Question 12415: Audit Edges and Nonedges without Replacing Bijectivity

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Original question and givens
Definition and scope
Step-by-step derivation
Reverse checks
Result
This question review and source binding

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R0023-12415 · q-20261013-advance-20261026-r06-graph-isomorphism-0040

Original question and givens

Use complete simple undirected graphs on Fin(6)={0,…,5} with source edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5]], target edges=[[0, 1], [1, 4], [3, 4], [2, 3], [2, 5]] and complete vertex map=[0, 1, 4, 3, 2, 5] in source-label order. Loops and duplicate undirected edges are forbidden. Isomorphism requires both bijectivity and adjacency iff on every unordered pair, including nonedges; neither obligation replaces the other. Give (mapping,all bad [source vertex a,source vertex b,source adjacent?,mapped target adjacent?] records with a<b in label order,adjacency_iff?).

G001 — Use complete simple undirected graphs on Fin(6)={0,…,5} with source edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5]], target edges=[[0, 1], [1, 4], [3, 4], [2, 3], [2, 5]] and complete vertex map=[0, 1, 4, 3, 2, 5] in source-label order. Loops and duplicate undirected edges are forbidden. Isomorphism requires both bijectivity and adjacency iff on every unordered pair, including nonedges; neither obligation replaces the other. Give (mapping,all bad [source vertex a,source vertex b,source adjacent?,mapped target adjacent?] records with a<b in label order,adjacency_iff?).

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

Step-by-step derivation

S001 · implies

Before

source graph:carrier Fin(6);raw undirected edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5]]

After

canonical simple edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 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

S002 · implies

Before

target graph:carrier Fin(6);raw undirected edges=[[0, 1], [1, 4], [3, 4], [2, 3], [2, 5]]

After

canonical simple edges=[[0, 1], [1, 4], [2, 3], [2, 5], [3, 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, S001

S003 · implies

Before

complete mapping=[0, 1, 4, 3, 2, 5];target label=0

After

source fiber=[0];exactly one=Yes

Conditions: none

For a complete total map between equally sized finite carriers,every target needs exactly one preimage.Typedness alone is not bijectivity.

Step identifiers and dependencies

Rule: finite_vertex_map_each_target_fiber · Uses: G001, S002

S004 · implies

Before

complete mapping=[0, 1, 4, 3, 2, 5];target label=1

After

source fiber=[1];exactly one=Yes

Conditions: none

For a complete total map between equally sized finite carriers,every target needs exactly one preimage.Typedness alone is not bijectivity.

Step identifiers and dependencies

Rule: finite_vertex_map_each_target_fiber · Uses: G001, S003

S005 · implies

Before

complete mapping=[0, 1, 4, 3, 2, 5];target label=2

After

source fiber=[4];exactly one=Yes

Conditions: none

For a complete total map between equally sized finite carriers,every target needs exactly one preimage.Typedness alone is not bijectivity.

Step identifiers and dependencies

Rule: finite_vertex_map_each_target_fiber · Uses: G001, S004

S006 · implies

Before

complete mapping=[0, 1, 4, 3, 2, 5];target label=3

After

source fiber=[3];exactly one=Yes

Conditions: none

For a complete total map between equally sized finite carriers,every target needs exactly one preimage.Typedness alone is not bijectivity.

Step identifiers and dependencies

Rule: finite_vertex_map_each_target_fiber · Uses: G001, S005

S007 · implies

Before

complete mapping=[0, 1, 4, 3, 2, 5];target label=4

After

source fiber=[2];exactly one=Yes

Conditions: none

For a complete total map between equally sized finite carriers,every target needs exactly one preimage.Typedness alone is not bijectivity.

Step identifiers and dependencies

Rule: finite_vertex_map_each_target_fiber · Uses: G001, S006

S008 · implies

Before

complete mapping=[0, 1, 4, 3, 2, 5];target label=5

After

source fiber=[5];exactly one=Yes

Conditions: none

For a complete total map between equally sized finite carriers,every target needs exactly one preimage.Typedness alone is not bijectivity.

Step identifiers and dependencies

Rule: finite_vertex_map_each_target_fiber · Uses: G001, S007

S009 · implies

Before

n=6;all target fibers=[[0], [1], [4], [3], [2], [5]]

After

bijective=Yes;empty carrier uses unique empty map

Conditions: none

The unique empty map is bijective on empty carriers.This checks the mapping obligation only,not adjacency.

Step identifiers and dependencies

Rule: whole_bijection_including_empty_carrier · Uses: G001, S008

S010 · implies

Before

source unordered pair=[0, 1];mapped pair=[0, 1]

After

source adjacent=Yes;target adjacent=Yes;iff=Yes

Conditions: none

Compare both truth values on every unordered source pair,including source nonedges.A collision maps to a target diagonal,which is not an edge in a simple graph.Retain the full adjacency failure list even if bijectivity fails.

Step identifiers and dependencies

Rule: graph_every_edge_and_nonedge_iff · Uses: G001, S009

S011 · implies

Before

source unordered pair=[0, 2];mapped pair=[0, 4]

After

source adjacent=No;target adjacent=No;iff=Yes

Conditions: none

Step identifiers and dependencies

Rule: graph_every_edge_and_nonedge_iff · Uses: G001, S010

S012 · implies

Before

source unordered pair=[0, 3];mapped pair=[0, 3]

After

source adjacent=No;target adjacent=No;iff=Yes

Conditions: none

Step identifiers and dependencies

Rule: graph_every_edge_and_nonedge_iff · Uses: G001, S011

S013 · implies

Before

source unordered pair=[0, 4];mapped pair=[0, 2]

After

source adjacent=No;target adjacent=No;iff=Yes

Conditions: none

Step identifiers and dependencies

Rule: graph_every_edge_and_nonedge_iff · Uses: G001, S012

S014 · implies

Before

source unordered pair=[0, 5];mapped pair=[0, 5]

After

source adjacent=No;target adjacent=No;iff=Yes

Conditions: none

Step identifiers and dependencies

Rule: graph_every_edge_and_nonedge_iff · Uses: G001, S013

S015 · implies

Before

source unordered pair=[1, 2];mapped pair=[1, 4]

After

source adjacent=Yes;target adjacent=Yes;iff=Yes

Conditions: none

Step identifiers and dependencies

Rule: graph_every_edge_and_nonedge_iff · Uses: G001, S014

S016 · implies

Before

source unordered pair=[1, 3];mapped pair=[1, 3]

After

source adjacent=No;target adjacent=No;iff=Yes

Conditions: none

Step identifiers and dependencies

Rule: graph_every_edge_and_nonedge_iff · Uses: G001, S015

S017 · implies

Before

source unordered pair=[1, 4];mapped pair=[1, 2]

After

source adjacent=No;target adjacent=No;iff=Yes

Conditions: none

Step identifiers and dependencies

Rule: graph_every_edge_and_nonedge_iff · Uses: G001, S016

S018 · implies

Before

source unordered pair=[1, 5];mapped pair=[1, 5]

After

source adjacent=No;target adjacent=No;iff=Yes

Conditions: none

Step identifiers and dependencies

Rule: graph_every_edge_and_nonedge_iff · Uses: G001, S017

S019 · implies

Before

source unordered pair=[2, 3];mapped pair=[4, 3]

After

source adjacent=Yes;target adjacent=Yes;iff=Yes

Conditions: none

Step identifiers and dependencies

Rule: graph_every_edge_and_nonedge_iff · Uses: G001, S018

S020 · implies

Before

source unordered pair=[2, 4];mapped pair=[4, 2]

After

source adjacent=No;target adjacent=No;iff=Yes

Conditions: none

Step identifiers and dependencies

Rule: graph_every_edge_and_nonedge_iff · Uses: G001, S019

S021 · implies

Before

source unordered pair=[2, 5];mapped pair=[4, 5]

After

source adjacent=No;target adjacent=No;iff=Yes

Conditions: none

Step identifiers and dependencies

Rule: graph_every_edge_and_nonedge_iff · Uses: G001, S020

S022 · implies

Before

source unordered pair=[3, 4];mapped pair=[3, 2]

After

source adjacent=Yes;target adjacent=Yes;iff=Yes

Conditions: none

Step identifiers and dependencies

Rule: graph_every_edge_and_nonedge_iff · Uses: G001, S021

S023 · implies

Before

source unordered pair=[3, 5];mapped pair=[3, 5]

After

source adjacent=No;target adjacent=No;iff=Yes

Conditions: none

Step identifiers and dependencies

Rule: graph_every_edge_and_nonedge_iff · Uses: G001, S022

S024 · implies

Before

source unordered pair=[4, 5];mapped pair=[2, 5]

After

source adjacent=Yes;target adjacent=Yes;iff=Yes

Conditions: none

Step identifiers and dependencies

Rule: graph_every_edge_and_nonedge_iff · Uses: G001, S023

S025 · implies

Before

mapping bijective=Yes;all bad adjacency pairs=[]

After

adjacency iff=Yes;specified map isomorphism=Yes

Conditions: none

Bijectivity and complete adjacency iff are separate obligations.Neither alone certifies isomorphism.This audits the declared map rather than searching all maps.

Step identifiers and dependencies

Rule: graph_bijection_and_adjacency_separate_conjunction · Uses: G001, S024

Reverse checks

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, 4, 3, 2, 5], [], Yes)

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: 55d20734dd1aab1e99071807530c71e5a8bbddc8b31470bd2cb62221582cbb07
candidate record SHA-256: acf3b6e1ac91c751f652f69c210788d50ad21afe2628058115137e439898a2cd
semantic review receipt SHA-256: f783e81f44ff0faa13dbf948657e9187244a59a809ebdf96f22cca24966649ff
question-first witness SHA-256: 49a50ed0550be55124d6562260c2ccf4fa6d3f316d7ad9d0d32174400e362e18
Lean: NOT_RUN · human review: NOT_PERFORMED

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