{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r07-finite-directed-graph-0001","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2cedebcc897ab257b52fbc3fe6d51258546b90dab4b1bce8c8d1538c115fd01e","family_id":"qf-mod39-digraph-qualification","seed":202611087001,"semantic_template":"qf-mod39-digraph-qualification/{\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,0]],\"n\":5}"},"fields":{"title":"A Directed Edge Is an Ordered Pair under the Declared Simple Convention","stem":"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).","answer":{"kind":"expression","value":"(5, [[0, 1], [1, 2], [2, 3], [3, 4], [4, 0]], Yes, Yes, ordered_pairs_and_no_parallel_duplicate_edges)","explanation":"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)."},"skills":["A Directed Edge Is an Ordered Pair under the Declared Simple Convention"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r07-finite-directed-graph-0002","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"372f33b522081990e9cb76268e936a9f976c5c090dd721138762531219dec370","family_id":"qf-mod39-digraph-qualification","seed":202611087002,"semantic_template":"qf-mod39-digraph-qualification/{\"edges\":[[0,1],[0,2],[1,2],[2,3],[3,4],[4,0]],\"n\":5}"},"fields":{"title":"A Directed Edge Is an Ordered Pair under the Declared Simple Convention","stem":"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).","answer":{"kind":"expression","value":"(5, [[0, 1], [0, 2], [1, 2], [2, 3], [3, 4], [4, 0]], Yes, Yes, ordered_pairs_and_no_parallel_duplicate_edges)","explanation":"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)."},"skills":["A Directed Edge Is an Ordered Pair under the Declared Simple Convention"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r07-finite-directed-graph-0003","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c0795aba921b0c19ca9144030dd0ae679e3fc93f5b8208e31eec831ccae0a85e","family_id":"qf-mod39-digraph-qualification","seed":202611087003,"semantic_template":"qf-mod39-digraph-qualification/{\"edges\":[[0,1],[1,2],[1,3],[2,3],[3,4],[4,0]],\"n\":5}"},"fields":{"title":"A Directed Edge Is an Ordered Pair under the Declared Simple Convention","stem":"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).","answer":{"kind":"expression","value":"(5, [[0, 1], [1, 2], [1, 3], [2, 3], [3, 4], [4, 0]], Yes, Yes, ordered_pairs_and_no_parallel_duplicate_edges)","explanation":"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)."},"skills":["A Directed Edge Is an Ordered Pair under the Declared Simple Convention"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r07-finite-directed-graph-0004","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9f9a8350f1e3e515af2c148a0da0eaeec11a20d7bb05d1418f6683c05f694740","family_id":"qf-mod39-digraph-qualification","seed":202611087004,"semantic_template":"qf-mod39-digraph-qualification/{\"edges\":[[0,1],[0,2],[1,2],[1,3],[2,3],[3,4],[4,0]],\"n\":5}"},"fields":{"title":"A Directed Edge Is an Ordered Pair under the Declared Simple Convention","stem":"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).","answer":{"kind":"expression","value":"(5, [[0, 1], [0, 2], [1, 2], [1, 3], [2, 3], [3, 4], [4, 0]], Yes, Yes, ordered_pairs_and_no_parallel_duplicate_edges)","explanation":"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)."},"skills":["A Directed Edge Is an Ordered Pair under the Declared Simple Convention"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r07-finite-directed-graph-0005","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ffba404d25d91615ce353055bd81e6b5f05f1580647361094375d6e3b8cf1344","family_id":"qf-mod39-digraph-qualification","seed":202611087005,"semantic_template":"qf-mod39-digraph-qualification/{\"edges\":[[0,1],[1,2],[2,3],[2,4],[3,4],[4,0]],\"n\":5}"},"fields":{"title":"A Directed Edge Is an Ordered Pair under the Declared Simple Convention","stem":"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).","answer":{"kind":"expression","value":"(5, [[0, 1], [1, 2], [2, 3], [2, 4], [3, 4], [4, 0]], Yes, Yes, ordered_pairs_and_no_parallel_duplicate_edges)","explanation":"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)."},"skills":["A Directed Edge Is an Ordered Pair under the Declared Simple Convention"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r07-finite-directed-graph-0006","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"48d7ff914137ab50643b33202948ea1555d74437cb2b16687928832833f2a580","family_id":"qf-mod39-digraph-qualification","seed":202611087006,"semantic_template":"qf-mod39-digraph-qualification/{\"edges\":[[0,1],[0,2],[1,2],[2,3],[2,4],[3,4],[4,0]],\"n\":5}"},"fields":{"title":"A Directed Edge Is an Ordered Pair under the Declared Simple Convention","stem":"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).","answer":{"kind":"expression","value":"(5, [[0, 1], [0, 2], [1, 2], [2, 3], [2, 4], [3, 4], [4, 0]], Yes, Yes, ordered_pairs_and_no_parallel_duplicate_edges)","explanation":"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)."},"skills":["A Directed Edge Is an Ordered Pair under the Declared Simple Convention"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r07-finite-directed-graph-0007","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0dd1d16c453ef4d4dca3f66f9e094df1a3366dcb1531f042c92b1b996f54f966","family_id":"qf-mod39-digraph-qualification","seed":202611087007,"semantic_template":"qf-mod39-digraph-qualification/{\"edges\":[[0,1],[1,2],[1,3],[2,3],[2,4],[3,4],[4,0]],\"n\":5}"},"fields":{"title":"A Directed Edge Is an Ordered Pair under the Declared Simple Convention","stem":"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).","answer":{"kind":"expression","value":"(5, [[0, 1], [1, 2], [1, 3], [2, 3], [2, 4], [3, 4], [4, 0]], Yes, Yes, ordered_pairs_and_no_parallel_duplicate_edges)","explanation":"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)."},"skills":["A Directed Edge Is an Ordered Pair under the Declared Simple Convention"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r07-finite-directed-graph-0008","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"715817f97d3afb6f787f660c124b2d268ffe4dada01f5f4ccd4b897fb629e53d","family_id":"qf-mod39-digraph-qualification","seed":202611087008,"semantic_template":"qf-mod39-digraph-qualification/{\"edges\":[[0,1],[0,2],[1,2],[1,3],[2,3],[2,4],[3,4],[4,0]],\"n\":5}"},"fields":{"title":"A Directed Edge Is an Ordered Pair under the Declared Simple Convention","stem":"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).","answer":{"kind":"expression","value":"(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)","explanation":"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)."},"skills":["A Directed Edge Is an Ordered Pair under the Declared Simple Convention"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r07-finite-directed-graph-0009","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6f7147d1efdb30238cf692a1f7843c159a14df3094e84552e7f28f67acbb45bb","family_id":"qf-mod39-digraph-qualification","seed":202611087009,"semantic_template":"qf-mod39-digraph-qualification/{\"edges\":[[0,1],[1,2],[2,3],[3,0],[3,4],[4,0]],\"n\":5}"},"fields":{"title":"A Directed Edge Is an Ordered Pair under the Declared Simple Convention","stem":"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).","answer":{"kind":"expression","value":"(5, [[0, 1], [1, 2], [2, 3], [3, 0], [3, 4], [4, 0]], Yes, Yes, ordered_pairs_and_no_parallel_duplicate_edges)","explanation":"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)."},"skills":["A Directed Edge Is an Ordered Pair under the Declared Simple Convention"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r07-finite-directed-graph-0010","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7feb7656aa635ad86af7da7cf4e6e620d72e7cc815494b2fd2436ab2fe0c2d80","family_id":"qf-mod39-digraph-qualification","seed":202611087010,"semantic_template":"qf-mod39-digraph-qualification/{\"edges\":[[0,1],[0,2],[1,2],[2,3],[3,0],[3,4],[4,0]],\"n\":5}"},"fields":{"title":"A Directed Edge Is an Ordered Pair under the Declared Simple Convention","stem":"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).","answer":{"kind":"expression","value":"(5, [[0, 1], [0, 2], [1, 2], [2, 3], [3, 0], [3, 4], [4, 0]], Yes, Yes, ordered_pairs_and_no_parallel_duplicate_edges)","explanation":"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)."},"skills":["A Directed Edge Is an Ordered Pair under the Declared Simple Convention"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r07-finite-directed-graph-0011","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"94af40d636008b5277893e79c18789651585afbe6e05b8d8416227320d3a4bbe","family_id":"qf-mod39-digraph-qualification","seed":202611087011,"semantic_template":"qf-mod39-digraph-qualification/{\"edges\":[[0,1],[1,2],[1,3],[2,3],[3,0],[3,4],[4,0]],\"n\":5}"},"fields":{"title":"A Directed Edge Is an Ordered Pair under the Declared Simple Convention","stem":"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).","answer":{"kind":"expression","value":"(5, [[0, 1], [1, 2], [1, 3], [2, 3], [3, 0], [3, 4], [4, 0]], Yes, Yes, ordered_pairs_and_no_parallel_duplicate_edges)","explanation":"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)."},"skills":["A Directed Edge Is an Ordered Pair under the Declared Simple Convention"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r07-finite-directed-graph-0012","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9d9dac2800f44a2bf126cec57f2d50fd97743eb9de619125508deb2f334eef9a","family_id":"qf-mod39-digraph-qualification","seed":202611087012,"semantic_template":"qf-mod39-digraph-qualification/{\"edges\":[[0,1],[0,2],[1,2],[1,3],[2,3],[3,0],[3,4],[4,0]],\"n\":5}"},"fields":{"title":"A Directed Edge Is an Ordered Pair under the Declared Simple Convention","stem":"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).","answer":{"kind":"expression","value":"(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)","explanation":"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)."},"skills":["A Directed Edge Is an Ordered Pair under the Declared Simple Convention"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r07-finite-directed-graph-0013","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"27af73f3abf51e4d169bda9365a53bf04a0a2668a4a81c4cfa5bf6fab94d146d","family_id":"qf-mod39-digraph-qualification","seed":202611087013,"semantic_template":"qf-mod39-digraph-qualification/{\"edges\":[[0,1],[1,2],[2,3],[2,4],[3,0],[3,4],[4,0]],\"n\":5}"},"fields":{"title":"A Directed Edge Is an Ordered Pair under the Declared Simple Convention","stem":"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).","answer":{"kind":"expression","value":"(5, [[0, 1], [1, 2], [2, 3], [2, 4], [3, 0], [3, 4], [4, 0]], Yes, Yes, ordered_pairs_and_no_parallel_duplicate_edges)","explanation":"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)."},"skills":["A Directed Edge Is an Ordered Pair under the Declared Simple Convention"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r07-finite-directed-graph-0014","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"54b92a29b53333386dc32e58da059b16f2fa5751076b287ef9258cfcd5d98626","family_id":"qf-mod39-digraph-qualification","seed":202611087014,"semantic_template":"qf-mod39-digraph-qualification/{\"edges\":[[0,1],[0,2],[1,2],[2,3],[2,4],[3,0],[3,4],[4,0]],\"n\":5}"},"fields":{"title":"A Directed Edge Is an Ordered Pair under the Declared Simple Convention","stem":"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).","answer":{"kind":"expression","value":"(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)","explanation":"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)."},"skills":["A Directed Edge Is an Ordered Pair under the Declared Simple Convention"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r07-finite-directed-graph-0015","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dec87ca458a42c943d34194668111517167a3a232ee26a778a3934a9081ab5a1","family_id":"qf-mod39-digraph-qualification","seed":202611087015,"semantic_template":"qf-mod39-digraph-qualification/{\"edges\":[[0,1],[1,2],[1,3],[2,3],[2,4],[3,0],[3,4],[4,0]],\"n\":5}"},"fields":{"title":"A Directed Edge Is an Ordered Pair under the Declared Simple Convention","stem":"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).","answer":{"kind":"expression","value":"(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)","explanation":"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)."},"skills":["A Directed Edge Is an Ordered Pair under the Declared Simple Convention"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r07-finite-directed-graph-0016","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8556d5a095788fc1bf0afdcad13e39717ad0fa08ed55e9b0d7d1238100ebe244","family_id":"qf-mod39-digraph-qualification","seed":202611087016,"semantic_template":"qf-mod39-digraph-qualification/{\"edges\":[[0,1],[0,2],[1,2],[1,3],[2,3],[2,4],[3,0],[3,4],[4,0]],\"n\":5}"},"fields":{"title":"A Directed Edge Is an Ordered Pair under the Declared Simple Convention","stem":"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).","answer":{"kind":"expression","value":"(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)","explanation":"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)."},"skills":["A Directed Edge Is an Ordered Pair under the Declared Simple Convention"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r07-finite-directed-graph-0017","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bc69f4bf6fe52e6c5d1f211578f0225de0fc6e95ec10dca690f39ed289296273","family_id":"qf-mod39-digraph-qualification","seed":202611087017,"semantic_template":"qf-mod39-digraph-qualification/{\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,0],[4,1]],\"n\":5}"},"fields":{"title":"A Directed Edge Is an Ordered Pair under the Declared Simple Convention","stem":"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).","answer":{"kind":"expression","value":"(5, [[0, 1], [1, 2], [2, 3], [3, 4], [4, 0], [4, 1]], Yes, Yes, ordered_pairs_and_no_parallel_duplicate_edges)","explanation":"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)."},"skills":["A Directed Edge Is an Ordered Pair under the Declared Simple Convention"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r07-finite-directed-graph-0018","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6f06b409a60f839b733291dcdd13cad329d214400fcba2814bc0745f748ded0d","family_id":"qf-mod39-digraph-qualification","seed":202611087018,"semantic_template":"qf-mod39-digraph-qualification/{\"edges\":[[0,1],[0,2],[1,2],[2,3],[3,4],[4,0],[4,1]],\"n\":5}"},"fields":{"title":"A Directed Edge Is an Ordered Pair under the Declared Simple Convention","stem":"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).","answer":{"kind":"expression","value":"(5, [[0, 1], [0, 2], [1, 2], [2, 3], [3, 4], [4, 0], [4, 1]], Yes, Yes, ordered_pairs_and_no_parallel_duplicate_edges)","explanation":"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)."},"skills":["A Directed Edge Is an Ordered Pair under the Declared Simple Convention"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r07-finite-directed-graph-0019","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"55bae9a6322a7c2ac4b63268b8b787fa5df17589f95adff2c09627c5ba567ecf","family_id":"qf-mod39-digraph-qualification","seed":202611087019,"semantic_template":"qf-mod39-digraph-qualification/{\"edges\":[[0,1],[1,2],[1,3],[2,3],[3,4],[4,0],[4,1]],\"n\":5}"},"fields":{"title":"A Directed Edge Is an Ordered Pair under the Declared Simple Convention","stem":"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).","answer":{"kind":"expression","value":"(5, [[0, 1], [1, 2], [1, 3], [2, 3], [3, 4], [4, 0], [4, 1]], Yes, Yes, ordered_pairs_and_no_parallel_duplicate_edges)","explanation":"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)."},"skills":["A Directed Edge Is an Ordered Pair under the Declared Simple Convention"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r07-finite-directed-graph-0020","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5fe1f41074882538c5bd06ea16ebe82b0ff863b153153b1f0bb786ec206f769a","family_id":"qf-mod39-digraph-qualification","seed":202611087020,"semantic_template":"qf-mod39-digraph-qualification/{\"edges\":[[0,1],[0,2],[1,2],[1,3],[2,3],[3,4],[4,0],[4,1]],\"n\":5}"},"fields":{"title":"A Directed Edge Is an Ordered Pair under the Declared Simple Convention","stem":"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).","answer":{"kind":"expression","value":"(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)","explanation":"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)."},"skills":["A Directed Edge Is an Ordered Pair under the Declared Simple Convention"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r07-finite-directed-graph-0021","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3c9404e66a4bf2b3a3718db98d75fb2900153a0a569fd46379c208b3f88cf44b","family_id":"qf-mod39-digraph-qualification","seed":202611087021,"semantic_template":"qf-mod39-digraph-qualification/{\"edges\":[[0,1],[1,2],[2,3],[2,4],[3,4],[4,0],[4,1]],\"n\":5}"},"fields":{"title":"A Directed Edge Is an Ordered Pair under the Declared Simple Convention","stem":"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).","answer":{"kind":"expression","value":"(5, [[0, 1], [1, 2], [2, 3], [2, 4], [3, 4], [4, 0], [4, 1]], Yes, Yes, ordered_pairs_and_no_parallel_duplicate_edges)","explanation":"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)."},"skills":["A Directed Edge Is an Ordered Pair under the Declared Simple Convention"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r07-finite-directed-graph-0022","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"221c86efe00107bb6cf7757fc82782a1ac064fa1090c0f74cd1330adf6d28f7e","family_id":"qf-mod39-digraph-qualification","seed":202611087022,"semantic_template":"qf-mod39-digraph-qualification/{\"edges\":[[0,1],[0,2],[1,2],[2,3],[2,4],[3,4],[4,0],[4,1]],\"n\":5}"},"fields":{"title":"A Directed Edge Is an Ordered Pair under the Declared Simple Convention","stem":"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).","answer":{"kind":"expression","value":"(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)","explanation":"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)."},"skills":["A Directed Edge Is an Ordered Pair under the Declared Simple Convention"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r07-finite-directed-graph-0023","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7495c5f73ddb8a9fff3c9c042623b99170f6e8029792a9b36427dc6e58bc630a","family_id":"qf-mod39-digraph-qualification","seed":202611087023,"semantic_template":"qf-mod39-digraph-qualification/{\"edges\":[[0,1],[1,2],[1,3],[2,3],[2,4],[3,4],[4,0],[4,1]],\"n\":5}"},"fields":{"title":"A Directed Edge Is an Ordered Pair under the Declared Simple Convention","stem":"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).","answer":{"kind":"expression","value":"(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)","explanation":"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)."},"skills":["A Directed Edge Is an Ordered Pair under the Declared Simple Convention"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r07-finite-directed-graph-0024","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"68ae5e3857c38292bcf252aa759211fcc9cb3c96df1061a43a5985dd69511367","family_id":"qf-mod39-digraph-qualification","seed":202611087024,"semantic_template":"qf-mod39-digraph-qualification/{\"edges\":[[0,1],[0,2],[1,2],[1,3],[2,3],[2,4],[3,4],[4,0],[4,1]],\"n\":5}"},"fields":{"title":"A Directed Edge Is an Ordered Pair under the Declared Simple Convention","stem":"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).","answer":{"kind":"expression","value":"(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)","explanation":"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)."},"skills":["A Directed Edge Is an Ordered Pair under the Declared Simple Convention"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r07-finite-directed-graph-0025","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f36488626c9afdaa1ae897b7e0e58558404d6d63c61ccd9523639cc57b9b68fb","family_id":"qf-mod39-digraph-qualification","seed":202611087025,"semantic_template":"qf-mod39-digraph-qualification/{\"edges\":[[0,1],[1,2],[2,3],[3,0],[3,4],[4,0],[4,1]],\"n\":5}"},"fields":{"title":"A Directed Edge Is an Ordered Pair under the Declared Simple Convention","stem":"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).","answer":{"kind":"expression","value":"(5, [[0, 1], [1, 2], [2, 3], [3, 0], [3, 4], [4, 0], [4, 1]], Yes, Yes, ordered_pairs_and_no_parallel_duplicate_edges)","explanation":"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)."},"skills":["A Directed Edge Is an Ordered Pair under the Declared Simple Convention"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r07-finite-directed-graph-0026","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"146eae595c87b18416d08cc0dce83120151faad568ffdfd1ab0d530728cf0f87","family_id":"qf-mod39-digraph-in-out-degree","seed":202611087026,"semantic_template":"qf-mod39-digraph-in-out-degree/{\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,0]],\"n\":5}"},"fields":{"title":"In-Degrees and Out-Degrees Each Sum to the Edge Count","stem":"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).","answer":{"kind":"expression","value":"([1, 1, 1, 1, 1], [1, 1, 1, 1, 1], 5, 5, 5, Yes)","explanation":"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)."},"skills":["In-Degrees and Out-Degrees Each Sum to the Edge Count"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r07-finite-directed-graph-0027","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ce0f38fa1bd683cf11d25159b41bd0663b9dbd9ed14836d2afd3b9ffdbf02f8c","family_id":"qf-mod39-digraph-in-out-degree","seed":202611087027,"semantic_template":"qf-mod39-digraph-in-out-degree/{\"edges\":[[0,1],[0,2],[1,2],[2,3],[3,4],[4,0]],\"n\":5}"},"fields":{"title":"In-Degrees and Out-Degrees Each Sum to the Edge Count","stem":"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).","answer":{"kind":"expression","value":"([2, 1, 1, 1, 1], [1, 1, 2, 1, 1], 6, 6, 6, Yes)","explanation":"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)."},"skills":["In-Degrees and Out-Degrees Each Sum to the Edge Count"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r07-finite-directed-graph-0028","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b4abc97341bb17b33dce0c5da661225dbb66cefd45e16f3181e33affe54c317e","family_id":"qf-mod39-digraph-in-out-degree","seed":202611087028,"semantic_template":"qf-mod39-digraph-in-out-degree/{\"edges\":[[0,1],[1,2],[1,3],[2,3],[3,4],[4,0]],\"n\":5}"},"fields":{"title":"In-Degrees and Out-Degrees Each Sum to the Edge Count","stem":"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).","answer":{"kind":"expression","value":"([1, 2, 1, 1, 1], [1, 1, 1, 2, 1], 6, 6, 6, Yes)","explanation":"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)."},"skills":["In-Degrees and Out-Degrees Each Sum to the Edge Count"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r07-finite-directed-graph-0029","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4055a0a145c14448ed5cad485b85ee11cbd1c62893bdf7ec94a9872083b2ca58","family_id":"qf-mod39-digraph-in-out-degree","seed":202611087029,"semantic_template":"qf-mod39-digraph-in-out-degree/{\"edges\":[[0,1],[0,2],[1,2],[1,3],[2,3],[3,4],[4,0]],\"n\":5}"},"fields":{"title":"In-Degrees and Out-Degrees Each Sum to the Edge Count","stem":"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).","answer":{"kind":"expression","value":"([2, 2, 1, 1, 1], [1, 1, 2, 2, 1], 7, 7, 7, Yes)","explanation":"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)."},"skills":["In-Degrees and Out-Degrees Each Sum to the Edge Count"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r07-finite-directed-graph-0030","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"63b8093f89f73f82da13da20b350896d687502f1bdfb5cb12ee9947465898a14","family_id":"qf-mod39-digraph-in-out-degree","seed":202611087030,"semantic_template":"qf-mod39-digraph-in-out-degree/{\"edges\":[[0,1],[1,2],[2,3],[2,4],[3,4],[4,0]],\"n\":5}"},"fields":{"title":"In-Degrees and Out-Degrees Each Sum to the Edge Count","stem":"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).","answer":{"kind":"expression","value":"([1, 1, 2, 1, 1], [1, 1, 1, 1, 2], 6, 6, 6, Yes)","explanation":"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)."},"skills":["In-Degrees and Out-Degrees Each Sum to the Edge Count"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r07-finite-directed-graph-0031","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4da9168afa7069bf1fe93bf093c2fbd111ba0a1ce442ce58c5511f3beb301a92","family_id":"qf-mod39-digraph-in-out-degree","seed":202611087031,"semantic_template":"qf-mod39-digraph-in-out-degree/{\"edges\":[[0,1],[0,2],[1,2],[2,3],[2,4],[3,4],[4,0]],\"n\":5}"},"fields":{"title":"In-Degrees and Out-Degrees Each Sum to the Edge Count","stem":"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).","answer":{"kind":"expression","value":"([2, 1, 2, 1, 1], [1, 1, 2, 1, 2], 7, 7, 7, Yes)","explanation":"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)."},"skills":["In-Degrees and Out-Degrees Each Sum to the Edge Count"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r07-finite-directed-graph-0032","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"08b6ac030e78217109ce8667d2bf656bf2ecec5517eade9ac11fa941fbf0a8e8","family_id":"qf-mod39-digraph-in-out-degree","seed":202611087032,"semantic_template":"qf-mod39-digraph-in-out-degree/{\"edges\":[[0,1],[1,2],[1,3],[2,3],[2,4],[3,4],[4,0]],\"n\":5}"},"fields":{"title":"In-Degrees and Out-Degrees Each Sum to the Edge Count","stem":"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).","answer":{"kind":"expression","value":"([1, 2, 2, 1, 1], [1, 1, 1, 2, 2], 7, 7, 7, Yes)","explanation":"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)."},"skills":["In-Degrees and Out-Degrees Each Sum to the Edge Count"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r07-finite-directed-graph-0033","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"42b23dd1a8be88cbffbbd44bd6ddccc4415e0ccfc5973cb25ef8ec9c2c9959f3","family_id":"qf-mod39-digraph-in-out-degree","seed":202611087033,"semantic_template":"qf-mod39-digraph-in-out-degree/{\"edges\":[[0,1],[0,2],[1,2],[1,3],[2,3],[2,4],[3,4],[4,0]],\"n\":5}"},"fields":{"title":"In-Degrees and Out-Degrees Each Sum to the Edge Count","stem":"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).","answer":{"kind":"expression","value":"([2, 2, 2, 1, 1], [1, 1, 2, 2, 2], 8, 8, 8, Yes)","explanation":"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)."},"skills":["In-Degrees and Out-Degrees Each Sum to the Edge Count"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r07-finite-directed-graph-0034","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8f2d8a1d65e84a31d0e836a883e43c842f3f9492da64901803764a74dbb00d4f","family_id":"qf-mod39-digraph-in-out-degree","seed":202611087034,"semantic_template":"qf-mod39-digraph-in-out-degree/{\"edges\":[[0,1],[1,2],[2,3],[3,0],[3,4],[4,0]],\"n\":5}"},"fields":{"title":"In-Degrees and Out-Degrees Each Sum to the Edge Count","stem":"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).","answer":{"kind":"expression","value":"([1, 1, 1, 2, 1], [2, 1, 1, 1, 1], 6, 6, 6, Yes)","explanation":"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)."},"skills":["In-Degrees and Out-Degrees Each Sum to the Edge Count"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r07-finite-directed-graph-0035","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"099be4f77c96a49c350bb3496a9948f2608505b337af1fc115eaab4750ca8207","family_id":"qf-mod39-digraph-in-out-degree","seed":202611087035,"semantic_template":"qf-mod39-digraph-in-out-degree/{\"edges\":[[0,1],[0,2],[1,2],[2,3],[3,0],[3,4],[4,0]],\"n\":5}"},"fields":{"title":"In-Degrees and Out-Degrees Each Sum to the Edge Count","stem":"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).","answer":{"kind":"expression","value":"([2, 1, 1, 2, 1], [2, 1, 2, 1, 1], 7, 7, 7, Yes)","explanation":"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)."},"skills":["In-Degrees and Out-Degrees Each Sum to the Edge Count"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r07-finite-directed-graph-0036","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6c86a4bb95fb8a41e0efcf2f6c0efdad0f053c0e0f3e053b1f1a1a3e5ca1dae1","family_id":"qf-mod39-digraph-in-out-degree","seed":202611087036,"semantic_template":"qf-mod39-digraph-in-out-degree/{\"edges\":[[0,1],[1,2],[1,3],[2,3],[3,0],[3,4],[4,0]],\"n\":5}"},"fields":{"title":"In-Degrees and Out-Degrees Each Sum to the Edge Count","stem":"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).","answer":{"kind":"expression","value":"([1, 2, 1, 2, 1], [2, 1, 1, 2, 1], 7, 7, 7, Yes)","explanation":"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)."},"skills":["In-Degrees and Out-Degrees Each Sum to the Edge Count"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r07-finite-directed-graph-0037","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"196a1b0a3abde0510b35ee32a996ca02551c446106073f930cff2fc94dba7e9d","family_id":"qf-mod39-digraph-in-out-degree","seed":202611087037,"semantic_template":"qf-mod39-digraph-in-out-degree/{\"edges\":[[0,1],[0,2],[1,2],[1,3],[2,3],[3,0],[3,4],[4,0]],\"n\":5}"},"fields":{"title":"In-Degrees and Out-Degrees Each Sum to the Edge Count","stem":"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).","answer":{"kind":"expression","value":"([2, 2, 1, 2, 1], [2, 1, 2, 2, 1], 8, 8, 8, Yes)","explanation":"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)."},"skills":["In-Degrees and Out-Degrees Each Sum to the Edge Count"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r07-finite-directed-graph-0038","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6e82faf895215d1320f5df873e7743c0438b2dbc5bbc2a4547c44a363eb32f6a","family_id":"qf-mod39-digraph-in-out-degree","seed":202611087038,"semantic_template":"qf-mod39-digraph-in-out-degree/{\"edges\":[[0,1],[1,2],[2,3],[2,4],[3,0],[3,4],[4,0]],\"n\":5}"},"fields":{"title":"In-Degrees and Out-Degrees Each Sum to the Edge Count","stem":"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).","answer":{"kind":"expression","value":"([1, 1, 2, 2, 1], [2, 1, 1, 1, 2], 7, 7, 7, Yes)","explanation":"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)."},"skills":["In-Degrees and Out-Degrees Each Sum to the Edge Count"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r07-finite-directed-graph-0039","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"64fb3d5a73b1d63e8223dac89e8ac3736fdb3500a51df6cdd908f554003d87e6","family_id":"qf-mod39-digraph-in-out-degree","seed":202611087039,"semantic_template":"qf-mod39-digraph-in-out-degree/{\"edges\":[[0,1],[0,2],[1,2],[2,3],[2,4],[3,0],[3,4],[4,0]],\"n\":5}"},"fields":{"title":"In-Degrees and Out-Degrees Each Sum to the Edge Count","stem":"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).","answer":{"kind":"expression","value":"([2, 1, 2, 2, 1], [2, 1, 2, 1, 2], 8, 8, 8, Yes)","explanation":"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)."},"skills":["In-Degrees and Out-Degrees Each Sum to the Edge Count"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r07-finite-directed-graph-0040","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9cc7af4160128eeb017814cdbcc1e1278e9ac5752551f15d733a40871f2023f2","family_id":"qf-mod39-digraph-in-out-degree","seed":202611087040,"semantic_template":"qf-mod39-digraph-in-out-degree/{\"edges\":[[0,1],[1,2],[1,3],[2,3],[2,4],[3,0],[3,4],[4,0]],\"n\":5}"},"fields":{"title":"In-Degrees and Out-Degrees Each Sum to the Edge Count","stem":"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).","answer":{"kind":"expression","value":"([1, 2, 2, 2, 1], [2, 1, 1, 2, 2], 8, 8, 8, Yes)","explanation":"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)."},"skills":["In-Degrees and Out-Degrees Each Sum to the Edge Count"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r07-finite-directed-graph-0041","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1d33bf6de00ba27f907148d051ec153f58ac8ac01c4db21e6023c50cc835ab95","family_id":"qf-mod39-digraph-in-out-degree","seed":202611087041,"semantic_template":"qf-mod39-digraph-in-out-degree/{\"edges\":[[0,1],[0,2],[1,2],[1,3],[2,3],[2,4],[3,0],[3,4],[4,0]],\"n\":5}"},"fields":{"title":"In-Degrees and Out-Degrees Each Sum to the Edge Count","stem":"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).","answer":{"kind":"expression","value":"([2, 2, 2, 2, 1], [2, 1, 2, 2, 2], 9, 9, 9, Yes)","explanation":"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)."},"skills":["In-Degrees and Out-Degrees Each Sum to the Edge Count"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r07-finite-directed-graph-0042","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e4f8ff5b74622c5da849dce2b802f19dbe442aa5700975d1ee0d2711f1d88ba7","family_id":"qf-mod39-digraph-in-out-degree","seed":202611087042,"semantic_template":"qf-mod39-digraph-in-out-degree/{\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,0],[4,1]],\"n\":5}"},"fields":{"title":"In-Degrees and Out-Degrees Each Sum to the Edge Count","stem":"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).","answer":{"kind":"expression","value":"([1, 1, 1, 1, 2], [1, 2, 1, 1, 1], 6, 6, 6, Yes)","explanation":"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)."},"skills":["In-Degrees and Out-Degrees Each Sum to the Edge Count"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r07-finite-directed-graph-0043","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"433006d722b0b7d8268fc156ff807d69ea41b7cd24cd622f7e5b97a4fc2c9c11","family_id":"qf-mod39-digraph-in-out-degree","seed":202611087043,"semantic_template":"qf-mod39-digraph-in-out-degree/{\"edges\":[[0,1],[0,2],[1,2],[2,3],[3,4],[4,0],[4,1]],\"n\":5}"},"fields":{"title":"In-Degrees and Out-Degrees Each Sum to the Edge Count","stem":"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).","answer":{"kind":"expression","value":"([2, 1, 1, 1, 2], [1, 2, 2, 1, 1], 7, 7, 7, Yes)","explanation":"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)."},"skills":["In-Degrees and Out-Degrees Each Sum to the Edge Count"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r07-finite-directed-graph-0044","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0098e6d4d9b83ab903579c5a63a166a1bef93ac0ecb48b787ee670a224b27285","family_id":"qf-mod39-digraph-in-out-degree","seed":202611087044,"semantic_template":"qf-mod39-digraph-in-out-degree/{\"edges\":[[0,1],[1,2],[1,3],[2,3],[3,4],[4,0],[4,1]],\"n\":5}"},"fields":{"title":"In-Degrees and Out-Degrees Each Sum to the Edge Count","stem":"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).","answer":{"kind":"expression","value":"([1, 2, 1, 1, 2], [1, 2, 1, 2, 1], 7, 7, 7, Yes)","explanation":"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)."},"skills":["In-Degrees and Out-Degrees Each Sum to the Edge Count"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r07-finite-directed-graph-0045","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ca9f2c75af9a4a91d72af77bc97c64750f26f522bc16b7dd0636393b02dbd699","family_id":"qf-mod39-digraph-in-out-degree","seed":202611087045,"semantic_template":"qf-mod39-digraph-in-out-degree/{\"edges\":[[0,1],[0,2],[1,2],[1,3],[2,3],[3,4],[4,0],[4,1]],\"n\":5}"},"fields":{"title":"In-Degrees and Out-Degrees Each Sum to the Edge Count","stem":"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).","answer":{"kind":"expression","value":"([2, 2, 1, 1, 2], [1, 2, 2, 2, 1], 8, 8, 8, Yes)","explanation":"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)."},"skills":["In-Degrees and Out-Degrees Each Sum to the Edge Count"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r07-finite-directed-graph-0046","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6d8f349c267da3b4d1c6bcd41fa5707216dfa1a64ce19d6bb7789ff053120ee8","family_id":"qf-mod39-digraph-in-out-degree","seed":202611087046,"semantic_template":"qf-mod39-digraph-in-out-degree/{\"edges\":[[0,1],[1,2],[2,3],[2,4],[3,4],[4,0],[4,1]],\"n\":5}"},"fields":{"title":"In-Degrees and Out-Degrees Each Sum to the Edge Count","stem":"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).","answer":{"kind":"expression","value":"([1, 1, 2, 1, 2], [1, 2, 1, 1, 2], 7, 7, 7, Yes)","explanation":"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)."},"skills":["In-Degrees and Out-Degrees Each Sum to the Edge Count"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r07-finite-directed-graph-0047","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ea21a3fa3e91ab6dbaea3e42fece09c98e60674f03ec723f5521d5c8e0593efc","family_id":"qf-mod39-digraph-in-out-degree","seed":202611087047,"semantic_template":"qf-mod39-digraph-in-out-degree/{\"edges\":[[0,1],[0,2],[1,2],[2,3],[2,4],[3,4],[4,0],[4,1]],\"n\":5}"},"fields":{"title":"In-Degrees and Out-Degrees Each Sum to the Edge Count","stem":"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).","answer":{"kind":"expression","value":"([2, 1, 2, 1, 2], [1, 2, 2, 1, 2], 8, 8, 8, Yes)","explanation":"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)."},"skills":["In-Degrees and Out-Degrees Each Sum to the Edge Count"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r07-finite-directed-graph-0048","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4b8b7cbf87a265087425d91495b7279106d2eb35c3351f071f8dd9527c4ac26c","family_id":"qf-mod39-digraph-in-out-degree","seed":202611087048,"semantic_template":"qf-mod39-digraph-in-out-degree/{\"edges\":[[0,1],[1,2],[1,3],[2,3],[2,4],[3,4],[4,0],[4,1]],\"n\":5}"},"fields":{"title":"In-Degrees and Out-Degrees Each Sum to the Edge Count","stem":"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).","answer":{"kind":"expression","value":"([1, 2, 2, 1, 2], [1, 2, 1, 2, 2], 8, 8, 8, Yes)","explanation":"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)."},"skills":["In-Degrees and Out-Degrees Each Sum to the Edge Count"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r07-finite-directed-graph-0049","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"20909c06e251855a05ad3b025e306c06ede7141d11bc389bba90c0d56a351b64","family_id":"qf-mod39-digraph-in-out-degree","seed":202611087049,"semantic_template":"qf-mod39-digraph-in-out-degree/{\"edges\":[[0,1],[0,2],[1,2],[1,3],[2,3],[2,4],[3,4],[4,0],[4,1]],\"n\":5}"},"fields":{"title":"In-Degrees and Out-Degrees Each Sum to the Edge Count","stem":"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).","answer":{"kind":"expression","value":"([2, 2, 2, 1, 2], [1, 2, 2, 2, 2], 9, 9, 9, Yes)","explanation":"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)."},"skills":["In-Degrees and Out-Degrees Each Sum to the Edge Count"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r07-finite-directed-graph-0050","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b21154785c601a8099d4408b9bf9fff357425afad860349ccce107f78eb40df7","family_id":"qf-mod39-digraph-in-out-degree","seed":202611087050,"semantic_template":"qf-mod39-digraph-in-out-degree/{\"edges\":[[0,1],[1,2],[2,3],[3,0],[3,4],[4,0],[4,1]],\"n\":5}"},"fields":{"title":"In-Degrees and Out-Degrees Each Sum to the Edge Count","stem":"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).","answer":{"kind":"expression","value":"([1, 1, 1, 2, 2], [2, 2, 1, 1, 1], 7, 7, 7, Yes)","explanation":"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)."},"skills":["In-Degrees and Out-Degrees Each Sum to the Edge Count"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r07-finite-directed-graph-0051","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4a8f81d3d77b4bdf78338cab454f691da757a2f7dff8e78bb71746878c67b1b6","family_id":"qf-mod39-digraph-directed-walk-audit","seed":202611087051,"semantic_template":"qf-mod39-digraph-directed-walk-audit/{\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,0]],\"n\":5,\"walk\":[0,1,2,3,4,0]}"},"fields":{"title":"Every Successive Step Must Follow Its Own Directed Edge","stem":"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).","answer":{"kind":"expression","value":"([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)","explanation":"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)."},"skills":["Every Successive Step Must Follow Its Own Directed Edge"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r07-finite-directed-graph-0052","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"acd2dc0c1a757f7756ba38b7810df58dfd0843b991c86770a70e87bb75d70594","family_id":"qf-mod39-digraph-directed-walk-audit","seed":202611087052,"semantic_template":"qf-mod39-digraph-directed-walk-audit/{\"edges\":[[0,1],[0,2],[1,2],[2,3],[3,4],[4,0]],\"n\":5,\"walk\":[0,1,2,3,4,0]}"},"fields":{"title":"Every Successive Step Must Follow Its Own Directed Edge","stem":"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).","answer":{"kind":"expression","value":"([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)","explanation":"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)."},"skills":["Every Successive Step Must Follow Its Own Directed Edge"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r07-finite-directed-graph-0053","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"54287f0169eb75b0b4ae76c59d9a1a558bdd05967313a2c6471444f81d1dcdd6","family_id":"qf-mod39-digraph-directed-walk-audit","seed":202611087053,"semantic_template":"qf-mod39-digraph-directed-walk-audit/{\"edges\":[[0,1],[1,2],[1,3],[2,3],[3,4],[4,0]],\"n\":5,\"walk\":[0,1,2,3,4,0]}"},"fields":{"title":"Every Successive Step Must Follow Its Own Directed Edge","stem":"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).","answer":{"kind":"expression","value":"([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)","explanation":"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)."},"skills":["Every Successive Step Must Follow Its Own Directed Edge"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r07-finite-directed-graph-0054","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1cfc5225852300877ea7759b9e9ecc3783562b89cacf528f7a2c86180ea2a19a","family_id":"qf-mod39-digraph-directed-walk-audit","seed":202611087054,"semantic_template":"qf-mod39-digraph-directed-walk-audit/{\"edges\":[[0,1],[0,2],[1,2],[1,3],[2,3],[3,4],[4,0]],\"n\":5,\"walk\":[0,1,2,3,4,0]}"},"fields":{"title":"Every Successive Step Must Follow Its Own Directed Edge","stem":"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).","answer":{"kind":"expression","value":"([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)","explanation":"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)."},"skills":["Every Successive Step Must Follow Its Own Directed Edge"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r07-finite-directed-graph-0055","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"084886d29ff5258d98cf2f04effa501e5f297d5fe9f8275190987b09293f5fab","family_id":"qf-mod39-digraph-directed-walk-audit","seed":202611087055,"semantic_template":"qf-mod39-digraph-directed-walk-audit/{\"edges\":[[0,1],[1,2],[2,3],[2,4],[3,4],[4,0]],\"n\":5,\"walk\":[0,1,2,3,4,0]}"},"fields":{"title":"Every Successive Step Must Follow Its Own Directed Edge","stem":"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).","answer":{"kind":"expression","value":"([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)","explanation":"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)."},"skills":["Every Successive Step Must Follow Its Own Directed Edge"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r07-finite-directed-graph-0056","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0db6b72f2a7ab85266451b8d179fc050fbdb4069b5d6d864e281be287c5476f7","family_id":"qf-mod39-digraph-directed-walk-audit","seed":202611087056,"semantic_template":"qf-mod39-digraph-directed-walk-audit/{\"edges\":[[0,1],[0,2],[1,2],[2,3],[2,4],[3,4],[4,0]],\"n\":5,\"walk\":[0,1,2,3,4,0]}"},"fields":{"title":"Every Successive Step Must Follow Its Own Directed Edge","stem":"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).","answer":{"kind":"expression","value":"([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)","explanation":"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)."},"skills":["Every Successive Step Must Follow Its Own Directed Edge"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r07-finite-directed-graph-0057","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"02da5879aca1e1658cf03cd9a390acb97bf0e7a86624ef67acbe7e04bfbb559c","family_id":"qf-mod39-digraph-directed-walk-audit","seed":202611087057,"semantic_template":"qf-mod39-digraph-directed-walk-audit/{\"edges\":[[0,1],[1,2],[1,3],[2,3],[2,4],[3,4],[4,0]],\"n\":5,\"walk\":[0,1,2,3,4,0]}"},"fields":{"title":"Every Successive Step Must Follow Its Own Directed Edge","stem":"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).","answer":{"kind":"expression","value":"([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)","explanation":"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)."},"skills":["Every Successive Step Must Follow Its Own Directed Edge"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r07-finite-directed-graph-0058","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dc13df74c20b16098520af351377c153f758beef2a3b328197a1c95aa1d71aea","family_id":"qf-mod39-digraph-directed-walk-audit","seed":202611087058,"semantic_template":"qf-mod39-digraph-directed-walk-audit/{\"edges\":[[0,1],[0,2],[1,2],[1,3],[2,3],[2,4],[3,4],[4,0]],\"n\":5,\"walk\":[0,1,2,3,4,0]}"},"fields":{"title":"Every Successive Step Must Follow Its Own Directed Edge","stem":"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).","answer":{"kind":"expression","value":"([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)","explanation":"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)."},"skills":["Every Successive Step Must Follow Its Own Directed Edge"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r07-finite-directed-graph-0059","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"83f7e3630e703cffffce32a02bce8c50ed653c141971b9c39047ff46107623f6","family_id":"qf-mod39-digraph-directed-walk-audit","seed":202611087059,"semantic_template":"qf-mod39-digraph-directed-walk-audit/{\"edges\":[[0,1],[1,2],[2,3],[3,0],[3,4],[4,0]],\"n\":5,\"walk\":[0,1,2,3,4,0]}"},"fields":{"title":"Every Successive Step Must Follow Its Own Directed Edge","stem":"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).","answer":{"kind":"expression","value":"([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)","explanation":"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)."},"skills":["Every Successive Step Must Follow Its Own Directed Edge"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r07-finite-directed-graph-0060","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"413747ba9dcf9747804a1050798fa23949cca9987741fab6974b39e9f2892db0","family_id":"qf-mod39-digraph-directed-walk-audit","seed":202611087060,"semantic_template":"qf-mod39-digraph-directed-walk-audit/{\"edges\":[[0,1],[0,2],[1,2],[2,3],[3,0],[3,4],[4,0]],\"n\":5,\"walk\":[0,1,2,3,4,0]}"},"fields":{"title":"Every Successive Step Must Follow Its Own Directed Edge","stem":"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).","answer":{"kind":"expression","value":"([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)","explanation":"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)."},"skills":["Every Successive Step Must Follow Its Own Directed Edge"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r07-finite-directed-graph-0061","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8d71a5c535faaa0fb3fe2ad310d104e1a3874344c646c781708ef7dd9a0b6996","family_id":"qf-mod39-digraph-directed-walk-audit","seed":202611087061,"semantic_template":"qf-mod39-digraph-directed-walk-audit/{\"edges\":[[0,1],[1,2],[1,3],[2,3],[3,0],[3,4],[4,0]],\"n\":5,\"walk\":[0,1,2,3,4,0]}"},"fields":{"title":"Every Successive Step Must Follow Its Own Directed Edge","stem":"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).","answer":{"kind":"expression","value":"([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)","explanation":"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)."},"skills":["Every Successive Step Must Follow Its Own Directed Edge"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r07-finite-directed-graph-0062","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"940a8f292942b57c50f80cc7946298d4c5201b3a1f6b92f0d32ba9bfa24edb1e","family_id":"qf-mod39-digraph-directed-walk-audit","seed":202611087062,"semantic_template":"qf-mod39-digraph-directed-walk-audit/{\"edges\":[[0,1],[0,2],[1,2],[1,3],[2,3],[3,0],[3,4],[4,0]],\"n\":5,\"walk\":[0,1,2,3,4,0]}"},"fields":{"title":"Every Successive Step Must Follow Its Own Directed Edge","stem":"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).","answer":{"kind":"expression","value":"([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)","explanation":"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)."},"skills":["Every Successive Step Must Follow Its Own Directed Edge"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r07-finite-directed-graph-0063","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6a3bc0fe3e2c24707dcf5f75fbbfaf1d025e372f3de26dd75cfbeb8c1970f650","family_id":"qf-mod39-digraph-directed-walk-audit","seed":202611087063,"semantic_template":"qf-mod39-digraph-directed-walk-audit/{\"edges\":[[0,1],[1,2],[2,3],[2,4],[3,0],[3,4],[4,0]],\"n\":5,\"walk\":[0,1,2,3,4,0]}"},"fields":{"title":"Every Successive Step Must Follow Its Own Directed Edge","stem":"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).","answer":{"kind":"expression","value":"([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)","explanation":"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)."},"skills":["Every Successive Step Must Follow Its Own Directed Edge"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r07-finite-directed-graph-0064","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"58d21f17592fb4fee46409627e1d8132c708a10e72aeffe9e4da138b85a63b69","family_id":"qf-mod39-digraph-directed-walk-audit","seed":202611087064,"semantic_template":"qf-mod39-digraph-directed-walk-audit/{\"edges\":[[0,1],[0,2],[1,2],[2,3],[2,4],[3,0],[3,4],[4,0]],\"n\":5,\"walk\":[0,1,2,3,4,0]}"},"fields":{"title":"Every Successive Step Must Follow Its Own Directed Edge","stem":"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).","answer":{"kind":"expression","value":"([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)","explanation":"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)."},"skills":["Every Successive Step Must Follow Its Own Directed Edge"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r07-finite-directed-graph-0065","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"363ee32f1793731a78a4ce6a184eaa850b29e30474a9eded6f17fb5559e9a5f8","family_id":"qf-mod39-digraph-directed-walk-audit","seed":202611087065,"semantic_template":"qf-mod39-digraph-directed-walk-audit/{\"edges\":[[0,1],[1,2],[1,3],[2,3],[2,4],[3,0],[3,4],[4,0]],\"n\":5,\"walk\":[0,1,2,3,4,0]}"},"fields":{"title":"Every Successive Step Must Follow Its Own Directed Edge","stem":"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).","answer":{"kind":"expression","value":"([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)","explanation":"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)."},"skills":["Every Successive Step Must Follow Its Own Directed Edge"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r07-finite-directed-graph-0066","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9ac4395e059e886352db218cd5d4d2f5bb61fb364116973bf77d3887fbc713f1","family_id":"qf-mod39-digraph-directed-walk-audit","seed":202611087066,"semantic_template":"qf-mod39-digraph-directed-walk-audit/{\"edges\":[[0,1],[0,2],[1,2],[1,3],[2,3],[2,4],[3,0],[3,4],[4,0]],\"n\":5,\"walk\":[0,1,2,3,4,0]}"},"fields":{"title":"Every Successive Step Must Follow Its Own Directed Edge","stem":"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).","answer":{"kind":"expression","value":"([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)","explanation":"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)."},"skills":["Every Successive Step Must Follow Its Own Directed Edge"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r07-finite-directed-graph-0067","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d6882b69605353b12208dd97623310e67bccc7da233847409428f38e997b5bb7","family_id":"qf-mod39-digraph-directed-walk-audit","seed":202611087067,"semantic_template":"qf-mod39-digraph-directed-walk-audit/{\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,0],[4,1]],\"n\":5,\"walk\":[0,1,2,3,4,0]}"},"fields":{"title":"Every Successive Step Must Follow Its Own Directed Edge","stem":"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).","answer":{"kind":"expression","value":"([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)","explanation":"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)."},"skills":["Every Successive Step Must Follow Its Own Directed Edge"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r07-finite-directed-graph-0068","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"52459363c8e76ceeb3845f70f9c8a67c446c8686c8df473e5e50de1c2fee1321","family_id":"qf-mod39-digraph-directed-walk-audit","seed":202611087068,"semantic_template":"qf-mod39-digraph-directed-walk-audit/{\"edges\":[[0,1],[0,2],[1,2],[2,3],[3,4],[4,0],[4,1]],\"n\":5,\"walk\":[0,1,2,3,4,0]}"},"fields":{"title":"Every Successive Step Must Follow Its Own Directed Edge","stem":"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).","answer":{"kind":"expression","value":"([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)","explanation":"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)."},"skills":["Every Successive Step Must Follow Its Own Directed Edge"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r07-finite-directed-graph-0069","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d1fbc7cf7385b677dad2fd99def17eb7a98a1017720428ab90e2c15d56664dfa","family_id":"qf-mod39-digraph-directed-walk-audit","seed":202611087069,"semantic_template":"qf-mod39-digraph-directed-walk-audit/{\"edges\":[[0,1],[1,2],[1,3],[2,3],[3,4],[4,0],[4,1]],\"n\":5,\"walk\":[0,1,2,3,4,0]}"},"fields":{"title":"Every Successive Step Must Follow Its Own Directed Edge","stem":"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).","answer":{"kind":"expression","value":"([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)","explanation":"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)."},"skills":["Every Successive Step Must Follow Its Own Directed Edge"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r07-finite-directed-graph-0070","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"08a864760da3242d6591874673aa53ceffbb718782a7e283064f68f0a00ba540","family_id":"qf-mod39-digraph-directed-walk-audit","seed":202611087070,"semantic_template":"qf-mod39-digraph-directed-walk-audit/{\"edges\":[[0,1],[0,2],[1,2],[1,3],[2,3],[3,4],[4,0],[4,1]],\"n\":5,\"walk\":[0,1,2,3,4,0]}"},"fields":{"title":"Every Successive Step Must Follow Its Own Directed Edge","stem":"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).","answer":{"kind":"expression","value":"([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)","explanation":"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)."},"skills":["Every Successive Step Must Follow Its Own Directed Edge"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r07-finite-directed-graph-0071","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a5f9bf2605f07454d1444b2ef269c30c8300f9714540b14c0843cd18013d1daf","family_id":"qf-mod39-digraph-directed-walk-audit","seed":202611087071,"semantic_template":"qf-mod39-digraph-directed-walk-audit/{\"edges\":[[0,1],[1,2],[2,3],[2,4],[3,4],[4,0],[4,1]],\"n\":5,\"walk\":[0,1,2,3,4,0]}"},"fields":{"title":"Every Successive Step Must Follow Its Own Directed Edge","stem":"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).","answer":{"kind":"expression","value":"([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)","explanation":"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)."},"skills":["Every Successive Step Must Follow Its Own Directed Edge"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r07-finite-directed-graph-0072","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f8f1ac4088a4624dfff69a328a957928c76e100b42836f1de259d521e811fc31","family_id":"qf-mod39-digraph-directed-walk-audit","seed":202611087072,"semantic_template":"qf-mod39-digraph-directed-walk-audit/{\"edges\":[[0,1],[0,2],[1,2],[2,3],[2,4],[3,4],[4,0],[4,1]],\"n\":5,\"walk\":[0,1,2,3,4,0]}"},"fields":{"title":"Every Successive Step Must Follow Its Own Directed Edge","stem":"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).","answer":{"kind":"expression","value":"([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)","explanation":"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)."},"skills":["Every Successive Step Must Follow Its Own Directed Edge"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r07-finite-directed-graph-0073","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d84eb4460a8b54e6564b94c1b22a6f679252a89bee0536ce875cd491fab8c8ae","family_id":"qf-mod39-digraph-directed-walk-audit","seed":202611087073,"semantic_template":"qf-mod39-digraph-directed-walk-audit/{\"edges\":[[0,1],[1,2],[1,3],[2,3],[2,4],[3,4],[4,0],[4,1]],\"n\":5,\"walk\":[0,1,2,3,4,0]}"},"fields":{"title":"Every Successive Step Must Follow Its Own Directed Edge","stem":"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).","answer":{"kind":"expression","value":"([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)","explanation":"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)."},"skills":["Every Successive Step Must Follow Its Own Directed Edge"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r07-finite-directed-graph-0074","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d477e539cd8fc7f8266c2629f4f8f02567d632c68496daeea1c2a198853b166b","family_id":"qf-mod39-digraph-directed-walk-audit","seed":202611087074,"semantic_template":"qf-mod39-digraph-directed-walk-audit/{\"edges\":[[0,1],[0,2],[1,2],[1,3],[2,3],[2,4],[3,4],[4,0],[4,1]],\"n\":5,\"walk\":[0,1,2,3,4,0]}"},"fields":{"title":"Every Successive Step Must Follow Its Own Directed Edge","stem":"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).","answer":{"kind":"expression","value":"([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)","explanation":"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)."},"skills":["Every Successive Step Must Follow Its Own Directed Edge"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r07-finite-directed-graph-0075","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"430f0ad2a85617247e3930fee5a38e17d60c8c9898e6f51fb2323e99945e5006","family_id":"qf-mod39-digraph-directed-walk-audit","seed":202611087075,"semantic_template":"qf-mod39-digraph-directed-walk-audit/{\"edges\":[[0,1],[1,2],[2,3],[3,0],[3,4],[4,0],[4,1]],\"n\":5,\"walk\":[0,1,2,3,4,0]}"},"fields":{"title":"Every Successive Step Must Follow Its Own Directed Edge","stem":"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).","answer":{"kind":"expression","value":"([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)","explanation":"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)."},"skills":["Every Successive Step Must Follow Its Own Directed Edge"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r07-finite-directed-graph-0076","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"697f014c50e1dbde7d2b9635fcbc5ab9a8ad02df1b0199598cab8c3de42d2b2f","family_id":"qf-mod39-digraph-loop-boundary","seed":202611087076,"semantic_template":"qf-mod39-digraph-loop-boundary/{\"edges\":[[0,0],[0,1],[1,2],[2,3],[3,4],[4,0]],\"n\":5}"},"fields":{"title":"A Loop Fails This Loop-Free Model, Not Every Possible Digraph Convention","stem":"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).","answer":{"kind":"expression","value":"([[0, 0]], No, [2, 1, 1, 1, 1], [2, 1, 1, 1, 1], loops_rejected_by_declared_loop_free_simple_digraph_convention)","explanation":"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)."},"skills":["A Loop Fails This Loop-Free Model, Not Every Possible Digraph Convention"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r07-finite-directed-graph-0077","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"98b7bfeb0e0ae8071b7720a44bb23ac380724bdaed041694f001c74817d0645e","family_id":"qf-mod39-digraph-loop-boundary","seed":202611087077,"semantic_template":"qf-mod39-digraph-loop-boundary/{\"edges\":[[0,0],[0,1],[0,2],[1,2],[2,3],[3,4],[4,0]],\"n\":5}"},"fields":{"title":"A Loop Fails This Loop-Free Model, Not Every Possible Digraph Convention","stem":"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).","answer":{"kind":"expression","value":"([[0, 0]], No, [3, 1, 1, 1, 1], [2, 1, 2, 1, 1], loops_rejected_by_declared_loop_free_simple_digraph_convention)","explanation":"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)."},"skills":["A Loop Fails This Loop-Free Model, Not Every Possible Digraph Convention"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r07-finite-directed-graph-0078","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6fead8ae055074d2760c4a0eced0ccbfec914a653f1afd20c6e71609f5d6296a","family_id":"qf-mod39-digraph-loop-boundary","seed":202611087078,"semantic_template":"qf-mod39-digraph-loop-boundary/{\"edges\":[[0,0],[0,1],[1,2],[1,3],[2,3],[3,4],[4,0]],\"n\":5}"},"fields":{"title":"A Loop Fails This Loop-Free Model, Not Every Possible Digraph Convention","stem":"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).","answer":{"kind":"expression","value":"([[0, 0]], No, [2, 2, 1, 1, 1], [2, 1, 1, 2, 1], loops_rejected_by_declared_loop_free_simple_digraph_convention)","explanation":"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)."},"skills":["A Loop Fails This Loop-Free Model, Not Every Possible Digraph Convention"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r07-finite-directed-graph-0079","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0f7fe4dd24ec6340ac09ce3f388076b499593b234fc3abea701f2428af8576b2","family_id":"qf-mod39-digraph-loop-boundary","seed":202611087079,"semantic_template":"qf-mod39-digraph-loop-boundary/{\"edges\":[[0,0],[0,1],[0,2],[1,2],[1,3],[2,3],[3,4],[4,0]],\"n\":5}"},"fields":{"title":"A Loop Fails This Loop-Free Model, Not Every Possible Digraph Convention","stem":"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).","answer":{"kind":"expression","value":"([[0, 0]], No, [3, 2, 1, 1, 1], [2, 1, 2, 2, 1], loops_rejected_by_declared_loop_free_simple_digraph_convention)","explanation":"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)."},"skills":["A Loop Fails This Loop-Free Model, Not Every Possible Digraph Convention"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r07-finite-directed-graph-0080","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9763b6ee223d67a25051cb49d4c63096123bcd739c7a36d32d634836da6afd84","family_id":"qf-mod39-digraph-loop-boundary","seed":202611087080,"semantic_template":"qf-mod39-digraph-loop-boundary/{\"edges\":[[0,0],[0,1],[1,2],[2,3],[2,4],[3,4],[4,0]],\"n\":5}"},"fields":{"title":"A Loop Fails This Loop-Free Model, Not Every Possible Digraph Convention","stem":"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).","answer":{"kind":"expression","value":"([[0, 0]], No, [2, 1, 2, 1, 1], [2, 1, 1, 1, 2], loops_rejected_by_declared_loop_free_simple_digraph_convention)","explanation":"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)."},"skills":["A Loop Fails This Loop-Free Model, Not Every Possible Digraph Convention"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r07-finite-directed-graph-0081","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"eaaede85c3ace6fc98d1962f56e419e4f896037f11c8b457b0332334e1526f95","family_id":"qf-mod39-digraph-loop-boundary","seed":202611087081,"semantic_template":"qf-mod39-digraph-loop-boundary/{\"edges\":[[0,0],[0,1],[0,2],[1,2],[2,3],[2,4],[3,4],[4,0]],\"n\":5}"},"fields":{"title":"A Loop Fails This Loop-Free Model, Not Every Possible Digraph Convention","stem":"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).","answer":{"kind":"expression","value":"([[0, 0]], No, [3, 1, 2, 1, 1], [2, 1, 2, 1, 2], loops_rejected_by_declared_loop_free_simple_digraph_convention)","explanation":"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)."},"skills":["A Loop Fails This Loop-Free Model, Not Every Possible Digraph Convention"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r07-finite-directed-graph-0082","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dbab3310fc7f15fdf6395170eccfbbe77016bed7e3fe3d43a5236483985b3da7","family_id":"qf-mod39-digraph-loop-boundary","seed":202611087082,"semantic_template":"qf-mod39-digraph-loop-boundary/{\"edges\":[[0,0],[0,1],[1,2],[1,3],[2,3],[2,4],[3,4],[4,0]],\"n\":5}"},"fields":{"title":"A Loop Fails This Loop-Free Model, Not Every Possible Digraph Convention","stem":"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).","answer":{"kind":"expression","value":"([[0, 0]], No, [2, 2, 2, 1, 1], [2, 1, 1, 2, 2], loops_rejected_by_declared_loop_free_simple_digraph_convention)","explanation":"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)."},"skills":["A Loop Fails This Loop-Free Model, Not Every Possible Digraph Convention"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r07-finite-directed-graph-0083","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4cbe9061983e39d56d258c53288df9024b3a05282f12c2132f86d731d573e57d","family_id":"qf-mod39-digraph-loop-boundary","seed":202611087083,"semantic_template":"qf-mod39-digraph-loop-boundary/{\"edges\":[[0,0],[0,1],[0,2],[1,2],[1,3],[2,3],[2,4],[3,4],[4,0]],\"n\":5}"},"fields":{"title":"A Loop Fails This Loop-Free Model, Not Every Possible Digraph Convention","stem":"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).","answer":{"kind":"expression","value":"([[0, 0]], No, [3, 2, 2, 1, 1], [2, 1, 2, 2, 2], loops_rejected_by_declared_loop_free_simple_digraph_convention)","explanation":"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)."},"skills":["A Loop Fails This Loop-Free Model, Not Every Possible Digraph Convention"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r07-finite-directed-graph-0084","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ad6a4b8422cf1dfeb325cff3f5312d2bef62a314268f6684c04e19db0c8e5cdd","family_id":"qf-mod39-digraph-loop-boundary","seed":202611087084,"semantic_template":"qf-mod39-digraph-loop-boundary/{\"edges\":[[0,0],[0,1],[1,2],[2,3],[3,0],[3,4],[4,0]],\"n\":5}"},"fields":{"title":"A Loop Fails This Loop-Free Model, Not Every Possible Digraph Convention","stem":"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).","answer":{"kind":"expression","value":"([[0, 0]], No, [2, 1, 1, 2, 1], [3, 1, 1, 1, 1], loops_rejected_by_declared_loop_free_simple_digraph_convention)","explanation":"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)."},"skills":["A Loop Fails This Loop-Free Model, Not Every Possible Digraph Convention"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r07-finite-directed-graph-0085","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b8a168ca4cffd7278de7a6fee74c4d859e0949b93ece736634a061c11033db80","family_id":"qf-mod39-digraph-loop-boundary","seed":202611087085,"semantic_template":"qf-mod39-digraph-loop-boundary/{\"edges\":[[0,0],[0,1],[0,2],[1,2],[2,3],[3,0],[3,4],[4,0]],\"n\":5}"},"fields":{"title":"A Loop Fails This Loop-Free Model, Not Every Possible Digraph Convention","stem":"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).","answer":{"kind":"expression","value":"([[0, 0]], No, [3, 1, 1, 2, 1], [3, 1, 2, 1, 1], loops_rejected_by_declared_loop_free_simple_digraph_convention)","explanation":"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)."},"skills":["A Loop Fails This Loop-Free Model, Not Every Possible Digraph Convention"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r07-finite-directed-graph-0086","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a1f77fcd484683d9bb9d6ab106a40cf520eab4808f937ac01bff8a73d347fdaa","family_id":"qf-mod39-digraph-loop-boundary","seed":202611087086,"semantic_template":"qf-mod39-digraph-loop-boundary/{\"edges\":[[0,0],[0,1],[1,2],[1,3],[2,3],[3,0],[3,4],[4,0]],\"n\":5}"},"fields":{"title":"A Loop Fails This Loop-Free Model, Not Every Possible Digraph Convention","stem":"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).","answer":{"kind":"expression","value":"([[0, 0]], No, [2, 2, 1, 2, 1], [3, 1, 1, 2, 1], loops_rejected_by_declared_loop_free_simple_digraph_convention)","explanation":"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)."},"skills":["A Loop Fails This Loop-Free Model, Not Every Possible Digraph Convention"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r07-finite-directed-graph-0087","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4612e36cd4bd4c7728f244b8331c6b8712fbde89ca82cd1a6436883a91cedf09","family_id":"qf-mod39-digraph-loop-boundary","seed":202611087087,"semantic_template":"qf-mod39-digraph-loop-boundary/{\"edges\":[[0,0],[0,1],[0,2],[1,2],[1,3],[2,3],[3,0],[3,4],[4,0]],\"n\":5}"},"fields":{"title":"A Loop Fails This Loop-Free Model, Not Every Possible Digraph Convention","stem":"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).","answer":{"kind":"expression","value":"([[0, 0]], No, [3, 2, 1, 2, 1], [3, 1, 2, 2, 1], loops_rejected_by_declared_loop_free_simple_digraph_convention)","explanation":"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)."},"skills":["A Loop Fails This Loop-Free Model, Not Every Possible Digraph Convention"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r07-finite-directed-graph-0088","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"965aff0edb3a441c7cbcadc47a5bef3e86ef6abc629a147c920594f1e32bc07e","family_id":"qf-mod39-digraph-loop-boundary","seed":202611087088,"semantic_template":"qf-mod39-digraph-loop-boundary/{\"edges\":[[0,0],[0,1],[1,2],[2,3],[2,4],[3,0],[3,4],[4,0]],\"n\":5}"},"fields":{"title":"A Loop Fails This Loop-Free Model, Not Every Possible Digraph Convention","stem":"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).","answer":{"kind":"expression","value":"([[0, 0]], No, [2, 1, 2, 2, 1], [3, 1, 1, 1, 2], loops_rejected_by_declared_loop_free_simple_digraph_convention)","explanation":"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)."},"skills":["A Loop Fails This Loop-Free Model, Not Every Possible Digraph Convention"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r07-finite-directed-graph-0089","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b10f64439b493f260bc7b7a330dce3638df5119d919838b49cffd93fc42dc31c","family_id":"qf-mod39-digraph-loop-boundary","seed":202611087089,"semantic_template":"qf-mod39-digraph-loop-boundary/{\"edges\":[[0,0],[0,1],[0,2],[1,2],[2,3],[2,4],[3,0],[3,4],[4,0]],\"n\":5}"},"fields":{"title":"A Loop Fails This Loop-Free Model, Not Every Possible Digraph Convention","stem":"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).","answer":{"kind":"expression","value":"([[0, 0]], No, [3, 1, 2, 2, 1], [3, 1, 2, 1, 2], loops_rejected_by_declared_loop_free_simple_digraph_convention)","explanation":"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)."},"skills":["A Loop Fails This Loop-Free Model, Not Every Possible Digraph Convention"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r07-finite-directed-graph-0090","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"843a068949dfdb1a13ec51868dd13d0f4d0d144868a7b819e667dad0a3556639","family_id":"qf-mod39-digraph-loop-boundary","seed":202611087090,"semantic_template":"qf-mod39-digraph-loop-boundary/{\"edges\":[[0,0],[0,1],[1,2],[1,3],[2,3],[2,4],[3,0],[3,4],[4,0]],\"n\":5}"},"fields":{"title":"A Loop Fails This Loop-Free Model, Not Every Possible Digraph Convention","stem":"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).","answer":{"kind":"expression","value":"([[0, 0]], No, [2, 2, 2, 2, 1], [3, 1, 1, 2, 2], loops_rejected_by_declared_loop_free_simple_digraph_convention)","explanation":"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)."},"skills":["A Loop Fails This Loop-Free Model, Not Every Possible Digraph Convention"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r07-finite-directed-graph-0091","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e2f3de184e9998b8c8242ea5445cc9d3d248a9f6ac1cd24a9eefdf1955e53c24","family_id":"qf-mod39-digraph-loop-boundary","seed":202611087091,"semantic_template":"qf-mod39-digraph-loop-boundary/{\"edges\":[[0,0],[0,1],[0,2],[1,2],[1,3],[2,3],[2,4],[3,0],[3,4],[4,0]],\"n\":5}"},"fields":{"title":"A Loop Fails This Loop-Free Model, Not Every Possible Digraph Convention","stem":"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).","answer":{"kind":"expression","value":"([[0, 0]], No, [3, 2, 2, 2, 1], [3, 1, 2, 2, 2], loops_rejected_by_declared_loop_free_simple_digraph_convention)","explanation":"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)."},"skills":["A Loop Fails This Loop-Free Model, Not Every Possible Digraph Convention"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r07-finite-directed-graph-0092","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"43599dc8eeb9ea7d578a1b54614ed3928680c88be20f80ca53ce7e83c26e11a3","family_id":"qf-mod39-digraph-loop-boundary","seed":202611087092,"semantic_template":"qf-mod39-digraph-loop-boundary/{\"edges\":[[0,0],[0,1],[1,2],[2,3],[3,4],[4,0],[4,1]],\"n\":5}"},"fields":{"title":"A Loop Fails This Loop-Free Model, Not Every Possible Digraph Convention","stem":"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).","answer":{"kind":"expression","value":"([[0, 0]], No, [2, 1, 1, 1, 2], [2, 2, 1, 1, 1], loops_rejected_by_declared_loop_free_simple_digraph_convention)","explanation":"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)."},"skills":["A Loop Fails This Loop-Free Model, Not Every Possible Digraph Convention"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r07-finite-directed-graph-0093","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4e8328ba10998cfea2aeb370251bdadec4abade2460f58ecf1c49f0096dd32bf","family_id":"qf-mod39-digraph-loop-boundary","seed":202611087093,"semantic_template":"qf-mod39-digraph-loop-boundary/{\"edges\":[[0,0],[0,1],[0,2],[1,2],[2,3],[3,4],[4,0],[4,1]],\"n\":5}"},"fields":{"title":"A Loop Fails This Loop-Free Model, Not Every Possible Digraph Convention","stem":"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).","answer":{"kind":"expression","value":"([[0, 0]], No, [3, 1, 1, 1, 2], [2, 2, 2, 1, 1], loops_rejected_by_declared_loop_free_simple_digraph_convention)","explanation":"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)."},"skills":["A Loop Fails This Loop-Free Model, Not Every Possible Digraph Convention"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r07-finite-directed-graph-0094","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d20e0788ec3237fbe2af44b627f9ed8bac2c37c99b3f512b8d69fd2d5dbaa5bf","family_id":"qf-mod39-digraph-loop-boundary","seed":202611087094,"semantic_template":"qf-mod39-digraph-loop-boundary/{\"edges\":[[0,0],[0,1],[1,2],[1,3],[2,3],[3,4],[4,0],[4,1]],\"n\":5}"},"fields":{"title":"A Loop Fails This Loop-Free Model, Not Every Possible Digraph Convention","stem":"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).","answer":{"kind":"expression","value":"([[0, 0]], No, [2, 2, 1, 1, 2], [2, 2, 1, 2, 1], loops_rejected_by_declared_loop_free_simple_digraph_convention)","explanation":"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)."},"skills":["A Loop Fails This Loop-Free Model, Not Every Possible Digraph Convention"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r07-finite-directed-graph-0095","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"292775fb349805dd284223871ee3b35619842e674f1bd37de874dea60bd40603","family_id":"qf-mod39-digraph-loop-boundary","seed":202611087095,"semantic_template":"qf-mod39-digraph-loop-boundary/{\"edges\":[[0,0],[0,1],[0,2],[1,2],[1,3],[2,3],[3,4],[4,0],[4,1]],\"n\":5}"},"fields":{"title":"A Loop Fails This Loop-Free Model, Not Every Possible Digraph Convention","stem":"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).","answer":{"kind":"expression","value":"([[0, 0]], No, [3, 2, 1, 1, 2], [2, 2, 2, 2, 1], loops_rejected_by_declared_loop_free_simple_digraph_convention)","explanation":"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)."},"skills":["A Loop Fails This Loop-Free Model, Not Every Possible Digraph Convention"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r07-finite-directed-graph-0096","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0b33e0fb6bd3d537e44a73818c73cdeadc79177e80da412d878de6c8aac9a650","family_id":"qf-mod39-digraph-loop-boundary","seed":202611087096,"semantic_template":"qf-mod39-digraph-loop-boundary/{\"edges\":[[0,0],[0,1],[1,2],[2,3],[2,4],[3,4],[4,0],[4,1]],\"n\":5}"},"fields":{"title":"A Loop Fails This Loop-Free Model, Not Every Possible Digraph Convention","stem":"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).","answer":{"kind":"expression","value":"([[0, 0]], No, [2, 1, 2, 1, 2], [2, 2, 1, 1, 2], loops_rejected_by_declared_loop_free_simple_digraph_convention)","explanation":"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)."},"skills":["A Loop Fails This Loop-Free Model, Not Every Possible Digraph Convention"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r07-finite-directed-graph-0097","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1aa30c6f205488f362d29af319e4f7365bfdb18050ebd1ba680bc71af9e1aedb","family_id":"qf-mod39-digraph-loop-boundary","seed":202611087097,"semantic_template":"qf-mod39-digraph-loop-boundary/{\"edges\":[[0,0],[0,1],[0,2],[1,2],[2,3],[2,4],[3,4],[4,0],[4,1]],\"n\":5}"},"fields":{"title":"A Loop Fails This Loop-Free Model, Not Every Possible Digraph Convention","stem":"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).","answer":{"kind":"expression","value":"([[0, 0]], No, [3, 1, 2, 1, 2], [2, 2, 2, 1, 2], loops_rejected_by_declared_loop_free_simple_digraph_convention)","explanation":"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)."},"skills":["A Loop Fails This Loop-Free Model, Not Every Possible Digraph Convention"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r07-finite-directed-graph-0098","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4cee61ac6b6f674522cd8ffe42c57c32cd218fc72508f69063b73280e08a47ad","family_id":"qf-mod39-digraph-loop-boundary","seed":202611087098,"semantic_template":"qf-mod39-digraph-loop-boundary/{\"edges\":[[0,0],[0,1],[1,2],[1,3],[2,3],[2,4],[3,4],[4,0],[4,1]],\"n\":5}"},"fields":{"title":"A Loop Fails This Loop-Free Model, Not Every Possible Digraph Convention","stem":"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).","answer":{"kind":"expression","value":"([[0, 0]], No, [2, 2, 2, 1, 2], [2, 2, 1, 2, 2], loops_rejected_by_declared_loop_free_simple_digraph_convention)","explanation":"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)."},"skills":["A Loop Fails This Loop-Free Model, Not Every Possible Digraph Convention"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r07-finite-directed-graph-0099","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5b2fa0d7d474e5ac6b0323b45e56aa11f355563cfd5f91c21655849a024effca","family_id":"qf-mod39-digraph-loop-boundary","seed":202611087099,"semantic_template":"qf-mod39-digraph-loop-boundary/{\"edges\":[[0,0],[0,1],[0,2],[1,2],[1,3],[2,3],[2,4],[3,4],[4,0],[4,1]],\"n\":5}"},"fields":{"title":"A Loop Fails This Loop-Free Model, Not Every Possible Digraph Convention","stem":"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).","answer":{"kind":"expression","value":"([[0, 0]], No, [3, 2, 2, 1, 2], [2, 2, 2, 2, 2], loops_rejected_by_declared_loop_free_simple_digraph_convention)","explanation":"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)."},"skills":["A Loop Fails This Loop-Free Model, Not Every Possible Digraph Convention"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r07-finite-directed-graph-0100","batch_id":"batch-20261014-advance-20261108-r07-finite-directed-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4cc316422b79b072e7d31255edcc090df2180927af849f6dc7307c9d07434d00","family_id":"qf-mod39-digraph-loop-boundary","seed":202611087100,"semantic_template":"qf-mod39-digraph-loop-boundary/{\"edges\":[[0,0],[0,1],[1,2],[2,3],[3,0],[3,4],[4,0],[4,1]],\"n\":5}"},"fields":{"title":"A Loop Fails This Loop-Free Model, Not Every Possible Digraph Convention","stem":"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).","answer":{"kind":"expression","value":"([[0, 0]], No, [2, 1, 1, 2, 2], [3, 2, 1, 1, 1], loops_rejected_by_declared_loop_free_simple_digraph_convention)","explanation":"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)."},"skills":["A Loop Fails This Loop-Free Model, Not Every Possible Digraph Convention"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}