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Question 12459: Invariant Mismatch Can Refute; a Match Does Not Prove

On this page
  1. Original question and givens
  2. Definition and scope
  3. Step-by-step derivation
  4. Reverse checks
  5. Result
  6. This question review and source binding

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R0023-12459 · q-20261013-advance-20261026-r06-graph-isomorphism-0084

Original question and givens

Use complete simple undirected source graph with left_n=6, edges=[[0, 1], [0, 4]], and target with right_n=6, edges=[[0, 1], [0, 3], [0, 4], [1, 3], [1, 5]]. 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, 1], [0, 4]], and target with right_n=6, edges=[[0, 1], [0, 3], [0, 4], [1, 3], [1, 5]]. 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.

Step-by-step derivation

S001 · implies

Before

source graph:carrier Fin(6);raw undirected edges=[[0, 1], [0, 4]]

After

canonical simple edges=[[0, 1], [0, 4]];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 dependencies

Rule: complete_simple_graph_canonical_edges · Uses: G001

S002 · implies

Before

target graph:carrier Fin(6);raw undirected edges=[[0, 1], [0, 3], [0, 4], [1, 3], [1, 5]]

After

canonical simple edges=[[0, 1], [0, 3], [0, 4], [1, 3], [1, 5]];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 dependencies

Rule: complete_simple_graph_canonical_edges · Uses: G001, S001

S003 · implies

Before

source graph:vertex=0;complete canonical edges=[[0, 1], [0, 4]]

After

neighbors=[1, 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 dependencies

Rule: graph_every_vertex_degree_including_isolates · Uses: G001, S002

S004 · implies

Before

source graph:vertex=1;complete canonical edges=[[0, 1], [0, 4]]

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 dependencies

Rule: graph_every_vertex_degree_including_isolates · Uses: G001, S003

S005 · implies

Before

source graph:vertex=2;complete canonical edges=[[0, 1], [0, 4]]

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 dependencies

Rule: graph_every_vertex_degree_including_isolates · Uses: G001, S004

S006 · implies

Before

source graph:vertex=3;complete canonical edges=[[0, 1], [0, 4]]

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 dependencies

Rule: graph_every_vertex_degree_including_isolates · Uses: G001, S005

S007 · implies

Before

source graph:vertex=4;complete canonical edges=[[0, 1], [0, 4]]

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 dependencies

Rule: graph_every_vertex_degree_including_isolates · Uses: G001, S006

S008 · implies

Before

source graph:vertex=5;complete canonical edges=[[0, 1], [0, 4]]

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 dependencies

Rule: graph_every_vertex_degree_including_isolates · Uses: G001, S007

S009 · implies

Before

source graph:degrees in vertex order=[2, 1, 0, 0, 1, 0]

After

sorted degree multiset=[0, 0, 0, 1, 1, 2]

Conditions: none

Sort numerically into a multiset.Do not deduplicate repeated degrees;distinguish vertex order from invariant order.

Step identifiers and dependencies

Rule: graph_degree_multiset_keeps_repeated_values · Uses: G001, S008

S010 · implies

Before

target graph:vertex=0;complete canonical edges=[[0, 1], [0, 3], [0, 4], [1, 3], [1, 5]]

After

neighbors=[1, 3, 4];degree=3

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 dependencies

Rule: graph_every_vertex_degree_including_isolates · Uses: G001, S009

S011 · implies

Before

target graph:vertex=1;complete canonical edges=[[0, 1], [0, 3], [0, 4], [1, 3], [1, 5]]

After

neighbors=[0, 3, 5];degree=3

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 dependencies

Rule: graph_every_vertex_degree_including_isolates · Uses: G001, S010

S012 · implies

Before

target graph:vertex=2;complete canonical edges=[[0, 1], [0, 3], [0, 4], [1, 3], [1, 5]]

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 dependencies

Rule: graph_every_vertex_degree_including_isolates · Uses: G001, S011

S013 · implies

Before

target graph:vertex=3;complete canonical edges=[[0, 1], [0, 3], [0, 4], [1, 3], [1, 5]]

After

neighbors=[0, 1];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 dependencies

Rule: graph_every_vertex_degree_including_isolates · Uses: G001, S012

S014 · implies

Before

target graph:vertex=4;complete canonical edges=[[0, 1], [0, 3], [0, 4], [1, 3], [1, 5]]

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 dependencies

Rule: graph_every_vertex_degree_including_isolates · Uses: G001, S013

S015 · implies

Before

target graph:vertex=5;complete canonical edges=[[0, 1], [0, 3], [0, 4], [1, 3], [1, 5]]

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 dependencies

Rule: graph_every_vertex_degree_including_isolates · Uses: G001, S014

S016 · implies

Before

target graph:degrees in vertex order=[3, 3, 0, 2, 1, 1]

After

sorted degree multiset=[0, 1, 1, 2, 3, 3]

Conditions: none

Sort numerically into a multiset.Do not deduplicate repeated degrees;distinguish vertex order from invariant order.

Step identifiers and dependencies

Rule: graph_degree_multiset_keeps_repeated_values · Uses: G001, S015

S017 · implies

Before

source size=6;degree multiset=[0, 0, 0, 1, 1, 2];target size=6;degree multiset=[0, 1, 1, 2, 3, 3]

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 dependencies

Rule: 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, 1, 1, 2], 6, [0, 1, 1, 2, 3, 3], 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: 87509c263bf4e0b89c57c0d646169690bf178a2cd89de110787ed261861034cd
  • candidate record SHA-256: 28cb056a73b038e9afad5c7f1ad38e7f7242531c2741727872ab0e322da58a18
  • semantic review receipt SHA-256: f783e81f44ff0faa13dbf948657e9187244a59a809ebdf96f22cca24966649ff
  • question-first witness SHA-256: 49a50ed0550be55124d6562260c2ccf4fa6d3f316d7ad9d0d32174400e362e18
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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