Complete Vertex Bijections, Adjacency Iff and Invariant Boundaries
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This question bank has corresponding bilingual step-by-step derivations. Source candidate and English ai_candidate states are retained. AI review, finite computation and Lean formal proof are labelled separately.
batch-20261013-advance-20261026-r06-graph-isomorphism-foundations · 2026-10-13 · 100 questions
This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.
1. Check the Whole Finite Vertex Map for Bijectivity
The complete map on Fin(6)={0,…,5} has images=[0, 1, 2, 3, 4, 0] in source-label order, all in the target carrier of the same size. Give (n,mapping,bijective?). This family allows the unique empty map when n=0.
View answer and explanation
(6, [0, 1, 2, 3, 4, 0], No)
Check that every target label occurs exactly once. This is the mapping obligation alone: it makes no adjacency or graph-isomorphism claim. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [0, 1, 2, 3, 4, 0], No).
Reviewed step-by-step derivation
Graph isomorphism · qf-mod26-graphiso-bijection-audit · candidate
2. Check the Whole Finite Vertex Map for Bijectivity
The complete map on Fin(6)={0,…,5} has images=[0, 1, 2, 3, 5, 4] in source-label order, all in the target carrier of the same size. Give (n,mapping,bijective?). This family allows the unique empty map when n=0.
View answer and explanation
(6, [0, 1, 2, 3, 5, 4], Yes)
Check that every target label occurs exactly once. This is the mapping obligation alone: it makes no adjacency or graph-isomorphism claim. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [0, 1, 2, 3, 5, 4], Yes).
Reviewed step-by-step derivation
Graph isomorphism · qf-mod26-graphiso-bijection-audit · candidate
3. Check the Whole Finite Vertex Map for Bijectivity
The complete map on Fin(6)={0,…,5} has images=[0, 1, 2, 4, 3, 5] in source-label order, all in the target carrier of the same size. Give (n,mapping,bijective?). This family allows the unique empty map when n=0.
View answer and explanation
(6, [0, 1, 2, 4, 3, 5], Yes)
Check that every target label occurs exactly once. This is the mapping obligation alone: it makes no adjacency or graph-isomorphism claim. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [0, 1, 2, 4, 3, 5], Yes).
Reviewed step-by-step derivation
Graph isomorphism · qf-mod26-graphiso-bijection-audit · candidate
4. Check the Whole Finite Vertex Map for Bijectivity
The complete map on Fin(6)={0,…,5} has images=[0, 1, 2, 4, 5, 3] in source-label order, all in the target carrier of the same size. Give (n,mapping,bijective?). This family allows the unique empty map when n=0.
View answer and explanation
(6, [0, 1, 2, 4, 5, 3], Yes)
Check that every target label occurs exactly once. This is the mapping obligation alone: it makes no adjacency or graph-isomorphism claim. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [0, 1, 2, 4, 5, 3], Yes).
Reviewed step-by-step derivation
Graph isomorphism · qf-mod26-graphiso-bijection-audit · candidate
5. Check the Whole Finite Vertex Map for Bijectivity
The complete map on Fin(6)={0,…,5} has images=[0, 1, 2, 5, 3, 4] in source-label order, all in the target carrier of the same size. Give (n,mapping,bijective?). This family allows the unique empty map when n=0.
View answer and explanation
(6, [0, 1, 2, 5, 3, 4], Yes)
Check that every target label occurs exactly once. This is the mapping obligation alone: it makes no adjacency or graph-isomorphism claim. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [0, 1, 2, 5, 3, 4], Yes).
Reviewed step-by-step derivation
Graph isomorphism · qf-mod26-graphiso-bijection-audit · candidate
6. Check the Whole Finite Vertex Map for Bijectivity
The complete map on Fin(6)={0,…,5} has images=[0, 1, 2, 5, 4, 0] in source-label order, all in the target carrier of the same size. Give (n,mapping,bijective?). This family allows the unique empty map when n=0.
View answer and explanation
(6, [0, 1, 2, 5, 4, 0], No)
Check that every target label occurs exactly once. This is the mapping obligation alone: it makes no adjacency or graph-isomorphism claim. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [0, 1, 2, 5, 4, 0], No).
Reviewed step-by-step derivation
Graph isomorphism · qf-mod26-graphiso-bijection-audit · candidate
7. Check the Whole Finite Vertex Map for Bijectivity
The complete map on Fin(6)={0,…,5} has images=[0, 1, 3, 2, 4, 5] in source-label order, all in the target carrier of the same size. Give (n,mapping,bijective?). This family allows the unique empty map when n=0.
View answer and explanation
(6, [0, 1, 3, 2, 4, 5], Yes)
Check that every target label occurs exactly once. This is the mapping obligation alone: it makes no adjacency or graph-isomorphism claim. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [0, 1, 3, 2, 4, 5], Yes).
Reviewed step-by-step derivation
Graph isomorphism · qf-mod26-graphiso-bijection-audit · candidate
8. Check the Whole Finite Vertex Map for Bijectivity
The complete map on Fin(6)={0,…,5} has images=[0, 1, 3, 2, 5, 4] in source-label order, all in the target carrier of the same size. Give (n,mapping,bijective?). This family allows the unique empty map when n=0.
View answer and explanation
(6, [0, 1, 3, 2, 5, 4], Yes)
Check that every target label occurs exactly once. This is the mapping obligation alone: it makes no adjacency or graph-isomorphism claim. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [0, 1, 3, 2, 5, 4], Yes).
Reviewed step-by-step derivation
Graph isomorphism · qf-mod26-graphiso-bijection-audit · candidate
9. Check the Whole Finite Vertex Map for Bijectivity
The complete map on Fin(6)={0,…,5} has images=[0, 1, 3, 4, 2, 5] in source-label order, all in the target carrier of the same size. Give (n,mapping,bijective?). This family allows the unique empty map when n=0.
View answer and explanation
(6, [0, 1, 3, 4, 2, 5], Yes)
Check that every target label occurs exactly once. This is the mapping obligation alone: it makes no adjacency or graph-isomorphism claim. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [0, 1, 3, 4, 2, 5], Yes).
Reviewed step-by-step derivation
Graph isomorphism · qf-mod26-graphiso-bijection-audit · candidate
10. Check the Whole Finite Vertex Map for Bijectivity
The complete map on Fin(6)={0,…,5} has images=[0, 1, 3, 4, 5, 2] in source-label order, all in the target carrier of the same size. Give (n,mapping,bijective?). This family allows the unique empty map when n=0.
View answer and explanation
(6, [0, 1, 3, 4, 5, 2], Yes)
Check that every target label occurs exactly once. This is the mapping obligation alone: it makes no adjacency or graph-isomorphism claim. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [0, 1, 3, 4, 5, 2], Yes).
Reviewed step-by-step derivation
Graph isomorphism · qf-mod26-graphiso-bijection-audit · candidate
11. Check the Whole Finite Vertex Map for Bijectivity
The complete map on Fin(6)={0,…,5} has images=[0, 1, 3, 5, 2, 0] in source-label order, all in the target carrier of the same size. Give (n,mapping,bijective?). This family allows the unique empty map when n=0.
View answer and explanation
(6, [0, 1, 3, 5, 2, 0], No)
Check that every target label occurs exactly once. This is the mapping obligation alone: it makes no adjacency or graph-isomorphism claim. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [0, 1, 3, 5, 2, 0], No).
Reviewed step-by-step derivation
Graph isomorphism · qf-mod26-graphiso-bijection-audit · candidate
12. Check the Whole Finite Vertex Map for Bijectivity
The complete map on Fin(6)={0,…,5} has images=[0, 1, 3, 5, 4, 2] in source-label order, all in the target carrier of the same size. Give (n,mapping,bijective?). This family allows the unique empty map when n=0.
View answer and explanation
(6, [0, 1, 3, 5, 4, 2], Yes)
Check that every target label occurs exactly once. This is the mapping obligation alone: it makes no adjacency or graph-isomorphism claim. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [0, 1, 3, 5, 4, 2], Yes).
Reviewed step-by-step derivation
Graph isomorphism · qf-mod26-graphiso-bijection-audit · candidate
13. Check the Whole Finite Vertex Map for Bijectivity
The complete map on Fin(6)={0,…,5} has images=[0, 1, 4, 2, 3, 5] in source-label order, all in the target carrier of the same size. Give (n,mapping,bijective?). This family allows the unique empty map when n=0.
View answer and explanation
(6, [0, 1, 4, 2, 3, 5], Yes)
Check that every target label occurs exactly once. This is the mapping obligation alone: it makes no adjacency or graph-isomorphism claim. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [0, 1, 4, 2, 3, 5], Yes).
Reviewed step-by-step derivation
Graph isomorphism · qf-mod26-graphiso-bijection-audit · candidate
14. Check the Whole Finite Vertex Map for Bijectivity
The complete map on Fin(6)={0,…,5} has images=[0, 1, 4, 2, 5, 3] in source-label order, all in the target carrier of the same size. Give (n,mapping,bijective?). This family allows the unique empty map when n=0.
View answer and explanation
(6, [0, 1, 4, 2, 5, 3], Yes)
Check that every target label occurs exactly once. This is the mapping obligation alone: it makes no adjacency or graph-isomorphism claim. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [0, 1, 4, 2, 5, 3], Yes).
Reviewed step-by-step derivation
Graph isomorphism · qf-mod26-graphiso-bijection-audit · candidate
15. Check the Whole Finite Vertex Map for Bijectivity
The complete map on Fin(6)={0,…,5} has images=[0, 1, 4, 3, 2, 5] in source-label order, all in the target carrier of the same size. Give (n,mapping,bijective?). This family allows the unique empty map when n=0.
View answer and explanation
(6, [0, 1, 4, 3, 2, 5], Yes)
Check that every target label occurs exactly once. This is the mapping obligation alone: it makes no adjacency or graph-isomorphism claim. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [0, 1, 4, 3, 2, 5], Yes).
Reviewed step-by-step derivation
Graph isomorphism · qf-mod26-graphiso-bijection-audit · candidate
16. Check the Whole Finite Vertex Map for Bijectivity
The complete map on Fin(6)={0,…,5} has images=[0, 1, 4, 3, 5, 0] in source-label order, all in the target carrier of the same size. Give (n,mapping,bijective?). This family allows the unique empty map when n=0.
View answer and explanation
(6, [0, 1, 4, 3, 5, 0], No)
Check that every target label occurs exactly once. This is the mapping obligation alone: it makes no adjacency or graph-isomorphism claim. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [0, 1, 4, 3, 5, 0], No).
Reviewed step-by-step derivation
Graph isomorphism · qf-mod26-graphiso-bijection-audit · candidate
17. Check the Whole Finite Vertex Map for Bijectivity
The complete map on Fin(6)={0,…,5} has images=[0, 1, 4, 5, 2, 3] in source-label order, all in the target carrier of the same size. Give (n,mapping,bijective?). This family allows the unique empty map when n=0.
View answer and explanation
(6, [0, 1, 4, 5, 2, 3], Yes)
Check that every target label occurs exactly once. This is the mapping obligation alone: it makes no adjacency or graph-isomorphism claim. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [0, 1, 4, 5, 2, 3], Yes).
Reviewed step-by-step derivation
Graph isomorphism · qf-mod26-graphiso-bijection-audit · candidate
18. Check the Whole Finite Vertex Map for Bijectivity
The complete map on Fin(6)={0,…,5} has images=[0, 1, 4, 5, 3, 2] in source-label order, all in the target carrier of the same size. Give (n,mapping,bijective?). This family allows the unique empty map when n=0.
View answer and explanation
(6, [0, 1, 4, 5, 3, 2], Yes)
Check that every target label occurs exactly once. This is the mapping obligation alone: it makes no adjacency or graph-isomorphism claim. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [0, 1, 4, 5, 3, 2], Yes).
Reviewed step-by-step derivation
Graph isomorphism · qf-mod26-graphiso-bijection-audit · candidate
19. Check the Whole Finite Vertex Map for Bijectivity
The complete map on Fin(6)={0,…,5} has images=[0, 1, 5, 2, 3, 4] in source-label order, all in the target carrier of the same size. Give (n,mapping,bijective?). This family allows the unique empty map when n=0.
View answer and explanation
(6, [0, 1, 5, 2, 3, 4], Yes)
Check that every target label occurs exactly once. This is the mapping obligation alone: it makes no adjacency or graph-isomorphism claim. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [0, 1, 5, 2, 3, 4], Yes).
Reviewed step-by-step derivation
Graph isomorphism · qf-mod26-graphiso-bijection-audit · candidate
20. Check the Whole Finite Vertex Map for Bijectivity
The complete map on Fin(6)={0,…,5} has images=[0, 1, 5, 2, 4, 3] in source-label order, all in the target carrier of the same size. Give (n,mapping,bijective?). This family allows the unique empty map when n=0.
View answer and explanation
(6, [0, 1, 5, 2, 4, 3], Yes)
Check that every target label occurs exactly once. This is the mapping obligation alone: it makes no adjacency or graph-isomorphism claim. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [0, 1, 5, 2, 4, 3], Yes).
Reviewed step-by-step derivation
Graph isomorphism · qf-mod26-graphiso-bijection-audit · candidate
21. Check the Whole Finite Vertex Map for Bijectivity
The complete map on Fin(6)={0,…,5} has images=[0, 1, 5, 3, 2, 0] in source-label order, all in the target carrier of the same size. Give (n,mapping,bijective?). This family allows the unique empty map when n=0.
View answer and explanation
(6, [0, 1, 5, 3, 2, 0], No)
Check that every target label occurs exactly once. This is the mapping obligation alone: it makes no adjacency or graph-isomorphism claim. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [0, 1, 5, 3, 2, 0], No).
Reviewed step-by-step derivation
Graph isomorphism · qf-mod26-graphiso-bijection-audit · candidate
22. Check the Whole Finite Vertex Map for Bijectivity
The complete map on Fin(6)={0,…,5} has images=[0, 1, 5, 3, 4, 2] in source-label order, all in the target carrier of the same size. Give (n,mapping,bijective?). This family allows the unique empty map when n=0.
View answer and explanation
(6, [0, 1, 5, 3, 4, 2], Yes)
Check that every target label occurs exactly once. This is the mapping obligation alone: it makes no adjacency or graph-isomorphism claim. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [0, 1, 5, 3, 4, 2], Yes).
Reviewed step-by-step derivation
Graph isomorphism · qf-mod26-graphiso-bijection-audit · candidate
23. Check the Whole Finite Vertex Map for Bijectivity
The complete map on Fin(6)={0,…,5} has images=[0, 1, 5, 4, 2, 3] in source-label order, all in the target carrier of the same size. Give (n,mapping,bijective?). This family allows the unique empty map when n=0.
View answer and explanation
(6, [0, 1, 5, 4, 2, 3], Yes)
Check that every target label occurs exactly once. This is the mapping obligation alone: it makes no adjacency or graph-isomorphism claim. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [0, 1, 5, 4, 2, 3], Yes).
Reviewed step-by-step derivation
Graph isomorphism · qf-mod26-graphiso-bijection-audit · candidate
24. Check the Whole Finite Vertex Map for Bijectivity
The complete map on Fin(6)={0,…,5} has images=[0, 1, 5, 4, 3, 2] in source-label order, all in the target carrier of the same size. Give (n,mapping,bijective?). This family allows the unique empty map when n=0.
View answer and explanation
(6, [0, 1, 5, 4, 3, 2], Yes)
Check that every target label occurs exactly once. This is the mapping obligation alone: it makes no adjacency or graph-isomorphism claim. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [0, 1, 5, 4, 3, 2], Yes).
Reviewed step-by-step derivation
Graph isomorphism · qf-mod26-graphiso-bijection-audit · candidate
25. Check the Whole Finite Vertex Map for Bijectivity
The complete map on Fin(6)={0,…,5} has images=[0, 2, 1, 3, 4, 5] in source-label order, all in the target carrier of the same size. Give (n,mapping,bijective?). This family allows the unique empty map when n=0.
View answer and explanation
(6, [0, 2, 1, 3, 4, 5], Yes)
Check that every target label occurs exactly once. This is the mapping obligation alone: it makes no adjacency or graph-isomorphism claim. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [0, 2, 1, 3, 4, 5], Yes).
Reviewed step-by-step derivation
Graph isomorphism · qf-mod26-graphiso-bijection-audit · candidate
26. Audit Edges and Nonedges without Replacing Bijectivity
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, 2], [2, 3], [3, 4], [4, 5], [0, 2]] and complete vertex map=[0, 1, 2, 3, 4, 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?).
View answer and explanation
([0, 1, 2, 3, 4, 5], [[0, 2, No, Yes]], No)
Compare both adjacency truth values for every unordered source pair. This flag alone does not certify bijectivity or an isomorphism. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 5], [[0, 2, No, Yes]], No).
Reviewed step-by-step derivation
Graph isomorphism · qf-mod26-graphiso-adjacency-iff-audit · candidate
27. Audit Edges and Nonedges without Replacing Bijectivity
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, 2], [2, 3], [3, 5], [4, 5]] and complete vertex map=[0, 1, 2, 3, 5, 4] 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?).
View answer and explanation
([0, 1, 2, 3, 5, 4], [], Yes)
Compare both adjacency truth values for every unordered source pair. This flag alone does not certify bijectivity or an isomorphism. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([0, 1, 2, 3, 5, 4], [], Yes).
Reviewed step-by-step derivation
Graph isomorphism · qf-mod26-graphiso-adjacency-iff-audit · candidate
28. Audit Edges and Nonedges without Replacing Bijectivity
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, 2], [2, 4], [3, 4], [3, 5]] and complete vertex map=[0, 1, 2, 4, 3, 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?).
View answer and explanation
([0, 1, 2, 4, 3, 5], [], Yes)
Compare both adjacency truth values for every unordered source pair. This flag alone does not certify bijectivity or an isomorphism. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([0, 1, 2, 4, 3, 5], [], Yes).
Reviewed step-by-step derivation
Graph isomorphism · qf-mod26-graphiso-adjacency-iff-audit · candidate
29. Audit Edges and Nonedges without Replacing Bijectivity
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, 2], [2, 4], [4, 5], [3, 5]] and complete vertex map=[0, 1, 2, 4, 5, 3] 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?).
View answer and explanation
([0, 1, 2, 4, 5, 3], [], Yes)
Compare both adjacency truth values for every unordered source pair. This flag alone does not certify bijectivity or an isomorphism. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([0, 1, 2, 4, 5, 3], [], Yes).
Reviewed step-by-step derivation
Graph isomorphism · qf-mod26-graphiso-adjacency-iff-audit · candidate
30. Audit Edges and Nonedges without Replacing Bijectivity
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, 2], [2, 5], [3, 5], [3, 4]] and complete vertex map=[0, 1, 2, 5, 3, 4] 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?).
View answer and explanation
([0, 1, 2, 5, 3, 4], [], Yes)
Compare both adjacency truth values for every unordered source pair. This flag alone does not certify bijectivity or an isomorphism. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([0, 1, 2, 5, 3, 4], [], Yes).
Reviewed step-by-step derivation
Graph isomorphism · qf-mod26-graphiso-adjacency-iff-audit · candidate
31. Audit Edges and Nonedges without Replacing Bijectivity
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, 2], [2, 5], [4, 5], [3, 4], [0, 2]] and complete vertex map=[0, 1, 2, 5, 4, 3] 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?).
View answer and explanation
([0, 1, 2, 5, 4, 3], [[0, 2, No, Yes]], No)
Compare both adjacency truth values for every unordered source pair. This flag alone does not certify bijectivity or an isomorphism. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([0, 1, 2, 5, 4, 3], [[0, 2, No, Yes]], No).
Reviewed step-by-step derivation
Graph isomorphism · qf-mod26-graphiso-adjacency-iff-audit · candidate
32. Audit Edges and Nonedges without Replacing Bijectivity
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, 3], [2, 3], [2, 4], [4, 5]] and complete vertex map=[0, 1, 3, 2, 4, 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?).
View answer and explanation
([0, 1, 3, 2, 4, 5], [], Yes)
Compare both adjacency truth values for every unordered source pair. This flag alone does not certify bijectivity or an isomorphism. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([0, 1, 3, 2, 4, 5], [], Yes).
Reviewed step-by-step derivation
Graph isomorphism · qf-mod26-graphiso-adjacency-iff-audit · candidate
33. Audit Edges and Nonedges without Replacing Bijectivity
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, 3], [2, 3], [2, 5], [4, 5]] and complete vertex map=[0, 1, 3, 2, 5, 4] 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?).
View answer and explanation
([0, 1, 3, 2, 5, 4], [], Yes)
Compare both adjacency truth values for every unordered source pair. This flag alone does not certify bijectivity or an isomorphism. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([0, 1, 3, 2, 5, 4], [], Yes).
Reviewed step-by-step derivation
Graph isomorphism · qf-mod26-graphiso-adjacency-iff-audit · candidate
34. Audit Edges and Nonedges without Replacing Bijectivity
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, 3], [3, 4], [2, 4], [2, 5]] and complete vertex map=[0, 1, 3, 4, 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?).
View answer and explanation
([0, 1, 3, 4, 2, 5], [], Yes)
Compare both adjacency truth values for every unordered source pair. This flag alone does not certify bijectivity or an isomorphism. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([0, 1, 3, 4, 2, 5], [], Yes).
Reviewed step-by-step derivation
Graph isomorphism · qf-mod26-graphiso-adjacency-iff-audit · candidate
35. Audit Edges and Nonedges without Replacing Bijectivity
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, 3], [3, 4], [4, 5], [2, 5]] and complete vertex map=[0, 1, 3, 4, 5, 2] 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?).
View answer and explanation
([0, 1, 3, 4, 5, 2], [], Yes)
Compare both adjacency truth values for every unordered source pair. This flag alone does not certify bijectivity or an isomorphism. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([0, 1, 3, 4, 5, 2], [], Yes).
Reviewed step-by-step derivation
Graph isomorphism · qf-mod26-graphiso-adjacency-iff-audit · candidate
36. Audit Edges and Nonedges without Replacing Bijectivity
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, 3], [3, 5], [2, 5], [2, 4], [0, 2]] and complete vertex map=[0, 1, 3, 5, 2, 4] 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?).
View answer and explanation
([0, 1, 3, 5, 2, 4], [[0, 4, No, Yes]], No)
Compare both adjacency truth values for every unordered source pair. This flag alone does not certify bijectivity or an isomorphism. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([0, 1, 3, 5, 2, 4], [[0, 4, No, Yes]], No).
Reviewed step-by-step derivation
Graph isomorphism · qf-mod26-graphiso-adjacency-iff-audit · candidate
37. Audit Edges and Nonedges without Replacing Bijectivity
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, 3], [3, 5], [4, 5], [2, 4]] and complete vertex map=[0, 1, 3, 5, 4, 2] 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?).
View answer and explanation
([0, 1, 3, 5, 4, 2], [], Yes)
Compare both adjacency truth values for every unordered source pair. This flag alone does not certify bijectivity or an isomorphism. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([0, 1, 3, 5, 4, 2], [], Yes).
Reviewed step-by-step derivation
Graph isomorphism · qf-mod26-graphiso-adjacency-iff-audit · candidate
38. Audit Edges and Nonedges without Replacing Bijectivity
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], [2, 4], [2, 3], [3, 5]] and complete vertex map=[0, 1, 4, 2, 3, 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?).
View answer and explanation
([0, 1, 4, 2, 3, 5], [], Yes)
Compare both adjacency truth values for every unordered source pair. This flag alone does not certify bijectivity or an isomorphism. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([0, 1, 4, 2, 3, 5], [], Yes).
Reviewed step-by-step derivation
Graph isomorphism · qf-mod26-graphiso-adjacency-iff-audit · candidate
39. Audit Edges and Nonedges without Replacing Bijectivity
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], [2, 4], [2, 5], [3, 5]] and complete vertex map=[0, 1, 4, 2, 5, 3] 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?).
View answer and explanation
([0, 1, 4, 2, 5, 3], [], Yes)
Compare both adjacency truth values for every unordered source pair. This flag alone does not certify bijectivity or an isomorphism. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([0, 1, 4, 2, 5, 3], [], Yes).
Reviewed step-by-step derivation
Graph isomorphism · qf-mod26-graphiso-adjacency-iff-audit · candidate
40. Audit Edges and Nonedges without Replacing Bijectivity
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?).
View answer and explanation
([0, 1, 4, 3, 2, 5], [], Yes)
Compare both adjacency truth values for every unordered source pair. This flag alone does not certify bijectivity or an isomorphism. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([0, 1, 4, 3, 2, 5], [], Yes).
Reviewed step-by-step derivation
Graph isomorphism · qf-mod26-graphiso-adjacency-iff-audit · candidate
41. Audit Edges and Nonedges without Replacing Bijectivity
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], [3, 5], [2, 5], [0, 2]] and complete vertex map=[0, 1, 4, 3, 5, 2] 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?).
View answer and explanation
([0, 1, 4, 3, 5, 2], [[0, 5, No, Yes]], No)
Compare both adjacency truth values for every unordered source pair. This flag alone does not certify bijectivity or an isomorphism. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([0, 1, 4, 3, 5, 2], [[0, 5, No, Yes]], No).
Reviewed step-by-step derivation
Graph isomorphism · qf-mod26-graphiso-adjacency-iff-audit · candidate
42. Audit Edges and Nonedges without Replacing Bijectivity
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], [4, 5], [2, 5], [2, 3]] and complete vertex map=[0, 1, 4, 5, 2, 3] 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?).
View answer and explanation
([0, 1, 4, 5, 2, 3], [], Yes)
Compare both adjacency truth values for every unordered source pair. This flag alone does not certify bijectivity or an isomorphism. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([0, 1, 4, 5, 2, 3], [], Yes).
Reviewed step-by-step derivation
Graph isomorphism · qf-mod26-graphiso-adjacency-iff-audit · candidate
43. Audit Edges and Nonedges without Replacing Bijectivity
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], [4, 5], [3, 5], [2, 3]] and complete vertex map=[0, 1, 4, 5, 3, 2] 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?).
View answer and explanation
([0, 1, 4, 5, 3, 2], [], Yes)
Compare both adjacency truth values for every unordered source pair. This flag alone does not certify bijectivity or an isomorphism. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([0, 1, 4, 5, 3, 2], [], Yes).
Reviewed step-by-step derivation
Graph isomorphism · qf-mod26-graphiso-adjacency-iff-audit · candidate
44. Audit Edges and Nonedges without Replacing Bijectivity
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, 5], [2, 5], [2, 3], [3, 4]] and complete vertex map=[0, 1, 5, 2, 3, 4] 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?).
View answer and explanation
([0, 1, 5, 2, 3, 4], [], Yes)
Compare both adjacency truth values for every unordered source pair. This flag alone does not certify bijectivity or an isomorphism. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([0, 1, 5, 2, 3, 4], [], Yes).
Reviewed step-by-step derivation
Graph isomorphism · qf-mod26-graphiso-adjacency-iff-audit · candidate
45. Audit Edges and Nonedges without Replacing Bijectivity
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, 5], [2, 5], [2, 4], [3, 4]] and complete vertex map=[0, 1, 5, 2, 4, 3] 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?).
View answer and explanation
([0, 1, 5, 2, 4, 3], [], Yes)
Compare both adjacency truth values for every unordered source pair. This flag alone does not certify bijectivity or an isomorphism. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([0, 1, 5, 2, 4, 3], [], Yes).
Reviewed step-by-step derivation
Graph isomorphism · qf-mod26-graphiso-adjacency-iff-audit · candidate
46. Audit Edges and Nonedges without Replacing Bijectivity
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, 5], [3, 5], [2, 3], [2, 4], [0, 2]] and complete vertex map=[0, 1, 5, 3, 2, 4] 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?).
View answer and explanation
([0, 1, 5, 3, 2, 4], [[0, 4, No, Yes]], No)
Compare both adjacency truth values for every unordered source pair. This flag alone does not certify bijectivity or an isomorphism. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([0, 1, 5, 3, 2, 4], [[0, 4, No, Yes]], No).
Reviewed step-by-step derivation
Graph isomorphism · qf-mod26-graphiso-adjacency-iff-audit · candidate
47. Audit Edges and Nonedges without Replacing Bijectivity
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, 5], [3, 5], [3, 4], [2, 4]] and complete vertex map=[0, 1, 5, 3, 4, 2] 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?).
View answer and explanation
([0, 1, 5, 3, 4, 2], [], Yes)
Compare both adjacency truth values for every unordered source pair. This flag alone does not certify bijectivity or an isomorphism. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([0, 1, 5, 3, 4, 2], [], Yes).
Reviewed step-by-step derivation
Graph isomorphism · qf-mod26-graphiso-adjacency-iff-audit · candidate
48. Audit Edges and Nonedges without Replacing Bijectivity
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, 5], [4, 5], [2, 4], [2, 3]] and complete vertex map=[0, 1, 5, 4, 2, 3] 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?).
View answer and explanation
([0, 1, 5, 4, 2, 3], [], Yes)
Compare both adjacency truth values for every unordered source pair. This flag alone does not certify bijectivity or an isomorphism. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([0, 1, 5, 4, 2, 3], [], Yes).
Reviewed step-by-step derivation
Graph isomorphism · qf-mod26-graphiso-adjacency-iff-audit · candidate
49. Audit Edges and Nonedges without Replacing Bijectivity
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, 5], [4, 5], [3, 4], [2, 3]] and complete vertex map=[0, 1, 5, 4, 3, 2] 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?).
View answer and explanation
([0, 1, 5, 4, 3, 2], [], Yes)
Compare both adjacency truth values for every unordered source pair. This flag alone does not certify bijectivity or an isomorphism. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([0, 1, 5, 4, 3, 2], [], Yes).
Reviewed step-by-step derivation
Graph isomorphism · qf-mod26-graphiso-adjacency-iff-audit · candidate
50. Audit Edges and Nonedges without Replacing Bijectivity
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, 2], [1, 2], [1, 3], [3, 4], [4, 5]] and complete vertex map=[0, 2, 1, 3, 4, 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?).
View answer and explanation
([0, 2, 1, 3, 4, 5], [], Yes)
Compare both adjacency truth values for every unordered source pair. This flag alone does not certify bijectivity or an isomorphism. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([0, 2, 1, 3, 4, 5], [], Yes).
Reviewed step-by-step derivation
Graph isomorphism · qf-mod26-graphiso-adjacency-iff-audit · candidate
51. Conjoin Bijectivity with the Complete Adjacency-Iff Audit
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, 2], [2, 3], [3, 4], [4, 5], [0, 2]] and complete vertex map=[0, 1, 2, 3, 4, 0] 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 (n,mapping,bijective?,bad [a,b,source adjacent?,mapped target adjacent?] records,isomorphism?).
View answer and explanation
(6, [0, 1, 2, 3, 4, 0], No, [[0, 2, No, Yes], [1, 5, No, Yes], [2, 5, No, Yes], [4, 5, Yes, No]], No)
Evaluate the two obligations independently, retaining bad-pair records even if bijectivity fails. Only their conjunction certifies this declared map as an isomorphism. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [0, 1, 2, 3, 4, 0], No, [[0, 2, No, Yes], [1, 5, No, Yes], [2, 5, No, Yes], [4, 5, Yes, No]], No).
Reviewed step-by-step derivation
Graph isomorphism · qf-mod26-graphiso-qualification · candidate
52. Conjoin Bijectivity with the Complete Adjacency-Iff Audit
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, 2], [2, 3], [3, 5], [4, 5]] and complete vertex map=[0, 1, 2, 3, 5, 4] 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 (n,mapping,bijective?,bad [a,b,source adjacent?,mapped target adjacent?] records,isomorphism?).
View answer and explanation
(6, [0, 1, 2, 3, 5, 4], Yes, [], Yes)
Evaluate the two obligations independently, retaining bad-pair records even if bijectivity fails. Only their conjunction certifies this declared map as an isomorphism. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [0, 1, 2, 3, 5, 4], Yes, [], Yes).
Reviewed step-by-step derivation
Graph isomorphism · qf-mod26-graphiso-qualification · candidate
53. Conjoin Bijectivity with the Complete Adjacency-Iff Audit
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, 2], [2, 4], [3, 4], [3, 5]] and complete vertex map=[0, 1, 2, 4, 3, 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 (n,mapping,bijective?,bad [a,b,source adjacent?,mapped target adjacent?] records,isomorphism?).
View answer and explanation
(6, [0, 1, 2, 4, 3, 5], Yes, [], Yes)
Evaluate the two obligations independently, retaining bad-pair records even if bijectivity fails. Only their conjunction certifies this declared map as an isomorphism. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [0, 1, 2, 4, 3, 5], Yes, [], Yes).
Reviewed step-by-step derivation
Graph isomorphism · qf-mod26-graphiso-qualification · candidate
54. Conjoin Bijectivity with the Complete Adjacency-Iff Audit
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, 2], [2, 4], [4, 5], [3, 5]] and complete vertex map=[0, 1, 2, 4, 5, 3] 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 (n,mapping,bijective?,bad [a,b,source adjacent?,mapped target adjacent?] records,isomorphism?).
View answer and explanation
(6, [0, 1, 2, 4, 5, 3], Yes, [], Yes)
Evaluate the two obligations independently, retaining bad-pair records even if bijectivity fails. Only their conjunction certifies this declared map as an isomorphism. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [0, 1, 2, 4, 5, 3], Yes, [], Yes).
Reviewed step-by-step derivation
Graph isomorphism · qf-mod26-graphiso-qualification · candidate
55. Conjoin Bijectivity with the Complete Adjacency-Iff Audit
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, 2], [2, 5], [3, 5], [3, 4]] and complete vertex map=[0, 1, 2, 5, 3, 4] 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 (n,mapping,bijective?,bad [a,b,source adjacent?,mapped target adjacent?] records,isomorphism?).
View answer and explanation
(6, [0, 1, 2, 5, 3, 4], Yes, [], Yes)
Evaluate the two obligations independently, retaining bad-pair records even if bijectivity fails. Only their conjunction certifies this declared map as an isomorphism. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [0, 1, 2, 5, 3, 4], Yes, [], Yes).
Reviewed step-by-step derivation
Graph isomorphism · qf-mod26-graphiso-qualification · candidate
56. Conjoin Bijectivity with the Complete Adjacency-Iff Audit
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, 2], [2, 5], [4, 5], [3, 4], [0, 2]] and complete vertex map=[0, 1, 2, 5, 4, 3] 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 (n,mapping,bijective?,bad [a,b,source adjacent?,mapped target adjacent?] records,isomorphism?).
View answer and explanation
(6, [0, 1, 2, 5, 4, 3], Yes, [[0, 2, No, Yes]], No)
Evaluate the two obligations independently, retaining bad-pair records even if bijectivity fails. Only their conjunction certifies this declared map as an isomorphism. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [0, 1, 2, 5, 4, 3], Yes, [[0, 2, No, Yes]], No).
Reviewed step-by-step derivation
Graph isomorphism · qf-mod26-graphiso-qualification · candidate
57. Conjoin Bijectivity with the Complete Adjacency-Iff Audit
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, 3], [2, 3], [2, 4], [4, 5]] and complete vertex map=[0, 1, 3, 2, 4, 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 (n,mapping,bijective?,bad [a,b,source adjacent?,mapped target adjacent?] records,isomorphism?).
View answer and explanation
(6, [0, 1, 3, 2, 4, 5], Yes, [], Yes)
Evaluate the two obligations independently, retaining bad-pair records even if bijectivity fails. Only their conjunction certifies this declared map as an isomorphism. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [0, 1, 3, 2, 4, 5], Yes, [], Yes).
Reviewed step-by-step derivation
Graph isomorphism · qf-mod26-graphiso-qualification · candidate
58. Conjoin Bijectivity with the Complete Adjacency-Iff Audit
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, 3], [2, 3], [2, 5], [4, 5]] and complete vertex map=[0, 1, 3, 2, 5, 0] 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 (n,mapping,bijective?,bad [a,b,source adjacent?,mapped target adjacent?] records,isomorphism?).
View answer and explanation
(6, [0, 1, 3, 2, 5, 0], No, [[1, 5, No, Yes], [4, 5, Yes, No]], No)
Evaluate the two obligations independently, retaining bad-pair records even if bijectivity fails. Only their conjunction certifies this declared map as an isomorphism. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [0, 1, 3, 2, 5, 0], No, [[1, 5, No, Yes], [4, 5, Yes, No]], No).
Reviewed step-by-step derivation
Graph isomorphism · qf-mod26-graphiso-qualification · candidate
59. Conjoin Bijectivity with the Complete Adjacency-Iff Audit
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, 3], [3, 4], [2, 4], [2, 5]] and complete vertex map=[0, 1, 3, 4, 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 (n,mapping,bijective?,bad [a,b,source adjacent?,mapped target adjacent?] records,isomorphism?).
View answer and explanation
(6, [0, 1, 3, 4, 2, 5], Yes, [], Yes)
Evaluate the two obligations independently, retaining bad-pair records even if bijectivity fails. Only their conjunction certifies this declared map as an isomorphism. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [0, 1, 3, 4, 2, 5], Yes, [], Yes).
Reviewed step-by-step derivation
Graph isomorphism · qf-mod26-graphiso-qualification · candidate
60. Conjoin Bijectivity with the Complete Adjacency-Iff Audit
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, 3], [3, 4], [4, 5], [2, 5]] and complete vertex map=[0, 1, 3, 4, 5, 2] 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 (n,mapping,bijective?,bad [a,b,source adjacent?,mapped target adjacent?] records,isomorphism?).
View answer and explanation
(6, [0, 1, 3, 4, 5, 2], Yes, [], Yes)
Evaluate the two obligations independently, retaining bad-pair records even if bijectivity fails. Only their conjunction certifies this declared map as an isomorphism. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [0, 1, 3, 4, 5, 2], Yes, [], Yes).
Reviewed step-by-step derivation
Graph isomorphism · qf-mod26-graphiso-qualification · candidate
61. Conjoin Bijectivity with the Complete Adjacency-Iff Audit
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, 3], [3, 5], [2, 5], [2, 4], [0, 2]] and complete vertex map=[0, 1, 3, 5, 2, 4] 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 (n,mapping,bijective?,bad [a,b,source adjacent?,mapped target adjacent?] records,isomorphism?).
View answer and explanation
(6, [0, 1, 3, 5, 2, 4], Yes, [[0, 4, No, Yes]], No)
Evaluate the two obligations independently, retaining bad-pair records even if bijectivity fails. Only their conjunction certifies this declared map as an isomorphism. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [0, 1, 3, 5, 2, 4], Yes, [[0, 4, No, Yes]], No).
Reviewed step-by-step derivation
Graph isomorphism · qf-mod26-graphiso-qualification · candidate
62. Conjoin Bijectivity with the Complete Adjacency-Iff Audit
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, 3], [3, 5], [4, 5], [2, 4]] and complete vertex map=[0, 1, 3, 5, 4, 2] 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 (n,mapping,bijective?,bad [a,b,source adjacent?,mapped target adjacent?] records,isomorphism?).
View answer and explanation
(6, [0, 1, 3, 5, 4, 2], Yes, [], Yes)
Evaluate the two obligations independently, retaining bad-pair records even if bijectivity fails. Only their conjunction certifies this declared map as an isomorphism. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [0, 1, 3, 5, 4, 2], Yes, [], Yes).
Reviewed step-by-step derivation
Graph isomorphism · qf-mod26-graphiso-qualification · candidate
63. Conjoin Bijectivity with the Complete Adjacency-Iff Audit
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], [2, 4], [2, 3], [3, 5]] and complete vertex map=[0, 1, 4, 2, 3, 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 (n,mapping,bijective?,bad [a,b,source adjacent?,mapped target adjacent?] records,isomorphism?).
View answer and explanation
(6, [0, 1, 4, 2, 3, 5], Yes, [], Yes)
Evaluate the two obligations independently, retaining bad-pair records even if bijectivity fails. Only their conjunction certifies this declared map as an isomorphism. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [0, 1, 4, 2, 3, 5], Yes, [], Yes).
Reviewed step-by-step derivation
Graph isomorphism · qf-mod26-graphiso-qualification · candidate
64. Conjoin Bijectivity with the Complete Adjacency-Iff Audit
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], [2, 4], [2, 5], [3, 5]] and complete vertex map=[0, 1, 4, 2, 5, 3] 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 (n,mapping,bijective?,bad [a,b,source adjacent?,mapped target adjacent?] records,isomorphism?).
View answer and explanation
(6, [0, 1, 4, 2, 5, 3], Yes, [], Yes)
Evaluate the two obligations independently, retaining bad-pair records even if bijectivity fails. Only their conjunction certifies this declared map as an isomorphism. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [0, 1, 4, 2, 5, 3], Yes, [], Yes).
Reviewed step-by-step derivation
Graph isomorphism · qf-mod26-graphiso-qualification · candidate
65. Conjoin Bijectivity with the Complete Adjacency-Iff Audit
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, 0] 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 (n,mapping,bijective?,bad [a,b,source adjacent?,mapped target adjacent?] records,isomorphism?).
View answer and explanation
(6, [0, 1, 4, 3, 2, 0], No, [[1, 5, No, Yes], [4, 5, Yes, No]], No)
Evaluate the two obligations independently, retaining bad-pair records even if bijectivity fails. Only their conjunction certifies this declared map as an isomorphism. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [0, 1, 4, 3, 2, 0], No, [[1, 5, No, Yes], [4, 5, Yes, No]], No).
Reviewed step-by-step derivation
Graph isomorphism · qf-mod26-graphiso-qualification · candidate
66. Conjoin Bijectivity with the Complete Adjacency-Iff Audit
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], [3, 5], [2, 5], [0, 2]] and complete vertex map=[0, 1, 4, 3, 5, 2] 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 (n,mapping,bijective?,bad [a,b,source adjacent?,mapped target adjacent?] records,isomorphism?).
View answer and explanation
(6, [0, 1, 4, 3, 5, 2], Yes, [[0, 5, No, Yes]], No)
Evaluate the two obligations independently, retaining bad-pair records even if bijectivity fails. Only their conjunction certifies this declared map as an isomorphism. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [0, 1, 4, 3, 5, 2], Yes, [[0, 5, No, Yes]], No).
Reviewed step-by-step derivation
Graph isomorphism · qf-mod26-graphiso-qualification · candidate
67. Conjoin Bijectivity with the Complete Adjacency-Iff Audit
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], [4, 5], [2, 5], [2, 3]] and complete vertex map=[0, 1, 4, 5, 2, 3] 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 (n,mapping,bijective?,bad [a,b,source adjacent?,mapped target adjacent?] records,isomorphism?).
View answer and explanation
(6, [0, 1, 4, 5, 2, 3], Yes, [], Yes)
Evaluate the two obligations independently, retaining bad-pair records even if bijectivity fails. Only their conjunction certifies this declared map as an isomorphism. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [0, 1, 4, 5, 2, 3], Yes, [], Yes).
Reviewed step-by-step derivation
Graph isomorphism · qf-mod26-graphiso-qualification · candidate
68. Conjoin Bijectivity with the Complete Adjacency-Iff Audit
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], [4, 5], [3, 5], [2, 3]] and complete vertex map=[0, 1, 4, 5, 3, 2] 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 (n,mapping,bijective?,bad [a,b,source adjacent?,mapped target adjacent?] records,isomorphism?).
View answer and explanation
(6, [0, 1, 4, 5, 3, 2], Yes, [], Yes)
Evaluate the two obligations independently, retaining bad-pair records even if bijectivity fails. Only their conjunction certifies this declared map as an isomorphism. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [0, 1, 4, 5, 3, 2], Yes, [], Yes).
Reviewed step-by-step derivation
Graph isomorphism · qf-mod26-graphiso-qualification · candidate
69. Conjoin Bijectivity with the Complete Adjacency-Iff Audit
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, 5], [2, 5], [2, 3], [3, 4]] and complete vertex map=[0, 1, 5, 2, 3, 4] 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 (n,mapping,bijective?,bad [a,b,source adjacent?,mapped target adjacent?] records,isomorphism?).
View answer and explanation
(6, [0, 1, 5, 2, 3, 4], Yes, [], Yes)
Evaluate the two obligations independently, retaining bad-pair records even if bijectivity fails. Only their conjunction certifies this declared map as an isomorphism. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [0, 1, 5, 2, 3, 4], Yes, [], Yes).
Reviewed step-by-step derivation
Graph isomorphism · qf-mod26-graphiso-qualification · candidate
70. Conjoin Bijectivity with the Complete Adjacency-Iff Audit
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, 5], [2, 5], [2, 4], [3, 4]] and complete vertex map=[0, 1, 5, 2, 4, 3] 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 (n,mapping,bijective?,bad [a,b,source adjacent?,mapped target adjacent?] records,isomorphism?).
View answer and explanation
(6, [0, 1, 5, 2, 4, 3], Yes, [], Yes)
Evaluate the two obligations independently, retaining bad-pair records even if bijectivity fails. Only their conjunction certifies this declared map as an isomorphism. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [0, 1, 5, 2, 4, 3], Yes, [], Yes).
Reviewed step-by-step derivation
Graph isomorphism · qf-mod26-graphiso-qualification · candidate
71. Conjoin Bijectivity with the Complete Adjacency-Iff Audit
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, 5], [3, 5], [2, 3], [2, 4], [0, 2]] and complete vertex map=[0, 1, 5, 3, 2, 4] 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 (n,mapping,bijective?,bad [a,b,source adjacent?,mapped target adjacent?] records,isomorphism?).
View answer and explanation
(6, [0, 1, 5, 3, 2, 4], Yes, [[0, 4, No, Yes]], No)
Evaluate the two obligations independently, retaining bad-pair records even if bijectivity fails. Only their conjunction certifies this declared map as an isomorphism. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [0, 1, 5, 3, 2, 4], Yes, [[0, 4, No, Yes]], No).
Reviewed step-by-step derivation
Graph isomorphism · qf-mod26-graphiso-qualification · candidate
72. Conjoin Bijectivity with the Complete Adjacency-Iff Audit
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, 5], [3, 5], [3, 4], [2, 4]] and complete vertex map=[0, 1, 5, 3, 4, 0] 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 (n,mapping,bijective?,bad [a,b,source adjacent?,mapped target adjacent?] records,isomorphism?).
View answer and explanation
(6, [0, 1, 5, 3, 4, 0], No, [[1, 5, No, Yes], [4, 5, Yes, No]], No)
Evaluate the two obligations independently, retaining bad-pair records even if bijectivity fails. Only their conjunction certifies this declared map as an isomorphism. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [0, 1, 5, 3, 4, 0], No, [[1, 5, No, Yes], [4, 5, Yes, No]], No).
Reviewed step-by-step derivation
Graph isomorphism · qf-mod26-graphiso-qualification · candidate
73. Conjoin Bijectivity with the Complete Adjacency-Iff Audit
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, 5], [4, 5], [2, 4], [2, 3]] and complete vertex map=[0, 1, 5, 4, 2, 3] 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 (n,mapping,bijective?,bad [a,b,source adjacent?,mapped target adjacent?] records,isomorphism?).
View answer and explanation
(6, [0, 1, 5, 4, 2, 3], Yes, [], Yes)
Evaluate the two obligations independently, retaining bad-pair records even if bijectivity fails. Only their conjunction certifies this declared map as an isomorphism. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [0, 1, 5, 4, 2, 3], Yes, [], Yes).
Reviewed step-by-step derivation
Graph isomorphism · qf-mod26-graphiso-qualification · candidate
74. Conjoin Bijectivity with the Complete Adjacency-Iff Audit
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, 5], [4, 5], [3, 4], [2, 3]] and complete vertex map=[0, 1, 5, 4, 3, 2] 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 (n,mapping,bijective?,bad [a,b,source adjacent?,mapped target adjacent?] records,isomorphism?).
View answer and explanation
(6, [0, 1, 5, 4, 3, 2], Yes, [], Yes)
Evaluate the two obligations independently, retaining bad-pair records even if bijectivity fails. Only their conjunction certifies this declared map as an isomorphism. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [0, 1, 5, 4, 3, 2], Yes, [], Yes).
Reviewed step-by-step derivation
Graph isomorphism · qf-mod26-graphiso-qualification · candidate
75. Conjoin Bijectivity with the Complete Adjacency-Iff Audit
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, 2], [1, 2], [1, 3], [3, 4], [4, 5]] and complete vertex map=[0, 2, 1, 3, 4, 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 (n,mapping,bijective?,bad [a,b,source adjacent?,mapped target adjacent?] records,isomorphism?).
View answer and explanation
(6, [0, 2, 1, 3, 4, 5], Yes, [], Yes)
Evaluate the two obligations independently, retaining bad-pair records even if bijectivity fails. Only their conjunction certifies this declared map as an isomorphism. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [0, 2, 1, 3, 4, 5], Yes, [], Yes).
Reviewed step-by-step derivation
Graph isomorphism · qf-mod26-graphiso-qualification · candidate
76. Invariant Mismatch Can Refute; a Match Does Not Prove
Use complete simple undirected source graph with left_n=6, edges=[[0, 1]], and target with right_n=6, edges=[[0, 1], [0, 3], [1, 2]]. 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.
View answer and explanation
(6, [0, 0, 0, 0, 1, 1], 6, [0, 0, 1, 1, 2, 2], Yes, nonisomorphic_certificate)
Different vertex counts or degree multisets refute isomorphism. Equal counts and multisets are necessary conditions only; no mapping has been supplied or checked. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [0, 0, 0, 0, 1, 1], 6, [0, 0, 1, 1, 2, 2], Yes, nonisomorphic_certificate).
Reviewed step-by-step derivation
Graph isomorphism · qf-mod26-graphiso-invariant-mismatch · candidate
77. Invariant Mismatch Can Refute; a Match Does Not Prove
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.
View answer and explanation
(6, [0, 0, 0, 0, 1, 1], 6, [0, 1, 1, 1, 1, 2], Yes, nonisomorphic_certificate)
Different vertex counts or degree multisets refute isomorphism. Equal counts and multisets are necessary conditions only; no mapping has been supplied or checked. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [0, 0, 0, 0, 1, 1], 6, [0, 1, 1, 1, 1, 2], Yes, nonisomorphic_certificate).
Reviewed step-by-step derivation
Graph isomorphism · qf-mod26-graphiso-invariant-mismatch · candidate
78. Invariant Mismatch Can Refute; a Match Does Not Prove
Use complete simple undirected source graph with left_n=6, edges=[[0, 1], [0, 2]], and target with right_n=6, edges=[[0, 1], [0, 2], [0, 3], [0, 4], [1, 2], [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.
View answer and explanation
(6, [0, 0, 0, 1, 1, 2], 6, [0, 1, 2, 2, 3, 4], Yes, nonisomorphic_certificate)
Different vertex counts or degree multisets refute isomorphism. Equal counts and multisets are necessary conditions only; no mapping has been supplied or checked. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [0, 0, 0, 1, 1, 2], 6, [0, 1, 2, 2, 3, 4], Yes, nonisomorphic_certificate).
Reviewed step-by-step derivation
Graph isomorphism · qf-mod26-graphiso-invariant-mismatch · candidate
79. Invariant Mismatch Can Refute; a Match Does Not Prove
Use complete simple undirected source graph with left_n=6, edges=[[0, 3]], and target with right_n=6, edges=[[0, 3], [0, 5], [1, 4]]. 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.
View answer and explanation
(6, [0, 0, 0, 0, 1, 1], 6, [0, 1, 1, 1, 1, 2], Yes, nonisomorphic_certificate)
Different vertex counts or degree multisets refute isomorphism. Equal counts and multisets are necessary conditions only; no mapping has been supplied or checked. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [0, 0, 0, 0, 1, 1], 6, [0, 1, 1, 1, 1, 2], Yes, nonisomorphic_certificate).
Reviewed step-by-step derivation
Graph isomorphism · qf-mod26-graphiso-invariant-mismatch · candidate
80. Invariant Mismatch Can Refute; a Match Does Not Prove
Use complete simple undirected source graph with left_n=6, edges=[[0, 1], [0, 3]], and target with right_n=6, edges=[[0, 1], [0, 4], [0, 5], [1, 2], [1, 4]]. 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.
View answer and explanation
(6, [0, 0, 0, 1, 1, 2], 6, [0, 1, 1, 2, 3, 3], Yes, nonisomorphic_certificate)
Different vertex counts or degree multisets refute isomorphism. Equal counts and multisets are necessary conditions only; no mapping has been supplied or checked. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [0, 0, 0, 1, 1, 2], 6, [0, 1, 1, 2, 3, 3], Yes, nonisomorphic_certificate).
Reviewed step-by-step derivation
Graph isomorphism · qf-mod26-graphiso-invariant-mismatch · candidate
81. Invariant Mismatch Can Refute; a Match Does Not Prove
Use complete simple undirected source graph with left_n=6, edges=[[0, 2], [0, 3]], and target with right_n=6, edges=[[0, 2], [0, 3], [0, 4], [0, 5], [1, 3], [1, 4]]. 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.
View answer and explanation
(6, [0, 0, 0, 1, 1, 2], 6, [1, 1, 2, 2, 2, 4], Yes, nonisomorphic_certificate)
Different vertex counts or degree multisets refute isomorphism. Equal counts and multisets are necessary conditions only; no mapping has been supplied or checked. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [0, 0, 0, 1, 1, 2], 6, [1, 1, 2, 2, 2, 4], Yes, nonisomorphic_certificate).
Reviewed step-by-step derivation
Graph isomorphism · qf-mod26-graphiso-invariant-mismatch · candidate
82. Invariant Mismatch Can Refute; a Match Does Not Prove
Use complete simple undirected source graph with left_n=6, edges=[[0, 1], [0, 2], [0, 3]], and target with right_n=6, edges=[[0, 1], [0, 2], [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.
View answer and explanation
(6, [0, 0, 1, 1, 1, 3], 6, [0, 0, 1, 1, 2, 2], Yes, nonisomorphic_certificate)
Different vertex counts or degree multisets refute isomorphism. Equal counts and multisets are necessary conditions only; no mapping has been supplied or checked. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [0, 0, 1, 1, 1, 3], 6, [0, 0, 1, 1, 2, 2], Yes, nonisomorphic_certificate).
Reviewed step-by-step derivation
Graph isomorphism · qf-mod26-graphiso-invariant-mismatch · candidate
83. Invariant Mismatch Can Refute; a Match Does Not Prove
Use complete simple undirected source graph with left_n=6, edges=[[0, 4]], and target with right_n=6, edges=[[0, 4], [1, 2], [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.
View answer and explanation
(6, [0, 0, 0, 0, 1, 1], 6, [0, 1, 1, 1, 1, 2], Yes, nonisomorphic_certificate)
Different vertex counts or degree multisets refute isomorphism. Equal counts and multisets are necessary conditions only; no mapping has been supplied or checked. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [0, 0, 0, 0, 1, 1], 6, [0, 1, 1, 1, 1, 2], Yes, nonisomorphic_certificate).
Reviewed step-by-step derivation
Graph isomorphism · qf-mod26-graphiso-invariant-mismatch · candidate
84. Invariant Mismatch Can Refute; a Match Does Not Prove
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.
View answer and explanation
(6, [0, 0, 0, 1, 1, 2], 6, [0, 1, 1, 2, 3, 3], Yes, nonisomorphic_certificate)
Different vertex counts or degree multisets refute isomorphism. Equal counts and multisets are necessary conditions only; no mapping has been supplied or checked. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [0, 0, 0, 1, 1, 2], 6, [0, 1, 1, 2, 3, 3], Yes, nonisomorphic_certificate).
Reviewed step-by-step derivation
Graph isomorphism · qf-mod26-graphiso-invariant-mismatch · candidate
85. Invariant Mismatch Can Refute; a Match Does Not Prove
Use complete simple undirected source graph with left_n=6, edges=[[0, 2], [0, 4]], and target with right_n=6, edges=[[0, 2], [0, 5], [1, 2], [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.
View answer and explanation
(6, [0, 0, 0, 1, 1, 2], 6, [0, 1, 2, 2, 2, 3], Yes, nonisomorphic_certificate)
Different vertex counts or degree multisets refute isomorphism. Equal counts and multisets are necessary conditions only; no mapping has been supplied or checked. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [0, 0, 0, 1, 1, 2], 6, [0, 1, 2, 2, 2, 3], Yes, nonisomorphic_certificate).
Reviewed step-by-step derivation
Graph isomorphism · qf-mod26-graphiso-invariant-mismatch · candidate
86. Invariant Mismatch Can Refute; a Match Does Not Prove
Use complete simple undirected source graph with left_n=6, edges=[[0, 1], [0, 2], [0, 4]], and target with right_n=6, edges=[[0, 1], [0, 2], [0, 3], [0, 5], [1, 4], [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.
View answer and explanation
(6, [0, 0, 1, 1, 1, 3], 6, [1, 1, 1, 2, 3, 4], Yes, nonisomorphic_certificate)
Different vertex counts or degree multisets refute isomorphism. Equal counts and multisets are necessary conditions only; no mapping has been supplied or checked. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [0, 0, 1, 1, 1, 3], 6, [1, 1, 1, 2, 3, 4], Yes, nonisomorphic_certificate).
Reviewed step-by-step derivation
Graph isomorphism · qf-mod26-graphiso-invariant-mismatch · candidate
87. Invariant Mismatch Can Refute; a Match Does Not Prove
Use complete simple undirected source graph with left_n=6, edges=[[0, 3], [0, 4]], and target with right_n=6, edges=[[0, 3], [0, 4], [0, 5], [1, 2], [1, 4], [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.
View answer and explanation
(6, [0, 0, 0, 1, 1, 2], 6, [1, 1, 2, 2, 3, 3], Yes, nonisomorphic_certificate)
Different vertex counts or degree multisets refute isomorphism. Equal counts and multisets are necessary conditions only; no mapping has been supplied or checked. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [0, 0, 0, 1, 1, 2], 6, [1, 1, 2, 2, 3, 3], Yes, nonisomorphic_certificate).
Reviewed step-by-step derivation
Graph isomorphism · qf-mod26-graphiso-invariant-mismatch · candidate
88. Invariant Mismatch Can Refute; a Match Does Not Prove
Use complete simple undirected source graph with left_n=6, edges=[[0, 1], [0, 3], [0, 4]], and target with right_n=6, edges=[[0, 1], [1, 2], [1, 3], [1, 4], [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.
View answer and explanation
(6, [0, 0, 1, 1, 1, 3], 6, [1, 1, 1, 1, 1, 5], Yes, nonisomorphic_certificate)
Different vertex counts or degree multisets refute isomorphism. Equal counts and multisets are necessary conditions only; no mapping has been supplied or checked. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [0, 0, 1, 1, 1, 3], 6, [1, 1, 1, 1, 1, 5], Yes, nonisomorphic_certificate).
Reviewed step-by-step derivation
Graph isomorphism · qf-mod26-graphiso-invariant-mismatch · candidate
89. Invariant Mismatch Can Refute; a Match Does Not Prove
Use complete simple undirected source graph with left_n=6, edges=[[0, 2], [0, 3], [0, 4]], and target with right_n=6, edges=[[0, 2], [0, 3], [2, 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.
View answer and explanation
(6, [0, 0, 1, 1, 1, 3], 6, [0, 0, 0, 2, 2, 2], Yes, nonisomorphic_certificate)
Different vertex counts or degree multisets refute isomorphism. Equal counts and multisets are necessary conditions only; no mapping has been supplied or checked. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [0, 0, 1, 1, 1, 3], 6, [0, 0, 0, 2, 2, 2], Yes, nonisomorphic_certificate).
Reviewed step-by-step derivation
Graph isomorphism · qf-mod26-graphiso-invariant-mismatch · candidate
90. Invariant Mismatch Can Refute; a Match Does Not Prove
Use complete simple undirected source graph with left_n=6, edges=[[0, 1], [0, 2], [0, 3], [0, 4]], and target with right_n=6, edges=[[0, 1], [0, 2], [0, 4], [1, 2], [2, 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.
View answer and explanation
(6, [0, 1, 1, 1, 1, 4], 6, [0, 1, 1, 2, 3, 3], Yes, nonisomorphic_certificate)
Different vertex counts or degree multisets refute isomorphism. Equal counts and multisets are necessary conditions only; no mapping has been supplied or checked. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [0, 1, 1, 1, 1, 4], 6, [0, 1, 1, 2, 3, 3], Yes, nonisomorphic_certificate).
Reviewed step-by-step derivation
Graph isomorphism · qf-mod26-graphiso-invariant-mismatch · candidate
91. Invariant Mismatch Can Refute; a Match Does Not Prove
Use complete simple undirected source graph with left_n=6, edges=[[0, 5]], and target with right_n=6, edges=[[0, 5], [1, 3], [2, 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.
View answer and explanation
(6, [0, 0, 0, 0, 1, 1], 6, [0, 1, 1, 1, 1, 2], Yes, nonisomorphic_certificate)
Different vertex counts or degree multisets refute isomorphism. Equal counts and multisets are necessary conditions only; no mapping has been supplied or checked. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [0, 0, 0, 0, 1, 1], 6, [0, 1, 1, 1, 1, 2], Yes, nonisomorphic_certificate).
Reviewed step-by-step derivation
Graph isomorphism · qf-mod26-graphiso-invariant-mismatch · candidate
92. Invariant Mismatch Can Refute; a Match Does Not Prove
Use complete simple undirected source graph with left_n=6, edges=[[0, 1], [0, 5]], and target with right_n=6, edges=[[0, 1], [0, 3], [0, 5], [1, 2], [1, 3], [2, 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.
View answer and explanation
(6, [0, 0, 0, 1, 1, 2], 6, [0, 1, 2, 3, 3, 3], Yes, nonisomorphic_certificate)
Different vertex counts or degree multisets refute isomorphism. Equal counts and multisets are necessary conditions only; no mapping has been supplied or checked. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [0, 0, 0, 1, 1, 2], 6, [0, 1, 2, 3, 3, 3], Yes, nonisomorphic_certificate).
Reviewed step-by-step derivation
Graph isomorphism · qf-mod26-graphiso-invariant-mismatch · candidate
93. Invariant Mismatch Can Refute; a Match Does Not Prove
Use complete simple undirected source graph with left_n=6, edges=[[0, 2], [0, 5]], and target with right_n=6, edges=[[0, 2], [0, 4], [0, 5], [1, 4], [2, 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.
View answer and explanation
(6, [0, 0, 0, 1, 1, 2], 6, [1, 1, 1, 2, 2, 3], Yes, nonisomorphic_certificate)
Different vertex counts or degree multisets refute isomorphism. Equal counts and multisets are necessary conditions only; no mapping has been supplied or checked. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [0, 0, 0, 1, 1, 2], 6, [1, 1, 1, 2, 2, 3], Yes, nonisomorphic_certificate).
Reviewed step-by-step derivation
Graph isomorphism · qf-mod26-graphiso-invariant-mismatch · candidate
94. Invariant Mismatch Can Refute; a Match Does Not Prove
Use complete simple undirected source graph with left_n=6, edges=[[0, 1], [0, 2], [0, 5]], and target with right_n=6, edges=[[0, 1], [0, 2], [0, 3], [0, 4], [0, 5], [1, 2], [1, 4], [2, 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.
View answer and explanation
(6, [0, 0, 1, 1, 1, 3], 6, [1, 2, 2, 3, 3, 5], Yes, nonisomorphic_certificate)
Different vertex counts or degree multisets refute isomorphism. Equal counts and multisets are necessary conditions only; no mapping has been supplied or checked. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [0, 0, 1, 1, 1, 3], 6, [1, 2, 2, 3, 3, 5], Yes, nonisomorphic_certificate).
Reviewed step-by-step derivation
Graph isomorphism · qf-mod26-graphiso-invariant-mismatch · candidate
95. Invariant Mismatch Can Refute; a Match Does Not Prove
Use complete simple undirected source graph with left_n=6, edges=[[0, 3], [0, 5]], and target with right_n=6, edges=[[0, 3], [1, 2], [1, 3], [1, 4], [2, 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.
View answer and explanation
(6, [0, 0, 0, 1, 1, 2], 6, [0, 1, 1, 2, 3, 3], Yes, nonisomorphic_certificate)
Different vertex counts or degree multisets refute isomorphism. Equal counts and multisets are necessary conditions only; no mapping has been supplied or checked. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [0, 0, 0, 1, 1, 2], 6, [0, 1, 1, 2, 3, 3], Yes, nonisomorphic_certificate).
Reviewed step-by-step derivation
Graph isomorphism · qf-mod26-graphiso-invariant-mismatch · candidate
96. Invariant Mismatch Can Refute; a Match Does Not Prove
Use complete simple undirected source graph with left_n=6, edges=[[0, 1], [0, 3], [0, 5]], and target with right_n=6, edges=[[0, 1], [0, 4], [1, 5], [2, 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.
View answer and explanation
(6, [0, 0, 1, 1, 1, 3], 6, [1, 1, 1, 1, 2, 2], Yes, nonisomorphic_certificate)
Different vertex counts or degree multisets refute isomorphism. Equal counts and multisets are necessary conditions only; no mapping has been supplied or checked. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [0, 0, 1, 1, 1, 3], 6, [1, 1, 1, 1, 2, 2], Yes, nonisomorphic_certificate).
Reviewed step-by-step derivation
Graph isomorphism · qf-mod26-graphiso-invariant-mismatch · candidate
97. Invariant Mismatch Can Refute; a Match Does Not Prove
Use complete simple undirected source graph with left_n=6, edges=[[0, 2], [0, 3], [0, 5]], and target with right_n=6, edges=[[0, 2], [0, 3], [0, 4], [1, 2], [1, 5], [2, 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.
View answer and explanation
(6, [0, 0, 1, 1, 1, 3], 6, [1, 1, 2, 2, 3, 3], Yes, nonisomorphic_certificate)
Different vertex counts or degree multisets refute isomorphism. Equal counts and multisets are necessary conditions only; no mapping has been supplied or checked. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [0, 0, 1, 1, 1, 3], 6, [1, 1, 2, 2, 3, 3], Yes, nonisomorphic_certificate).
Reviewed step-by-step derivation
Graph isomorphism · qf-mod26-graphiso-invariant-mismatch · candidate
98. Invariant Mismatch Can Refute; a Match Does Not Prove
Use complete simple undirected source graph with left_n=6, edges=[[0, 1], [0, 2], [0, 3], [0, 5]], and target with right_n=6, edges=[[0, 1], [0, 2], [0, 5], [1, 3], [1, 5], [2, 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.
View answer and explanation
(6, [0, 1, 1, 1, 1, 4], 6, [0, 2, 2, 2, 3, 3], Yes, nonisomorphic_certificate)
Different vertex counts or degree multisets refute isomorphism. Equal counts and multisets are necessary conditions only; no mapping has been supplied or checked. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [0, 1, 1, 1, 1, 4], 6, [0, 2, 2, 2, 3, 3], Yes, nonisomorphic_certificate).
Reviewed step-by-step derivation
Graph isomorphism · qf-mod26-graphiso-invariant-mismatch · candidate
99. Invariant Mismatch Can Refute; a Match Does Not Prove
Use complete simple undirected source graph with left_n=6, edges=[[0, 4], [0, 5]], and target with right_n=6, edges=[[0, 4], [0, 5], [1, 2], [1, 3], [1, 5], [2, 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.
View answer and explanation
(6, [0, 0, 0, 1, 1, 2], 6, [1, 2, 2, 2, 2, 3], Yes, nonisomorphic_certificate)
Different vertex counts or degree multisets refute isomorphism. Equal counts and multisets are necessary conditions only; no mapping has been supplied or checked. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [0, 0, 0, 1, 1, 2], 6, [1, 2, 2, 2, 2, 3], Yes, nonisomorphic_certificate).
Reviewed step-by-step derivation
Graph isomorphism · qf-mod26-graphiso-invariant-mismatch · candidate
100. Invariant Mismatch Can Refute; a Match Does Not Prove
Use complete simple undirected source graph with left_n=6, edges=[[0, 1], [0, 4], [0, 5]], and target with right_n=6, edges=[[0, 1], [0, 3], [0, 4], [0, 5], [1, 4], [1, 5], [2, 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.
View answer and explanation
(6, [0, 0, 1, 1, 1, 3], 6, [1, 2, 2, 2, 3, 4], Yes, nonisomorphic_certificate)
Different vertex counts or degree multisets refute isomorphism. Equal counts and multisets are necessary conditions only; no mapping has been supplied or checked. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [0, 0, 1, 1, 1, 3], 6, [1, 2, 2, 2, 3, 4], Yes, nonisomorphic_certificate).
Reviewed step-by-step derivation
Graph isomorphism · qf-mod26-graphiso-invariant-mismatch · candidate
Canonical question JSONL · English JSONL · Validation report