# Question 12452: Invariant Mismatch Can Refute; a Match Does Not Prove [Return to catalogue page 63](https://ocme.evemisslab.com/en/questions/page/63/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261013-advance-20261026-r06-graph-isomorphism-foundations/#q-20261013-advance-20261026-r06-graph-isomorphism-0077) 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. R0023-12452 · q-20261013-advance-20261026-r06-graph-isomorphism-0077 ## Original question and givens Use complete simple undirected source graph with left_n=6, edges=[[0, 2]], and target with right_n=6, edges=[[0, 2], [0, 4], [1, 3]]. Give (left_n,left_degree_multiset,right_n,right_degree_multiset,mismatch?,conclusion). Sort each degree multiset increasingly. The conclusion is nonisomorphic_certificate for a size or degree mismatch, otherwise necessary-conditions-do-not-prove-isomorphism. - G001 — Use complete simple undirected source graph with left_n=6, edges=[[0, 2]], and target with right_n=6, edges=[[0, 2], [0, 4], [1, 3]]. Give (left_n,left_degree_multiset,right_n,right_degree_multiset,mismatch?,conclusion). Sort each degree multiset increasingly. The conclusion is nonisomorphic_certificate for a size or degree mismatch, otherwise necessary-conditions-do-not-prove-isomorphism. ## 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-graph-isomorphism](https://ocme.evemisslab.com/en/mko/mko-graph-isomorphism/) ## Step-by-step derivation S001 · impliesBefore source graph:carrier Fin(6);raw undirected edges=[[0, 2]] After canonical simple edges=[[0, 2]];loops and repeated undirected edges absent Conditions: none Normalize each edge with its smaller endpoint first,then sort.Reversed duplicates remain invalid rather than being silently removed.Preserve every isolated vertex. Step identifiers and dependenciesRule: complete_simple_graph_canonical_edges · Uses: G001 S002 · impliesBefore target graph:carrier Fin(6);raw undirected edges=[[0, 2], [0, 4], [1, 3]] After canonical simple edges=[[0, 2], [0, 4], [1, 3]];loops and repeated undirected edges absent Conditions: none Normalize each edge with its smaller endpoint first,then sort.Reversed duplicates remain invalid rather than being silently removed.Preserve every isolated vertex. Step identifiers and dependenciesRule: complete_simple_graph_canonical_edges · Uses: G001, S001 S003 · impliesBefore source graph:vertex=0;complete canonical edges=[[0, 2]] After neighbors=[2];degree=1 Conditions: none In a loop-free duplicate-free graph,the neighbor count is the degree.Vertices absent from every edge still have degree0. Step identifiers and dependenciesRule: graph_every_vertex_degree_including_isolates · Uses: G001, S002 S004 · impliesBefore source graph:vertex=1;complete canonical edges=[[0, 2]] After neighbors=[];degree=0 Conditions: none In a loop-free duplicate-free graph,the neighbor count is the degree.Vertices absent from every edge still have degree0. Step identifiers and dependenciesRule: graph_every_vertex_degree_including_isolates · Uses: G001, S003 S005 · impliesBefore source graph:vertex=2;complete canonical edges=[[0, 2]] After neighbors=[0];degree=1 Conditions: none In a loop-free duplicate-free graph,the neighbor count is the degree.Vertices absent from every edge still have degree0. Step identifiers and dependenciesRule: graph_every_vertex_degree_including_isolates · Uses: G001, S004 S006 · impliesBefore source graph:vertex=3;complete canonical edges=[[0, 2]] After neighbors=[];degree=0 Conditions: none In a loop-free duplicate-free graph,the neighbor count is the degree.Vertices absent from every edge still have degree0. Step identifiers and dependenciesRule: graph_every_vertex_degree_including_isolates · Uses: G001, S005 S007 · impliesBefore source graph:vertex=4;complete canonical edges=[[0, 2]] After neighbors=[];degree=0 Conditions: none In a loop-free duplicate-free graph,the neighbor count is the degree.Vertices absent from every edge still have degree0. Step identifiers and dependenciesRule: graph_every_vertex_degree_including_isolates · Uses: G001, S006 S008 · impliesBefore source graph:vertex=5;complete canonical edges=[[0, 2]] After neighbors=[];degree=0 Conditions: none In a loop-free duplicate-free graph,the neighbor count is the degree.Vertices absent from every edge still have degree0. Step identifiers and dependenciesRule: graph_every_vertex_degree_including_isolates · Uses: G001, S007 S009 · impliesBefore source graph:degrees in vertex order=[1, 0, 1, 0, 0, 0] After sorted degree multiset=[0, 0, 0, 0, 1, 1] Conditions: none Sort numerically into a multiset.Do not deduplicate repeated degrees;distinguish vertex order from invariant order. Step identifiers and dependenciesRule: graph_degree_multiset_keeps_repeated_values · Uses: G001, S008 S010 · impliesBefore target graph:vertex=0;complete canonical edges=[[0, 2], [0, 4], [1, 3]] After neighbors=[2, 4];degree=2 Conditions: none In a loop-free duplicate-free graph,the neighbor count is the degree.Vertices absent from every edge still have degree0. Step identifiers and dependenciesRule: graph_every_vertex_degree_including_isolates · Uses: G001, S009 S011 · impliesBefore target graph:vertex=1;complete canonical edges=[[0, 2], [0, 4], [1, 3]] After neighbors=[3];degree=1 Conditions: none In a loop-free duplicate-free graph,the neighbor count is the degree.Vertices absent from every edge still have degree0. Step identifiers and dependenciesRule: graph_every_vertex_degree_including_isolates · Uses: G001, S010 S012 · impliesBefore target graph:vertex=2;complete canonical edges=[[0, 2], [0, 4], [1, 3]] After neighbors=[0];degree=1 Conditions: none In a loop-free duplicate-free graph,the neighbor count is the degree.Vertices absent from every edge still have degree0. Step identifiers and dependenciesRule: graph_every_vertex_degree_including_isolates · Uses: G001, S011 S013 · impliesBefore target graph:vertex=3;complete canonical edges=[[0, 2], [0, 4], [1, 3]] After neighbors=[1];degree=1 Conditions: none In a loop-free duplicate-free graph,the neighbor count is the degree.Vertices absent from every edge still have degree0. Step identifiers and dependenciesRule: graph_every_vertex_degree_including_isolates · Uses: G001, S012 S014 · impliesBefore target graph:vertex=4;complete canonical edges=[[0, 2], [0, 4], [1, 3]] After neighbors=[0];degree=1 Conditions: none In a loop-free duplicate-free graph,the neighbor count is the degree.Vertices absent from every edge still have degree0. Step identifiers and dependenciesRule: graph_every_vertex_degree_including_isolates · Uses: G001, S013 S015 · impliesBefore target graph:vertex=5;complete canonical edges=[[0, 2], [0, 4], [1, 3]] After neighbors=[];degree=0 Conditions: none In a loop-free duplicate-free graph,the neighbor count is the degree.Vertices absent from every edge still have degree0. Step identifiers and dependenciesRule: graph_every_vertex_degree_including_isolates · Uses: G001, S014 S016 · impliesBefore target graph:degrees in vertex order=[2, 1, 1, 1, 1, 0] After sorted degree multiset=[0, 1, 1, 1, 1, 2] Conditions: none Sort numerically into a multiset.Do not deduplicate repeated degrees;distinguish vertex order from invariant order. Step identifiers and dependenciesRule: graph_degree_multiset_keeps_repeated_values · Uses: G001, S015 S017 · impliesBefore source size=6;degree multiset=[0, 0, 0, 0, 1, 1];target size=6;degree multiset=[0, 1, 1, 1, 1, 2] After invariant mismatch=Yes;conclusion=nonisomorphic_certificate Conditions: none An isomorphism preserves vertex count and bijects each vertex's neighbors with its image's neighbors,hence preserves the degree multiset.Any mismatch refutes it.A match passes necessary conditions only and supplies no isomorphism map. Step identifiers and dependenciesRule: graph_necessary_invariants_not_sufficient_match · Uses: G001, S016 ## Reverse checks - 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 (6, [0, 0, 0, 0, 1, 1], 6, [0, 1, 1, 1, 1, 2], Yes, nonisomorphic_certificate) 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: 5646442fab8de1854b88522f17b66c5b1afb4ba052b08791bc0e4b2623facd8b - candidate record SHA-256: 06fe6a03fe35bac4982cd116d719d78fcd7fbc29d990f67d63b483687a324784 - semantic review receipt SHA-256: f783e81f44ff0faa13dbf948657e9187244a59a809ebdf96f22cca24966649ff - question-first witness SHA-256: 49a50ed0550be55124d6562260c2ccf4fa6d3f316d7ad9d0d32174400e362e18 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261013-advance-20261026-r06-graph-isomorphism-0077.json)