pairs and update only for a strictly larger valid size.Ties retain the first numeric bitmask rather than lexicographically sorting vertex lists;sampling cannot replace a row. Step identifiers and dependenciesRule: clique_every_bitmask_exact_extrema_strict_tie_order · Uses: G001, S012 S014 · impliesBefore next ORIGINAL vertex bitmask=11;decoded increasing vertices=[0, 1, 3];ALL pair rows=[[0, 1, Yes], [0, 3, No], [1, 3, No]];previous best bitmasks=[7, 9] After clique=No;independent=No;size=3;strict-larger ONLY update gives best clique=[0, 1, 2] at mask7,best independent=[0, 3] at mask9 Conditions: none Each mask denotes exactly one subset of the original carrier.Audit all pairs and update only for a strictly larger valid size.Ties retain the first numeric bitmask rather than lexicographically sorting vertex lists;sampling cannot replace a row. Step identifiers and dependenciesRule: clique_every_bitmask_exact_extrema_strict_tie_order · Uses: G001, S013 S015 · impliesBefore next ORIGINAL vertex bitmask=12;decoded increasing vertices=[2, 3];ALL pair rows=[[2, 3, No]];previous best bitmasks=[7, 9] After clique=No;independent=Yes;size=2;strict-larger ONLY update gives best clique=[0, 1, 2] at mask7,best independent=[0, 3] at mask9 Conditions: none Each mask denotes exactly one subset of the original carrier.Audit all pairs and update only for a strictly larger valid size.Ties retain the first numeric bitmask rather than lexicographically sorting vertex lists;sampling cannot replace a row. Step identifiers and dependenciesRule: clique_every_bitmask_exact_extrema_strict_tie_order · Uses: G001, S014 S016 · impliesBefore next ORIGINAL vertex bitmask=13;decoded increasing vertices=[0, 2, 3];ALL pair rows=[[0, 2, Yes], [0, 3, No], [2, 3, No]];previous best bitmasks=[7, 9] After clique=No;independent=No;size=3;strict-larger ONLY update gives best clique=[0, 1, 2] at mask7,best independent=[0, 3] at mask9 Conditions: none Each mask denotes exactly one subset of the original carrier.Audit all pairs and update only for a strictly larger valid size.Ties retain the first numeric bitmask rather than lexicographically sorting vertex lists;sampling cannot replace a row. Step identifiers and dependenciesRule: clique_every_bitmask_exact_extrema_strict_tie_order · Uses: G001, S015 S017 · impliesBefore next ORIGINAL vertex bitmask=14;decoded increasing vertices=[1, 2, 3];ALL pair rows=[[1, 2, Yes], [1, 3, No], [2, 3, No]];previous best bitmasks=[7, 9] After clique=No;independent=No;size=3;strict-larger ONLY update gives best clique=[0, 1, 2] at mask7,best independent=[0, 3] at mask9 Conditions: none Each mask denotes exactly one subset of the original carrier.Audit all pairs and update only for a strictly larger valid size.Ties retain the first numeric bitmask rather than lexicographically sorting vertex lists;sampling cannot replace a row. Step identifiers and dependenciesRule: clique_every_bitmask_exact_extrema_strict_tie_order · Uses: G001, S016 S018 · impliesBefore next ORIGINAL vertex bitmask=15;decoded increasing vertices=[0, 1, 2, 3];ALL pair rows=[[0, 1, Yes], [0, 2, Yes], [0, 3, No], [1, 2, Yes], [1, 3, No], [2, 3, No]];previous best bitmasks=[7, 9] After clique=No;independent=No;size=4;strict-larger ONLY update gives best clique=[0, 1, 2] at mask7,best independent=[0, 3] at mask9 Conditions: none Each mask denotes exactly one subset of the original carrier.Audit all pairs and update only for a strictly larger valid size.Ties retain the first numeric bitmask rather than lexicographically sorting vertex lists;sampling cannot replace a row. Step identifiers and dependenciesRule: clique_every_bitmask_exact_extrema_strict_tie_order · Uses: G001, S017 S019 · impliesBefore next ORIGINAL vertex bitmask=16;decoded increasing vertices=[4];ALL pair rows=[];previous best bitmasks=[7, 9] After clique=Yes;independent=Yes;size=1;strict-larger ONLY update gives best clique=[0, 1, 2] at mask7,best independent=[0, 3] at mask9 Conditions: none Each mask denotes exactly one subset of the original carrier.Audit all pairs and update only for a strictly larger valid size.Ties retain the first numeric bitmask rather than lexicographically sorting vertex lists;sampling cannot replace a row. Step identifiers and dependenciesRule: clique_every_bitmask_exact_extrema_strict_tie_order · Uses: G001, S018 S020 · impliesBefore next ORIGINAL vertex bitmask=17;decoded increasing vertices=[0, 4];ALL pair rows=[[0, 4, No]];previous best bitmasks=[7, 9] After clique=No;independent=Yes;size=2;strict-larger ONLY update gives best clique=[0, 1, 2] at mask7,best independent=[0, 3] at mask9 Conditions: none Each mask denotes exactly one subset of the original carrier.Audit all pairs and update only for a strictly larger valid size.Ties retain the first numeric bitmask rather than lexicographically sorting vertex lists;sampling cannot replace a row. Step identifiers and dependenciesRule: clique_every_bitmask_exact_extrema_strict_tie_order · Uses: G001, S019 S021 · impliesBefore next ORIGINAL vertex bitmask=18;decoded increasing vertices=[1, 4];ALL pair rows=[[1, 4, No]];previous best bitmasks=[7, 9] After clique=No;independent=Yes;size=2;strict-larger ONLY update gives best clique=[0, 1, 2] at mask7,best independent=[0, 3] at mask9 Conditions: none Each mask denotes exactly one subset of the original carrier.Audit all pairs and update only for a strictly larger valid size.Ties retain the first numeric bitmask rather than lexicographically sorting vertex lists;sampling cannot replace a row. Step identifiers and dependenciesRule: clique_every_bitmask_exact_extrema_strict_tie_order · Uses: G001, S020 S022 · impliesBefore next ORIGINAL vertex bitmask=19;decoded increasing vertices=[0, 1, 4];ALL pair rows=[[0, 1, Yes], [0, 4, No], [1, 4, No]];previous best bitmasks=[7, 9] After clique=No;independent=No;size=3;strict-larger ONLY update gives best clique=[0, 1, 2] at mask7,best independent=[0, 3] at mask9 Conditions: none Each mask denotes exactly one subset of the original carrier.Audit all pairs and update only for a strictly larger valid size.Ties retain the first numeric bitmask rather than lexicographically sorting vertex lists;sampling cannot replace a row. Step identifiers and dependenciesRule: clique_every_bitmask_exact_extrema_strict_tie_order · Uses: G001, S021 S023 · impliesBefore next ORIGINAL vertex bitmask=20;decoded increasing vertices=[2, 4];ALL pair rows=[[2, 4, Yes]];previous best bitmasks=[7, 9] After clique=Yes;independent=No;size=2;strict-larger ONLY update gives best clique=[0, 1, 2] at mask7,best independent=[0, 3] at mask9 Conditions: none Each mask denotes exactly one subset of the original carrier.Audit all pairs and update only for a strictly larger valid size.Ties retain the first numeric bitmask rather than lexicographically sorting vertex lists;sampling cannot replace a row. Step identifiers and dependenciesRule: clique_every_bitmask_exact_extrema_strict_tie_order · Uses: G001, S022 S024 · impliesBefore next ORIGINAL vertex bitmask=21;decoded increasing vertices=[0, 2, 4];ALL pair rows=[[0, 2, Yes], [0, 4, No], [2, 4, Yes]];previous best bitmasks=[7, 9] After clique=No;independent=No;size=3;strict-larger ONLY update gives best clique=[0, 1, 2] at mask7,best independent=[0, 3] at mask9 Conditions: none Each mask denotes exactly one subset of the original carrier.Audit all pairs and update only for a strictly larger valid size.Ties retain the first numeric bitmask rather than lexicographically sorting vertex lists;sampling cannot replace a row. Step identifiers and dependenciesRule: clique_every_bitmask_exact_extrema_strict_tie_order · Uses: G001, S023 S025 · impliesBefore next ORIGINAL vertex bitmask=22;decoded increasing vertices=[1, 2, 4];ALL pair rows=[[1, 2, Yes], [1, 4, No], [2, 4, Yes]];previous best bitmasks=[7, 9] After clique=No;independent=No;size=3;strict-larger ONLY update gives best clique=[0, 1, 2] at mask7,best independent=[0, 3] at mask9 Conditions: none Each mask denotes exactly one subset of the original carrier.Audit all pairs and update only for a strictly larger valid size.Ties retain the first numeric bitmask rather than lexicographically sorting vertex lists;sampling cannot replace a row. Step identifiers and dependenciesRule: clique_every_bitmask_exact_extrema_strict_tie_order · Uses: G001, S024 S026 · impliesBefore next ORIGINAL vertex bitmask=23;decoded increasing vertices=[0, 1, 2, 4];ALL pair rows=[[0, 1, Yes], [0, 2, Yes], [0, 4, No], [1, 2, Yes], [1, 4, No], [2, 4, Yes]];previous best bitmasks=[7, 9] After clique=No;independent=No;size=4;strict-larger ONLY update gives best clique=[0, 1, 2] at mask7,best independent=[0, 3] at mask9 Conditions: none Each mask denotes exactly one subset of the original carrier.Audit all pairs and update only for a strictly larger valid size.Ties retain the first numeric bitmask rather than lexicographically sorting vertex lists;sampling cannot replace a row. Step identifiers and dependenciesRule: clique_every_bitmask_exact_extrema_strict_tie_order · Uses: G001, S025 S027 · impliesBefore next ORIGINAL vertex bitmask=24;decoded increasing vertices=[3, 4];ALL pair rows=[[3, 4, Yes]];previous best bitmasks=[7, 9] After clique=Yes;independent=No;size=2;strict-larger ONLY update gives best clique=[0, 1, 2] at mask7,best independent=[0, 3] at mask9 Conditions: none Each mask denotes exactly one subset of the original carrier.Audit all pairs and update only for a strictly larger valid size.Ties retain the first numeric bitmask rather than lexicographically sorting vertex lists;sampling cannot replace a row. Step identifiers and dependenciesRule: clique_every_bitmask_exact_extrema_strict_tie_order · Uses: G001, S026 S028 · impliesBefore next ORIGINAL vertex bitmask=25;decoded increasing vertices=[0, 3, 4];ALL pair rows=[[0, 3, No], [0, 4, No], [3, 4, Yes]];previous best bitmasks=[7, 9] After clique=No;independent=No;size=3;strict-larger ONLY update gives best clique=[0, 1, 2] at mask7,best independent=[0, 3] at mask9 Conditions: none Each mask denotes exactly one subset of the original carrier.Audit all pairs and update only for a strictly larger valid size.Ties retain the first numeric bitmask rather than lexicographically sorting vertex lists;sampling cannot replace a row. Step identifiers and dependenciesRule: clique_every_bitmask_exact_extrema_strict_tie_order · Uses: G001, S027 S029 · impliesBefore next ORIGINAL vertex bitmask=26;decoded increasing vertices=[1, 3, 4];ALL pair rows=[[1, 3, No], [1, 4, No], [3, 4, Yes]];previous best bitmasks=[7, 9] After clique=No;independent=No;size=3;strict-larger ONLY update gives best clique=[0, 1, 2] at mask7,best independent=[0, 3] at mask9 Conditions: none Each mask denotes exactly one subset of the original carrier.Audit all pairs and update only for a strictly larger valid size.Ties retain the first numeric bitmask rather than lexicographically sorting vertex lists;sampling cannot replace a row. Step identifiers and dependenciesRule: clique_every_bitmask_exact_extrema_strict_tie_order · Uses: G001, S028 S030 · impliesBefore next ORIGINAL vertex bitmask=27;decoded increasing vertices=[0, 1, 3, 4];ALL pair rows=[[0, 1, Yes], [0, 3, No], [0, 4, No], [1, 3, No], [1, 4, No], [3, 4, Yes]];previous best bitmasks=[7, 9] After clique=No;independent=No;size=4;strict-larger ONLY update gives best clique=[0, 1, 2] at mask7,best independent=[0, 3] at mask9 Conditions: none Each mask denotes exactly one subset of the original carrier.Audit all pairs and update only for a strictly larger valid size.Ties retain the first numeric bitmask rather than lexicographically sorting vertex lists;sampling cannot replace a row. Step identifiers and dependenciesRule: clique_every_bitmask_exact_extrema_strict_tie_order · Uses: G001, S029 S031 · impliesBefore next ORIGINAL vertex bitmask=28;decoded increasing vertices=[2, 3, 4];ALL pair rows=[[2, 3, No], [2, 4, Yes], [3, 4, Yes]];previous best bitmasks=[7, 9] After clique=No;independent=No;size=3;strict-larger ONLY update gives best clique=[0, 1, 2] at mask7,best independent=[0, 3] at mask9 Conditions: none Each mask denotes exactly one subset of the original carrier.Audit all pairs and update only for a strictly larger valid size.Ties retain the first numeric bitmask rather than lexicographically sorting vertex lists;sampling cannot replace a row. Step identifiers and dependenciesRule: clique_every_bitmask_exact_extrema_strict_tie_order · Uses: G001, S030 S032 · impliesBefore next ORIGINAL vertex bitmask=29;decoded increasing vertices=[0, 2, 3, 4];ALL pair rows=[[0, 2, Yes], [0, 3, No], [0, 4, No], [2, 3, No], [2, 4, Yes], [3, 4, Yes]];previous best bitmasks=[7, 9] After clique=No;independent=No;size=4;strict-larger ONLY update gives best clique=[0, 1, 2] at mask7,best independent=[0, 3] at mask9 Conditions: none Each mask denotes exactly one subset of the original carrier.Audit all pairs and update only for a strictly larger valid size.Ties retain the first numeric bitmask rather than lexicographically sorting vertex lists;sampling cannot replace a row. Step identifiers and dependenciesRule: clique_every_bitmask_exact_extrema_strict_tie_order · Uses: G001, S031 S033 · impliesBefore next ORIGINAL vertex bitmask=30;decoded increasing vertices=[1, 2, 3, 4];ALL pair rows=[[1, 2, Yes], [1, 3, No], [1, 4, No], [2, 3, No], [2, 4, Yes], [3, 4, Yes]];previous best bitmasks=[7, 9] After clique=No;independent=No;size=4;strict-larger ONLY update gives best clique=[0, 1, 2] at mask7,best independent=[0, 3] at mask9 Conditions: none Each mask denotes exactly one subset of the original carrier.Audit all pairs and update only for a strictly larger valid size.Ties retain the first numeric bitmask rather than lexicographically sorting vertex lists;sampling cannot replace a row. Step identifiers and dependenciesRule: clique_every_bitmask_exact_extrema_strict_tie_order · Uses: G001, S032 S034 · impliesBefore next ORIGINAL vertex bitmask=31;decoded increasing vertices=[0, 1, 2, 3, 4];ALL pair rows=[[0, 1, Yes], [0, 2, Yes], [0, 3, No], [0, 4, No], [1, 2, Yes], [1, 3, No], [1, 4, No], [2, 3, No], [2, 4, Yes], [3, 4, Yes]];previous best bitmasks=[7, 9] After clique=No;independent=No;size=5;strict-larger ONLY update gives best clique=[0, 1, 2] at mask7,best independent=[0, 3] at mask9 Conditions: none Each mask denotes exactly one subset of the original carrier.Audit all pairs and update only for a strictly larger valid size.Ties retain the first numeric bitmask rather than lexicographically sorting vertex lists;sampling cannot replace a row. Step identifiers and dependenciesRule: clique_every_bitmask_exact_extrema_strict_tie_order · Uses: G001, S033 S035 · impliesBefore next ORIGINAL vertex bitmask=32;decoded increasing vertices=[5];ALL pair rows=[];previous best bitmasks=[7, 9] After clique=Yes;independent=Yes;size=1;strict-larger ONLY update gives best clique=[0, 1, 2] at mask7,best independent=[0, 3] at mask9 Conditions: none Each mask denotes exactly one subset of the original carrier.Audit all pairs and update only for a strictly larger valid size.Ties retain the first numeric bitmask rather than lexicographically sorting vertex lists;sampling cannot replace a row. Step identifiers and dependenciesRule: clique_every_bitmask_exact_extrema_strict_tie_order · Uses: G001, S034 S036 · impliesBefore next ORIGINAL vertex bitmask=33;decoded increasing vertices=[0, 5];ALL pair rows=[[0, 5, No]];previous best bitmasks=[7, 9] After clique=No;independent=Yes;size=2;strict-larger ONLY update gives best clique=[0, 1, 2] at mask7,best independent=[0, 3] at mask9 Conditions: none Each mask denotes exactly one subset of the original carrier.Audit all pairs and update only for a strictly larger valid size.Ties retain the first numeric bitmask rather than lexicographically sorting vertex lists;sampling cannot replace a row. Step identifiers and dependenciesRule: clique_every_bitmask_exact_extrema_strict_tie_order · Uses: G001, S035 S037 · impliesBefore next ORIGINAL vertex bitmask=34;decoded increasing vertices=[1, 5];ALL pair rows=[[1, 5, No]];previous best bitmasks=[7, 9] After clique=No;independent=Yes;size=2;strict-larger ONLY update gives best clique=[0, 1, 2] at mask7,best independent=[0, 3] at mask9 Conditions: none Each mask denotes exactly one subset of the original carrier.Audit all pairs and update only for a strictly larger valid size.Ties retain the first numeric bitmask rather than lexicographically sorting vertex lists;sampling cannot replace a row. Step identifiers and dependenciesRule: clique_every_bitmask_exact_extrema_strict_tie_order · Uses: G001, S036 S038 · impliesBefore next ORIGINAL vertex bitmask=35;decoded increasing vertices=[0, 1, 5];ALL pair rows=[[0, 1, Yes], [0, 5, No], [1, 5, No]];previous best bitmasks=[7, 9] After clique=No;independent=No;size=3;strict-larger ONLY update gives best clique=[0, 1, 2] at mask7,best independent=[0, 3] at mask9 Conditions: none Each mask denotes exactly one subset of the original carrier.Audit all pairs and update only for a strictly larger valid size.Ties retain the first numeric bitmask rather than lexicographically sorting vertex lists;sampling cannot replace a row. Step identifiers and dependenciesRule: clique_every_bitmask_exact_extrema_strict_tie_order · Uses: G001, S037 S039 · impliesBefore next ORIGINAL vertex bitmask=36;decoded increasing vertices=[2, 5];ALL pair rows=[[2, 5, No]];previous best bitmasks=[7, 9] After clique=No;independent=Yes;size=2;strict-larger ONLY update gives best clique=[0, 1, 2] at mask7,best independent=[0, 3] at mask9 Conditions: none Each mask denotes exactly one subset of the original carrier.Audit all pairs and update only for a strictly larger valid size.Ties retain the first numeric bitmask rather than lexicographically sorting vertex lists;sampling cannot replace a row. Step identifiers and dependenciesRule: clique_every_bitmask_exact_extrema_strict_tie_order · Uses: G001, S038 S040 · impliesBefore next ORIGINAL vertex bitmask=37;decoded increasing vertices=[0, 2, 5];ALL pair rows=[[0, 2, Yes], [0, 5, No], [2, 5, No]];previous best bitmasks=[7, 9] After clique=No;independent=No;size=3;strict-larger ONLY update gives best clique=[0, 1, 2] at mask7,best independent=[0, 3] at mask9 Conditions: none Each mask denotes exactly one subset of the original carrier.Audit all pairs and update only for a strictly larger valid size.Ties retain the first numeric bitmask rather than lexicographically sorting vertex lists;sampling cannot replace a row. Step identifiers and dependenciesRule: clique_every_bitmask_exact_extrema_strict_tie_order · Uses: G001, S039 S041 · impliesBefore next ORIGINAL vertex bitmask=38;decoded increasing vertices=[1, 2, 5];ALL pair rows=[[1, 2, Yes], [1, 5, No], [2, 5, No]];previous best bitmasks=[7, 9] After clique=No;independent=No;size=3;strict-larger ONLY update gives best clique=[0, 1, 2] at mask7,best independent=[0, 3] at mask9 Conditions: none Each mask denotes exactly one subset of the original carrier.Audit all pairs and update only for a strictly larger valid size.Ties retain the first numeric bitmask rather than lexicographically sorting vertex lists;sampling cannot replace a row. Step identifiers and dependenciesRule: clique_every_bitmask_exact_extrema_strict_tie_order · Uses: G001, S040 S042 · impliesBefore next ORIGINAL vertex bitmask=39;decoded increasing vertices=[0, 1, 2, 5];ALL pair rows=[[0, 1, Yes], [0, 2, Yes], [0, 5, No], [1, 2, Yes], [1, 5, No], [2, 5, No]];previous best bitmasks=[7, 9] After clique=No;independent=No;size=4;strict-larger ONLY update gives best clique=[0, 1, 2] at mask7,best independent=[0, 3] at mask9 Conditions: none Each mask denotes exactly one subset of the original carrier.Audit all pairs and update only for a strictly larger valid size.Ties retain the first numeric bitmask rather than lexicographically sorting vertex lists;sampling cannot replace a row. Step identifiers and dependenciesRule: clique_every_bitmask_exact_extrema_strict_tie_order · Uses: G001, S041 S043 · impliesBefore next ORIGINAL vertex bitmask=40;decoded increasing vertices=[3, 5];ALL pair rows=[[3, 5, No]];previous best bitmasks=[7, 9] After clique=No;independent=Yes;size=2;strict-larger ONLY update gives best clique=[0, 1, 2] at mask7,best independent=[0, 3] at mask9 Conditions: none Each mask denotes exactly one subset of the original carrier.Audit all pairs and update only for a strictly larger valid size.Ties retain the first numeric bitmask rather than lexicographically sorting vertex lists;sampling cannot replace a row. Step identifiers and dependenciesRule: clique_every_bitmask_exact_extrema_strict_tie_order · Uses: G001, S042 S044 · impliesBefore next ORIGINAL vertex bitmask=41;decoded increasing vertices=[0, 3, 5];ALL pair rows=[[0, 3, No], [0, 5, No], [3, 5, No]];previous best bitmasks=[7, 9] After clique=No;independent=Yes;size=3;strict-larger ONLY update gives best clique=[0, 1, 2] at mask7,best independent=[0, 3, 5] at mask41 Conditions: none Each mask denotes exactly one subset of the original carrier.Audit all pairs and update only for a strictly larger valid size.Ties retain the first numeric bitmask rather than lexicographically sorting vertex lists;sampling cannot replace a row. Step identifiers and dependenciesRule: clique_every_bitmask_exact_extrema_strict_tie_order · Uses: G001, S043 S045 · impliesBefore next ORIGINAL vertex bitmask=42;decoded increasing vertices=[1, 3, 5];ALL pair rows=[[1, 3, No], [1, 5, No], [3, 5, No]];previous best bitmasks=[7, 41] After clique=No;independent=Yes;size=3;strict-larger ONLY update gives best clique=[0, 1, 2] at mask7,best independent=[0, 3, 5] at mask41 Conditions: none Each mask denotes exactly one subset of the original carrier.Audit all pairs and update only for a strictly larger valid size.Ties retain the first numeric bitmask rather than lexicographically sorting vertex lists;sampling cannot replace a row. Step identifiers and dependenciesRule: clique_every_bitmask_exact_extrema_strict_tie_order · Uses: G001, S044 S046 · impliesBefore next ORIGINAL vertex bitmask=43;decoded increasing vertices=[0, 1, 3, 5];ALL pair rows=[[0, 1, Yes], [0, 3, No], [0, 5, No], [1, 3, No], [1, 5, No], [3, 5, No]];previous best bitmasks=[7, 41] After clique=No;independent=No;size=4;strict-larger ONLY update gives best clique=[0, 1, 2] at mask7,best independent=[0, 3, 5] at mask41 Conditions: none Each mask denotes exactly one subset of the original carrier.Audit all pairs and update only for a strictly larger valid size.Ties retain the first numeric bitmask rather than lexicographically sorting vertex lists;sampling cannot replace a row. Step identifiers and dependenciesRule: clique_every_bitmask_exact_extrema_strict_tie_order · Uses: G001, S045 S047 · impliesBefore next ORIGINAL vertex bitmask=44;decoded increasing vertices=[2, 3, 5];ALL pair rows=[[2, 3, No], [2, 5, No], [3, 5, No]];previous best bitmasks=[7, 41] After clique=No;independent=Yes;size=3;strict-larger ONLY update gives best clique=[0, 1, 2] at mask7,best independent=[0, 3, 5] at mask41 Conditions: none Each mask denotes exactly one subset of the original carrier.Audit all pairs and update only for a strictly larger valid size.Ties retain the first numeric bitmask rather than lexicographically sorting vertex lists;sampling cannot replace a row. Step identifiers and dependenciesRule: clique_every_bitmask_exact_extrema_strict_tie_order · Uses: G001, S046 S048 · impliesBefore next ORIGINAL vertex bitmask=45;decoded increasing vertices=[0, 2, 3, 5];ALL pair rows=[[0, 2, Yes], [0, 3, No], [0, 5, No], [2, 3, No], [2, 5, No], [3, 5, No]];previous best bitmasks=[7, 41] After clique=No;independent=No;size=4;strict-larger ONLY update gives best clique=[0, 1, 2] at mask7,best independent=[0, 3, 5] at mask41 Conditions: none Each mask denotes exactly one subset of the original carrier.Audit all pairs and update only for a strictly larger valid size.Ties retain the first numeric bitmask rather than lexicographically sorting vertex lists;sampling cannot replace a row. Step identifiers and dependenciesRule: clique_every_bitmask_exact_extrema_strict_tie_order · Uses: G001, S047 S049 · impliesBefore next ORIGINAL vertex bitmask=46;decoded increasing vertices=[1, 2, 3, 5];ALL pair rows=[[1, 2, Yes], [1, 3, No], [1, 5, No], [2, 3, No], [2, 5, No], [3, 5, No]];previous best bitmasks=[7, 41] After clique=No;independent=No;size=4;strict-larger ONLY update gives best clique=[0, 1, 2] at mask7,best independent=[0, 3, 5] at mask41 Conditions: none Each mask denotes exactly one subset of the original carrier.Audit all pairs and update only for a strictly larger valid size.Ties retain the first numeric bitmask rather than lexicographically sorting vertex lists;sampling cannot replace a row. Step identifiers and dependenciesRule: clique_every_bitmask_exact_extrema_strict_tie_order · Uses: G001, S048 S050 · impliesBefore next ORIGINAL vertex bitmask=47;decoded increasing vertices=[0, 1, 2, 3, 5];ALL pair rows=[[0, 1, Yes], [0, 2, Yes], [0, 3, No], [0, 5, No], [1, 2, Yes], [1, 3, No], [1, 5, No], [2, 3, No], [2, 5, No], [3, 5, No]];previous best bitmasks=[7, 41] After clique=No;independent=No;size=5;strict-larger ONLY update gives best clique=[0, 1, 2] at mask7,best independent=[0, 3, 5] at mask41 Conditions: none Each mask denotes exactly one subset of the original carrier.Audit all pairs and update only for a strictly larger valid size.Ties retain the first numeric bitmask rather than lexicographically sorting vertex lists;sampling cannot replace a row. Step identifiers and dependenciesRule: clique_every_bitmask_exact_extrema_strict_tie_order · Uses: G001, S049 S051 · impliesBefore next ORIGINAL vertex bitmask=48;decoded increasing vertices=[4, 5];ALL pair rows=[[4, 5, No]];previous best bitmasks=[7, 41] After clique=No;independent=Yes;size=2;strict-larger ONLY update gives best clique=[0, 1, 2] at mask7,best independent=[0, 3, 5] at mask41 Conditions: none Each mask denotes exactly one subset of the original carrier.Audit all pairs and update only for a strictly larger valid size.Ties retain the first numeric bitmask rather than lexicographically sorting vertex lists;sampling cannot replace a row. Step identifiers and dependenciesRule: clique_every_bitmask_exact_extrema_strict_tie_order · Uses: G001, S050 S052 · impliesBefore next ORIGINAL vertex bitmask=49;decoded increasing vertices=[0, 4, 5];ALL pair rows=[[0, 4, No], [0, 5, No], [4, 5, No]];previous best bitmasks=[7, 41] After clique=No;independent=Yes;size=3;strict-larger ONLY update gives best clique=[0, 1, 2] at mask7,best independent=[0, 3, 5] at mask41 Conditions: none Each mask denotes exactly one subset of the original carrier.Audit all pairs and update only for a strictly larger valid size.Ties retain the first numeric bitmask rather than lexicographically sorting vertex lists;sampling cannot replace a row. Step identifiers and dependenciesRule: clique_every_bitmask_exact_extrema_strict_tie_order · Uses: G001, S051 S053 · impliesBefore next ORIGINAL vertex bitmask=50;decoded increasing vertices=[1, 4, 5];ALL pair rows=[[1, 4, No], [1, 5, No], [4, 5, No]];previous best bitmasks=[7, 41] After clique=No;independent=Yes;size=3;strict-larger ONLY update gives best clique=[0, 1, 2] at mask7,best independent=[0, 3, 5] at mask41 Conditions: none Each mask denotes exactly one subset of the original carrier.Audit all pairs and update only for a strictly larger valid size.Ties retain the first numeric bitmask rather than lexicographically sorting vertex lists;sampling cannot replace a row. Step identifiers and dependenciesRule: clique_every_bitmask_exact_extrema_strict_tie_order · Uses: G001, S052 S054 · impliesBefore next ORIGINAL vertex bitmask=51;decoded increasing vertices=[0, 1, 4, 5];ALL pair rows=[[0, 1, Yes], [0, 4, No], [0, 5, No], [1, 4, No], [1, 5, No], [4, 5, No]];previous best bitmasks=[7, 41] After clique=No;independent=No;size=4;strict-larger ONLY update gives best clique=[0, 1, 2] at mask7,best independent=[0, 3, 5] at mask41 Conditions: none Each mask denotes exactly one subset of the original carrier.Audit all pairs and update only for a strictly larger valid size.Ties retain the first numeric bitmask rather than lexicographically sorting vertex lists;sampling cannot replace a row. Step identifiers and dependenciesRule: clique_every_bitmask_exact_extrema_strict_tie_order · Uses: G001, S053 S055 · impliesBefore next ORIGINAL vertex bitmask=52;decoded increasing vertices=[2, 4, 5];ALL pair rows=[[2, 4, Yes], [2, 5, No], [4, 5, No]];previous best bitmasks=[7, 41] After clique=No;independent=No;size=3;strict-larger ONLY update gives best clique=[0, 1, 2] at mask7,best independent=[0, 3, 5] at mask41 Conditions: none Each mask denotes exactly one subset of the original carrier.Audit all pairs and update only for a strictly larger valid size.Ties retain the first numeric bitmask rather than lexicographically sorting vertex lists;sampling cannot replace a row. Step identifiers and dependenciesRule: clique_every_bitmask_exact_extrema_strict_tie_order · Uses: G001, S054 S056 · impliesBefore next ORIGINAL vertex bitmask=53;decoded increasing vertices=[0, 2, 4, 5];ALL pair rows=[[0, 2, Yes], [0, 4, No], [0, 5, No], [2, 4, Yes], [2, 5, No], [4, 5, No]];previous best bitmasks=[7, 41] After clique=No;independent=No;size=4;strict-larger ONLY update gives best clique=[0, 1, 2] at mask7,best independent=[0, 3, 5] at mask41 Conditions: none Each mask denotes exactly one subset of the original carrier.Audit all pairs and update only for a strictly larger valid size.Ties retain the first numeric bitmask rather than lexicographically sorting vertex lists;sampling cannot replace a row. Step identifiers and dependenciesRule: clique_every_bitmask_exact_extrema_strict_tie_order · Uses: G001, S055 S057 · impliesBefore next ORIGINAL vertex bitmask=54;decoded increasing vertices=[1, 2, 4, 5];ALL pair rows=[[1, 2, Yes], [1, 4, No], [1, 5, No], [2, 4, Yes], [2, 5, No], [4, 5, No]];previous best bitmasks=[7, 41] After clique=No;independent=No;size=4;strict-larger ONLY update gives best clique=[0, 1, 2] at mask7,best independent=[0, 3, 5] at mask41 Conditions: none Each mask denotes exactly one subset of the original carrier.Audit all pairs and update only for a strictly larger valid size.Ties retain the first numeric bitmask rather than lexicographically sorting vertex lists;sampling cannot replace a row. Step identifiers and dependenciesRule: clique_every_bitmask_exact_extrema_strict_tie_order · Uses: G001, S056 S058 · impliesBefore next ORIGINAL vertex bitmask=55;decoded increasing vertices=[0, 1, 2, 4, 5];ALL pair rows=[[0, 1, Yes], [0, 2, Yes], [0, 4, No], [0, 5, No], [1, 2, Yes], [1, 4, No], [1, 5, No], [2, 4, Yes], [2, 5, No], [4, 5, No]];previous best bitmasks=[7, 41] After clique=No;independent=No;size=5;strict-larger ONLY update gives best clique=[0, 1, 2] at mask7,best independent=[0, 3, 5] at mask41 Conditions: none Each mask denotes exactly one subset of the original carrier.Audit all pairs and update only for a strictly larger valid size.Ties retain the first numeric bitmask rather than lexicographically sorting vertex lists;sampling cannot replace a row. Step identifiers and dependenciesRule: clique_every_bitmask_exact_extrema_strict_tie_order · Uses: G001, S057 S059 · impliesBefore next ORIGINAL vertex bitmask=56;decoded increasing vertices=[3, 4, 5];ALL pair rows=[[3, 4, Yes], [3, 5, No], [4, 5, No]];previous best bitmasks=[7, 41] After clique=No;independent=No;size=3;strict-larger ONLY update gives best clique=[0, 1, 2] at mask7,best independent=[0, 3, 5] at mask41 Conditions: none Each mask denotes exactly one subset of the original carrier.Audit all pairs and update only for a strictly larger valid size.Ties retain the first numeric bitmask rather than lexicographically sorting vertex lists;sampling cannot replace a row. Step identifiers and dependenciesRule: clique_every_bitmask_exact_extrema_strict_tie_order · Uses: G001, S058 S060 · impliesBefore next ORIGINAL vertex bitmask=57;decoded increasing vertices=[0, 3, 4, 5];ALL pair rows=[[0, 3, No], [0, 4, No], [0, 5, No], [3, 4, Yes], [3, 5, No], [4, 5, No]];previous best bitmasks=[7, 41] After clique=No;independent=No;size=4;strict-larger ONLY update gives best clique=[0, 1, 2] at mask7,best independent=[0, 3, 5] at mask41 Conditions: none Each mask denotes exactly one subset of the original carrier.Audit all pairs and update only for a strictly larger valid size.Ties retain the first numeric bitmask rather than lexicographically sorting vertex lists;sampling cannot replace a row. Step identifiers and dependenciesRule: clique_every_bitmask_exact_extrema_strict_tie_order · Uses: G001, S059 S061 · impliesBefore next ORIGINAL vertex bitmask=58;decoded increasing vertices=[1, 3, 4, 5];ALL pair rows=[[1, 3, No], [1, 4, No], [1, 5, No], [3, 4, Yes], [3, 5, No], [4, 5, No]];previous best bitmasks=[7, 41] After clique=No;independent=No;size=4;strict-larger ONLY update gives best clique=[0, 1, 2] at mask7,best independent=[0, 3, 5] at mask41 Conditions: none Each mask denotes exactly one subset of the original carrier.Audit all pairs and update only for a strictly larger valid size.Ties retain the first numeric bitmask rather than lexicographically sorting vertex lists;sampling cannot replace a row. Step identifiers and dependenciesRule: clique_every_bitmask_exact_extrema_strict_tie_order · Uses: G001, S060 S062 · impliesBefore next ORIGINAL vertex bitmask=59;decoded increasing vertices=[0, 1, 3, 4, 5];ALL pair rows=[[0, 1, Yes], [0, 3, No], [0, 4, No], [0, 5, No], [1, 3, No], [1, 4, No], [1, 5, No], [3, 4, Yes], [3, 5, No], [4, 5, No]];previous best bitmasks=[7, 41] After clique=No;independent=No;size=5;strict-larger ONLY update gives best clique=[0, 1, 2] at mask7,best independent=[0, 3, 5] at mask41 Conditions: none Each mask denotes exactly one subset of the original carrier.Audit all pairs and update only for a strictly larger valid size.Ties retain the first numeric bitmask rather than lexicographically sorting vertex lists;sampling cannot replace a row. Step identifiers and dependenciesRule: clique_every_bitmask_exact_extrema_strict_tie_order · Uses: G001, S061 S063 · impliesBefore next ORIGINAL vertex bitmask=60;decoded increasing vertices=[2, 3, 4, 5];ALL pair rows=[[2, 3, No], [2, 4, Yes], [2, 5, No], [3, 4, Yes], [3, 5, No], [4, 5, No]];previous best bitmasks=[7, 41] After clique=No;independent=No;size=4;strict-larger ONLY update gives best clique=[0, 1, 2] at mask7,best independent=[0, 3, 5] at mask41 Conditions: none Each mask denotes exactly one subset of the original carrier.Audit all pairs and update only for a strictly larger valid size.Ties retain the first numeric bitmask rather than lexicographically sorting vertex lists;sampling cannot replace a row. Step identifiers and dependenciesRule: clique_every_bitmask_exact_extrema_strict_tie_order · Uses: G001, S062 S064 · impliesBefore next ORIGINAL vertex bitmask=61;decoded increasing vertices=[0, 2, 3, 4, 5];ALL pair rows=[[0, 2, Yes], [0, 3, No], [0, 4, No], [0, 5, No], [2, 3, No], [2, 4, Yes], [2, 5, No], [3, 4, Yes], [3, 5, No], [4, 5, No]];previous best bitmasks=[7, 41] After clique=No;independent=No;size=5;strict-larger ONLY update gives best clique=[0, 1, 2] at mask7,best independent=[0, 3, 5] at mask41 Conditions: none Each mask denotes exactly one subset of the original carrier.Audit all pairs and update only for a strictly larger valid size.Ties retain the first numeric bitmask rather than lexicographically sorting vertex lists;sampling cannot replace a row. Step identifiers and dependenciesRule: clique_every_bitmask_exact_extrema_strict_tie_order · Uses: G001, S063 S065 · impliesBefore next ORIGINAL vertex bitmask=62;decoded increasing vertices=[1, 2, 3, 4, 5];ALL pair rows=[[1, 2, Yes], [1, 3, No], [1, 4, No], [1, 5, No], [2, 3, No], [2, 4, Yes], [2, 5, No], [3, 4, Yes], [3, 5, No], [4, 5, No]];previous best bitmasks=[7, 41] After clique=No;independent=No;size=5;strict-larger ONLY update gives best clique=[0, 1, 2] at mask7,best independent=[0, 3, 5] at mask41 Conditions: none Each mask denotes exactly one subset of the original carrier.Audit all pairs and update only for a strictly larger valid size.Ties retain the first numeric bitmask rather than lexicographically sorting vertex lists;sampling cannot replace a row. Step identifiers and dependenciesRule: clique_every_bitmask_exact_extrema_strict_tie_order · Uses: G001, S064 S066 · impliesBefore next ORIGINAL vertex bitmask=63;decoded increasing vertices=[0, 1, 2, 3, 4, 5];ALL pair rows=[[0, 1, Yes], [0, 2, Yes], [0, 3, No], [0, 4, No], [0, 5, No], [1, 2, Yes], [1, 3, No], [1, 4, No], [1, 5, No], [2, 3, No], [2, 4, Yes], [2, 5, No], [3, 4, Yes], [3, 5, No], [4, 5, No]];previous best bitmasks=[7, 41] After clique=No;independent=No;size=6;strict-larger ONLY update gives best clique=[0, 1, 2] at mask7,best independent=[0, 3, 5] at mask41 Conditions: none Each mask denotes exactly one subset of the original carrier.Audit all pairs and update only for a strictly larger valid size.Ties retain the first numeric bitmask rather than lexicographically sorting vertex lists;sampling cannot replace a row. Step identifiers and dependenciesRule: clique_every_bitmask_exact_extrema_strict_tie_order · Uses: G001, S065 S067 · impliesBefore ALL masks0..63 processed once;best clique=[0, 1, 2] at mask7;best independent=[0, 3, 5] at mask41 After every possible subset covered;both actual witnesses valid;no valid subset larger;omega=3,alpha=3;FIRST numeric-mask ties retained Conditions: none Complete subset coverage gives global upper bounds,and the valid witnesses attain them.Exact extrema require both bounds;inclusion-maximality alone is insufficient.On an empty carrier both extrema are0. Step identifiers and dependenciesRule: clique_global_upper_and_attained_lower_not_maximal_only · Uses: G001, S066 ## Reverse checks - S067 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S066 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S065 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S064 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S063 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S062 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S061 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S060 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S059 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S058 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S057 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S056 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S055 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S054 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S053 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S052 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S051 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S050 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S049 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S048 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S047 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S046 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S045 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S044 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S043 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S042 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S041 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S040 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S039 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S038 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S037 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S036 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S035 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S034 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S033 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S032 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S031 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S030 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S029 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S028 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S027 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S026 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S025 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S024 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S023 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S022 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S021 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S020 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S019 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S018 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S017 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S016 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S015 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S014 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S013 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S012 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S011 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S010 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S009 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S008 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S007 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. ## Result ([0, 1, 2], 3, [0, 3, 5], 3, Yes, exhaustive_all_vertex_subsets) The full substitutions and premise checks give the original answer; source identity,format and status are unchanged. ## This question review and source binding Review date: 2026-10-26 Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof. Source hashes and review receipts - input commit: 0f7be61e2c22317a4075561dc822c87d881a50db - original question SHA-256: 4336edd751e6a6fa364630580a245800aa2047233991acf64b4f3f91e3ae5a4d - candidate record SHA-256: b0dc0fbe7db32066f41fdda9beea5a6b793ff35a7e9109b656ffea947a393c39 - semantic review receipt SHA-256: d165cad30849a3aba67390a59c572c0d965269d4b405e43df36da19e28d85612 - question-first witness SHA-256: c222ca47a07939c019865a1e4d9ce633205cdee68db92bf96a96f5683a62ebc7 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261014-advance-20261106-r02-clique-and-independent-set-0055.json)