# Complete Vertex Bijections, Adjacency Iff and Invariant Boundaries

This batch has passed Schema, deduplication, source-snapshot, and independent-answer recomputation checks; it is still not marked as human-reviewed.

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r06-graph-isomorphism-0001/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r06-graph-isomorphism-0002/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r06-graph-isomorphism-0003/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r06-graph-isomorphism-0004/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r06-graph-isomorphism-0005/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r06-graph-isomorphism-0006/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r06-graph-isomorphism-0007/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r06-graph-isomorphism-0008/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r06-graph-isomorphism-0009/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r06-graph-isomorphism-0010/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r06-graph-isomorphism-0011/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r06-graph-isomorphism-0012/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r06-graph-isomorphism-0013/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r06-graph-isomorphism-0014/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r06-graph-isomorphism-0015/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r06-graph-isomorphism-0016/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r06-graph-isomorphism-0017/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r06-graph-isomorphism-0018/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r06-graph-isomorphism-0019/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r06-graph-isomorphism-0020/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r06-graph-isomorphism-0021/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r06-graph-isomorphism-0022/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r06-graph-isomorphism-0023/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r06-graph-isomorphism-0024/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r06-graph-isomorphism-0025/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r06-graph-isomorphism-0026/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r06-graph-isomorphism-0027/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r06-graph-isomorphism-0028/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r06-graph-isomorphism-0029/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r06-graph-isomorphism-0030/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r06-graph-isomorphism-0031/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r06-graph-isomorphism-0032/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r06-graph-isomorphism-0033/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r06-graph-isomorphism-0034/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r06-graph-isomorphism-0035/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r06-graph-isomorphism-0036/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r06-graph-isomorphism-0037/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r06-graph-isomorphism-0038/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r06-graph-isomorphism-0039/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r06-graph-isomorphism-0040/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r06-graph-isomorphism-0041/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r06-graph-isomorphism-0042/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r06-graph-isomorphism-0043/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r06-graph-isomorphism-0044/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r06-graph-isomorphism-0045/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r06-graph-isomorphism-0046/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r06-graph-isomorphism-0047/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r06-graph-isomorphism-0048/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r06-graph-isomorphism-0049/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r06-graph-isomorphism-0050/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r06-graph-isomorphism-0051/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r06-graph-isomorphism-0052/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r06-graph-isomorphism-0053/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r06-graph-isomorphism-0054/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r06-graph-isomorphism-0055/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r06-graph-isomorphism-0056/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r06-graph-isomorphism-0057/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r06-graph-isomorphism-0058/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r06-graph-isomorphism-0059/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r06-graph-isomorphism-0060/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r06-graph-isomorphism-0061/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r06-graph-isomorphism-0062/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r06-graph-isomorphism-0063/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r06-graph-isomorphism-0064/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r06-graph-isomorphism-0065/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r06-graph-isomorphism-0066/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r06-graph-isomorphism-0067/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r06-graph-isomorphism-0068/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r06-graph-isomorphism-0069/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r06-graph-isomorphism-0070/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r06-graph-isomorphism-0071/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r06-graph-isomorphism-0072/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r06-graph-isomorphism-0073/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r06-graph-isomorphism-0074/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r06-graph-isomorphism-0075/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r06-graph-isomorphism-0076/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r06-graph-isomorphism-0077/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r06-graph-isomorphism-0078/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r06-graph-isomorphism-0079/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r06-graph-isomorphism-0080/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r06-graph-isomorphism-0081/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r06-graph-isomorphism-0082/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r06-graph-isomorphism-0083/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r06-graph-isomorphism-0084/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r06-graph-isomorphism-0085/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r06-graph-isomorphism-0086/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r06-graph-isomorphism-0087/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r06-graph-isomorphism-0088/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r06-graph-isomorphism-0089/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r06-graph-isomorphism-0090/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r06-graph-isomorphism-0091/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r06-graph-isomorphism-0092/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r06-graph-isomorphism-0093/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r06-graph-isomorphism-0094/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r06-graph-isomorphism-0095/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r06-graph-isomorphism-0096/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r06-graph-isomorphism-0097/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r06-graph-isomorphism-0098/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r06-graph-isomorphism-0099/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r06-graph-isomorphism-0100/)

Graph isomorphism · qf-mod26-graphiso-invariant-mismatch · candidate



[Canonical question JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20261013-advance-20261026-r06-graph-isomorphism-foundations/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20261013-advance-20261026-r06-graph-isomorphism-foundations.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/batches/batch-20261013-advance-20261026-r06-graph-isomorphism-foundations/validation-report.json)