# Oriented Edge Sets, Separate Degree Totals and Loop-Free Boundaries

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This question bank has corresponding bilingual step-by-step derivations. Source candidate and English ai_candidate states are retained. AI review, finite computation and Lean formal proof are labelled separately.

batch-20261014-advance-20261108-r07-finite-directed-graph-foundations · 2026-10-14 · 100 questions

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.

Field order uses a source-bound presentation supplement; canonical JSONL and mathematical answers are unchanged.

[Field-order and source-binding data](https://ocme.evemisslab.com/data/i18n/question-output-protocols.json)



## 1. A Directed Edge Is an Ordered Pair under the Declared Simple Convention

Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 0]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Give (vertex_count, edges, loop_free, simple_directed_set_qualified, convention).

View answer and explanation(5, [[0, 1], [1, 2], [2, 3], [3, 4], [4, 0]], Yes, Yes, ordered_pairs_and_no_parallel_duplicate_edges)

Validate every endpoint, canonical edge order, uniqueness and absence of loops. Opposite oriented edges may both be present and remain distinct; no undirected connectivity or degree convention is inferred. The exact result, retaining the requested order, is (5, [[0, 1], [1, 2], [2, 3], [3, 4], [4, 0]], Yes, Yes, ordered_pairs_and_no_parallel_duplicate_edges).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r07-finite-directed-graph-0001/)

Finite directed graph · qf-mod39-digraph-qualification · candidate





## 2. A Directed Edge Is an Ordered Pair under the Declared Simple Convention

Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 1], [0, 2], [1, 2], [2, 3], [3, 4], [4, 0]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Give (vertex_count, edges, loop_free, simple_directed_set_qualified, convention).

View answer and explanation(5, [[0, 1], [0, 2], [1, 2], [2, 3], [3, 4], [4, 0]], Yes, Yes, ordered_pairs_and_no_parallel_duplicate_edges)

Validate every endpoint, canonical edge order, uniqueness and absence of loops. Opposite oriented edges may both be present and remain distinct; no undirected connectivity or degree convention is inferred. The exact result, retaining the requested order, is (5, [[0, 1], [0, 2], [1, 2], [2, 3], [3, 4], [4, 0]], Yes, Yes, ordered_pairs_and_no_parallel_duplicate_edges).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r07-finite-directed-graph-0002/)

Finite directed graph · qf-mod39-digraph-qualification · candidate





## 3. A Directed Edge Is an Ordered Pair under the Declared Simple Convention

Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 1], [1, 2], [1, 3], [2, 3], [3, 4], [4, 0]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Give (vertex_count, edges, loop_free, simple_directed_set_qualified, convention).

View answer and explanation(5, [[0, 1], [1, 2], [1, 3], [2, 3], [3, 4], [4, 0]], Yes, Yes, ordered_pairs_and_no_parallel_duplicate_edges)

Validate every endpoint, canonical edge order, uniqueness and absence of loops. Opposite oriented edges may both be present and remain distinct; no undirected connectivity or degree convention is inferred. The exact result, retaining the requested order, is (5, [[0, 1], [1, 2], [1, 3], [2, 3], [3, 4], [4, 0]], Yes, Yes, ordered_pairs_and_no_parallel_duplicate_edges).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r07-finite-directed-graph-0003/)

Finite directed graph · qf-mod39-digraph-qualification · candidate





## 4. A Directed Edge Is an Ordered Pair under the Declared Simple Convention

Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 1], [0, 2], [1, 2], [1, 3], [2, 3], [3, 4], [4, 0]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Give (vertex_count, edges, loop_free, simple_directed_set_qualified, convention).

View answer and explanation(5, [[0, 1], [0, 2], [1, 2], [1, 3], [2, 3], [3, 4], [4, 0]], Yes, Yes, ordered_pairs_and_no_parallel_duplicate_edges)

Validate every endpoint, canonical edge order, uniqueness and absence of loops. Opposite oriented edges may both be present and remain distinct; no undirected connectivity or degree convention is inferred. The exact result, retaining the requested order, is (5, [[0, 1], [0, 2], [1, 2], [1, 3], [2, 3], [3, 4], [4, 0]], Yes, Yes, ordered_pairs_and_no_parallel_duplicate_edges).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r07-finite-directed-graph-0004/)

Finite directed graph · qf-mod39-digraph-qualification · candidate





## 5. A Directed Edge Is an Ordered Pair under the Declared Simple Convention

Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 1], [1, 2], [2, 3], [2, 4], [3, 4], [4, 0]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Give (vertex_count, edges, loop_free, simple_directed_set_qualified, convention).

View answer and explanation(5, [[0, 1], [1, 2], [2, 3], [2, 4], [3, 4], [4, 0]], Yes, Yes, ordered_pairs_and_no_parallel_duplicate_edges)

Validate every endpoint, canonical edge order, uniqueness and absence of loops. Opposite oriented edges may both be present and remain distinct; no undirected connectivity or degree convention is inferred. The exact result, retaining the requested order, is (5, [[0, 1], [1, 2], [2, 3], [2, 4], [3, 4], [4, 0]], Yes, Yes, ordered_pairs_and_no_parallel_duplicate_edges).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r07-finite-directed-graph-0005/)

Finite directed graph · qf-mod39-digraph-qualification · candidate





## 6. A Directed Edge Is an Ordered Pair under the Declared Simple Convention

Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 1], [0, 2], [1, 2], [2, 3], [2, 4], [3, 4], [4, 0]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Give (vertex_count, edges, loop_free, simple_directed_set_qualified, convention).

View answer and explanation(5, [[0, 1], [0, 2], [1, 2], [2, 3], [2, 4], [3, 4], [4, 0]], Yes, Yes, ordered_pairs_and_no_parallel_duplicate_edges)

Validate every endpoint, canonical edge order, uniqueness and absence of loops. Opposite oriented edges may both be present and remain distinct; no undirected connectivity or degree convention is inferred. The exact result, retaining the requested order, is (5, [[0, 1], [0, 2], [1, 2], [2, 3], [2, 4], [3, 4], [4, 0]], Yes, Yes, ordered_pairs_and_no_parallel_duplicate_edges).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r07-finite-directed-graph-0006/)

Finite directed graph · qf-mod39-digraph-qualification · candidate





## 7. A Directed Edge Is an Ordered Pair under the Declared Simple Convention

Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 1], [1, 2], [1, 3], [2, 3], [2, 4], [3, 4], [4, 0]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Give (vertex_count, edges, loop_free, simple_directed_set_qualified, convention).

View answer and explanation(5, [[0, 1], [1, 2], [1, 3], [2, 3], [2, 4], [3, 4], [4, 0]], Yes, Yes, ordered_pairs_and_no_parallel_duplicate_edges)

Validate every endpoint, canonical edge order, uniqueness and absence of loops. Opposite oriented edges may both be present and remain distinct; no undirected connectivity or degree convention is inferred. The exact result, retaining the requested order, is (5, [[0, 1], [1, 2], [1, 3], [2, 3], [2, 4], [3, 4], [4, 0]], Yes, Yes, ordered_pairs_and_no_parallel_duplicate_edges).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r07-finite-directed-graph-0007/)

Finite directed graph · qf-mod39-digraph-qualification · candidate





## 8. A Directed Edge Is an Ordered Pair under the Declared Simple Convention

Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 1], [0, 2], [1, 2], [1, 3], [2, 3], [2, 4], [3, 4], [4, 0]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Give (vertex_count, edges, loop_free, simple_directed_set_qualified, convention).

View answer and explanation(5, [[0, 1], [0, 2], [1, 2], [1, 3], [2, 3], [2, 4], [3, 4], [4, 0]], Yes, Yes, ordered_pairs_and_no_parallel_duplicate_edges)

Validate every endpoint, canonical edge order, uniqueness and absence of loops. Opposite oriented edges may both be present and remain distinct; no undirected connectivity or degree convention is inferred. The exact result, retaining the requested order, is (5, [[0, 1], [0, 2], [1, 2], [1, 3], [2, 3], [2, 4], [3, 4], [4, 0]], Yes, Yes, ordered_pairs_and_no_parallel_duplicate_edges).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r07-finite-directed-graph-0008/)

Finite directed graph · qf-mod39-digraph-qualification · candidate





## 9. A Directed Edge Is an Ordered Pair under the Declared Simple Convention

Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 1], [1, 2], [2, 3], [3, 0], [3, 4], [4, 0]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Give (vertex_count, edges, loop_free, simple_directed_set_qualified, convention).

View answer and explanation(5, [[0, 1], [1, 2], [2, 3], [3, 0], [3, 4], [4, 0]], Yes, Yes, ordered_pairs_and_no_parallel_duplicate_edges)

Validate every endpoint, canonical edge order, uniqueness and absence of loops. Opposite oriented edges may both be present and remain distinct; no undirected connectivity or degree convention is inferred. The exact result, retaining the requested order, is (5, [[0, 1], [1, 2], [2, 3], [3, 0], [3, 4], [4, 0]], Yes, Yes, ordered_pairs_and_no_parallel_duplicate_edges).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r07-finite-directed-graph-0009/)

Finite directed graph · qf-mod39-digraph-qualification · candidate





## 10. A Directed Edge Is an Ordered Pair under the Declared Simple Convention

Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 1], [0, 2], [1, 2], [2, 3], [3, 0], [3, 4], [4, 0]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Give (vertex_count, edges, loop_free, simple_directed_set_qualified, convention).

View answer and explanation(5, [[0, 1], [0, 2], [1, 2], [2, 3], [3, 0], [3, 4], [4, 0]], Yes, Yes, ordered_pairs_and_no_parallel_duplicate_edges)

Validate every endpoint, canonical edge order, uniqueness and absence of loops. Opposite oriented edges may both be present and remain distinct; no undirected connectivity or degree convention is inferred. The exact result, retaining the requested order, is (5, [[0, 1], [0, 2], [1, 2], [2, 3], [3, 0], [3, 4], [4, 0]], Yes, Yes, ordered_pairs_and_no_parallel_duplicate_edges).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r07-finite-directed-graph-0010/)

Finite directed graph · qf-mod39-digraph-qualification · candidate





## 11. A Directed Edge Is an Ordered Pair under the Declared Simple Convention

Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 1], [1, 2], [1, 3], [2, 3], [3, 0], [3, 4], [4, 0]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Give (vertex_count, edges, loop_free, simple_directed_set_qualified, convention).

View answer and explanation(5, [[0, 1], [1, 2], [1, 3], [2, 3], [3, 0], [3, 4], [4, 0]], Yes, Yes, ordered_pairs_and_no_parallel_duplicate_edges)

Validate every endpoint, canonical edge order, uniqueness and absence of loops. Opposite oriented edges may both be present and remain distinct; no undirected connectivity or degree convention is inferred. The exact result, retaining the requested order, is (5, [[0, 1], [1, 2], [1, 3], [2, 3], [3, 0], [3, 4], [4, 0]], Yes, Yes, ordered_pairs_and_no_parallel_duplicate_edges).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r07-finite-directed-graph-0011/)

Finite directed graph · qf-mod39-digraph-qualification · candidate





## 12. A Directed Edge Is an Ordered Pair under the Declared Simple Convention

Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 1], [0, 2], [1, 2], [1, 3], [2, 3], [3, 0], [3, 4], [4, 0]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Give (vertex_count, edges, loop_free, simple_directed_set_qualified, convention).

View answer and explanation(5, [[0, 1], [0, 2], [1, 2], [1, 3], [2, 3], [3, 0], [3, 4], [4, 0]], Yes, Yes, ordered_pairs_and_no_parallel_duplicate_edges)

Validate every endpoint, canonical edge order, uniqueness and absence of loops. Opposite oriented edges may both be present and remain distinct; no undirected connectivity or degree convention is inferred. The exact result, retaining the requested order, is (5, [[0, 1], [0, 2], [1, 2], [1, 3], [2, 3], [3, 0], [3, 4], [4, 0]], Yes, Yes, ordered_pairs_and_no_parallel_duplicate_edges).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r07-finite-directed-graph-0012/)

Finite directed graph · qf-mod39-digraph-qualification · candidate





## 13. A Directed Edge Is an Ordered Pair under the Declared Simple Convention

Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 1], [1, 2], [2, 3], [2, 4], [3, 0], [3, 4], [4, 0]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Give (vertex_count, edges, loop_free, simple_directed_set_qualified, convention).

View answer and explanation(5, [[0, 1], [1, 2], [2, 3], [2, 4], [3, 0], [3, 4], [4, 0]], Yes, Yes, ordered_pairs_and_no_parallel_duplicate_edges)

Validate every endpoint, canonical edge order, uniqueness and absence of loops. Opposite oriented edges may both be present and remain distinct; no undirected connectivity or degree convention is inferred. The exact result, retaining the requested order, is (5, [[0, 1], [1, 2], [2, 3], [2, 4], [3, 0], [3, 4], [4, 0]], Yes, Yes, ordered_pairs_and_no_parallel_duplicate_edges).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r07-finite-directed-graph-0013/)

Finite directed graph · qf-mod39-digraph-qualification · candidate





## 14. A Directed Edge Is an Ordered Pair under the Declared Simple Convention

Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 1], [0, 2], [1, 2], [2, 3], [2, 4], [3, 0], [3, 4], [4, 0]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Give (vertex_count, edges, loop_free, simple_directed_set_qualified, convention).

View answer and explanation(5, [[0, 1], [0, 2], [1, 2], [2, 3], [2, 4], [3, 0], [3, 4], [4, 0]], Yes, Yes, ordered_pairs_and_no_parallel_duplicate_edges)

Validate every endpoint, canonical edge order, uniqueness and absence of loops. Opposite oriented edges may both be present and remain distinct; no undirected connectivity or degree convention is inferred. The exact result, retaining the requested order, is (5, [[0, 1], [0, 2], [1, 2], [2, 3], [2, 4], [3, 0], [3, 4], [4, 0]], Yes, Yes, ordered_pairs_and_no_parallel_duplicate_edges).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r07-finite-directed-graph-0014/)

Finite directed graph · qf-mod39-digraph-qualification · candidate





## 15. A Directed Edge Is an Ordered Pair under the Declared Simple Convention

Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 1], [1, 2], [1, 3], [2, 3], [2, 4], [3, 0], [3, 4], [4, 0]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Give (vertex_count, edges, loop_free, simple_directed_set_qualified, convention).

View answer and explanation(5, [[0, 1], [1, 2], [1, 3], [2, 3], [2, 4], [3, 0], [3, 4], [4, 0]], Yes, Yes, ordered_pairs_and_no_parallel_duplicate_edges)

Validate every endpoint, canonical edge order, uniqueness and absence of loops. Opposite oriented edges may both be present and remain distinct; no undirected connectivity or degree convention is inferred. The exact result, retaining the requested order, is (5, [[0, 1], [1, 2], [1, 3], [2, 3], [2, 4], [3, 0], [3, 4], [4, 0]], Yes, Yes, ordered_pairs_and_no_parallel_duplicate_edges).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r07-finite-directed-graph-0015/)

Finite directed graph · qf-mod39-digraph-qualification · candidate





## 16. A Directed Edge Is an Ordered Pair under the Declared Simple Convention

Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 1], [0, 2], [1, 2], [1, 3], [2, 3], [2, 4], [3, 0], [3, 4], [4, 0]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Give (vertex_count, edges, loop_free, simple_directed_set_qualified, convention).

View answer and explanation(5, [[0, 1], [0, 2], [1, 2], [1, 3], [2, 3], [2, 4], [3, 0], [3, 4], [4, 0]], Yes, Yes, ordered_pairs_and_no_parallel_duplicate_edges)

Validate every endpoint, canonical edge order, uniqueness and absence of loops. Opposite oriented edges may both be present and remain distinct; no undirected connectivity or degree convention is inferred. The exact result, retaining the requested order, is (5, [[0, 1], [0, 2], [1, 2], [1, 3], [2, 3], [2, 4], [3, 0], [3, 4], [4, 0]], Yes, Yes, ordered_pairs_and_no_parallel_duplicate_edges).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r07-finite-directed-graph-0016/)

Finite directed graph · qf-mod39-digraph-qualification · candidate





## 17. A Directed Edge Is an Ordered Pair under the Declared Simple Convention

Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 0], [4, 1]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Give (vertex_count, edges, loop_free, simple_directed_set_qualified, convention).

View answer and explanation(5, [[0, 1], [1, 2], [2, 3], [3, 4], [4, 0], [4, 1]], Yes, Yes, ordered_pairs_and_no_parallel_duplicate_edges)

Validate every endpoint, canonical edge order, uniqueness and absence of loops. Opposite oriented edges may both be present and remain distinct; no undirected connectivity or degree convention is inferred. The exact result, retaining the requested order, is (5, [[0, 1], [1, 2], [2, 3], [3, 4], [4, 0], [4, 1]], Yes, Yes, ordered_pairs_and_no_parallel_duplicate_edges).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r07-finite-directed-graph-0017/)

Finite directed graph · qf-mod39-digraph-qualification · candidate





## 18. A Directed Edge Is an Ordered Pair under the Declared Simple Convention

Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 1], [0, 2], [1, 2], [2, 3], [3, 4], [4, 0], [4, 1]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Give (vertex_count, edges, loop_free, simple_directed_set_qualified, convention).

View answer and explanation(5, [[0, 1], [0, 2], [1, 2], [2, 3], [3, 4], [4, 0], [4, 1]], Yes, Yes, ordered_pairs_and_no_parallel_duplicate_edges)

Validate every endpoint, canonical edge order, uniqueness and absence of loops. Opposite oriented edges may both be present and remain distinct; no undirected connectivity or degree convention is inferred. The exact result, retaining the requested order, is (5, [[0, 1], [0, 2], [1, 2], [2, 3], [3, 4], [4, 0], [4, 1]], Yes, Yes, ordered_pairs_and_no_parallel_duplicate_edges).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r07-finite-directed-graph-0018/)

Finite directed graph · qf-mod39-digraph-qualification · candidate





## 19. A Directed Edge Is an Ordered Pair under the Declared Simple Convention

Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 1], [1, 2], [1, 3], [2, 3], [3, 4], [4, 0], [4, 1]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Give (vertex_count, edges, loop_free, simple_directed_set_qualified, convention).

View answer and explanation(5, [[0, 1], [1, 2], [1, 3], [2, 3], [3, 4], [4, 0], [4, 1]], Yes, Yes, ordered_pairs_and_no_parallel_duplicate_edges)

Validate every endpoint, canonical edge order, uniqueness and absence of loops. Opposite oriented edges may both be present and remain distinct; no undirected connectivity or degree convention is inferred. The exact result, retaining the requested order, is (5, [[0, 1], [1, 2], [1, 3], [2, 3], [3, 4], [4, 0], [4, 1]], Yes, Yes, ordered_pairs_and_no_parallel_duplicate_edges).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r07-finite-directed-graph-0019/)

Finite directed graph · qf-mod39-digraph-qualification · candidate





## 20. A Directed Edge Is an Ordered Pair under the Declared Simple Convention

Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 1], [0, 2], [1, 2], [1, 3], [2, 3], [3, 4], [4, 0], [4, 1]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Give (vertex_count, edges, loop_free, simple_directed_set_qualified, convention).

View answer and explanation(5, [[0, 1], [0, 2], [1, 2], [1, 3], [2, 3], [3, 4], [4, 0], [4, 1]], Yes, Yes, ordered_pairs_and_no_parallel_duplicate_edges)

Validate every endpoint, canonical edge order, uniqueness and absence of loops. Opposite oriented edges may both be present and remain distinct; no undirected connectivity or degree convention is inferred. The exact result, retaining the requested order, is (5, [[0, 1], [0, 2], [1, 2], [1, 3], [2, 3], [3, 4], [4, 0], [4, 1]], Yes, Yes, ordered_pairs_and_no_parallel_duplicate_edges).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r07-finite-directed-graph-0020/)

Finite directed graph · qf-mod39-digraph-qualification · candidate





## 21. A Directed Edge Is an Ordered Pair under the Declared Simple Convention

Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 1], [1, 2], [2, 3], [2, 4], [3, 4], [4, 0], [4, 1]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Give (vertex_count, edges, loop_free, simple_directed_set_qualified, convention).

View answer and explanation(5, [[0, 1], [1, 2], [2, 3], [2, 4], [3, 4], [4, 0], [4, 1]], Yes, Yes, ordered_pairs_and_no_parallel_duplicate_edges)

Validate every endpoint, canonical edge order, uniqueness and absence of loops. Opposite oriented edges may both be present and remain distinct; no undirected connectivity or degree convention is inferred. The exact result, retaining the requested order, is (5, [[0, 1], [1, 2], [2, 3], [2, 4], [3, 4], [4, 0], [4, 1]], Yes, Yes, ordered_pairs_and_no_parallel_duplicate_edges).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r07-finite-directed-graph-0021/)

Finite directed graph · qf-mod39-digraph-qualification · candidate





## 22. A Directed Edge Is an Ordered Pair under the Declared Simple Convention

Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 1], [0, 2], [1, 2], [2, 3], [2, 4], [3, 4], [4, 0], [4, 1]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Give (vertex_count, edges, loop_free, simple_directed_set_qualified, convention).

View answer and explanation(5, [[0, 1], [0, 2], [1, 2], [2, 3], [2, 4], [3, 4], [4, 0], [4, 1]], Yes, Yes, ordered_pairs_and_no_parallel_duplicate_edges)

Validate every endpoint, canonical edge order, uniqueness and absence of loops. Opposite oriented edges may both be present and remain distinct; no undirected connectivity or degree convention is inferred. The exact result, retaining the requested order, is (5, [[0, 1], [0, 2], [1, 2], [2, 3], [2, 4], [3, 4], [4, 0], [4, 1]], Yes, Yes, ordered_pairs_and_no_parallel_duplicate_edges).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r07-finite-directed-graph-0022/)

Finite directed graph · qf-mod39-digraph-qualification · candidate





## 23. A Directed Edge Is an Ordered Pair under the Declared Simple Convention

Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 1], [1, 2], [1, 3], [2, 3], [2, 4], [3, 4], [4, 0], [4, 1]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Give (vertex_count, edges, loop_free, simple_directed_set_qualified, convention).

View answer and explanation(5, [[0, 1], [1, 2], [1, 3], [2, 3], [2, 4], [3, 4], [4, 0], [4, 1]], Yes, Yes, ordered_pairs_and_no_parallel_duplicate_edges)

Validate every endpoint, canonical edge order, uniqueness and absence of loops. Opposite oriented edges may both be present and remain distinct; no undirected connectivity or degree convention is inferred. The exact result, retaining the requested order, is (5, [[0, 1], [1, 2], [1, 3], [2, 3], [2, 4], [3, 4], [4, 0], [4, 1]], Yes, Yes, ordered_pairs_and_no_parallel_duplicate_edges).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r07-finite-directed-graph-0023/)

Finite directed graph · qf-mod39-digraph-qualification · candidate





## 24. A Directed Edge Is an Ordered Pair under the Declared Simple Convention

Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 1], [0, 2], [1, 2], [1, 3], [2, 3], [2, 4], [3, 4], [4, 0], [4, 1]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Give (vertex_count, edges, loop_free, simple_directed_set_qualified, convention).

View answer and explanation(5, [[0, 1], [0, 2], [1, 2], [1, 3], [2, 3], [2, 4], [3, 4], [4, 0], [4, 1]], Yes, Yes, ordered_pairs_and_no_parallel_duplicate_edges)

Validate every endpoint, canonical edge order, uniqueness and absence of loops. Opposite oriented edges may both be present and remain distinct; no undirected connectivity or degree convention is inferred. The exact result, retaining the requested order, is (5, [[0, 1], [0, 2], [1, 2], [1, 3], [2, 3], [2, 4], [3, 4], [4, 0], [4, 1]], Yes, Yes, ordered_pairs_and_no_parallel_duplicate_edges).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r07-finite-directed-graph-0024/)

Finite directed graph · qf-mod39-digraph-qualification · candidate





## 25. A Directed Edge Is an Ordered Pair under the Declared Simple Convention

Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 1], [1, 2], [2, 3], [3, 0], [3, 4], [4, 0], [4, 1]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Give (vertex_count, edges, loop_free, simple_directed_set_qualified, convention).

View answer and explanation(5, [[0, 1], [1, 2], [2, 3], [3, 0], [3, 4], [4, 0], [4, 1]], Yes, Yes, ordered_pairs_and_no_parallel_duplicate_edges)

Validate every endpoint, canonical edge order, uniqueness and absence of loops. Opposite oriented edges may both be present and remain distinct; no undirected connectivity or degree convention is inferred. The exact result, retaining the requested order, is (5, [[0, 1], [1, 2], [2, 3], [3, 0], [3, 4], [4, 0], [4, 1]], Yes, Yes, ordered_pairs_and_no_parallel_duplicate_edges).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r07-finite-directed-graph-0025/)

Finite directed graph · qf-mod39-digraph-qualification · candidate





## 26. In-Degrees and Out-Degrees Each Sum to the Edge Count

Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 0]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Give (out_degrees, in_degrees, total_out, total_in, edge_count, equal_counts).

View answer and explanation([1, 1, 1, 1, 1], [1, 1, 1, 1, 1], 5, 5, 5, Yes)

Count outgoing and incoming edges separately at each vertex in index order. Both totals equal |E|, not the undirected2|E| degree-sum formula. The exact result, retaining the requested order, is ([1, 1, 1, 1, 1], [1, 1, 1, 1, 1], 5, 5, 5, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r07-finite-directed-graph-0026/)

Finite directed graph · qf-mod39-digraph-in-out-degree · candidate





## 27. In-Degrees and Out-Degrees Each Sum to the Edge Count

Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 1], [0, 2], [1, 2], [2, 3], [3, 4], [4, 0]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Give (out_degrees, in_degrees, total_out, total_in, edge_count, equal_counts).

View answer and explanation([2, 1, 1, 1, 1], [1, 1, 2, 1, 1], 6, 6, 6, Yes)

Count outgoing and incoming edges separately at each vertex in index order. Both totals equal |E|, not the undirected2|E| degree-sum formula. The exact result, retaining the requested order, is ([2, 1, 1, 1, 1], [1, 1, 2, 1, 1], 6, 6, 6, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r07-finite-directed-graph-0027/)

Finite directed graph · qf-mod39-digraph-in-out-degree · candidate





## 28. In-Degrees and Out-Degrees Each Sum to the Edge Count

Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 1], [1, 2], [1, 3], [2, 3], [3, 4], [4, 0]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Give (out_degrees, in_degrees, total_out, total_in, edge_count, equal_counts).

View answer and explanation([1, 2, 1, 1, 1], [1, 1, 1, 2, 1], 6, 6, 6, Yes)

Count outgoing and incoming edges separately at each vertex in index order. Both totals equal |E|, not the undirected2|E| degree-sum formula. The exact result, retaining the requested order, is ([1, 2, 1, 1, 1], [1, 1, 1, 2, 1], 6, 6, 6, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r07-finite-directed-graph-0028/)

Finite directed graph · qf-mod39-digraph-in-out-degree · candidate





## 29. In-Degrees and Out-Degrees Each Sum to the Edge Count

Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 1], [0, 2], [1, 2], [1, 3], [2, 3], [3, 4], [4, 0]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Give (out_degrees, in_degrees, total_out, total_in, edge_count, equal_counts).

View answer and explanation([2, 2, 1, 1, 1], [1, 1, 2, 2, 1], 7, 7, 7, Yes)

Count outgoing and incoming edges separately at each vertex in index order. Both totals equal |E|, not the undirected2|E| degree-sum formula. The exact result, retaining the requested order, is ([2, 2, 1, 1, 1], [1, 1, 2, 2, 1], 7, 7, 7, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r07-finite-directed-graph-0029/)

Finite directed graph · qf-mod39-digraph-in-out-degree · candidate





## 30. In-Degrees and Out-Degrees Each Sum to the Edge Count

Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 1], [1, 2], [2, 3], [2, 4], [3, 4], [4, 0]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Give (out_degrees, in_degrees, total_out, total_in, edge_count, equal_counts).

View answer and explanation([1, 1, 2, 1, 1], [1, 1, 1, 1, 2], 6, 6, 6, Yes)

Count outgoing and incoming edges separately at each vertex in index order. Both totals equal |E|, not the undirected2|E| degree-sum formula. The exact result, retaining the requested order, is ([1, 1, 2, 1, 1], [1, 1, 1, 1, 2], 6, 6, 6, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r07-finite-directed-graph-0030/)

Finite directed graph · qf-mod39-digraph-in-out-degree · candidate





## 31. In-Degrees and Out-Degrees Each Sum to the Edge Count

Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 1], [0, 2], [1, 2], [2, 3], [2, 4], [3, 4], [4, 0]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Give (out_degrees, in_degrees, total_out, total_in, edge_count, equal_counts).

View answer and explanation([2, 1, 2, 1, 1], [1, 1, 2, 1, 2], 7, 7, 7, Yes)

Count outgoing and incoming edges separately at each vertex in index order. Both totals equal |E|, not the undirected2|E| degree-sum formula. The exact result, retaining the requested order, is ([2, 1, 2, 1, 1], [1, 1, 2, 1, 2], 7, 7, 7, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r07-finite-directed-graph-0031/)

Finite directed graph · qf-mod39-digraph-in-out-degree · candidate





## 32. In-Degrees and Out-Degrees Each Sum to the Edge Count

Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 1], [1, 2], [1, 3], [2, 3], [2, 4], [3, 4], [4, 0]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Give (out_degrees, in_degrees, total_out, total_in, edge_count, equal_counts).

View answer and explanation([1, 2, 2, 1, 1], [1, 1, 1, 2, 2], 7, 7, 7, Yes)

Count outgoing and incoming edges separately at each vertex in index order. Both totals equal |E|, not the undirected2|E| degree-sum formula. The exact result, retaining the requested order, is ([1, 2, 2, 1, 1], [1, 1, 1, 2, 2], 7, 7, 7, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r07-finite-directed-graph-0032/)

Finite directed graph · qf-mod39-digraph-in-out-degree · candidate





## 33. In-Degrees and Out-Degrees Each Sum to the Edge Count

Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 1], [0, 2], [1, 2], [1, 3], [2, 3], [2, 4], [3, 4], [4, 0]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Give (out_degrees, in_degrees, total_out, total_in, edge_count, equal_counts).

View answer and explanation([2, 2, 2, 1, 1], [1, 1, 2, 2, 2], 8, 8, 8, Yes)

Count outgoing and incoming edges separately at each vertex in index order. Both totals equal |E|, not the undirected2|E| degree-sum formula. The exact result, retaining the requested order, is ([2, 2, 2, 1, 1], [1, 1, 2, 2, 2], 8, 8, 8, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r07-finite-directed-graph-0033/)

Finite directed graph · qf-mod39-digraph-in-out-degree · candidate





## 34. In-Degrees and Out-Degrees Each Sum to the Edge Count

Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 1], [1, 2], [2, 3], [3, 0], [3, 4], [4, 0]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Give (out_degrees, in_degrees, total_out, total_in, edge_count, equal_counts).

View answer and explanation([1, 1, 1, 2, 1], [2, 1, 1, 1, 1], 6, 6, 6, Yes)

Count outgoing and incoming edges separately at each vertex in index order. Both totals equal |E|, not the undirected2|E| degree-sum formula. The exact result, retaining the requested order, is ([1, 1, 1, 2, 1], [2, 1, 1, 1, 1], 6, 6, 6, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r07-finite-directed-graph-0034/)

Finite directed graph · qf-mod39-digraph-in-out-degree · candidate





## 35. In-Degrees and Out-Degrees Each Sum to the Edge Count

Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 1], [0, 2], [1, 2], [2, 3], [3, 0], [3, 4], [4, 0]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Give (out_degrees, in_degrees, total_out, total_in, edge_count, equal_counts).

View answer and explanation([2, 1, 1, 2, 1], [2, 1, 2, 1, 1], 7, 7, 7, Yes)

Count outgoing and incoming edges separately at each vertex in index order. Both totals equal |E|, not the undirected2|E| degree-sum formula. The exact result, retaining the requested order, is ([2, 1, 1, 2, 1], [2, 1, 2, 1, 1], 7, 7, 7, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r07-finite-directed-graph-0035/)

Finite directed graph · qf-mod39-digraph-in-out-degree · candidate





## 36. In-Degrees and Out-Degrees Each Sum to the Edge Count

Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 1], [1, 2], [1, 3], [2, 3], [3, 0], [3, 4], [4, 0]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Give (out_degrees, in_degrees, total_out, total_in, edge_count, equal_counts).

View answer and explanation([1, 2, 1, 2, 1], [2, 1, 1, 2, 1], 7, 7, 7, Yes)

Count outgoing and incoming edges separately at each vertex in index order. Both totals equal |E|, not the undirected2|E| degree-sum formula. The exact result, retaining the requested order, is ([1, 2, 1, 2, 1], [2, 1, 1, 2, 1], 7, 7, 7, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r07-finite-directed-graph-0036/)

Finite directed graph · qf-mod39-digraph-in-out-degree · candidate





## 37. In-Degrees and Out-Degrees Each Sum to the Edge Count

Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 1], [0, 2], [1, 2], [1, 3], [2, 3], [3, 0], [3, 4], [4, 0]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Give (out_degrees, in_degrees, total_out, total_in, edge_count, equal_counts).

View answer and explanation([2, 2, 1, 2, 1], [2, 1, 2, 2, 1], 8, 8, 8, Yes)

Count outgoing and incoming edges separately at each vertex in index order. Both totals equal |E|, not the undirected2|E| degree-sum formula. The exact result, retaining the requested order, is ([2, 2, 1, 2, 1], [2, 1, 2, 2, 1], 8, 8, 8, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r07-finite-directed-graph-0037/)

Finite directed graph · qf-mod39-digraph-in-out-degree · candidate





## 38. In-Degrees and Out-Degrees Each Sum to the Edge Count

Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 1], [1, 2], [2, 3], [2, 4], [3, 0], [3, 4], [4, 0]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Give (out_degrees, in_degrees, total_out, total_in, edge_count, equal_counts).

View answer and explanation([1, 1, 2, 2, 1], [2, 1, 1, 1, 2], 7, 7, 7, Yes)

Count outgoing and incoming edges separately at each vertex in index order. Both totals equal |E|, not the undirected2|E| degree-sum formula. The exact result, retaining the requested order, is ([1, 1, 2, 2, 1], [2, 1, 1, 1, 2], 7, 7, 7, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r07-finite-directed-graph-0038/)

Finite directed graph · qf-mod39-digraph-in-out-degree · candidate





## 39. In-Degrees and Out-Degrees Each Sum to the Edge Count

Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 1], [0, 2], [1, 2], [2, 3], [2, 4], [3, 0], [3, 4], [4, 0]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Give (out_degrees, in_degrees, total_out, total_in, edge_count, equal_counts).

View answer and explanation([2, 1, 2, 2, 1], [2, 1, 2, 1, 2], 8, 8, 8, Yes)

Count outgoing and incoming edges separately at each vertex in index order. Both totals equal |E|, not the undirected2|E| degree-sum formula. The exact result, retaining the requested order, is ([2, 1, 2, 2, 1], [2, 1, 2, 1, 2], 8, 8, 8, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r07-finite-directed-graph-0039/)

Finite directed graph · qf-mod39-digraph-in-out-degree · candidate





## 40. In-Degrees and Out-Degrees Each Sum to the Edge Count

Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 1], [1, 2], [1, 3], [2, 3], [2, 4], [3, 0], [3, 4], [4, 0]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Give (out_degrees, in_degrees, total_out, total_in, edge_count, equal_counts).

View answer and explanation([1, 2, 2, 2, 1], [2, 1, 1, 2, 2], 8, 8, 8, Yes)

Count outgoing and incoming edges separately at each vertex in index order. Both totals equal |E|, not the undirected2|E| degree-sum formula. The exact result, retaining the requested order, is ([1, 2, 2, 2, 1], [2, 1, 1, 2, 2], 8, 8, 8, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r07-finite-directed-graph-0040/)

Finite directed graph · qf-mod39-digraph-in-out-degree · candidate





## 41. In-Degrees and Out-Degrees Each Sum to the Edge Count

Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 1], [0, 2], [1, 2], [1, 3], [2, 3], [2, 4], [3, 0], [3, 4], [4, 0]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Give (out_degrees, in_degrees, total_out, total_in, edge_count, equal_counts).

View answer and explanation([2, 2, 2, 2, 1], [2, 1, 2, 2, 2], 9, 9, 9, Yes)

Count outgoing and incoming edges separately at each vertex in index order. Both totals equal |E|, not the undirected2|E| degree-sum formula. The exact result, retaining the requested order, is ([2, 2, 2, 2, 1], [2, 1, 2, 2, 2], 9, 9, 9, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r07-finite-directed-graph-0041/)

Finite directed graph · qf-mod39-digraph-in-out-degree · candidate





## 42. In-Degrees and Out-Degrees Each Sum to the Edge Count

Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 0], [4, 1]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Give (out_degrees, in_degrees, total_out, total_in, edge_count, equal_counts).

View answer and explanation([1, 1, 1, 1, 2], [1, 2, 1, 1, 1], 6, 6, 6, Yes)

Count outgoing and incoming edges separately at each vertex in index order. Both totals equal |E|, not the undirected2|E| degree-sum formula. The exact result, retaining the requested order, is ([1, 1, 1, 1, 2], [1, 2, 1, 1, 1], 6, 6, 6, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r07-finite-directed-graph-0042/)

Finite directed graph · qf-mod39-digraph-in-out-degree · candidate





## 43. In-Degrees and Out-Degrees Each Sum to the Edge Count

Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 1], [0, 2], [1, 2], [2, 3], [3, 4], [4, 0], [4, 1]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Give (out_degrees, in_degrees, total_out, total_in, edge_count, equal_counts).

View answer and explanation([2, 1, 1, 1, 2], [1, 2, 2, 1, 1], 7, 7, 7, Yes)

Count outgoing and incoming edges separately at each vertex in index order. Both totals equal |E|, not the undirected2|E| degree-sum formula. The exact result, retaining the requested order, is ([2, 1, 1, 1, 2], [1, 2, 2, 1, 1], 7, 7, 7, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r07-finite-directed-graph-0043/)

Finite directed graph · qf-mod39-digraph-in-out-degree · candidate





## 44. In-Degrees and Out-Degrees Each Sum to the Edge Count

Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 1], [1, 2], [1, 3], [2, 3], [3, 4], [4, 0], [4, 1]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Give (out_degrees, in_degrees, total_out, total_in, edge_count, equal_counts).

View answer and explanation([1, 2, 1, 1, 2], [1, 2, 1, 2, 1], 7, 7, 7, Yes)

Count outgoing and incoming edges separately at each vertex in index order. Both totals equal |E|, not the undirected2|E| degree-sum formula. The exact result, retaining the requested order, is ([1, 2, 1, 1, 2], [1, 2, 1, 2, 1], 7, 7, 7, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r07-finite-directed-graph-0044/)

Finite directed graph · qf-mod39-digraph-in-out-degree · candidate





## 45. In-Degrees and Out-Degrees Each Sum to the Edge Count

Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 1], [0, 2], [1, 2], [1, 3], [2, 3], [3, 4], [4, 0], [4, 1]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Give (out_degrees, in_degrees, total_out, total_in, edge_count, equal_counts).

View answer and explanation([2, 2, 1, 1, 2], [1, 2, 2, 2, 1], 8, 8, 8, Yes)

Count outgoing and incoming edges separately at each vertex in index order. Both totals equal |E|, not the undirected2|E| degree-sum formula. The exact result, retaining the requested order, is ([2, 2, 1, 1, 2], [1, 2, 2, 2, 1], 8, 8, 8, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r07-finite-directed-graph-0045/)

Finite directed graph · qf-mod39-digraph-in-out-degree · candidate





## 46. In-Degrees and Out-Degrees Each Sum to the Edge Count

Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 1], [1, 2], [2, 3], [2, 4], [3, 4], [4, 0], [4, 1]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Give (out_degrees, in_degrees, total_out, total_in, edge_count, equal_counts).

View answer and explanation([1, 1, 2, 1, 2], [1, 2, 1, 1, 2], 7, 7, 7, Yes)

Count outgoing and incoming edges separately at each vertex in index order. Both totals equal |E|, not the undirected2|E| degree-sum formula. The exact result, retaining the requested order, is ([1, 1, 2, 1, 2], [1, 2, 1, 1, 2], 7, 7, 7, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r07-finite-directed-graph-0046/)

Finite directed graph · qf-mod39-digraph-in-out-degree · candidate





## 47. In-Degrees and Out-Degrees Each Sum to the Edge Count

Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 1], [0, 2], [1, 2], [2, 3], [2, 4], [3, 4], [4, 0], [4, 1]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Give (out_degrees, in_degrees, total_out, total_in, edge_count, equal_counts).

View answer and explanation([2, 1, 2, 1, 2], [1, 2, 2, 1, 2], 8, 8, 8, Yes)

Count outgoing and incoming edges separately at each vertex in index order. Both totals equal |E|, not the undirected2|E| degree-sum formula. The exact result, retaining the requested order, is ([2, 1, 2, 1, 2], [1, 2, 2, 1, 2], 8, 8, 8, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r07-finite-directed-graph-0047/)

Finite directed graph · qf-mod39-digraph-in-out-degree · candidate





## 48. In-Degrees and Out-Degrees Each Sum to the Edge Count

Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 1], [1, 2], [1, 3], [2, 3], [2, 4], [3, 4], [4, 0], [4, 1]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Give (out_degrees, in_degrees, total_out, total_in, edge_count, equal_counts).

View answer and explanation([1, 2, 2, 1, 2], [1, 2, 1, 2, 2], 8, 8, 8, Yes)

Count outgoing and incoming edges separately at each vertex in index order. Both totals equal |E|, not the undirected2|E| degree-sum formula. The exact result, retaining the requested order, is ([1, 2, 2, 1, 2], [1, 2, 1, 2, 2], 8, 8, 8, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r07-finite-directed-graph-0048/)

Finite directed graph · qf-mod39-digraph-in-out-degree · candidate





## 49. In-Degrees and Out-Degrees Each Sum to the Edge Count

Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 1], [0, 2], [1, 2], [1, 3], [2, 3], [2, 4], [3, 4], [4, 0], [4, 1]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Give (out_degrees, in_degrees, total_out, total_in, edge_count, equal_counts).

View answer and explanation([2, 2, 2, 1, 2], [1, 2, 2, 2, 2], 9, 9, 9, Yes)

Count outgoing and incoming edges separately at each vertex in index order. Both totals equal |E|, not the undirected2|E| degree-sum formula. The exact result, retaining the requested order, is ([2, 2, 2, 1, 2], [1, 2, 2, 2, 2], 9, 9, 9, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r07-finite-directed-graph-0049/)

Finite directed graph · qf-mod39-digraph-in-out-degree · candidate





## 50. In-Degrees and Out-Degrees Each Sum to the Edge Count

Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 1], [1, 2], [2, 3], [3, 0], [3, 4], [4, 0], [4, 1]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Give (out_degrees, in_degrees, total_out, total_in, edge_count, equal_counts).

View answer and explanation([1, 1, 1, 2, 2], [2, 2, 1, 1, 1], 7, 7, 7, Yes)

Count outgoing and incoming edges separately at each vertex in index order. Both totals equal |E|, not the undirected2|E| degree-sum formula. The exact result, retaining the requested order, is ([1, 1, 1, 2, 2], [2, 2, 1, 1, 1], 7, 7, 7, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r07-finite-directed-graph-0050/)

Finite directed graph · qf-mod39-digraph-in-out-degree · candidate





## 51. Every Successive Step Must Follow Its Own Directed Edge

Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 0]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Supplied vertex list walk=[0, 1, 2, 3, 4, 0]. Give (walk, step_checks, all_oriented_steps_present, boundary).

This flag checks successive oriented edges only. An empty list has no step to refute; that alone does not certify a separately required nonempty-walk convention.

View answer and explanation([0, 1, 2, 3, 4, 0], [[0, 1, Yes], [1, 2, Yes], [2, 3, Yes], [3, 4, Yes], [4, 0, Yes]], Yes, edge_orientation_must_match)

For each consecutive pair report its oriented-edge membership and their conjunction. This is a successive-step audit, not an extra simple-path, distinct-vertex or nonempty-walk certificate. An empty list has no step to refute and makes that conjunction vacuously Yes. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 0], [[0, 1, Yes], [1, 2, Yes], [2, 3, Yes], [3, 4, Yes], [4, 0, Yes]], Yes, edge_orientation_must_match).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r07-finite-directed-graph-0051/)

Finite directed graph · qf-mod39-digraph-directed-walk-audit · candidate





## 52. Every Successive Step Must Follow Its Own Directed Edge

Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 1], [0, 2], [1, 2], [2, 3], [3, 4], [4, 0]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Supplied vertex list walk=[0, 1, 2, 3, 4, 0]. Give (walk, step_checks, all_oriented_steps_present, boundary).

This flag checks successive oriented edges only. An empty list has no step to refute; that alone does not certify a separately required nonempty-walk convention.

View answer and explanation([0, 1, 2, 3, 4, 0], [[0, 1, Yes], [1, 2, Yes], [2, 3, Yes], [3, 4, Yes], [4, 0, Yes]], Yes, edge_orientation_must_match)

For each consecutive pair report its oriented-edge membership and their conjunction. This is a successive-step audit, not an extra simple-path, distinct-vertex or nonempty-walk certificate. An empty list has no step to refute and makes that conjunction vacuously Yes. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 0], [[0, 1, Yes], [1, 2, Yes], [2, 3, Yes], [3, 4, Yes], [4, 0, Yes]], Yes, edge_orientation_must_match).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r07-finite-directed-graph-0052/)

Finite directed graph · qf-mod39-digraph-directed-walk-audit · candidate





## 53. Every Successive Step Must Follow Its Own Directed Edge

Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 1], [1, 2], [1, 3], [2, 3], [3, 4], [4, 0]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Supplied vertex list walk=[0, 1, 2, 3, 4, 0]. Give (walk, step_checks, all_oriented_steps_present, boundary).

This flag checks successive oriented edges only. An empty list has no step to refute; that alone does not certify a separately required nonempty-walk convention.

View answer and explanation([0, 1, 2, 3, 4, 0], [[0, 1, Yes], [1, 2, Yes], [2, 3, Yes], [3, 4, Yes], [4, 0, Yes]], Yes, edge_orientation_must_match)

For each consecutive pair report its oriented-edge membership and their conjunction. This is a successive-step audit, not an extra simple-path, distinct-vertex or nonempty-walk certificate. An empty list has no step to refute and makes that conjunction vacuously Yes. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 0], [[0, 1, Yes], [1, 2, Yes], [2, 3, Yes], [3, 4, Yes], [4, 0, Yes]], Yes, edge_orientation_must_match).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r07-finite-directed-graph-0053/)

Finite directed graph · qf-mod39-digraph-directed-walk-audit · candidate





## 54. Every Successive Step Must Follow Its Own Directed Edge

Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 1], [0, 2], [1, 2], [1, 3], [2, 3], [3, 4], [4, 0]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Supplied vertex list walk=[0, 1, 2, 3, 4, 0]. Give (walk, step_checks, all_oriented_steps_present, boundary).

This flag checks successive oriented edges only. An empty list has no step to refute; that alone does not certify a separately required nonempty-walk convention.

View answer and explanation([0, 1, 2, 3, 4, 0], [[0, 1, Yes], [1, 2, Yes], [2, 3, Yes], [3, 4, Yes], [4, 0, Yes]], Yes, edge_orientation_must_match)

For each consecutive pair report its oriented-edge membership and their conjunction. This is a successive-step audit, not an extra simple-path, distinct-vertex or nonempty-walk certificate. An empty list has no step to refute and makes that conjunction vacuously Yes. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 0], [[0, 1, Yes], [1, 2, Yes], [2, 3, Yes], [3, 4, Yes], [4, 0, Yes]], Yes, edge_orientation_must_match).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r07-finite-directed-graph-0054/)

Finite directed graph · qf-mod39-digraph-directed-walk-audit · candidate





## 55. Every Successive Step Must Follow Its Own Directed Edge

Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 1], [1, 2], [2, 3], [2, 4], [3, 4], [4, 0]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Supplied vertex list walk=[0, 1, 2, 3, 4, 0]. Give (walk, step_checks, all_oriented_steps_present, boundary).

This flag checks successive oriented edges only. An empty list has no step to refute; that alone does not certify a separately required nonempty-walk convention.

View answer and explanation([0, 1, 2, 3, 4, 0], [[0, 1, Yes], [1, 2, Yes], [2, 3, Yes], [3, 4, Yes], [4, 0, Yes]], Yes, edge_orientation_must_match)

For each consecutive pair report its oriented-edge membership and their conjunction. This is a successive-step audit, not an extra simple-path, distinct-vertex or nonempty-walk certificate. An empty list has no step to refute and makes that conjunction vacuously Yes. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 0], [[0, 1, Yes], [1, 2, Yes], [2, 3, Yes], [3, 4, Yes], [4, 0, Yes]], Yes, edge_orientation_must_match).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r07-finite-directed-graph-0055/)

Finite directed graph · qf-mod39-digraph-directed-walk-audit · candidate





## 56. Every Successive Step Must Follow Its Own Directed Edge

Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 1], [0, 2], [1, 2], [2, 3], [2, 4], [3, 4], [4, 0]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Supplied vertex list walk=[0, 1, 2, 3, 4, 0]. Give (walk, step_checks, all_oriented_steps_present, boundary).

This flag checks successive oriented edges only. An empty list has no step to refute; that alone does not certify a separately required nonempty-walk convention.

View answer and explanation([0, 1, 2, 3, 4, 0], [[0, 1, Yes], [1, 2, Yes], [2, 3, Yes], [3, 4, Yes], [4, 0, Yes]], Yes, edge_orientation_must_match)

For each consecutive pair report its oriented-edge membership and their conjunction. This is a successive-step audit, not an extra simple-path, distinct-vertex or nonempty-walk certificate. An empty list has no step to refute and makes that conjunction vacuously Yes. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 0], [[0, 1, Yes], [1, 2, Yes], [2, 3, Yes], [3, 4, Yes], [4, 0, Yes]], Yes, edge_orientation_must_match).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r07-finite-directed-graph-0056/)

Finite directed graph · qf-mod39-digraph-directed-walk-audit · candidate





## 57. Every Successive Step Must Follow Its Own Directed Edge

Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 1], [1, 2], [1, 3], [2, 3], [2, 4], [3, 4], [4, 0]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Supplied vertex list walk=[0, 1, 2, 3, 4, 0]. Give (walk, step_checks, all_oriented_steps_present, boundary).

This flag checks successive oriented edges only. An empty list has no step to refute; that alone does not certify a separately required nonempty-walk convention.

View answer and explanation([0, 1, 2, 3, 4, 0], [[0, 1, Yes], [1, 2, Yes], [2, 3, Yes], [3, 4, Yes], [4, 0, Yes]], Yes, edge_orientation_must_match)

For each consecutive pair report its oriented-edge membership and their conjunction. This is a successive-step audit, not an extra simple-path, distinct-vertex or nonempty-walk certificate. An empty list has no step to refute and makes that conjunction vacuously Yes. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 0], [[0, 1, Yes], [1, 2, Yes], [2, 3, Yes], [3, 4, Yes], [4, 0, Yes]], Yes, edge_orientation_must_match).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r07-finite-directed-graph-0057/)

Finite directed graph · qf-mod39-digraph-directed-walk-audit · candidate





## 58. Every Successive Step Must Follow Its Own Directed Edge

Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 1], [0, 2], [1, 2], [1, 3], [2, 3], [2, 4], [3, 4], [4, 0]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Supplied vertex list walk=[0, 1, 2, 3, 4, 0]. Give (walk, step_checks, all_oriented_steps_present, boundary).

This flag checks successive oriented edges only. An empty list has no step to refute; that alone does not certify a separately required nonempty-walk convention.

View answer and explanation([0, 1, 2, 3, 4, 0], [[0, 1, Yes], [1, 2, Yes], [2, 3, Yes], [3, 4, Yes], [4, 0, Yes]], Yes, edge_orientation_must_match)

For each consecutive pair report its oriented-edge membership and their conjunction. This is a successive-step audit, not an extra simple-path, distinct-vertex or nonempty-walk certificate. An empty list has no step to refute and makes that conjunction vacuously Yes. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 0], [[0, 1, Yes], [1, 2, Yes], [2, 3, Yes], [3, 4, Yes], [4, 0, Yes]], Yes, edge_orientation_must_match).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r07-finite-directed-graph-0058/)

Finite directed graph · qf-mod39-digraph-directed-walk-audit · candidate





## 59. Every Successive Step Must Follow Its Own Directed Edge

Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 1], [1, 2], [2, 3], [3, 0], [3, 4], [4, 0]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Supplied vertex list walk=[0, 1, 2, 3, 4, 0]. Give (walk, step_checks, all_oriented_steps_present, boundary).

This flag checks successive oriented edges only. An empty list has no step to refute; that alone does not certify a separately required nonempty-walk convention.

View answer and explanation([0, 1, 2, 3, 4, 0], [[0, 1, Yes], [1, 2, Yes], [2, 3, Yes], [3, 4, Yes], [4, 0, Yes]], Yes, edge_orientation_must_match)

For each consecutive pair report its oriented-edge membership and their conjunction. This is a successive-step audit, not an extra simple-path, distinct-vertex or nonempty-walk certificate. An empty list has no step to refute and makes that conjunction vacuously Yes. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 0], [[0, 1, Yes], [1, 2, Yes], [2, 3, Yes], [3, 4, Yes], [4, 0, Yes]], Yes, edge_orientation_must_match).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r07-finite-directed-graph-0059/)

Finite directed graph · qf-mod39-digraph-directed-walk-audit · candidate





## 60. Every Successive Step Must Follow Its Own Directed Edge

Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 1], [0, 2], [1, 2], [2, 3], [3, 0], [3, 4], [4, 0]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Supplied vertex list walk=[0, 1, 2, 3, 4, 0]. Give (walk, step_checks, all_oriented_steps_present, boundary).

This flag checks successive oriented edges only. An empty list has no step to refute; that alone does not certify a separately required nonempty-walk convention.

View answer and explanation([0, 1, 2, 3, 4, 0], [[0, 1, Yes], [1, 2, Yes], [2, 3, Yes], [3, 4, Yes], [4, 0, Yes]], Yes, edge_orientation_must_match)

For each consecutive pair report its oriented-edge membership and their conjunction. This is a successive-step audit, not an extra simple-path, distinct-vertex or nonempty-walk certificate. An empty list has no step to refute and makes that conjunction vacuously Yes. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 0], [[0, 1, Yes], [1, 2, Yes], [2, 3, Yes], [3, 4, Yes], [4, 0, Yes]], Yes, edge_orientation_must_match).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r07-finite-directed-graph-0060/)

Finite directed graph · qf-mod39-digraph-directed-walk-audit · candidate





## 61. Every Successive Step Must Follow Its Own Directed Edge

Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 1], [1, 2], [1, 3], [2, 3], [3, 0], [3, 4], [4, 0]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Supplied vertex list walk=[0, 1, 2, 3, 4, 0]. Give (walk, step_checks, all_oriented_steps_present, boundary).

This flag checks successive oriented edges only. An empty list has no step to refute; that alone does not certify a separately required nonempty-walk convention.

View answer and explanation([0, 1, 2, 3, 4, 0], [[0, 1, Yes], [1, 2, Yes], [2, 3, Yes], [3, 4, Yes], [4, 0, Yes]], Yes, edge_orientation_must_match)

For each consecutive pair report its oriented-edge membership and their conjunction. This is a successive-step audit, not an extra simple-path, distinct-vertex or nonempty-walk certificate. An empty list has no step to refute and makes that conjunction vacuously Yes. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 0], [[0, 1, Yes], [1, 2, Yes], [2, 3, Yes], [3, 4, Yes], [4, 0, Yes]], Yes, edge_orientation_must_match).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r07-finite-directed-graph-0061/)

Finite directed graph · qf-mod39-digraph-directed-walk-audit · candidate





## 62. Every Successive Step Must Follow Its Own Directed Edge

Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 1], [0, 2], [1, 2], [1, 3], [2, 3], [3, 0], [3, 4], [4, 0]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Supplied vertex list walk=[0, 1, 2, 3, 4, 0]. Give (walk, step_checks, all_oriented_steps_present, boundary).

This flag checks successive oriented edges only. An empty list has no step to refute; that alone does not certify a separately required nonempty-walk convention.

View answer and explanation([0, 1, 2, 3, 4, 0], [[0, 1, Yes], [1, 2, Yes], [2, 3, Yes], [3, 4, Yes], [4, 0, Yes]], Yes, edge_orientation_must_match)

For each consecutive pair report its oriented-edge membership and their conjunction. This is a successive-step audit, not an extra simple-path, distinct-vertex or nonempty-walk certificate. An empty list has no step to refute and makes that conjunction vacuously Yes. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 0], [[0, 1, Yes], [1, 2, Yes], [2, 3, Yes], [3, 4, Yes], [4, 0, Yes]], Yes, edge_orientation_must_match).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r07-finite-directed-graph-0062/)

Finite directed graph · qf-mod39-digraph-directed-walk-audit · candidate





## 63. Every Successive Step Must Follow Its Own Directed Edge

Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 1], [1, 2], [2, 3], [2, 4], [3, 0], [3, 4], [4, 0]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Supplied vertex list walk=[0, 1, 2, 3, 4, 0]. Give (walk, step_checks, all_oriented_steps_present, boundary).

This flag checks successive oriented edges only. An empty list has no step to refute; that alone does not certify a separately required nonempty-walk convention.

View answer and explanation([0, 1, 2, 3, 4, 0], [[0, 1, Yes], [1, 2, Yes], [2, 3, Yes], [3, 4, Yes], [4, 0, Yes]], Yes, edge_orientation_must_match)

For each consecutive pair report its oriented-edge membership and their conjunction. This is a successive-step audit, not an extra simple-path, distinct-vertex or nonempty-walk certificate. An empty list has no step to refute and makes that conjunction vacuously Yes. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 0], [[0, 1, Yes], [1, 2, Yes], [2, 3, Yes], [3, 4, Yes], [4, 0, Yes]], Yes, edge_orientation_must_match).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r07-finite-directed-graph-0063/)

Finite directed graph · qf-mod39-digraph-directed-walk-audit · candidate





## 64. Every Successive Step Must Follow Its Own Directed Edge

Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 1], [0, 2], [1, 2], [2, 3], [2, 4], [3, 0], [3, 4], [4, 0]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Supplied vertex list walk=[0, 1, 2, 3, 4, 0]. Give (walk, step_checks, all_oriented_steps_present, boundary).

This flag checks successive oriented edges only. An empty list has no step to refute; that alone does not certify a separately required nonempty-walk convention.

View answer and explanation([0, 1, 2, 3, 4, 0], [[0, 1, Yes], [1, 2, Yes], [2, 3, Yes], [3, 4, Yes], [4, 0, Yes]], Yes, edge_orientation_must_match)

For each consecutive pair report its oriented-edge membership and their conjunction. This is a successive-step audit, not an extra simple-path, distinct-vertex or nonempty-walk certificate. An empty list has no step to refute and makes that conjunction vacuously Yes. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 0], [[0, 1, Yes], [1, 2, Yes], [2, 3, Yes], [3, 4, Yes], [4, 0, Yes]], Yes, edge_orientation_must_match).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r07-finite-directed-graph-0064/)

Finite directed graph · qf-mod39-digraph-directed-walk-audit · candidate





## 65. Every Successive Step Must Follow Its Own Directed Edge

Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 1], [1, 2], [1, 3], [2, 3], [2, 4], [3, 0], [3, 4], [4, 0]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Supplied vertex list walk=[0, 1, 2, 3, 4, 0]. Give (walk, step_checks, all_oriented_steps_present, boundary).

This flag checks successive oriented edges only. An empty list has no step to refute; that alone does not certify a separately required nonempty-walk convention.

View answer and explanation([0, 1, 2, 3, 4, 0], [[0, 1, Yes], [1, 2, Yes], [2, 3, Yes], [3, 4, Yes], [4, 0, Yes]], Yes, edge_orientation_must_match)

For each consecutive pair report its oriented-edge membership and their conjunction. This is a successive-step audit, not an extra simple-path, distinct-vertex or nonempty-walk certificate. An empty list has no step to refute and makes that conjunction vacuously Yes. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 0], [[0, 1, Yes], [1, 2, Yes], [2, 3, Yes], [3, 4, Yes], [4, 0, Yes]], Yes, edge_orientation_must_match).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r07-finite-directed-graph-0065/)

Finite directed graph · qf-mod39-digraph-directed-walk-audit · candidate





## 66. Every Successive Step Must Follow Its Own Directed Edge

Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 1], [0, 2], [1, 2], [1, 3], [2, 3], [2, 4], [3, 0], [3, 4], [4, 0]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Supplied vertex list walk=[0, 1, 2, 3, 4, 0]. Give (walk, step_checks, all_oriented_steps_present, boundary).

This flag checks successive oriented edges only. An empty list has no step to refute; that alone does not certify a separately required nonempty-walk convention.

View answer and explanation([0, 1, 2, 3, 4, 0], [[0, 1, Yes], [1, 2, Yes], [2, 3, Yes], [3, 4, Yes], [4, 0, Yes]], Yes, edge_orientation_must_match)

For each consecutive pair report its oriented-edge membership and their conjunction. This is a successive-step audit, not an extra simple-path, distinct-vertex or nonempty-walk certificate. An empty list has no step to refute and makes that conjunction vacuously Yes. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 0], [[0, 1, Yes], [1, 2, Yes], [2, 3, Yes], [3, 4, Yes], [4, 0, Yes]], Yes, edge_orientation_must_match).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r07-finite-directed-graph-0066/)

Finite directed graph · qf-mod39-digraph-directed-walk-audit · candidate





## 67. Every Successive Step Must Follow Its Own Directed Edge

Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 0], [4, 1]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Supplied vertex list walk=[0, 1, 2, 3, 4, 0]. Give (walk, step_checks, all_oriented_steps_present, boundary).

This flag checks successive oriented edges only. An empty list has no step to refute; that alone does not certify a separately required nonempty-walk convention.

View answer and explanation([0, 1, 2, 3, 4, 0], [[0, 1, Yes], [1, 2, Yes], [2, 3, Yes], [3, 4, Yes], [4, 0, Yes]], Yes, edge_orientation_must_match)

For each consecutive pair report its oriented-edge membership and their conjunction. This is a successive-step audit, not an extra simple-path, distinct-vertex or nonempty-walk certificate. An empty list has no step to refute and makes that conjunction vacuously Yes. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 0], [[0, 1, Yes], [1, 2, Yes], [2, 3, Yes], [3, 4, Yes], [4, 0, Yes]], Yes, edge_orientation_must_match).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r07-finite-directed-graph-0067/)

Finite directed graph · qf-mod39-digraph-directed-walk-audit · candidate





## 68. Every Successive Step Must Follow Its Own Directed Edge

Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 1], [0, 2], [1, 2], [2, 3], [3, 4], [4, 0], [4, 1]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Supplied vertex list walk=[0, 1, 2, 3, 4, 0]. Give (walk, step_checks, all_oriented_steps_present, boundary).

This flag checks successive oriented edges only. An empty list has no step to refute; that alone does not certify a separately required nonempty-walk convention.

View answer and explanation([0, 1, 2, 3, 4, 0], [[0, 1, Yes], [1, 2, Yes], [2, 3, Yes], [3, 4, Yes], [4, 0, Yes]], Yes, edge_orientation_must_match)

For each consecutive pair report its oriented-edge membership and their conjunction. This is a successive-step audit, not an extra simple-path, distinct-vertex or nonempty-walk certificate. An empty list has no step to refute and makes that conjunction vacuously Yes. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 0], [[0, 1, Yes], [1, 2, Yes], [2, 3, Yes], [3, 4, Yes], [4, 0, Yes]], Yes, edge_orientation_must_match).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r07-finite-directed-graph-0068/)

Finite directed graph · qf-mod39-digraph-directed-walk-audit · candidate





## 69. Every Successive Step Must Follow Its Own Directed Edge

Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 1], [1, 2], [1, 3], [2, 3], [3, 4], [4, 0], [4, 1]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Supplied vertex list walk=[0, 1, 2, 3, 4, 0]. Give (walk, step_checks, all_oriented_steps_present, boundary).

This flag checks successive oriented edges only. An empty list has no step to refute; that alone does not certify a separately required nonempty-walk convention.

View answer and explanation([0, 1, 2, 3, 4, 0], [[0, 1, Yes], [1, 2, Yes], [2, 3, Yes], [3, 4, Yes], [4, 0, Yes]], Yes, edge_orientation_must_match)

For each consecutive pair report its oriented-edge membership and their conjunction. This is a successive-step audit, not an extra simple-path, distinct-vertex or nonempty-walk certificate. An empty list has no step to refute and makes that conjunction vacuously Yes. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 0], [[0, 1, Yes], [1, 2, Yes], [2, 3, Yes], [3, 4, Yes], [4, 0, Yes]], Yes, edge_orientation_must_match).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r07-finite-directed-graph-0069/)

Finite directed graph · qf-mod39-digraph-directed-walk-audit · candidate





## 70. Every Successive Step Must Follow Its Own Directed Edge

Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 1], [0, 2], [1, 2], [1, 3], [2, 3], [3, 4], [4, 0], [4, 1]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Supplied vertex list walk=[0, 1, 2, 3, 4, 0]. Give (walk, step_checks, all_oriented_steps_present, boundary).

This flag checks successive oriented edges only. An empty list has no step to refute; that alone does not certify a separately required nonempty-walk convention.

View answer and explanation([0, 1, 2, 3, 4, 0], [[0, 1, Yes], [1, 2, Yes], [2, 3, Yes], [3, 4, Yes], [4, 0, Yes]], Yes, edge_orientation_must_match)

For each consecutive pair report its oriented-edge membership and their conjunction. This is a successive-step audit, not an extra simple-path, distinct-vertex or nonempty-walk certificate. An empty list has no step to refute and makes that conjunction vacuously Yes. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 0], [[0, 1, Yes], [1, 2, Yes], [2, 3, Yes], [3, 4, Yes], [4, 0, Yes]], Yes, edge_orientation_must_match).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r07-finite-directed-graph-0070/)

Finite directed graph · qf-mod39-digraph-directed-walk-audit · candidate





## 71. Every Successive Step Must Follow Its Own Directed Edge

Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 1], [1, 2], [2, 3], [2, 4], [3, 4], [4, 0], [4, 1]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Supplied vertex list walk=[0, 1, 2, 3, 4, 0]. Give (walk, step_checks, all_oriented_steps_present, boundary).

This flag checks successive oriented edges only. An empty list has no step to refute; that alone does not certify a separately required nonempty-walk convention.

View answer and explanation([0, 1, 2, 3, 4, 0], [[0, 1, Yes], [1, 2, Yes], [2, 3, Yes], [3, 4, Yes], [4, 0, Yes]], Yes, edge_orientation_must_match)

For each consecutive pair report its oriented-edge membership and their conjunction. This is a successive-step audit, not an extra simple-path, distinct-vertex or nonempty-walk certificate. An empty list has no step to refute and makes that conjunction vacuously Yes. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 0], [[0, 1, Yes], [1, 2, Yes], [2, 3, Yes], [3, 4, Yes], [4, 0, Yes]], Yes, edge_orientation_must_match).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r07-finite-directed-graph-0071/)

Finite directed graph · qf-mod39-digraph-directed-walk-audit · candidate





## 72. Every Successive Step Must Follow Its Own Directed Edge

Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 1], [0, 2], [1, 2], [2, 3], [2, 4], [3, 4], [4, 0], [4, 1]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Supplied vertex list walk=[0, 1, 2, 3, 4, 0]. Give (walk, step_checks, all_oriented_steps_present, boundary).

This flag checks successive oriented edges only. An empty list has no step to refute; that alone does not certify a separately required nonempty-walk convention.

View answer and explanation([0, 1, 2, 3, 4, 0], [[0, 1, Yes], [1, 2, Yes], [2, 3, Yes], [3, 4, Yes], [4, 0, Yes]], Yes, edge_orientation_must_match)

For each consecutive pair report its oriented-edge membership and their conjunction. This is a successive-step audit, not an extra simple-path, distinct-vertex or nonempty-walk certificate. An empty list has no step to refute and makes that conjunction vacuously Yes. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 0], [[0, 1, Yes], [1, 2, Yes], [2, 3, Yes], [3, 4, Yes], [4, 0, Yes]], Yes, edge_orientation_must_match).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r07-finite-directed-graph-0072/)

Finite directed graph · qf-mod39-digraph-directed-walk-audit · candidate





## 73. Every Successive Step Must Follow Its Own Directed Edge

Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 1], [1, 2], [1, 3], [2, 3], [2, 4], [3, 4], [4, 0], [4, 1]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Supplied vertex list walk=[0, 1, 2, 3, 4, 0]. Give (walk, step_checks, all_oriented_steps_present, boundary).

This flag checks successive oriented edges only. An empty list has no step to refute; that alone does not certify a separately required nonempty-walk convention.

View answer and explanation([0, 1, 2, 3, 4, 0], [[0, 1, Yes], [1, 2, Yes], [2, 3, Yes], [3, 4, Yes], [4, 0, Yes]], Yes, edge_orientation_must_match)

For each consecutive pair report its oriented-edge membership and their conjunction. This is a successive-step audit, not an extra simple-path, distinct-vertex or nonempty-walk certificate. An empty list has no step to refute and makes that conjunction vacuously Yes. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 0], [[0, 1, Yes], [1, 2, Yes], [2, 3, Yes], [3, 4, Yes], [4, 0, Yes]], Yes, edge_orientation_must_match).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r07-finite-directed-graph-0073/)

Finite directed graph · qf-mod39-digraph-directed-walk-audit · candidate





## 74. Every Successive Step Must Follow Its Own Directed Edge

Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 1], [0, 2], [1, 2], [1, 3], [2, 3], [2, 4], [3, 4], [4, 0], [4, 1]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Supplied vertex list walk=[0, 1, 2, 3, 4, 0]. Give (walk, step_checks, all_oriented_steps_present, boundary).

This flag checks successive oriented edges only. An empty list has no step to refute; that alone does not certify a separately required nonempty-walk convention.

View answer and explanation([0, 1, 2, 3, 4, 0], [[0, 1, Yes], [1, 2, Yes], [2, 3, Yes], [3, 4, Yes], [4, 0, Yes]], Yes, edge_orientation_must_match)

For each consecutive pair report its oriented-edge membership and their conjunction. This is a successive-step audit, not an extra simple-path, distinct-vertex or nonempty-walk certificate. An empty list has no step to refute and makes that conjunction vacuously Yes. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 0], [[0, 1, Yes], [1, 2, Yes], [2, 3, Yes], [3, 4, Yes], [4, 0, Yes]], Yes, edge_orientation_must_match).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r07-finite-directed-graph-0074/)

Finite directed graph · qf-mod39-digraph-directed-walk-audit · candidate





## 75. Every Successive Step Must Follow Its Own Directed Edge

Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 1], [1, 2], [2, 3], [3, 0], [3, 4], [4, 0], [4, 1]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Supplied vertex list walk=[0, 1, 2, 3, 4, 0]. Give (walk, step_checks, all_oriented_steps_present, boundary).

This flag checks successive oriented edges only. An empty list has no step to refute; that alone does not certify a separately required nonempty-walk convention.

View answer and explanation([0, 1, 2, 3, 4, 0], [[0, 1, Yes], [1, 2, Yes], [2, 3, Yes], [3, 4, Yes], [4, 0, Yes]], Yes, edge_orientation_must_match)

For each consecutive pair report its oriented-edge membership and their conjunction. This is a successive-step audit, not an extra simple-path, distinct-vertex or nonempty-walk certificate. An empty list has no step to refute and makes that conjunction vacuously Yes. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 0], [[0, 1, Yes], [1, 2, Yes], [2, 3, Yes], [3, 4, Yes], [4, 0, Yes]], Yes, edge_orientation_must_match).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r07-finite-directed-graph-0075/)

Finite directed graph · qf-mod39-digraph-directed-walk-audit · candidate





## 76. A Loop Fails This Loop-Free Model, Not Every Possible Digraph Convention

Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 0], [0, 1], [1, 2], [2, 3], [3, 4], [4, 0]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Give (loops, declared_model_qualified, out_degrees, in_degrees, boundary).

View answer and explanation([[0, 0]], No, [2, 1, 1, 1, 1], [2, 1, 1, 1, 1], loops_rejected_by_declared_loop_free_simple_digraph_convention)

Require at least one loop in this explicit boundary. Report those loops, No qualification under the declared main model and the supplied edge-list in/out counts. A loop contributes once to each directed count here; other loop/multigraph conventions require a different model. The exact result, retaining the requested order, is ([[0, 0]], No, [2, 1, 1, 1, 1], [2, 1, 1, 1, 1], loops_rejected_by_declared_loop_free_simple_digraph_convention).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r07-finite-directed-graph-0076/)

Finite directed graph · qf-mod39-digraph-loop-boundary · candidate





## 77. A Loop Fails This Loop-Free Model, Not Every Possible Digraph Convention

Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 0], [0, 1], [0, 2], [1, 2], [2, 3], [3, 4], [4, 0]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Give (loops, declared_model_qualified, out_degrees, in_degrees, boundary).

View answer and explanation([[0, 0]], No, [3, 1, 1, 1, 1], [2, 1, 2, 1, 1], loops_rejected_by_declared_loop_free_simple_digraph_convention)

Require at least one loop in this explicit boundary. Report those loops, No qualification under the declared main model and the supplied edge-list in/out counts. A loop contributes once to each directed count here; other loop/multigraph conventions require a different model. The exact result, retaining the requested order, is ([[0, 0]], No, [3, 1, 1, 1, 1], [2, 1, 2, 1, 1], loops_rejected_by_declared_loop_free_simple_digraph_convention).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r07-finite-directed-graph-0077/)

Finite directed graph · qf-mod39-digraph-loop-boundary · candidate





## 78. A Loop Fails This Loop-Free Model, Not Every Possible Digraph Convention

Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 0], [0, 1], [1, 2], [1, 3], [2, 3], [3, 4], [4, 0]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Give (loops, declared_model_qualified, out_degrees, in_degrees, boundary).

View answer and explanation([[0, 0]], No, [2, 2, 1, 1, 1], [2, 1, 1, 2, 1], loops_rejected_by_declared_loop_free_simple_digraph_convention)

Require at least one loop in this explicit boundary. Report those loops, No qualification under the declared main model and the supplied edge-list in/out counts. A loop contributes once to each directed count here; other loop/multigraph conventions require a different model. The exact result, retaining the requested order, is ([[0, 0]], No, [2, 2, 1, 1, 1], [2, 1, 1, 2, 1], loops_rejected_by_declared_loop_free_simple_digraph_convention).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r07-finite-directed-graph-0078/)

Finite directed graph · qf-mod39-digraph-loop-boundary · candidate





## 79. A Loop Fails This Loop-Free Model, Not Every Possible Digraph Convention

Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 0], [0, 1], [0, 2], [1, 2], [1, 3], [2, 3], [3, 4], [4, 0]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Give (loops, declared_model_qualified, out_degrees, in_degrees, boundary).

View answer and explanation([[0, 0]], No, [3, 2, 1, 1, 1], [2, 1, 2, 2, 1], loops_rejected_by_declared_loop_free_simple_digraph_convention)

Require at least one loop in this explicit boundary. Report those loops, No qualification under the declared main model and the supplied edge-list in/out counts. A loop contributes once to each directed count here; other loop/multigraph conventions require a different model. The exact result, retaining the requested order, is ([[0, 0]], No, [3, 2, 1, 1, 1], [2, 1, 2, 2, 1], loops_rejected_by_declared_loop_free_simple_digraph_convention).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r07-finite-directed-graph-0079/)

Finite directed graph · qf-mod39-digraph-loop-boundary · candidate





## 80. A Loop Fails This Loop-Free Model, Not Every Possible Digraph Convention

Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 0], [0, 1], [1, 2], [2, 3], [2, 4], [3, 4], [4, 0]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Give (loops, declared_model_qualified, out_degrees, in_degrees, boundary).

View answer and explanation([[0, 0]], No, [2, 1, 2, 1, 1], [2, 1, 1, 1, 2], loops_rejected_by_declared_loop_free_simple_digraph_convention)

Require at least one loop in this explicit boundary. Report those loops, No qualification under the declared main model and the supplied edge-list in/out counts. A loop contributes once to each directed count here; other loop/multigraph conventions require a different model. The exact result, retaining the requested order, is ([[0, 0]], No, [2, 1, 2, 1, 1], [2, 1, 1, 1, 2], loops_rejected_by_declared_loop_free_simple_digraph_convention).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r07-finite-directed-graph-0080/)

Finite directed graph · qf-mod39-digraph-loop-boundary · candidate





## 81. A Loop Fails This Loop-Free Model, Not Every Possible Digraph Convention

Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 0], [0, 1], [0, 2], [1, 2], [2, 3], [2, 4], [3, 4], [4, 0]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Give (loops, declared_model_qualified, out_degrees, in_degrees, boundary).

View answer and explanation([[0, 0]], No, [3, 1, 2, 1, 1], [2, 1, 2, 1, 2], loops_rejected_by_declared_loop_free_simple_digraph_convention)

Require at least one loop in this explicit boundary. Report those loops, No qualification under the declared main model and the supplied edge-list in/out counts. A loop contributes once to each directed count here; other loop/multigraph conventions require a different model. The exact result, retaining the requested order, is ([[0, 0]], No, [3, 1, 2, 1, 1], [2, 1, 2, 1, 2], loops_rejected_by_declared_loop_free_simple_digraph_convention).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r07-finite-directed-graph-0081/)

Finite directed graph · qf-mod39-digraph-loop-boundary · candidate





## 82. A Loop Fails This Loop-Free Model, Not Every Possible Digraph Convention

Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 0], [0, 1], [1, 2], [1, 3], [2, 3], [2, 4], [3, 4], [4, 0]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Give (loops, declared_model_qualified, out_degrees, in_degrees, boundary).

View answer and explanation([[0, 0]], No, [2, 2, 2, 1, 1], [2, 1, 1, 2, 2], loops_rejected_by_declared_loop_free_simple_digraph_convention)

Require at least one loop in this explicit boundary. Report those loops, No qualification under the declared main model and the supplied edge-list in/out counts. A loop contributes once to each directed count here; other loop/multigraph conventions require a different model. The exact result, retaining the requested order, is ([[0, 0]], No, [2, 2, 2, 1, 1], [2, 1, 1, 2, 2], loops_rejected_by_declared_loop_free_simple_digraph_convention).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r07-finite-directed-graph-0082/)

Finite directed graph · qf-mod39-digraph-loop-boundary · candidate





## 83. A Loop Fails This Loop-Free Model, Not Every Possible Digraph Convention

Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 0], [0, 1], [0, 2], [1, 2], [1, 3], [2, 3], [2, 4], [3, 4], [4, 0]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Give (loops, declared_model_qualified, out_degrees, in_degrees, boundary).

View answer and explanation([[0, 0]], No, [3, 2, 2, 1, 1], [2, 1, 2, 2, 2], loops_rejected_by_declared_loop_free_simple_digraph_convention)

Require at least one loop in this explicit boundary. Report those loops, No qualification under the declared main model and the supplied edge-list in/out counts. A loop contributes once to each directed count here; other loop/multigraph conventions require a different model. The exact result, retaining the requested order, is ([[0, 0]], No, [3, 2, 2, 1, 1], [2, 1, 2, 2, 2], loops_rejected_by_declared_loop_free_simple_digraph_convention).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r07-finite-directed-graph-0083/)

Finite directed graph · qf-mod39-digraph-loop-boundary · candidate





## 84. A Loop Fails This Loop-Free Model, Not Every Possible Digraph Convention

Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 0], [0, 1], [1, 2], [2, 3], [3, 0], [3, 4], [4, 0]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Give (loops, declared_model_qualified, out_degrees, in_degrees, boundary).

View answer and explanation([[0, 0]], No, [2, 1, 1, 2, 1], [3, 1, 1, 1, 1], loops_rejected_by_declared_loop_free_simple_digraph_convention)

Require at least one loop in this explicit boundary. Report those loops, No qualification under the declared main model and the supplied edge-list in/out counts. A loop contributes once to each directed count here; other loop/multigraph conventions require a different model. The exact result, retaining the requested order, is ([[0, 0]], No, [2, 1, 1, 2, 1], [3, 1, 1, 1, 1], loops_rejected_by_declared_loop_free_simple_digraph_convention).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r07-finite-directed-graph-0084/)

Finite directed graph · qf-mod39-digraph-loop-boundary · candidate





## 85. A Loop Fails This Loop-Free Model, Not Every Possible Digraph Convention

Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 0], [0, 1], [0, 2], [1, 2], [2, 3], [3, 0], [3, 4], [4, 0]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Give (loops, declared_model_qualified, out_degrees, in_degrees, boundary).

View answer and explanation([[0, 0]], No, [3, 1, 1, 2, 1], [3, 1, 2, 1, 1], loops_rejected_by_declared_loop_free_simple_digraph_convention)

Require at least one loop in this explicit boundary. Report those loops, No qualification under the declared main model and the supplied edge-list in/out counts. A loop contributes once to each directed count here; other loop/multigraph conventions require a different model. The exact result, retaining the requested order, is ([[0, 0]], No, [3, 1, 1, 2, 1], [3, 1, 2, 1, 1], loops_rejected_by_declared_loop_free_simple_digraph_convention).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r07-finite-directed-graph-0085/)

Finite directed graph · qf-mod39-digraph-loop-boundary · candidate





## 86. A Loop Fails This Loop-Free Model, Not Every Possible Digraph Convention

Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 0], [0, 1], [1, 2], [1, 3], [2, 3], [3, 0], [3, 4], [4, 0]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Give (loops, declared_model_qualified, out_degrees, in_degrees, boundary).

View answer and explanation([[0, 0]], No, [2, 2, 1, 2, 1], [3, 1, 1, 2, 1], loops_rejected_by_declared_loop_free_simple_digraph_convention)

Require at least one loop in this explicit boundary. Report those loops, No qualification under the declared main model and the supplied edge-list in/out counts. A loop contributes once to each directed count here; other loop/multigraph conventions require a different model. The exact result, retaining the requested order, is ([[0, 0]], No, [2, 2, 1, 2, 1], [3, 1, 1, 2, 1], loops_rejected_by_declared_loop_free_simple_digraph_convention).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r07-finite-directed-graph-0086/)

Finite directed graph · qf-mod39-digraph-loop-boundary · candidate





## 87. A Loop Fails This Loop-Free Model, Not Every Possible Digraph Convention

Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 0], [0, 1], [0, 2], [1, 2], [1, 3], [2, 3], [3, 0], [3, 4], [4, 0]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Give (loops, declared_model_qualified, out_degrees, in_degrees, boundary).

View answer and explanation([[0, 0]], No, [3, 2, 1, 2, 1], [3, 1, 2, 2, 1], loops_rejected_by_declared_loop_free_simple_digraph_convention)

Require at least one loop in this explicit boundary. Report those loops, No qualification under the declared main model and the supplied edge-list in/out counts. A loop contributes once to each directed count here; other loop/multigraph conventions require a different model. The exact result, retaining the requested order, is ([[0, 0]], No, [3, 2, 1, 2, 1], [3, 1, 2, 2, 1], loops_rejected_by_declared_loop_free_simple_digraph_convention).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r07-finite-directed-graph-0087/)

Finite directed graph · qf-mod39-digraph-loop-boundary · candidate





## 88. A Loop Fails This Loop-Free Model, Not Every Possible Digraph Convention

Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 0], [0, 1], [1, 2], [2, 3], [2, 4], [3, 0], [3, 4], [4, 0]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Give (loops, declared_model_qualified, out_degrees, in_degrees, boundary).

View answer and explanation([[0, 0]], No, [2, 1, 2, 2, 1], [3, 1, 1, 1, 2], loops_rejected_by_declared_loop_free_simple_digraph_convention)

Require at least one loop in this explicit boundary. Report those loops, No qualification under the declared main model and the supplied edge-list in/out counts. A loop contributes once to each directed count here; other loop/multigraph conventions require a different model. The exact result, retaining the requested order, is ([[0, 0]], No, [2, 1, 2, 2, 1], [3, 1, 1, 1, 2], loops_rejected_by_declared_loop_free_simple_digraph_convention).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r07-finite-directed-graph-0088/)

Finite directed graph · qf-mod39-digraph-loop-boundary · candidate





## 89. A Loop Fails This Loop-Free Model, Not Every Possible Digraph Convention

Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 0], [0, 1], [0, 2], [1, 2], [2, 3], [2, 4], [3, 0], [3, 4], [4, 0]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Give (loops, declared_model_qualified, out_degrees, in_degrees, boundary).

View answer and explanation([[0, 0]], No, [3, 1, 2, 2, 1], [3, 1, 2, 1, 2], loops_rejected_by_declared_loop_free_simple_digraph_convention)

Require at least one loop in this explicit boundary. Report those loops, No qualification under the declared main model and the supplied edge-list in/out counts. A loop contributes once to each directed count here; other loop/multigraph conventions require a different model. The exact result, retaining the requested order, is ([[0, 0]], No, [3, 1, 2, 2, 1], [3, 1, 2, 1, 2], loops_rejected_by_declared_loop_free_simple_digraph_convention).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r07-finite-directed-graph-0089/)

Finite directed graph · qf-mod39-digraph-loop-boundary · candidate





## 90. A Loop Fails This Loop-Free Model, Not Every Possible Digraph Convention

Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 0], [0, 1], [1, 2], [1, 3], [2, 3], [2, 4], [3, 0], [3, 4], [4, 0]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Give (loops, declared_model_qualified, out_degrees, in_degrees, boundary).

View answer and explanation([[0, 0]], No, [2, 2, 2, 2, 1], [3, 1, 1, 2, 2], loops_rejected_by_declared_loop_free_simple_digraph_convention)

Require at least one loop in this explicit boundary. Report those loops, No qualification under the declared main model and the supplied edge-list in/out counts. A loop contributes once to each directed count here; other loop/multigraph conventions require a different model. The exact result, retaining the requested order, is ([[0, 0]], No, [2, 2, 2, 2, 1], [3, 1, 1, 2, 2], loops_rejected_by_declared_loop_free_simple_digraph_convention).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r07-finite-directed-graph-0090/)

Finite directed graph · qf-mod39-digraph-loop-boundary · candidate





## 91. A Loop Fails This Loop-Free Model, Not Every Possible Digraph Convention

Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 0], [0, 1], [0, 2], [1, 2], [1, 3], [2, 3], [2, 4], [3, 0], [3, 4], [4, 0]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Give (loops, declared_model_qualified, out_degrees, in_degrees, boundary).

View answer and explanation([[0, 0]], No, [3, 2, 2, 2, 1], [3, 1, 2, 2, 2], loops_rejected_by_declared_loop_free_simple_digraph_convention)

Require at least one loop in this explicit boundary. Report those loops, No qualification under the declared main model and the supplied edge-list in/out counts. A loop contributes once to each directed count here; other loop/multigraph conventions require a different model. The exact result, retaining the requested order, is ([[0, 0]], No, [3, 2, 2, 2, 1], [3, 1, 2, 2, 2], loops_rejected_by_declared_loop_free_simple_digraph_convention).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r07-finite-directed-graph-0091/)

Finite directed graph · qf-mod39-digraph-loop-boundary · candidate





## 92. A Loop Fails This Loop-Free Model, Not Every Possible Digraph Convention

Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 0], [0, 1], [1, 2], [2, 3], [3, 4], [4, 0], [4, 1]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Give (loops, declared_model_qualified, out_degrees, in_degrees, boundary).

View answer and explanation([[0, 0]], No, [2, 1, 1, 1, 2], [2, 2, 1, 1, 1], loops_rejected_by_declared_loop_free_simple_digraph_convention)

Require at least one loop in this explicit boundary. Report those loops, No qualification under the declared main model and the supplied edge-list in/out counts. A loop contributes once to each directed count here; other loop/multigraph conventions require a different model. The exact result, retaining the requested order, is ([[0, 0]], No, [2, 1, 1, 1, 2], [2, 2, 1, 1, 1], loops_rejected_by_declared_loop_free_simple_digraph_convention).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r07-finite-directed-graph-0092/)

Finite directed graph · qf-mod39-digraph-loop-boundary · candidate





## 93. A Loop Fails This Loop-Free Model, Not Every Possible Digraph Convention

Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 0], [0, 1], [0, 2], [1, 2], [2, 3], [3, 4], [4, 0], [4, 1]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Give (loops, declared_model_qualified, out_degrees, in_degrees, boundary).

View answer and explanation([[0, 0]], No, [3, 1, 1, 1, 2], [2, 2, 2, 1, 1], loops_rejected_by_declared_loop_free_simple_digraph_convention)

Require at least one loop in this explicit boundary. Report those loops, No qualification under the declared main model and the supplied edge-list in/out counts. A loop contributes once to each directed count here; other loop/multigraph conventions require a different model. The exact result, retaining the requested order, is ([[0, 0]], No, [3, 1, 1, 1, 2], [2, 2, 2, 1, 1], loops_rejected_by_declared_loop_free_simple_digraph_convention).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r07-finite-directed-graph-0093/)

Finite directed graph · qf-mod39-digraph-loop-boundary · candidate





## 94. A Loop Fails This Loop-Free Model, Not Every Possible Digraph Convention

Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 0], [0, 1], [1, 2], [1, 3], [2, 3], [3, 4], [4, 0], [4, 1]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Give (loops, declared_model_qualified, out_degrees, in_degrees, boundary).

View answer and explanation([[0, 0]], No, [2, 2, 1, 1, 2], [2, 2, 1, 2, 1], loops_rejected_by_declared_loop_free_simple_digraph_convention)

Require at least one loop in this explicit boundary. Report those loops, No qualification under the declared main model and the supplied edge-list in/out counts. A loop contributes once to each directed count here; other loop/multigraph conventions require a different model. The exact result, retaining the requested order, is ([[0, 0]], No, [2, 2, 1, 1, 2], [2, 2, 1, 2, 1], loops_rejected_by_declared_loop_free_simple_digraph_convention).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r07-finite-directed-graph-0094/)

Finite directed graph · qf-mod39-digraph-loop-boundary · candidate





## 95. A Loop Fails This Loop-Free Model, Not Every Possible Digraph Convention

Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 0], [0, 1], [0, 2], [1, 2], [1, 3], [2, 3], [3, 4], [4, 0], [4, 1]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Give (loops, declared_model_qualified, out_degrees, in_degrees, boundary).

View answer and explanation([[0, 0]], No, [3, 2, 1, 1, 2], [2, 2, 2, 2, 1], loops_rejected_by_declared_loop_free_simple_digraph_convention)

Require at least one loop in this explicit boundary. Report those loops, No qualification under the declared main model and the supplied edge-list in/out counts. A loop contributes once to each directed count here; other loop/multigraph conventions require a different model. The exact result, retaining the requested order, is ([[0, 0]], No, [3, 2, 1, 1, 2], [2, 2, 2, 2, 1], loops_rejected_by_declared_loop_free_simple_digraph_convention).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r07-finite-directed-graph-0095/)

Finite directed graph · qf-mod39-digraph-loop-boundary · candidate





## 96. A Loop Fails This Loop-Free Model, Not Every Possible Digraph Convention

Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 0], [0, 1], [1, 2], [2, 3], [2, 4], [3, 4], [4, 0], [4, 1]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Give (loops, declared_model_qualified, out_degrees, in_degrees, boundary).

View answer and explanation([[0, 0]], No, [2, 1, 2, 1, 2], [2, 2, 1, 1, 2], loops_rejected_by_declared_loop_free_simple_digraph_convention)

Require at least one loop in this explicit boundary. Report those loops, No qualification under the declared main model and the supplied edge-list in/out counts. A loop contributes once to each directed count here; other loop/multigraph conventions require a different model. The exact result, retaining the requested order, is ([[0, 0]], No, [2, 1, 2, 1, 2], [2, 2, 1, 1, 2], loops_rejected_by_declared_loop_free_simple_digraph_convention).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r07-finite-directed-graph-0096/)

Finite directed graph · qf-mod39-digraph-loop-boundary · candidate





## 97. A Loop Fails This Loop-Free Model, Not Every Possible Digraph Convention

Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 0], [0, 1], [0, 2], [1, 2], [2, 3], [2, 4], [3, 4], [4, 0], [4, 1]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Give (loops, declared_model_qualified, out_degrees, in_degrees, boundary).

View answer and explanation([[0, 0]], No, [3, 1, 2, 1, 2], [2, 2, 2, 1, 2], loops_rejected_by_declared_loop_free_simple_digraph_convention)

Require at least one loop in this explicit boundary. Report those loops, No qualification under the declared main model and the supplied edge-list in/out counts. A loop contributes once to each directed count here; other loop/multigraph conventions require a different model. The exact result, retaining the requested order, is ([[0, 0]], No, [3, 1, 2, 1, 2], [2, 2, 2, 1, 2], loops_rejected_by_declared_loop_free_simple_digraph_convention).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r07-finite-directed-graph-0097/)

Finite directed graph · qf-mod39-digraph-loop-boundary · candidate





## 98. A Loop Fails This Loop-Free Model, Not Every Possible Digraph Convention

Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 0], [0, 1], [1, 2], [1, 3], [2, 3], [2, 4], [3, 4], [4, 0], [4, 1]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Give (loops, declared_model_qualified, out_degrees, in_degrees, boundary).

View answer and explanation([[0, 0]], No, [2, 2, 2, 1, 2], [2, 2, 1, 2, 2], loops_rejected_by_declared_loop_free_simple_digraph_convention)

Require at least one loop in this explicit boundary. Report those loops, No qualification under the declared main model and the supplied edge-list in/out counts. A loop contributes once to each directed count here; other loop/multigraph conventions require a different model. The exact result, retaining the requested order, is ([[0, 0]], No, [2, 2, 2, 1, 2], [2, 2, 1, 2, 2], loops_rejected_by_declared_loop_free_simple_digraph_convention).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r07-finite-directed-graph-0098/)

Finite directed graph · qf-mod39-digraph-loop-boundary · candidate





## 99. A Loop Fails This Loop-Free Model, Not Every Possible Digraph Convention

Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 0], [0, 1], [0, 2], [1, 2], [1, 3], [2, 3], [2, 4], [3, 4], [4, 0], [4, 1]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Give (loops, declared_model_qualified, out_degrees, in_degrees, boundary).

View answer and explanation([[0, 0]], No, [3, 2, 2, 1, 2], [2, 2, 2, 2, 2], loops_rejected_by_declared_loop_free_simple_digraph_convention)

Require at least one loop in this explicit boundary. Report those loops, No qualification under the declared main model and the supplied edge-list in/out counts. A loop contributes once to each directed count here; other loop/multigraph conventions require a different model. The exact result, retaining the requested order, is ([[0, 0]], No, [3, 2, 2, 1, 2], [2, 2, 2, 2, 2], loops_rejected_by_declared_loop_free_simple_digraph_convention).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r07-finite-directed-graph-0099/)

Finite directed graph · qf-mod39-digraph-loop-boundary · candidate





## 100. A Loop Fails This Loop-Free Model, Not Every Possible Digraph Convention

Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 0], [0, 1], [1, 2], [2, 3], [3, 0], [3, 4], [4, 0], [4, 1]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Give (loops, declared_model_qualified, out_degrees, in_degrees, boundary).

View answer and explanation([[0, 0]], No, [2, 1, 1, 2, 2], [3, 2, 1, 1, 1], loops_rejected_by_declared_loop_free_simple_digraph_convention)

Require at least one loop in this explicit boundary. Report those loops, No qualification under the declared main model and the supplied edge-list in/out counts. A loop contributes once to each directed count here; other loop/multigraph conventions require a different model. The exact result, retaining the requested order, is ([[0, 0]], No, [2, 1, 1, 2, 2], [3, 2, 1, 1, 1], loops_rejected_by_declared_loop_free_simple_digraph_convention).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r07-finite-directed-graph-0100/)

Finite directed graph · qf-mod39-digraph-loop-boundary · candidate



[Canonical question JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20261014-advance-20261108-r07-finite-directed-graph-foundations/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20261014-advance-20261108-r07-finite-directed-graph-foundations.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/batches/batch-20261014-advance-20261108-r07-finite-directed-graph-foundations/validation-report.json)