{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r04-eulerian-trail-0001","batch_id":"batch-20261014-advance-20261105-r04-eulerian-trail-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7e9f02f49f22e56e6f60cde387597150100c5165bb27a0bb9f80c1795fd9243f","family_id":"qf-mod36-euler-trail-audit","seed":202611054001,"semantic_template":"qf-mod36-euler-trail-audit/{\"edges\":[[0,1],[0,2],[1,2]],\"n\":3,\"vertex_walk\":[0,1,2,0]}"},"fields":{"title":"Revisiting Vertices Is Allowed; Repeating an Edge Is Not","stem":"Use the complete finite simple undirected graph on Fin(3)={0,…,2}, with canonical sorted edges=[[0, 1], [0, 2], [1, 2]] and zero-based edge indices in that order. The edge support contains exactly the nonisolated vertices. A trail may revisit vertices but may not repeat an edge; an Euler trail uses every edge once. Isolated vertices do not invalidate connected edge support, and no Hamiltonian conclusion follows. Supplied vertex_walk=[0, 1, 2, 0]. Give (vertex_walk, used_edge_indices, length_and_adjacency, recorded_edge_indices_unique, recorded_edges_cover_exactly_once, vertex_revisited, valid_euler_trail, boundary).","answer":{"kind":"expression","value":"([0, 1, 2, 0], [0, 2, 1], Yes, Yes, Yes, Yes, Yes, trail_allows_repeated_vertices_not_edges)","explanation":"Record edge indices for steps found in the parent graph. Check walk length and all adjacencies separately from uniqueness and exact coverage of those recorded edge indices. The final validity requires both the length/adjacency and complete single-use edge checks; repeated vertices alone do not refute an Euler trail. The exact result, retaining the requested order, is ([0, 1, 2, 0], [0, 2, 1], Yes, Yes, Yes, Yes, Yes, trail_allows_repeated_vertices_not_edges)."},"skills":["Revisiting Vertices Is Allowed; Repeating an Edge Is Not"],"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-20261105-r04-eulerian-trail-0002","batch_id":"batch-20261014-advance-20261105-r04-eulerian-trail-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ab75719e9ed5214d3d55ddafa408f70fdf6860ef10c4f82f7ea24438ad99ee7e","family_id":"qf-mod36-euler-trail-audit","seed":202611054002,"semantic_template":"qf-mod36-euler-trail-audit/{\"edges\":[[0,1],[0,3],[1,2],[2,3]],\"n\":4,\"vertex_walk\":[0,1,2,3,0]}"},"fields":{"title":"Revisiting Vertices Is Allowed; Repeating an Edge Is Not","stem":"Use the complete finite simple undirected graph on Fin(4)={0,…,3}, with canonical sorted edges=[[0, 1], [0, 3], [1, 2], [2, 3]] and zero-based edge indices in that order. The edge support contains exactly the nonisolated vertices. A trail may revisit vertices but may not repeat an edge; an Euler trail uses every edge once. Isolated vertices do not invalidate connected edge support, and no Hamiltonian conclusion follows. Supplied vertex_walk=[0, 1, 2, 3, 0]. Give (vertex_walk, used_edge_indices, length_and_adjacency, recorded_edge_indices_unique, recorded_edges_cover_exactly_once, vertex_revisited, valid_euler_trail, boundary).","answer":{"kind":"expression","value":"([0, 1, 2, 3, 0], [0, 2, 3, 1], Yes, Yes, Yes, Yes, Yes, trail_allows_repeated_vertices_not_edges)","explanation":"Record edge indices for steps found in the parent graph. Check walk length and all adjacencies separately from uniqueness and exact coverage of those recorded edge indices. The final validity requires both the length/adjacency and complete single-use edge checks; repeated vertices alone do not refute an Euler trail. The exact result, retaining the requested order, is ([0, 1, 2, 3, 0], [0, 2, 3, 1], Yes, Yes, Yes, Yes, Yes, trail_allows_repeated_vertices_not_edges)."},"skills":["Revisiting Vertices Is Allowed; Repeating an Edge Is Not"],"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-20261105-r04-eulerian-trail-0003","batch_id":"batch-20261014-advance-20261105-r04-eulerian-trail-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"34637ac7812d918ac53574ea12c3fdd7936241639b16713ac52d4b84b5250058","family_id":"qf-mod36-euler-trail-audit","seed":202611054003,"semantic_template":"qf-mod36-euler-trail-audit/{\"edges\":[[0,1],[0,4],[1,2],[2,3],[3,4]],\"n\":5,\"vertex_walk\":[0,1,2,3,4,0]}"},"fields":{"title":"Revisiting Vertices Is Allowed; Repeating an Edge Is Not","stem":"Use the complete finite simple undirected graph on Fin(5)={0,…,4}, with canonical sorted edges=[[0, 1], [0, 4], [1, 2], [2, 3], [3, 4]] and zero-based edge indices in that order. The edge support contains exactly the nonisolated vertices. A trail may revisit vertices but may not repeat an edge; an Euler trail uses every edge once. Isolated vertices do not invalidate connected edge support, and no Hamiltonian conclusion follows. Supplied vertex_walk=[0, 1, 2, 3, 4, 0]. Give (vertex_walk, used_edge_indices, length_and_adjacency, recorded_edge_indices_unique, recorded_edges_cover_exactly_once, vertex_revisited, valid_euler_trail, boundary).","answer":{"kind":"expression","value":"([0, 1, 2, 3, 4, 0], [0, 2, 3, 4, 1], Yes, Yes, Yes, Yes, Yes, trail_allows_repeated_vertices_not_edges)","explanation":"Record edge indices for steps found in the parent graph. Check walk length and all adjacencies separately from uniqueness and exact coverage of those recorded edge indices. The final validity requires both the length/adjacency and complete single-use edge checks; repeated vertices alone do not refute an Euler trail. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 0], [0, 2, 3, 4, 1], Yes, Yes, Yes, Yes, Yes, trail_allows_repeated_vertices_not_edges)."},"skills":["Revisiting Vertices Is Allowed; Repeating an Edge Is Not"],"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-20261105-r04-eulerian-trail-0004","batch_id":"batch-20261014-advance-20261105-r04-eulerian-trail-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ee2d75c555d705cb9dab4c841d39b88dba92f07916d33bf0c36773de3580a43f","family_id":"qf-mod36-euler-trail-audit","seed":202611054004,"semantic_template":"qf-mod36-euler-trail-audit/{\"edges\":[[0,1],[0,5],[1,2],[2,3],[3,4],[4,5]],\"n\":6,\"vertex_walk\":[0,1,2,3,4,5,0]}"},"fields":{"title":"Revisiting Vertices Is Allowed; Repeating an Edge Is Not","stem":"Use the complete finite simple undirected graph on Fin(6)={0,…,5}, with canonical sorted edges=[[0, 1], [0, 5], [1, 2], [2, 3], [3, 4], [4, 5]] and zero-based edge indices in that order. The edge support contains exactly the nonisolated vertices. A trail may revisit vertices but may not repeat an edge; an Euler trail uses every edge once. Isolated vertices do not invalidate connected edge support, and no Hamiltonian conclusion follows. Supplied vertex_walk=[0, 1, 2, 3, 4, 5, 0]. Give (vertex_walk, used_edge_indices, length_and_adjacency, recorded_edge_indices_unique, recorded_edges_cover_exactly_once, vertex_revisited, valid_euler_trail, boundary).","answer":{"kind":"expression","value":"([0, 1, 2, 3, 4, 5, 0], [0, 2, 3, 4, 5, 1], Yes, Yes, Yes, Yes, Yes, trail_allows_repeated_vertices_not_edges)","explanation":"Record edge indices for steps found in the parent graph. Check walk length and all adjacencies separately from uniqueness and exact coverage of those recorded edge indices. The final validity requires both the length/adjacency and complete single-use edge checks; repeated vertices alone do not refute an Euler trail. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 5, 0], [0, 2, 3, 4, 5, 1], Yes, Yes, Yes, Yes, Yes, trail_allows_repeated_vertices_not_edges)."},"skills":["Revisiting Vertices Is Allowed; Repeating an Edge Is Not"],"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-20261105-r04-eulerian-trail-0005","batch_id":"batch-20261014-advance-20261105-r04-eulerian-trail-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"db06579696ecb09bd04efbf20137584e155272e5b549aa7743edbd64673ddba0","family_id":"qf-mod36-euler-trail-audit","seed":202611054005,"semantic_template":"qf-mod36-euler-trail-audit/{\"edges\":[[0,1],[0,6],[1,2],[2,3],[3,4],[4,5],[5,6]],\"n\":7,\"vertex_walk\":[0,1,2,3,4,5,6,0]}"},"fields":{"title":"Revisiting Vertices Is Allowed; Repeating an Edge Is Not","stem":"Use the complete finite simple undirected graph on Fin(7)={0,…,6}, with canonical sorted edges=[[0, 1], [0, 6], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6]] and zero-based edge indices in that order. The edge support contains exactly the nonisolated vertices. A trail may revisit vertices but may not repeat an edge; an Euler trail uses every edge once. Isolated vertices do not invalidate connected edge support, and no Hamiltonian conclusion follows. Supplied vertex_walk=[0, 1, 2, 3, 4, 5, 6, 0]. Give (vertex_walk, used_edge_indices, length_and_adjacency, recorded_edge_indices_unique, recorded_edges_cover_exactly_once, vertex_revisited, valid_euler_trail, boundary).","answer":{"kind":"expression","value":"([0, 1, 2, 3, 4, 5, 6, 0], [0, 2, 3, 4, 5, 6, 1], Yes, Yes, Yes, Yes, Yes, trail_allows_repeated_vertices_not_edges)","explanation":"Record edge indices for steps found in the parent graph. Check walk length and all adjacencies separately from uniqueness and exact coverage of those recorded edge indices. The final validity requires both the length/adjacency and complete single-use edge checks; repeated vertices alone do not refute an Euler trail. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 5, 6, 0], [0, 2, 3, 4, 5, 6, 1], Yes, Yes, Yes, Yes, Yes, trail_allows_repeated_vertices_not_edges)."},"skills":["Revisiting Vertices Is Allowed; Repeating an Edge Is Not"],"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-20261105-r04-eulerian-trail-0006","batch_id":"batch-20261014-advance-20261105-r04-eulerian-trail-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"060c09450856028e14afc58f21f3de3dcc84053d56aee827c69a2c95735ac630","family_id":"qf-mod36-euler-trail-audit","seed":202611054006,"semantic_template":"qf-mod36-euler-trail-audit/{\"edges\":[[0,1],[0,7],[1,2],[2,3],[3,4],[4,5],[5,6],[6,7]],\"n\":8,\"vertex_walk\":[0,1,2,3,4,5,6,7,0]}"},"fields":{"title":"Revisiting Vertices Is Allowed; Repeating an Edge Is Not","stem":"Use the complete finite simple undirected graph on Fin(8)={0,…,7}, with canonical sorted edges=[[0, 1], [0, 7], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7]] and zero-based edge indices in that order. The edge support contains exactly the nonisolated vertices. A trail may revisit vertices but may not repeat an edge; an Euler trail uses every edge once. Isolated vertices do not invalidate connected edge support, and no Hamiltonian conclusion follows. Supplied vertex_walk=[0, 1, 2, 3, 4, 5, 6, 7, 0]. Give (vertex_walk, used_edge_indices, length_and_adjacency, recorded_edge_indices_unique, recorded_edges_cover_exactly_once, vertex_revisited, valid_euler_trail, boundary).","answer":{"kind":"expression","value":"([0, 1, 2, 3, 4, 5, 6, 7, 0], [0, 2, 3, 4, 5, 6, 7, 1], Yes, Yes, Yes, Yes, Yes, trail_allows_repeated_vertices_not_edges)","explanation":"Record edge indices for steps found in the parent graph. Check walk length and all adjacencies separately from uniqueness and exact coverage of those recorded edge indices. The final validity requires both the length/adjacency and complete single-use edge checks; repeated vertices alone do not refute an Euler trail. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 5, 6, 7, 0], [0, 2, 3, 4, 5, 6, 7, 1], Yes, Yes, Yes, Yes, Yes, trail_allows_repeated_vertices_not_edges)."},"skills":["Revisiting Vertices Is Allowed; Repeating an Edge Is Not"],"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-20261105-r04-eulerian-trail-0007","batch_id":"batch-20261014-advance-20261105-r04-eulerian-trail-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c61c0d77691e0a594b7f87f4f2780e0831dfe2127b384b82e26c824619d36751","family_id":"qf-mod36-euler-trail-audit","seed":202611054007,"semantic_template":"qf-mod36-euler-trail-audit/{\"edges\":[[0,1],[0,8],[1,2],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8]],\"n\":9,\"vertex_walk\":[0,1,2,3,4,5,6,7,8,0]}"},"fields":{"title":"Revisiting Vertices Is Allowed; Repeating an Edge Is Not","stem":"Use the complete finite simple undirected graph on Fin(9)={0,…,8}, with canonical sorted edges=[[0, 1], [0, 8], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7], [7, 8]] and zero-based edge indices in that order. The edge support contains exactly the nonisolated vertices. A trail may revisit vertices but may not repeat an edge; an Euler trail uses every edge once. Isolated vertices do not invalidate connected edge support, and no Hamiltonian conclusion follows. Supplied vertex_walk=[0, 1, 2, 3, 4, 5, 6, 7, 8, 0]. Give (vertex_walk, used_edge_indices, length_and_adjacency, recorded_edge_indices_unique, recorded_edges_cover_exactly_once, vertex_revisited, valid_euler_trail, boundary).","answer":{"kind":"expression","value":"([0, 1, 2, 3, 4, 5, 6, 7, 8, 0], [0, 2, 3, 4, 5, 6, 7, 8, 1], Yes, Yes, Yes, Yes, Yes, trail_allows_repeated_vertices_not_edges)","explanation":"Record edge indices for steps found in the parent graph. Check walk length and all adjacencies separately from uniqueness and exact coverage of those recorded edge indices. The final validity requires both the length/adjacency and complete single-use edge checks; repeated vertices alone do not refute an Euler trail. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 5, 6, 7, 8, 0], [0, 2, 3, 4, 5, 6, 7, 8, 1], Yes, Yes, Yes, Yes, Yes, trail_allows_repeated_vertices_not_edges)."},"skills":["Revisiting Vertices Is Allowed; Repeating an Edge Is Not"],"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-20261105-r04-eulerian-trail-0008","batch_id":"batch-20261014-advance-20261105-r04-eulerian-trail-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0e379a23ee8bb543e1c0f7f5a44540b0eaa35d54d5a734324621e0194b32fce5","family_id":"qf-mod36-euler-trail-audit","seed":202611054008,"semantic_template":"qf-mod36-euler-trail-audit/{\"edges\":[[0,1],[0,9],[1,2],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9]],\"n\":10,\"vertex_walk\":[0,1,2,3,4,5,6,7,8,9,0]}"},"fields":{"title":"Revisiting Vertices Is Allowed; Repeating an Edge Is Not","stem":"Use the complete finite simple undirected graph on Fin(10)={0,…,9}, with canonical sorted edges=[[0, 1], [0, 9], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7], [7, 8], [8, 9]] and zero-based edge indices in that order. The edge support contains exactly the nonisolated vertices. A trail may revisit vertices but may not repeat an edge; an Euler trail uses every edge once. Isolated vertices do not invalidate connected edge support, and no Hamiltonian conclusion follows. Supplied vertex_walk=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0]. Give (vertex_walk, used_edge_indices, length_and_adjacency, recorded_edge_indices_unique, recorded_edges_cover_exactly_once, vertex_revisited, valid_euler_trail, boundary).","answer":{"kind":"expression","value":"([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0], [0, 2, 3, 4, 5, 6, 7, 8, 9, 1], Yes, Yes, Yes, Yes, Yes, trail_allows_repeated_vertices_not_edges)","explanation":"Record edge indices for steps found in the parent graph. Check walk length and all adjacencies separately from uniqueness and exact coverage of those recorded edge indices. The final validity requires both the length/adjacency and complete single-use edge checks; repeated vertices alone do not refute an Euler trail. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0], [0, 2, 3, 4, 5, 6, 7, 8, 9, 1], Yes, Yes, Yes, Yes, Yes, trail_allows_repeated_vertices_not_edges)."},"skills":["Revisiting Vertices Is Allowed; Repeating an Edge Is Not"],"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-20261105-r04-eulerian-trail-0009","batch_id":"batch-20261014-advance-20261105-r04-eulerian-trail-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"86735cc20a33988aa1df1e0d3426a9270b26013ce7f92d8a97e0db817ed20540","family_id":"qf-mod36-euler-trail-audit","seed":202611054009,"semantic_template":"qf-mod36-euler-trail-audit/{\"edges\":[[0,1],[0,10],[1,2],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10]],\"n\":11,\"vertex_walk\":[0,1,2,3,4,5,6,7,8,9,10,0]}"},"fields":{"title":"Revisiting Vertices Is Allowed; Repeating an Edge Is Not","stem":"Use the complete finite simple undirected graph on Fin(11)={0,…,10}, with canonical sorted edges=[[0, 1], [0, 10], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7], [7, 8], [8, 9], [9, 10]] and zero-based edge indices in that order. The edge support contains exactly the nonisolated vertices. A trail may revisit vertices but may not repeat an edge; an Euler trail uses every edge once. Isolated vertices do not invalidate connected edge support, and no Hamiltonian conclusion follows. Supplied vertex_walk=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 0]. Give (vertex_walk, used_edge_indices, length_and_adjacency, recorded_edge_indices_unique, recorded_edges_cover_exactly_once, vertex_revisited, valid_euler_trail, boundary).","answer":{"kind":"expression","value":"([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 0], [0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1], Yes, Yes, Yes, Yes, Yes, trail_allows_repeated_vertices_not_edges)","explanation":"Record edge indices for steps found in the parent graph. Check walk length and all adjacencies separately from uniqueness and exact coverage of those recorded edge indices. The final validity requires both the length/adjacency and complete single-use edge checks; repeated vertices alone do not refute an Euler trail. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 0], [0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1], Yes, Yes, Yes, Yes, Yes, trail_allows_repeated_vertices_not_edges)."},"skills":["Revisiting Vertices Is Allowed; Repeating an Edge Is Not"],"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-20261105-r04-eulerian-trail-0010","batch_id":"batch-20261014-advance-20261105-r04-eulerian-trail-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c854ae6e406eb640a2abded3066cf78c39a8aa00df7351205c4f0e65f85d3d41","family_id":"qf-mod36-euler-trail-audit","seed":202611054010,"semantic_template":"qf-mod36-euler-trail-audit/{\"edges\":[[0,1],[0,11],[1,2],[10,11],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10]],\"n\":12,\"vertex_walk\":[0,1,2,3,4,5,6,7,8,9,10,11,0]}"},"fields":{"title":"Revisiting Vertices Is Allowed; Repeating an Edge Is Not","stem":"Use the complete finite simple undirected graph on Fin(12)={0,…,11}, with canonical sorted edges=[[0, 1], [0, 11], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7], [7, 8], [8, 9], [9, 10], [10, 11]] and zero-based edge indices in that order. The edge support contains exactly the nonisolated vertices. A trail may revisit vertices but may not repeat an edge; an Euler trail uses every edge once. Isolated vertices do not invalidate connected edge support, and no Hamiltonian conclusion follows. Supplied vertex_walk=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 0]. Give (vertex_walk, used_edge_indices, length_and_adjacency, recorded_edge_indices_unique, recorded_edges_cover_exactly_once, vertex_revisited, valid_euler_trail, boundary).","answer":{"kind":"expression","value":"([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 0], [0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 1], Yes, Yes, Yes, Yes, Yes, trail_allows_repeated_vertices_not_edges)","explanation":"Record edge indices for steps found in the parent graph. Check walk length and all adjacencies separately from uniqueness and exact coverage of those recorded edge indices. The final validity requires both the length/adjacency and complete single-use edge checks; repeated vertices alone do not refute an Euler trail. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 0], [0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 1], Yes, Yes, Yes, Yes, Yes, trail_allows_repeated_vertices_not_edges)."},"skills":["Revisiting Vertices Is Allowed; Repeating an Edge Is Not"],"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-20261105-r04-eulerian-trail-0011","batch_id":"batch-20261014-advance-20261105-r04-eulerian-trail-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"65a6378c5bb876f30002628536aba4ed833767951399798f432aa98b1aed7861","family_id":"qf-mod36-euler-trail-audit","seed":202611054011,"semantic_template":"qf-mod36-euler-trail-audit/{\"edges\":[[0,1],[0,12],[1,2],[10,11],[11,12],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10]],\"n\":13,\"vertex_walk\":[0,1,2,3,4,5,6,7,8,9,10,11,12,0]}"},"fields":{"title":"Revisiting Vertices Is Allowed; Repeating an Edge Is Not","stem":"Use the complete finite simple undirected graph on Fin(13)={0,…,12}, with canonical sorted edges=[[0, 1], [0, 12], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7], [7, 8], [8, 9], [9, 10], [10, 11], [11, 12]] and zero-based edge indices in that order. The edge support contains exactly the nonisolated vertices. A trail may revisit vertices but may not repeat an edge; an Euler trail uses every edge once. Isolated vertices do not invalidate connected edge support, and no Hamiltonian conclusion follows. Supplied vertex_walk=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 0]. Give (vertex_walk, used_edge_indices, length_and_adjacency, recorded_edge_indices_unique, recorded_edges_cover_exactly_once, vertex_revisited, valid_euler_trail, boundary).","answer":{"kind":"expression","value":"([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 0], [0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 1], Yes, Yes, Yes, Yes, Yes, trail_allows_repeated_vertices_not_edges)","explanation":"Record edge indices for steps found in the parent graph. Check walk length and all adjacencies separately from uniqueness and exact coverage of those recorded edge indices. The final validity requires both the length/adjacency and complete single-use edge checks; repeated vertices alone do not refute an Euler trail. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 0], [0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 1], Yes, Yes, Yes, Yes, Yes, trail_allows_repeated_vertices_not_edges)."},"skills":["Revisiting Vertices Is Allowed; Repeating an Edge Is Not"],"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-20261105-r04-eulerian-trail-0012","batch_id":"batch-20261014-advance-20261105-r04-eulerian-trail-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cc5130afdc8eaea4af0e5ff19d5e88e2a09c18256f4679b6688034ce4f8e1c76","family_id":"qf-mod36-euler-trail-audit","seed":202611054012,"semantic_template":"qf-mod36-euler-trail-audit/{\"edges\":[[0,1],[0,13],[1,2],[10,11],[11,12],[12,13],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10]],\"n\":14,\"vertex_walk\":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,0]}"},"fields":{"title":"Revisiting Vertices Is Allowed; Repeating an Edge Is Not","stem":"Use the complete finite simple undirected graph on Fin(14)={0,…,13}, with canonical sorted edges=[[0, 1], [0, 13], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7], [7, 8], [8, 9], [9, 10], [10, 11], [11, 12], [12, 13]] and zero-based edge indices in that order. The edge support contains exactly the nonisolated vertices. A trail may revisit vertices but may not repeat an edge; an Euler trail uses every edge once. Isolated vertices do not invalidate connected edge support, and no Hamiltonian conclusion follows. Supplied vertex_walk=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 0]. Give (vertex_walk, used_edge_indices, length_and_adjacency, recorded_edge_indices_unique, recorded_edges_cover_exactly_once, vertex_revisited, valid_euler_trail, boundary).","answer":{"kind":"expression","value":"([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 0], [0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 1], Yes, Yes, Yes, Yes, Yes, trail_allows_repeated_vertices_not_edges)","explanation":"Record edge indices for steps found in the parent graph. Check walk length and all adjacencies separately from uniqueness and exact coverage of those recorded edge indices. The final validity requires both the length/adjacency and complete single-use edge checks; repeated vertices alone do not refute an Euler trail. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 0], [0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 1], Yes, Yes, Yes, Yes, Yes, trail_allows_repeated_vertices_not_edges)."},"skills":["Revisiting Vertices Is Allowed; Repeating an Edge Is Not"],"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-20261105-r04-eulerian-trail-0013","batch_id":"batch-20261014-advance-20261105-r04-eulerian-trail-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"05a7340d56c956d1dee59bebc1b4c669087b0cafd322ec6439cb42e0f10cc69b","family_id":"qf-mod36-euler-trail-audit","seed":202611054013,"semantic_template":"qf-mod36-euler-trail-audit/{\"edges\":[[0,1],[0,14],[1,2],[10,11],[11,12],[12,13],[13,14],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10]],\"n\":15,\"vertex_walk\":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,0]}"},"fields":{"title":"Revisiting Vertices Is Allowed; Repeating an Edge Is Not","stem":"Use the complete finite simple undirected graph on Fin(15)={0,…,14}, with canonical sorted edges=[[0, 1], [0, 14], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7], [7, 8], [8, 9], [9, 10], [10, 11], [11, 12], [12, 13], [13, 14]] and zero-based edge indices in that order. The edge support contains exactly the nonisolated vertices. A trail may revisit vertices but may not repeat an edge; an Euler trail uses every edge once. Isolated vertices do not invalidate connected edge support, and no Hamiltonian conclusion follows. Supplied vertex_walk=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 0]. Give (vertex_walk, used_edge_indices, length_and_adjacency, recorded_edge_indices_unique, recorded_edges_cover_exactly_once, vertex_revisited, valid_euler_trail, boundary).","answer":{"kind":"expression","value":"([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 0], [0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 1], Yes, Yes, Yes, Yes, Yes, trail_allows_repeated_vertices_not_edges)","explanation":"Record edge indices for steps found in the parent graph. Check walk length and all adjacencies separately from uniqueness and exact coverage of those recorded edge indices. The final validity requires both the length/adjacency and complete single-use edge checks; repeated vertices alone do not refute an Euler trail. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 0], [0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 1], Yes, Yes, Yes, Yes, Yes, trail_allows_repeated_vertices_not_edges)."},"skills":["Revisiting Vertices Is Allowed; Repeating an Edge Is Not"],"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-20261105-r04-eulerian-trail-0014","batch_id":"batch-20261014-advance-20261105-r04-eulerian-trail-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e7f30604c110cc6bd86c4cd7e9a2c7131cc7876f301fcb073cea9b50a6ef5640","family_id":"qf-mod36-euler-trail-audit","seed":202611054014,"semantic_template":"qf-mod36-euler-trail-audit/{\"edges\":[[0,1],[0,15],[1,2],[10,11],[11,12],[12,13],[13,14],[14,15],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10]],\"n\":16,\"vertex_walk\":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,0]}"},"fields":{"title":"Revisiting Vertices Is Allowed; Repeating an Edge Is Not","stem":"Use the complete finite simple undirected graph on Fin(16)={0,…,15}, with canonical sorted edges=[[0, 1], [0, 15], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7], [7, 8], [8, 9], [9, 10], [10, 11], [11, 12], [12, 13], [13, 14], [14, 15]] and zero-based edge indices in that order. The edge support contains exactly the nonisolated vertices. A trail may revisit vertices but may not repeat an edge; an Euler trail uses every edge once. Isolated vertices do not invalidate connected edge support, and no Hamiltonian conclusion follows. Supplied vertex_walk=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0]. Give (vertex_walk, used_edge_indices, length_and_adjacency, recorded_edge_indices_unique, recorded_edges_cover_exactly_once, vertex_revisited, valid_euler_trail, boundary).","answer":{"kind":"expression","value":"([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0], [0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 1], Yes, Yes, Yes, Yes, Yes, trail_allows_repeated_vertices_not_edges)","explanation":"Record edge indices for steps found in the parent graph. Check walk length and all adjacencies separately from uniqueness and exact coverage of those recorded edge indices. The final validity requires both the length/adjacency and complete single-use edge checks; repeated vertices alone do not refute an Euler trail. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0], [0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 1], Yes, Yes, Yes, Yes, Yes, trail_allows_repeated_vertices_not_edges)."},"skills":["Revisiting Vertices Is Allowed; Repeating an Edge Is Not"],"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-20261105-r04-eulerian-trail-0015","batch_id":"batch-20261014-advance-20261105-r04-eulerian-trail-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"284d3161497799b8e0c110ff5bc44cadc8ba1ddb9fb0a80862f98c675fb51e9b","family_id":"qf-mod36-euler-trail-audit","seed":202611054015,"semantic_template":"qf-mod36-euler-trail-audit/{\"edges\":[[0,1],[0,16],[1,2],[10,11],[11,12],[12,13],[13,14],[14,15],[15,16],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10]],\"n\":17,\"vertex_walk\":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,0]}"},"fields":{"title":"Revisiting Vertices Is Allowed; Repeating an Edge Is Not","stem":"Use the complete finite simple undirected graph on Fin(17)={0,…,16}, with canonical sorted edges=[[0, 1], [0, 16], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7], [7, 8], [8, 9], [9, 10], [10, 11], [11, 12], [12, 13], [13, 14], [14, 15], [15, 16]] and zero-based edge indices in that order. The edge support contains exactly the nonisolated vertices. A trail may revisit vertices but may not repeat an edge; an Euler trail uses every edge once. Isolated vertices do not invalidate connected edge support, and no Hamiltonian conclusion follows. Supplied vertex_walk=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 0]. Give (vertex_walk, used_edge_indices, length_and_adjacency, recorded_edge_indices_unique, recorded_edges_cover_exactly_once, vertex_revisited, valid_euler_trail, boundary).","answer":{"kind":"expression","value":"([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 0], [0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 1], Yes, Yes, Yes, Yes, Yes, trail_allows_repeated_vertices_not_edges)","explanation":"Record edge indices for steps found in the parent graph. Check walk length and all adjacencies separately from uniqueness and exact coverage of those recorded edge indices. The final validity requires both the length/adjacency and complete single-use edge checks; repeated vertices alone do not refute an Euler trail. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 0], [0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 1], Yes, Yes, Yes, Yes, Yes, trail_allows_repeated_vertices_not_edges)."},"skills":["Revisiting Vertices Is Allowed; Repeating an Edge Is Not"],"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-20261105-r04-eulerian-trail-0016","batch_id":"batch-20261014-advance-20261105-r04-eulerian-trail-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8e19a7b383fb8cb6f69f284c31f65e8b5d4ee37eb42b02da2be42b05d67e710c","family_id":"qf-mod36-euler-trail-audit","seed":202611054016,"semantic_template":"qf-mod36-euler-trail-audit/{\"edges\":[[0,1],[0,17],[1,2],[10,11],[11,12],[12,13],[13,14],[14,15],[15,16],[16,17],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10]],\"n\":18,\"vertex_walk\":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,0]}"},"fields":{"title":"Revisiting Vertices Is Allowed; Repeating an Edge Is Not","stem":"Use the complete finite simple undirected graph on Fin(18)={0,…,17}, with canonical sorted edges=[[0, 1], [0, 17], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7], [7, 8], [8, 9], [9, 10], [10, 11], [11, 12], [12, 13], [13, 14], [14, 15], [15, 16], [16, 17]] and zero-based edge indices in that order. The edge support contains exactly the nonisolated vertices. A trail may revisit vertices but may not repeat an edge; an Euler trail uses every edge once. Isolated vertices do not invalidate connected edge support, and no Hamiltonian conclusion follows. Supplied vertex_walk=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0]. Give (vertex_walk, used_edge_indices, length_and_adjacency, recorded_edge_indices_unique, recorded_edges_cover_exactly_once, vertex_revisited, valid_euler_trail, boundary).","answer":{"kind":"expression","value":"([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0], [0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 1], Yes, Yes, Yes, Yes, Yes, trail_allows_repeated_vertices_not_edges)","explanation":"Record edge indices for steps found in the parent graph. Check walk length and all adjacencies separately from uniqueness and exact coverage of those recorded edge indices. The final validity requires both the length/adjacency and complete single-use edge checks; repeated vertices alone do not refute an Euler trail. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0], [0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 1], Yes, Yes, Yes, Yes, Yes, trail_allows_repeated_vertices_not_edges)."},"skills":["Revisiting Vertices Is Allowed; Repeating an Edge Is Not"],"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-20261105-r04-eulerian-trail-0017","batch_id":"batch-20261014-advance-20261105-r04-eulerian-trail-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4f02fedd8ec840025fff08e66be1c1164b38bd25f18b0d054678194ddbb59ae9","family_id":"qf-mod36-euler-trail-audit","seed":202611054017,"semantic_template":"qf-mod36-euler-trail-audit/{\"edges\":[[0,1],[0,18],[1,2],[10,11],[11,12],[12,13],[13,14],[14,15],[15,16],[16,17],[17,18],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10]],\"n\":19,\"vertex_walk\":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,0]}"},"fields":{"title":"Revisiting Vertices Is Allowed; Repeating an Edge Is Not","stem":"Use the complete finite simple undirected graph on Fin(19)={0,…,18}, with canonical sorted edges=[[0, 1], [0, 18], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7], [7, 8], [8, 9], [9, 10], [10, 11], [11, 12], [12, 13], [13, 14], [14, 15], [15, 16], [16, 17], [17, 18]] and zero-based edge indices in that order. The edge support contains exactly the nonisolated vertices. A trail may revisit vertices but may not repeat an edge; an Euler trail uses every edge once. Isolated vertices do not invalidate connected edge support, and no Hamiltonian conclusion follows. Supplied vertex_walk=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 0]. Give (vertex_walk, used_edge_indices, length_and_adjacency, recorded_edge_indices_unique, recorded_edges_cover_exactly_once, vertex_revisited, valid_euler_trail, boundary).","answer":{"kind":"expression","value":"([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 0], [0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 1], Yes, Yes, Yes, Yes, Yes, trail_allows_repeated_vertices_not_edges)","explanation":"Record edge indices for steps found in the parent graph. Check walk length and all adjacencies separately from uniqueness and exact coverage of those recorded edge indices. The final validity requires both the length/adjacency and complete single-use edge checks; repeated vertices alone do not refute an Euler trail. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 0], [0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 1], Yes, Yes, Yes, Yes, Yes, trail_allows_repeated_vertices_not_edges)."},"skills":["Revisiting Vertices Is Allowed; Repeating an Edge Is Not"],"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-20261105-r04-eulerian-trail-0018","batch_id":"batch-20261014-advance-20261105-r04-eulerian-trail-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"965298d8f89416d9bbd1fe0c4d1b144eaf6b8ed67c96a33178e744ec6b73df9f","family_id":"qf-mod36-euler-trail-audit","seed":202611054018,"semantic_template":"qf-mod36-euler-trail-audit/{\"edges\":[[0,1],[0,19],[1,2],[10,11],[11,12],[12,13],[13,14],[14,15],[15,16],[16,17],[17,18],[18,19],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10]],\"n\":20,\"vertex_walk\":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0]}"},"fields":{"title":"Revisiting Vertices Is Allowed; Repeating an Edge Is Not","stem":"Use the complete finite simple undirected graph on Fin(20)={0,…,19}, with canonical sorted edges=[[0, 1], [0, 19], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7], [7, 8], [8, 9], [9, 10], [10, 11], [11, 12], [12, 13], [13, 14], [14, 15], [15, 16], [16, 17], [17, 18], [18, 19]] and zero-based edge indices in that order. The edge support contains exactly the nonisolated vertices. A trail may revisit vertices but may not repeat an edge; an Euler trail uses every edge once. Isolated vertices do not invalidate connected edge support, and no Hamiltonian conclusion follows. Supplied vertex_walk=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0]. Give (vertex_walk, used_edge_indices, length_and_adjacency, recorded_edge_indices_unique, recorded_edges_cover_exactly_once, vertex_revisited, valid_euler_trail, boundary).","answer":{"kind":"expression","value":"([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0], [0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 1], Yes, Yes, Yes, Yes, Yes, trail_allows_repeated_vertices_not_edges)","explanation":"Record edge indices for steps found in the parent graph. Check walk length and all adjacencies separately from uniqueness and exact coverage of those recorded edge indices. The final validity requires both the length/adjacency and complete single-use edge checks; repeated vertices alone do not refute an Euler trail. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0], [0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 1], Yes, Yes, Yes, Yes, Yes, trail_allows_repeated_vertices_not_edges)."},"skills":["Revisiting Vertices Is Allowed; Repeating an Edge Is Not"],"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-20261105-r04-eulerian-trail-0019","batch_id":"batch-20261014-advance-20261105-r04-eulerian-trail-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a3a83c05ba6f63ba83e247a35580b54407357c8bb9ebffac1074747351b5f1e9","family_id":"qf-mod36-euler-trail-audit","seed":202611054019,"semantic_template":"qf-mod36-euler-trail-audit/{\"edges\":[[0,1],[0,20],[1,2],[10,11],[11,12],[12,13],[13,14],[14,15],[15,16],[16,17],[17,18],[18,19],[19,20],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10]],\"n\":21,\"vertex_walk\":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,0]}"},"fields":{"title":"Revisiting Vertices Is Allowed; Repeating an Edge Is Not","stem":"Use the complete finite simple undirected graph on Fin(21)={0,…,20}, with canonical sorted edges=[[0, 1], [0, 20], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7], [7, 8], [8, 9], [9, 10], [10, 11], [11, 12], [12, 13], [13, 14], [14, 15], [15, 16], [16, 17], [17, 18], [18, 19], [19, 20]] and zero-based edge indices in that order. The edge support contains exactly the nonisolated vertices. A trail may revisit vertices but may not repeat an edge; an Euler trail uses every edge once. Isolated vertices do not invalidate connected edge support, and no Hamiltonian conclusion follows. Supplied vertex_walk=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0]. Give (vertex_walk, used_edge_indices, length_and_adjacency, recorded_edge_indices_unique, recorded_edges_cover_exactly_once, vertex_revisited, valid_euler_trail, boundary).","answer":{"kind":"expression","value":"([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0], [0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 1], Yes, Yes, Yes, Yes, Yes, trail_allows_repeated_vertices_not_edges)","explanation":"Record edge indices for steps found in the parent graph. Check walk length and all adjacencies separately from uniqueness and exact coverage of those recorded edge indices. The final validity requires both the length/adjacency and complete single-use edge checks; repeated vertices alone do not refute an Euler trail. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0], [0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 1], Yes, Yes, Yes, Yes, Yes, trail_allows_repeated_vertices_not_edges)."},"skills":["Revisiting Vertices Is Allowed; Repeating an Edge Is Not"],"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-20261105-r04-eulerian-trail-0020","batch_id":"batch-20261014-advance-20261105-r04-eulerian-trail-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c162e6b3acd3a4c9385322ede05329cef68fbb951874a7a0b1c6abec7b85bf57","family_id":"qf-mod36-euler-trail-audit","seed":202611054020,"semantic_template":"qf-mod36-euler-trail-audit/{\"edges\":[[0,1],[0,21],[1,2],[10,11],[11,12],[12,13],[13,14],[14,15],[15,16],[16,17],[17,18],[18,19],[19,20],[2,3],[20,21],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10]],\"n\":22,\"vertex_walk\":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0]}"},"fields":{"title":"Revisiting Vertices Is Allowed; Repeating an Edge Is Not","stem":"Use the complete finite simple undirected graph on Fin(22)={0,…,21}, with canonical sorted edges=[[0, 1], [0, 21], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7], [7, 8], [8, 9], [9, 10], [10, 11], [11, 12], [12, 13], [13, 14], [14, 15], [15, 16], [16, 17], [17, 18], [18, 19], [19, 20], [20, 21]] and zero-based edge indices in that order. The edge support contains exactly the nonisolated vertices. A trail may revisit vertices but may not repeat an edge; an Euler trail uses every edge once. Isolated vertices do not invalidate connected edge support, and no Hamiltonian conclusion follows. Supplied vertex_walk=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0]. Give (vertex_walk, used_edge_indices, length_and_adjacency, recorded_edge_indices_unique, recorded_edges_cover_exactly_once, vertex_revisited, valid_euler_trail, boundary).","answer":{"kind":"expression","value":"([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0], [0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 1], Yes, Yes, Yes, Yes, Yes, trail_allows_repeated_vertices_not_edges)","explanation":"Record edge indices for steps found in the parent graph. Check walk length and all adjacencies separately from uniqueness and exact coverage of those recorded edge indices. The final validity requires both the length/adjacency and complete single-use edge checks; repeated vertices alone do not refute an Euler trail. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0], [0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 1], Yes, Yes, Yes, Yes, Yes, trail_allows_repeated_vertices_not_edges)."},"skills":["Revisiting Vertices Is Allowed; Repeating an Edge Is Not"],"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-20261105-r04-eulerian-trail-0021","batch_id":"batch-20261014-advance-20261105-r04-eulerian-trail-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"30b1e1e9ab798b4a3be87543a0ee9cb1dcc3fe32b758999892640173e2bd5b78","family_id":"qf-mod36-euler-trail-audit","seed":202611054021,"semantic_template":"qf-mod36-euler-trail-audit/{\"edges\":[[0,1],[0,22],[1,2],[10,11],[11,12],[12,13],[13,14],[14,15],[15,16],[16,17],[17,18],[18,19],[19,20],[2,3],[20,21],[21,22],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10]],\"n\":23,\"vertex_walk\":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0]}"},"fields":{"title":"Revisiting Vertices Is Allowed; Repeating an Edge Is Not","stem":"Use the complete finite simple undirected graph on Fin(23)={0,…,22}, with canonical sorted edges=[[0, 1], [0, 22], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7], [7, 8], [8, 9], [9, 10], [10, 11], [11, 12], [12, 13], [13, 14], [14, 15], [15, 16], [16, 17], [17, 18], [18, 19], [19, 20], [20, 21], [21, 22]] and zero-based edge indices in that order. The edge support contains exactly the nonisolated vertices. A trail may revisit vertices but may not repeat an edge; an Euler trail uses every edge once. Isolated vertices do not invalidate connected edge support, and no Hamiltonian conclusion follows. Supplied vertex_walk=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0]. Give (vertex_walk, used_edge_indices, length_and_adjacency, recorded_edge_indices_unique, recorded_edges_cover_exactly_once, vertex_revisited, valid_euler_trail, boundary).","answer":{"kind":"expression","value":"([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0], [0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 1], Yes, Yes, Yes, Yes, Yes, trail_allows_repeated_vertices_not_edges)","explanation":"Record edge indices for steps found in the parent graph. Check walk length and all adjacencies separately from uniqueness and exact coverage of those recorded edge indices. The final validity requires both the length/adjacency and complete single-use edge checks; repeated vertices alone do not refute an Euler trail. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0], [0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 1], Yes, Yes, Yes, Yes, Yes, trail_allows_repeated_vertices_not_edges)."},"skills":["Revisiting Vertices Is Allowed; Repeating an Edge Is Not"],"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-20261105-r04-eulerian-trail-0022","batch_id":"batch-20261014-advance-20261105-r04-eulerian-trail-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a14dc5ba059334e8e69165691b0b61e880e6fdea047ca43edb3b83f6e6a011aa","family_id":"qf-mod36-euler-trail-audit","seed":202611054022,"semantic_template":"qf-mod36-euler-trail-audit/{\"edges\":[[0,1],[0,23],[1,2],[10,11],[11,12],[12,13],[13,14],[14,15],[15,16],[16,17],[17,18],[18,19],[19,20],[2,3],[20,21],[21,22],[22,23],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10]],\"n\":24,\"vertex_walk\":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0]}"},"fields":{"title":"Revisiting Vertices Is Allowed; Repeating an Edge Is Not","stem":"Use the complete finite simple undirected graph on Fin(24)={0,…,23}, with canonical sorted edges=[[0, 1], [0, 23], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7], [7, 8], [8, 9], [9, 10], [10, 11], [11, 12], [12, 13], [13, 14], [14, 15], [15, 16], [16, 17], [17, 18], [18, 19], [19, 20], [20, 21], [21, 22], [22, 23]] and zero-based edge indices in that order. The edge support contains exactly the nonisolated vertices. A trail may revisit vertices but may not repeat an edge; an Euler trail uses every edge once. Isolated vertices do not invalidate connected edge support, and no Hamiltonian conclusion follows. Supplied vertex_walk=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0]. Give (vertex_walk, used_edge_indices, length_and_adjacency, recorded_edge_indices_unique, recorded_edges_cover_exactly_once, vertex_revisited, valid_euler_trail, boundary).","answer":{"kind":"expression","value":"([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0], [0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 1], Yes, Yes, Yes, Yes, Yes, trail_allows_repeated_vertices_not_edges)","explanation":"Record edge indices for steps found in the parent graph. Check walk length and all adjacencies separately from uniqueness and exact coverage of those recorded edge indices. The final validity requires both the length/adjacency and complete single-use edge checks; repeated vertices alone do not refute an Euler trail. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0], [0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 1], Yes, Yes, Yes, Yes, Yes, trail_allows_repeated_vertices_not_edges)."},"skills":["Revisiting Vertices Is Allowed; Repeating an Edge Is Not"],"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-20261105-r04-eulerian-trail-0023","batch_id":"batch-20261014-advance-20261105-r04-eulerian-trail-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6e76703f3d0ae8737fc90596d58264de91a5e557836bcea023f132bbb35172fb","family_id":"qf-mod36-euler-trail-audit","seed":202611054023,"semantic_template":"qf-mod36-euler-trail-audit/{\"edges\":[[0,1],[0,24],[1,2],[10,11],[11,12],[12,13],[13,14],[14,15],[15,16],[16,17],[17,18],[18,19],[19,20],[2,3],[20,21],[21,22],[22,23],[23,24],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10]],\"n\":25,\"vertex_walk\":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0]}"},"fields":{"title":"Revisiting Vertices Is Allowed; Repeating an Edge Is Not","stem":"Use the complete finite simple undirected graph on Fin(25)={0,…,24}, with canonical sorted edges=[[0, 1], [0, 24], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7], [7, 8], [8, 9], [9, 10], [10, 11], [11, 12], [12, 13], [13, 14], [14, 15], [15, 16], [16, 17], [17, 18], [18, 19], [19, 20], [20, 21], [21, 22], [22, 23], [23, 24]] and zero-based edge indices in that order. The edge support contains exactly the nonisolated vertices. A trail may revisit vertices but may not repeat an edge; an Euler trail uses every edge once. Isolated vertices do not invalidate connected edge support, and no Hamiltonian conclusion follows. Supplied vertex_walk=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0]. Give (vertex_walk, used_edge_indices, length_and_adjacency, recorded_edge_indices_unique, recorded_edges_cover_exactly_once, vertex_revisited, valid_euler_trail, boundary).","answer":{"kind":"expression","value":"([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0], [0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 1], Yes, Yes, Yes, Yes, Yes, trail_allows_repeated_vertices_not_edges)","explanation":"Record edge indices for steps found in the parent graph. Check walk length and all adjacencies separately from uniqueness and exact coverage of those recorded edge indices. The final validity requires both the length/adjacency and complete single-use edge checks; repeated vertices alone do not refute an Euler trail. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0], [0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 1], Yes, Yes, Yes, Yes, Yes, trail_allows_repeated_vertices_not_edges)."},"skills":["Revisiting Vertices Is Allowed; Repeating an Edge Is Not"],"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-20261105-r04-eulerian-trail-0024","batch_id":"batch-20261014-advance-20261105-r04-eulerian-trail-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"21af828809e3e99e9cb6bbb437b0ae2a221bf547d73854a51d95b78aed42d9f4","family_id":"qf-mod36-euler-trail-audit","seed":202611054024,"semantic_template":"qf-mod36-euler-trail-audit/{\"edges\":[[0,1],[0,25],[1,2],[10,11],[11,12],[12,13],[13,14],[14,15],[15,16],[16,17],[17,18],[18,19],[19,20],[2,3],[20,21],[21,22],[22,23],[23,24],[24,25],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10]],\"n\":26,\"vertex_walk\":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0]}"},"fields":{"title":"Revisiting Vertices Is Allowed; Repeating an Edge Is Not","stem":"Use the complete finite simple undirected graph on Fin(26)={0,…,25}, with canonical sorted edges=[[0, 1], [0, 25], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7], [7, 8], [8, 9], [9, 10], [10, 11], [11, 12], [12, 13], [13, 14], [14, 15], [15, 16], [16, 17], [17, 18], [18, 19], [19, 20], [20, 21], [21, 22], [22, 23], [23, 24], [24, 25]] and zero-based edge indices in that order. The edge support contains exactly the nonisolated vertices. A trail may revisit vertices but may not repeat an edge; an Euler trail uses every edge once. Isolated vertices do not invalidate connected edge support, and no Hamiltonian conclusion follows. Supplied vertex_walk=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0]. Give (vertex_walk, used_edge_indices, length_and_adjacency, recorded_edge_indices_unique, recorded_edges_cover_exactly_once, vertex_revisited, valid_euler_trail, boundary).","answer":{"kind":"expression","value":"([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0], [0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 1], Yes, Yes, Yes, Yes, Yes, trail_allows_repeated_vertices_not_edges)","explanation":"Record edge indices for steps found in the parent graph. Check walk length and all adjacencies separately from uniqueness and exact coverage of those recorded edge indices. The final validity requires both the length/adjacency and complete single-use edge checks; repeated vertices alone do not refute an Euler trail. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0], [0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 1], Yes, Yes, Yes, Yes, Yes, trail_allows_repeated_vertices_not_edges)."},"skills":["Revisiting Vertices Is Allowed; Repeating an Edge Is Not"],"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-20261105-r04-eulerian-trail-0025","batch_id":"batch-20261014-advance-20261105-r04-eulerian-trail-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3e8e594994ef7da211fa2cf8180873040870d170a235e75df66c1544f6c2ffdc","family_id":"qf-mod36-euler-trail-audit","seed":202611054025,"semantic_template":"qf-mod36-euler-trail-audit/{\"edges\":[[0,1],[0,26],[1,2],[10,11],[11,12],[12,13],[13,14],[14,15],[15,16],[16,17],[17,18],[18,19],[19,20],[2,3],[20,21],[21,22],[22,23],[23,24],[24,25],[25,26],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10]],\"n\":27,\"vertex_walk\":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0]}"},"fields":{"title":"Revisiting Vertices Is Allowed; Repeating an Edge Is Not","stem":"Use the complete finite simple undirected graph on Fin(27)={0,…,26}, with canonical sorted edges=[[0, 1], [0, 26], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7], [7, 8], [8, 9], [9, 10], [10, 11], [11, 12], [12, 13], [13, 14], [14, 15], [15, 16], [16, 17], [17, 18], [18, 19], [19, 20], [20, 21], [21, 22], [22, 23], [23, 24], [24, 25], [25, 26]] and zero-based edge indices in that order. The edge support contains exactly the nonisolated vertices. A trail may revisit vertices but may not repeat an edge; an Euler trail uses every edge once. Isolated vertices do not invalidate connected edge support, and no Hamiltonian conclusion follows. Supplied vertex_walk=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0]. Give (vertex_walk, used_edge_indices, length_and_adjacency, recorded_edge_indices_unique, recorded_edges_cover_exactly_once, vertex_revisited, valid_euler_trail, boundary).","answer":{"kind":"expression","value":"([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0], [0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 1], Yes, Yes, Yes, Yes, Yes, trail_allows_repeated_vertices_not_edges)","explanation":"Record edge indices for steps found in the parent graph. Check walk length and all adjacencies separately from uniqueness and exact coverage of those recorded edge indices. The final validity requires both the length/adjacency and complete single-use edge checks; repeated vertices alone do not refute an Euler trail. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0], [0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 1], Yes, Yes, Yes, Yes, Yes, trail_allows_repeated_vertices_not_edges)."},"skills":["Revisiting Vertices Is Allowed; Repeating an Edge Is Not"],"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-20261105-r04-eulerian-trail-0026","batch_id":"batch-20261014-advance-20261105-r04-eulerian-trail-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4eaa9d4c02d5e00481729469de15f04981e82142f833799a83c7a6710eeaecde","family_id":"qf-mod36-euler-odd-degree-classification","seed":202611054026,"semantic_template":"qf-mod36-euler-odd-degree-classification/{\"edges\":[[0,1]],\"n\":2}"},"fields":{"title":"Degree Parity Needs Connected Nonisolated Support","stem":"Use the complete finite simple undirected graph on Fin(2)={0,…,1}, with canonical sorted edges=[[0, 1]] and zero-based edge indices in that order. The edge support contains exactly the nonisolated vertices. A trail may revisit vertices but may not repeat an edge; an Euler trail uses every edge once. Isolated vertices do not invalidate connected edge support, and no Hamiltonian conclusion follows. Give (degrees, odd_vertices, edge_support_connected, euler_type, criterion).","answer":{"kind":"expression","value":"([1, 1], [0, 1], Yes, open_euler_trail, zero_or_two_odd_vertices_on_connected_edge_support)","explanation":"For a nonzero-edge graph, compute every degree, all odd vertices and edge-support connectivity. Connected support with zero odd vertices gives a circuit, with two gives an open Euler trail; otherwise none. Parity alone does not connect separated edge components. The exact result, retaining the requested order, is ([1, 1], [0, 1], Yes, open_euler_trail, zero_or_two_odd_vertices_on_connected_edge_support)."},"skills":["Degree Parity Needs Connected Nonisolated Support"],"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-20261105-r04-eulerian-trail-0027","batch_id":"batch-20261014-advance-20261105-r04-eulerian-trail-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8b46def8d1d7b9df8f2c11821cc0d49a9cbaea4cb620efb21346475396a43e43","family_id":"qf-mod36-euler-odd-degree-classification","seed":202611054027,"semantic_template":"qf-mod36-euler-odd-degree-classification/{\"edges\":[[0,1],[1,2]],\"n\":3}"},"fields":{"title":"Degree Parity Needs Connected Nonisolated Support","stem":"Use the complete finite simple undirected graph on Fin(3)={0,…,2}, with canonical sorted edges=[[0, 1], [1, 2]] and zero-based edge indices in that order. The edge support contains exactly the nonisolated vertices. A trail may revisit vertices but may not repeat an edge; an Euler trail uses every edge once. Isolated vertices do not invalidate connected edge support, and no Hamiltonian conclusion follows. Give (degrees, odd_vertices, edge_support_connected, euler_type, criterion).","answer":{"kind":"expression","value":"([1, 2, 1], [0, 2], Yes, open_euler_trail, zero_or_two_odd_vertices_on_connected_edge_support)","explanation":"For a nonzero-edge graph, compute every degree, all odd vertices and edge-support connectivity. Connected support with zero odd vertices gives a circuit, with two gives an open Euler trail; otherwise none. Parity alone does not connect separated edge components. The exact result, retaining the requested order, is ([1, 2, 1], [0, 2], Yes, open_euler_trail, zero_or_two_odd_vertices_on_connected_edge_support)."},"skills":["Degree Parity Needs Connected Nonisolated Support"],"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-20261105-r04-eulerian-trail-0028","batch_id":"batch-20261014-advance-20261105-r04-eulerian-trail-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c1361876677eb33e492b4f554dbd7d5a4a7ab4248ba444566d157dfedc76ba57","family_id":"qf-mod36-euler-odd-degree-classification","seed":202611054028,"semantic_template":"qf-mod36-euler-odd-degree-classification/{\"edges\":[[0,1],[1,2],[2,3]],\"n\":4}"},"fields":{"title":"Degree Parity Needs Connected Nonisolated Support","stem":"Use the complete finite simple undirected graph on Fin(4)={0,…,3}, with canonical sorted edges=[[0, 1], [1, 2], [2, 3]] and zero-based edge indices in that order. The edge support contains exactly the nonisolated vertices. A trail may revisit vertices but may not repeat an edge; an Euler trail uses every edge once. Isolated vertices do not invalidate connected edge support, and no Hamiltonian conclusion follows. Give (degrees, odd_vertices, edge_support_connected, euler_type, criterion).","answer":{"kind":"expression","value":"([1, 2, 2, 1], [0, 3], Yes, open_euler_trail, zero_or_two_odd_vertices_on_connected_edge_support)","explanation":"For a nonzero-edge graph, compute every degree, all odd vertices and edge-support connectivity. Connected support with zero odd vertices gives a circuit, with two gives an open Euler trail; otherwise none. Parity alone does not connect separated edge components. The exact result, retaining the requested order, is ([1, 2, 2, 1], [0, 3], Yes, open_euler_trail, zero_or_two_odd_vertices_on_connected_edge_support)."},"skills":["Degree Parity Needs Connected Nonisolated Support"],"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-20261105-r04-eulerian-trail-0029","batch_id":"batch-20261014-advance-20261105-r04-eulerian-trail-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fa1f2894133f8aab5a11c2156592570be32402fd7a6e160a1970e5df3fc286ef","family_id":"qf-mod36-euler-odd-degree-classification","seed":202611054029,"semantic_template":"qf-mod36-euler-odd-degree-classification/{\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"n\":5}"},"fields":{"title":"Degree Parity Needs Connected Nonisolated Support","stem":"Use the complete finite simple undirected graph on Fin(5)={0,…,4}, with canonical sorted edges=[[0, 1], [1, 2], [2, 3], [3, 4]] and zero-based edge indices in that order. The edge support contains exactly the nonisolated vertices. A trail may revisit vertices but may not repeat an edge; an Euler trail uses every edge once. Isolated vertices do not invalidate connected edge support, and no Hamiltonian conclusion follows. Give (degrees, odd_vertices, edge_support_connected, euler_type, criterion).","answer":{"kind":"expression","value":"([1, 2, 2, 2, 1], [0, 4], Yes, open_euler_trail, zero_or_two_odd_vertices_on_connected_edge_support)","explanation":"For a nonzero-edge graph, compute every degree, all odd vertices and edge-support connectivity. Connected support with zero odd vertices gives a circuit, with two gives an open Euler trail; otherwise none. Parity alone does not connect separated edge components. The exact result, retaining the requested order, is ([1, 2, 2, 2, 1], [0, 4], Yes, open_euler_trail, zero_or_two_odd_vertices_on_connected_edge_support)."},"skills":["Degree Parity Needs Connected Nonisolated Support"],"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-20261105-r04-eulerian-trail-0030","batch_id":"batch-20261014-advance-20261105-r04-eulerian-trail-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"eba353f22379e7e54763e750ec43b9c23c70eaf58a893c76848f4cc33ffd336d","family_id":"qf-mod36-euler-odd-degree-classification","seed":202611054030,"semantic_template":"qf-mod36-euler-odd-degree-classification/{\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"n\":6}"},"fields":{"title":"Degree Parity Needs Connected Nonisolated Support","stem":"Use the complete finite simple undirected graph on Fin(6)={0,…,5}, with canonical sorted edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5]] and zero-based edge indices in that order. The edge support contains exactly the nonisolated vertices. A trail may revisit vertices but may not repeat an edge; an Euler trail uses every edge once. Isolated vertices do not invalidate connected edge support, and no Hamiltonian conclusion follows. Give (degrees, odd_vertices, edge_support_connected, euler_type, criterion).","answer":{"kind":"expression","value":"([1, 2, 2, 2, 2, 1], [0, 5], Yes, open_euler_trail, zero_or_two_odd_vertices_on_connected_edge_support)","explanation":"For a nonzero-edge graph, compute every degree, all odd vertices and edge-support connectivity. Connected support with zero odd vertices gives a circuit, with two gives an open Euler trail; otherwise none. Parity alone does not connect separated edge components. The exact result, retaining the requested order, is ([1, 2, 2, 2, 2, 1], [0, 5], Yes, open_euler_trail, zero_or_two_odd_vertices_on_connected_edge_support)."},"skills":["Degree Parity Needs Connected Nonisolated Support"],"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-20261105-r04-eulerian-trail-0031","batch_id":"batch-20261014-advance-20261105-r04-eulerian-trail-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"257f1a926ab222ffed757070fae47290b6c5bd70aa05c03c572e5a15613fc53a","family_id":"qf-mod36-euler-odd-degree-classification","seed":202611054031,"semantic_template":"qf-mod36-euler-odd-degree-classification/{\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5],[5,6]],\"n\":7}"},"fields":{"title":"Degree Parity Needs Connected Nonisolated Support","stem":"Use the complete finite simple undirected graph on Fin(7)={0,…,6}, with canonical sorted edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6]] and zero-based edge indices in that order. The edge support contains exactly the nonisolated vertices. A trail may revisit vertices but may not repeat an edge; an Euler trail uses every edge once. Isolated vertices do not invalidate connected edge support, and no Hamiltonian conclusion follows. Give (degrees, odd_vertices, edge_support_connected, euler_type, criterion).","answer":{"kind":"expression","value":"([1, 2, 2, 2, 2, 2, 1], [0, 6], Yes, open_euler_trail, zero_or_two_odd_vertices_on_connected_edge_support)","explanation":"For a nonzero-edge graph, compute every degree, all odd vertices and edge-support connectivity. Connected support with zero odd vertices gives a circuit, with two gives an open Euler trail; otherwise none. Parity alone does not connect separated edge components. The exact result, retaining the requested order, is ([1, 2, 2, 2, 2, 2, 1], [0, 6], Yes, open_euler_trail, zero_or_two_odd_vertices_on_connected_edge_support)."},"skills":["Degree Parity Needs Connected Nonisolated Support"],"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-20261105-r04-eulerian-trail-0032","batch_id":"batch-20261014-advance-20261105-r04-eulerian-trail-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b2a93ad2f6f8d90a2a5c84bdefe999137ed8d80a830ac35b0456d4a2fb1c6ebf","family_id":"qf-mod36-euler-odd-degree-classification","seed":202611054032,"semantic_template":"qf-mod36-euler-odd-degree-classification/{\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5],[5,6],[6,7]],\"n\":8}"},"fields":{"title":"Degree Parity Needs Connected Nonisolated Support","stem":"Use the complete finite simple undirected graph on Fin(8)={0,…,7}, with canonical sorted edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7]] and zero-based edge indices in that order. The edge support contains exactly the nonisolated vertices. A trail may revisit vertices but may not repeat an edge; an Euler trail uses every edge once. Isolated vertices do not invalidate connected edge support, and no Hamiltonian conclusion follows. Give (degrees, odd_vertices, edge_support_connected, euler_type, criterion).","answer":{"kind":"expression","value":"([1, 2, 2, 2, 2, 2, 2, 1], [0, 7], Yes, open_euler_trail, zero_or_two_odd_vertices_on_connected_edge_support)","explanation":"For a nonzero-edge graph, compute every degree, all odd vertices and edge-support connectivity. Connected support with zero odd vertices gives a circuit, with two gives an open Euler trail; otherwise none. Parity alone does not connect separated edge components. The exact result, retaining the requested order, is ([1, 2, 2, 2, 2, 2, 2, 1], [0, 7], Yes, open_euler_trail, zero_or_two_odd_vertices_on_connected_edge_support)."},"skills":["Degree Parity Needs Connected Nonisolated Support"],"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-20261105-r04-eulerian-trail-0033","batch_id":"batch-20261014-advance-20261105-r04-eulerian-trail-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"635741e746a5c200f463c7c4798b19eb203f275332fa92f34c726e350a6d8a2a","family_id":"qf-mod36-euler-odd-degree-classification","seed":202611054033,"semantic_template":"qf-mod36-euler-odd-degree-classification/{\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8]],\"n\":9}"},"fields":{"title":"Degree Parity Needs Connected Nonisolated Support","stem":"Use the complete finite simple undirected graph on Fin(9)={0,…,8}, with canonical sorted edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7], [7, 8]] and zero-based edge indices in that order. The edge support contains exactly the nonisolated vertices. A trail may revisit vertices but may not repeat an edge; an Euler trail uses every edge once. Isolated vertices do not invalidate connected edge support, and no Hamiltonian conclusion follows. Give (degrees, odd_vertices, edge_support_connected, euler_type, criterion).","answer":{"kind":"expression","value":"([1, 2, 2, 2, 2, 2, 2, 2, 1], [0, 8], Yes, open_euler_trail, zero_or_two_odd_vertices_on_connected_edge_support)","explanation":"For a nonzero-edge graph, compute every degree, all odd vertices and edge-support connectivity. Connected support with zero odd vertices gives a circuit, with two gives an open Euler trail; otherwise none. Parity alone does not connect separated edge components. The exact result, retaining the requested order, is ([1, 2, 2, 2, 2, 2, 2, 2, 1], [0, 8], Yes, open_euler_trail, zero_or_two_odd_vertices_on_connected_edge_support)."},"skills":["Degree Parity Needs Connected Nonisolated Support"],"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-20261105-r04-eulerian-trail-0034","batch_id":"batch-20261014-advance-20261105-r04-eulerian-trail-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6fe03bc02b071896d6f9b66742eeb0a37562f61e87a7ed963192f0b26ebab7fe","family_id":"qf-mod36-euler-odd-degree-classification","seed":202611054034,"semantic_template":"qf-mod36-euler-odd-degree-classification/{\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9]],\"n\":10}"},"fields":{"title":"Degree Parity Needs Connected Nonisolated Support","stem":"Use the complete finite simple undirected graph on Fin(10)={0,…,9}, with canonical sorted edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7], [7, 8], [8, 9]] and zero-based edge indices in that order. The edge support contains exactly the nonisolated vertices. A trail may revisit vertices but may not repeat an edge; an Euler trail uses every edge once. Isolated vertices do not invalidate connected edge support, and no Hamiltonian conclusion follows. Give (degrees, odd_vertices, edge_support_connected, euler_type, criterion).","answer":{"kind":"expression","value":"([1, 2, 2, 2, 2, 2, 2, 2, 2, 1], [0, 9], Yes, open_euler_trail, zero_or_two_odd_vertices_on_connected_edge_support)","explanation":"For a nonzero-edge graph, compute every degree, all odd vertices and edge-support connectivity. Connected support with zero odd vertices gives a circuit, with two gives an open Euler trail; otherwise none. Parity alone does not connect separated edge components. The exact result, retaining the requested order, is ([1, 2, 2, 2, 2, 2, 2, 2, 2, 1], [0, 9], Yes, open_euler_trail, zero_or_two_odd_vertices_on_connected_edge_support)."},"skills":["Degree Parity Needs Connected Nonisolated Support"],"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-20261105-r04-eulerian-trail-0035","batch_id":"batch-20261014-advance-20261105-r04-eulerian-trail-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c3afc26034eaed39582dd044b21db4ba0fbfa07d07e5b312f14c288060036067","family_id":"qf-mod36-euler-odd-degree-classification","seed":202611054035,"semantic_template":"qf-mod36-euler-odd-degree-classification/{\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10]],\"n\":11}"},"fields":{"title":"Degree Parity Needs Connected Nonisolated Support","stem":"Use the complete finite simple undirected graph on Fin(11)={0,…,10}, with canonical sorted edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7], [7, 8], [8, 9], [9, 10]] and zero-based edge indices in that order. The edge support contains exactly the nonisolated vertices. A trail may revisit vertices but may not repeat an edge; an Euler trail uses every edge once. Isolated vertices do not invalidate connected edge support, and no Hamiltonian conclusion follows. Give (degrees, odd_vertices, edge_support_connected, euler_type, criterion).","answer":{"kind":"expression","value":"([1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1], [0, 10], Yes, open_euler_trail, zero_or_two_odd_vertices_on_connected_edge_support)","explanation":"For a nonzero-edge graph, compute every degree, all odd vertices and edge-support connectivity. Connected support with zero odd vertices gives a circuit, with two gives an open Euler trail; otherwise none. Parity alone does not connect separated edge components. The exact result, retaining the requested order, is ([1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1], [0, 10], Yes, open_euler_trail, zero_or_two_odd_vertices_on_connected_edge_support)."},"skills":["Degree Parity Needs Connected Nonisolated Support"],"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-20261105-r04-eulerian-trail-0036","batch_id":"batch-20261014-advance-20261105-r04-eulerian-trail-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"13fddc9f91bcdd95c7ef8f1851e955e5144f063edf85c241b286bba674567621","family_id":"qf-mod36-euler-odd-degree-classification","seed":202611054036,"semantic_template":"qf-mod36-euler-odd-degree-classification/{\"edges\":[[0,1],[1,2],[10,11],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10]],\"n\":12}"},"fields":{"title":"Degree Parity Needs Connected Nonisolated Support","stem":"Use the complete finite simple undirected graph on Fin(12)={0,…,11}, with canonical sorted edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7], [7, 8], [8, 9], [9, 10], [10, 11]] and zero-based edge indices in that order. The edge support contains exactly the nonisolated vertices. A trail may revisit vertices but may not repeat an edge; an Euler trail uses every edge once. Isolated vertices do not invalidate connected edge support, and no Hamiltonian conclusion follows. Give (degrees, odd_vertices, edge_support_connected, euler_type, criterion).","answer":{"kind":"expression","value":"([1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1], [0, 11], Yes, open_euler_trail, zero_or_two_odd_vertices_on_connected_edge_support)","explanation":"For a nonzero-edge graph, compute every degree, all odd vertices and edge-support connectivity. Connected support with zero odd vertices gives a circuit, with two gives an open Euler trail; otherwise none. Parity alone does not connect separated edge components. The exact result, retaining the requested order, is ([1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1], [0, 11], Yes, open_euler_trail, zero_or_two_odd_vertices_on_connected_edge_support)."},"skills":["Degree Parity Needs Connected Nonisolated Support"],"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-20261105-r04-eulerian-trail-0037","batch_id":"batch-20261014-advance-20261105-r04-eulerian-trail-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0b71fb16066f7bb274c9b5c03d299fc430d794dc816161c05d35492cbd8fca6f","family_id":"qf-mod36-euler-odd-degree-classification","seed":202611054037,"semantic_template":"qf-mod36-euler-odd-degree-classification/{\"edges\":[[0,1],[1,2],[10,11],[11,12],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10]],\"n\":13}"},"fields":{"title":"Degree Parity Needs Connected Nonisolated Support","stem":"Use the complete finite simple undirected graph on Fin(13)={0,…,12}, with canonical sorted edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7], [7, 8], [8, 9], [9, 10], [10, 11], [11, 12]] and zero-based edge indices in that order. The edge support contains exactly the nonisolated vertices. A trail may revisit vertices but may not repeat an edge; an Euler trail uses every edge once. Isolated vertices do not invalidate connected edge support, and no Hamiltonian conclusion follows. Give (degrees, odd_vertices, edge_support_connected, euler_type, criterion).","answer":{"kind":"expression","value":"([1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1], [0, 12], Yes, open_euler_trail, zero_or_two_odd_vertices_on_connected_edge_support)","explanation":"For a nonzero-edge graph, compute every degree, all odd vertices and edge-support connectivity. Connected support with zero odd vertices gives a circuit, with two gives an open Euler trail; otherwise none. Parity alone does not connect separated edge components. The exact result, retaining the requested order, is ([1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1], [0, 12], Yes, open_euler_trail, zero_or_two_odd_vertices_on_connected_edge_support)."},"skills":["Degree Parity Needs Connected Nonisolated Support"],"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-20261105-r04-eulerian-trail-0038","batch_id":"batch-20261014-advance-20261105-r04-eulerian-trail-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"57285aa7c97c9ea88d37ddffa707f935892d77d35881e5e353d1ccadff5b6c5a","family_id":"qf-mod36-euler-odd-degree-classification","seed":202611054038,"semantic_template":"qf-mod36-euler-odd-degree-classification/{\"edges\":[[0,1],[1,2],[10,11],[11,12],[12,13],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10]],\"n\":14}"},"fields":{"title":"Degree Parity Needs Connected Nonisolated Support","stem":"Use the complete finite simple undirected graph on Fin(14)={0,…,13}, with canonical sorted edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7], [7, 8], [8, 9], [9, 10], [10, 11], [11, 12], [12, 13]] and zero-based edge indices in that order. The edge support contains exactly the nonisolated vertices. A trail may revisit vertices but may not repeat an edge; an Euler trail uses every edge once. Isolated vertices do not invalidate connected edge support, and no Hamiltonian conclusion follows. Give (degrees, odd_vertices, edge_support_connected, euler_type, criterion).","answer":{"kind":"expression","value":"([1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1], [0, 13], Yes, open_euler_trail, zero_or_two_odd_vertices_on_connected_edge_support)","explanation":"For a nonzero-edge graph, compute every degree, all odd vertices and edge-support connectivity. Connected support with zero odd vertices gives a circuit, with two gives an open Euler trail; otherwise none. Parity alone does not connect separated edge components. The exact result, retaining the requested order, is ([1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1], [0, 13], Yes, open_euler_trail, zero_or_two_odd_vertices_on_connected_edge_support)."},"skills":["Degree Parity Needs Connected Nonisolated Support"],"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-20261105-r04-eulerian-trail-0039","batch_id":"batch-20261014-advance-20261105-r04-eulerian-trail-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c793ebe899de4d0ea76b5b6f4f03f27227e5bd4e945f06a6dbaf0609aeb14a1b","family_id":"qf-mod36-euler-odd-degree-classification","seed":202611054039,"semantic_template":"qf-mod36-euler-odd-degree-classification/{\"edges\":[[0,1],[1,2],[10,11],[11,12],[12,13],[13,14],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10]],\"n\":15}"},"fields":{"title":"Degree Parity Needs Connected Nonisolated Support","stem":"Use the complete finite simple undirected graph on Fin(15)={0,…,14}, with canonical sorted edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7], [7, 8], [8, 9], [9, 10], [10, 11], [11, 12], [12, 13], [13, 14]] and zero-based edge indices in that order. The edge support contains exactly the nonisolated vertices. A trail may revisit vertices but may not repeat an edge; an Euler trail uses every edge once. Isolated vertices do not invalidate connected edge support, and no Hamiltonian conclusion follows. Give (degrees, odd_vertices, edge_support_connected, euler_type, criterion).","answer":{"kind":"expression","value":"([1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1], [0, 14], Yes, open_euler_trail, zero_or_two_odd_vertices_on_connected_edge_support)","explanation":"For a nonzero-edge graph, compute every degree, all odd vertices and edge-support connectivity. Connected support with zero odd vertices gives a circuit, with two gives an open Euler trail; otherwise none. Parity alone does not connect separated edge components. The exact result, retaining the requested order, is ([1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1], [0, 14], Yes, open_euler_trail, zero_or_two_odd_vertices_on_connected_edge_support)."},"skills":["Degree Parity Needs Connected Nonisolated Support"],"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-20261105-r04-eulerian-trail-0040","batch_id":"batch-20261014-advance-20261105-r04-eulerian-trail-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dca6d1fa0702996c80e892b207ad3823df48a8f1ffa965a3ddcf53b71147102b","family_id":"qf-mod36-euler-odd-degree-classification","seed":202611054040,"semantic_template":"qf-mod36-euler-odd-degree-classification/{\"edges\":[[0,1],[1,2],[10,11],[11,12],[12,13],[13,14],[14,15],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10]],\"n\":16}"},"fields":{"title":"Degree Parity Needs Connected Nonisolated Support","stem":"Use the complete finite simple undirected graph on Fin(16)={0,…,15}, with canonical sorted edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7], [7, 8], [8, 9], [9, 10], [10, 11], [11, 12], [12, 13], [13, 14], [14, 15]] and zero-based edge indices in that order. The edge support contains exactly the nonisolated vertices. A trail may revisit vertices but may not repeat an edge; an Euler trail uses every edge once. Isolated vertices do not invalidate connected edge support, and no Hamiltonian conclusion follows. Give (degrees, odd_vertices, edge_support_connected, euler_type, criterion).","answer":{"kind":"expression","value":"([1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1], [0, 15], Yes, open_euler_trail, zero_or_two_odd_vertices_on_connected_edge_support)","explanation":"For a nonzero-edge graph, compute every degree, all odd vertices and edge-support connectivity. Connected support with zero odd vertices gives a circuit, with two gives an open Euler trail; otherwise none. Parity alone does not connect separated edge components. The exact result, retaining the requested order, is ([1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1], [0, 15], Yes, open_euler_trail, zero_or_two_odd_vertices_on_connected_edge_support)."},"skills":["Degree Parity Needs Connected Nonisolated Support"],"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-20261105-r04-eulerian-trail-0041","batch_id":"batch-20261014-advance-20261105-r04-eulerian-trail-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d6725059b1fee95d6017758fd7aaca5aa706ee0fe7dcbb73073ac8e55cfc2a07","family_id":"qf-mod36-euler-odd-degree-classification","seed":202611054041,"semantic_template":"qf-mod36-euler-odd-degree-classification/{\"edges\":[[0,1],[1,2],[10,11],[11,12],[12,13],[13,14],[14,15],[15,16],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10]],\"n\":17}"},"fields":{"title":"Degree Parity Needs Connected Nonisolated Support","stem":"Use the complete finite simple undirected graph on Fin(17)={0,…,16}, with canonical sorted edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7], [7, 8], [8, 9], [9, 10], [10, 11], [11, 12], [12, 13], [13, 14], [14, 15], [15, 16]] and zero-based edge indices in that order. The edge support contains exactly the nonisolated vertices. A trail may revisit vertices but may not repeat an edge; an Euler trail uses every edge once. Isolated vertices do not invalidate connected edge support, and no Hamiltonian conclusion follows. Give (degrees, odd_vertices, edge_support_connected, euler_type, criterion).","answer":{"kind":"expression","value":"([1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1], [0, 16], Yes, open_euler_trail, zero_or_two_odd_vertices_on_connected_edge_support)","explanation":"For a nonzero-edge graph, compute every degree, all odd vertices and edge-support connectivity. Connected support with zero odd vertices gives a circuit, with two gives an open Euler trail; otherwise none. Parity alone does not connect separated edge components. The exact result, retaining the requested order, is ([1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1], [0, 16], Yes, open_euler_trail, zero_or_two_odd_vertices_on_connected_edge_support)."},"skills":["Degree Parity Needs Connected Nonisolated Support"],"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-20261105-r04-eulerian-trail-0042","batch_id":"batch-20261014-advance-20261105-r04-eulerian-trail-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"85186f736926df6bb4fdfe73d47e590b25a69c0750dfe1b6a4e63fbd6b1b0345","family_id":"qf-mod36-euler-odd-degree-classification","seed":202611054042,"semantic_template":"qf-mod36-euler-odd-degree-classification/{\"edges\":[[0,1],[1,2],[10,11],[11,12],[12,13],[13,14],[14,15],[15,16],[16,17],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10]],\"n\":18}"},"fields":{"title":"Degree Parity Needs Connected Nonisolated Support","stem":"Use the complete finite simple undirected graph on Fin(18)={0,…,17}, with canonical sorted edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7], [7, 8], [8, 9], [9, 10], [10, 11], [11, 12], [12, 13], [13, 14], [14, 15], [15, 16], [16, 17]] and zero-based edge indices in that order. The edge support contains exactly the nonisolated vertices. A trail may revisit vertices but may not repeat an edge; an Euler trail uses every edge once. Isolated vertices do not invalidate connected edge support, and no Hamiltonian conclusion follows. Give (degrees, odd_vertices, edge_support_connected, euler_type, criterion).","answer":{"kind":"expression","value":"([1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1], [0, 17], Yes, open_euler_trail, zero_or_two_odd_vertices_on_connected_edge_support)","explanation":"For a nonzero-edge graph, compute every degree, all odd vertices and edge-support connectivity. Connected support with zero odd vertices gives a circuit, with two gives an open Euler trail; otherwise none. Parity alone does not connect separated edge components. The exact result, retaining the requested order, is ([1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1], [0, 17], Yes, open_euler_trail, zero_or_two_odd_vertices_on_connected_edge_support)."},"skills":["Degree Parity Needs Connected Nonisolated Support"],"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-20261105-r04-eulerian-trail-0043","batch_id":"batch-20261014-advance-20261105-r04-eulerian-trail-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f3e95b4d816c57219215bc19111154645b26cb6b8d1cfe532607e8cdba93070c","family_id":"qf-mod36-euler-odd-degree-classification","seed":202611054043,"semantic_template":"qf-mod36-euler-odd-degree-classification/{\"edges\":[[0,1],[1,2],[10,11],[11,12],[12,13],[13,14],[14,15],[15,16],[16,17],[17,18],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10]],\"n\":19}"},"fields":{"title":"Degree Parity Needs Connected Nonisolated Support","stem":"Use the complete finite simple undirected graph on Fin(19)={0,…,18}, with canonical sorted edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7], [7, 8], [8, 9], [9, 10], [10, 11], [11, 12], [12, 13], [13, 14], [14, 15], [15, 16], [16, 17], [17, 18]] and zero-based edge indices in that order. The edge support contains exactly the nonisolated vertices. A trail may revisit vertices but may not repeat an edge; an Euler trail uses every edge once. Isolated vertices do not invalidate connected edge support, and no Hamiltonian conclusion follows. Give (degrees, odd_vertices, edge_support_connected, euler_type, criterion).","answer":{"kind":"expression","value":"([1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1], [0, 18], Yes, open_euler_trail, zero_or_two_odd_vertices_on_connected_edge_support)","explanation":"For a nonzero-edge graph, compute every degree, all odd vertices and edge-support connectivity. Connected support with zero odd vertices gives a circuit, with two gives an open Euler trail; otherwise none. Parity alone does not connect separated edge components. The exact result, retaining the requested order, is ([1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1], [0, 18], Yes, open_euler_trail, zero_or_two_odd_vertices_on_connected_edge_support)."},"skills":["Degree Parity Needs Connected Nonisolated Support"],"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-20261105-r04-eulerian-trail-0044","batch_id":"batch-20261014-advance-20261105-r04-eulerian-trail-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e43b2eb31e40f64f5613683aa782f15cb6273788fec34d14c6aa0616a1f95df3","family_id":"qf-mod36-euler-odd-degree-classification","seed":202611054044,"semantic_template":"qf-mod36-euler-odd-degree-classification/{\"edges\":[[0,1],[1,2],[10,11],[11,12],[12,13],[13,14],[14,15],[15,16],[16,17],[17,18],[18,19],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10]],\"n\":20}"},"fields":{"title":"Degree Parity Needs Connected Nonisolated Support","stem":"Use the complete finite simple undirected graph on Fin(20)={0,…,19}, with canonical sorted edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7], [7, 8], [8, 9], [9, 10], [10, 11], [11, 12], [12, 13], [13, 14], [14, 15], [15, 16], [16, 17], [17, 18], [18, 19]] and zero-based edge indices in that order. The edge support contains exactly the nonisolated vertices. A trail may revisit vertices but may not repeat an edge; an Euler trail uses every edge once. Isolated vertices do not invalidate connected edge support, and no Hamiltonian conclusion follows. Give (degrees, odd_vertices, edge_support_connected, euler_type, criterion).","answer":{"kind":"expression","value":"([1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1], [0, 19], Yes, open_euler_trail, zero_or_two_odd_vertices_on_connected_edge_support)","explanation":"For a nonzero-edge graph, compute every degree, all odd vertices and edge-support connectivity. Connected support with zero odd vertices gives a circuit, with two gives an open Euler trail; otherwise none. Parity alone does not connect separated edge components. The exact result, retaining the requested order, is ([1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1], [0, 19], Yes, open_euler_trail, zero_or_two_odd_vertices_on_connected_edge_support)."},"skills":["Degree Parity Needs Connected Nonisolated Support"],"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-20261105-r04-eulerian-trail-0045","batch_id":"batch-20261014-advance-20261105-r04-eulerian-trail-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"27ebef10abc17512aec21aa63e945ace3399ba78c08c0d4471b9c22a545f4b8c","family_id":"qf-mod36-euler-odd-degree-classification","seed":202611054045,"semantic_template":"qf-mod36-euler-odd-degree-classification/{\"edges\":[[0,1],[1,2],[10,11],[11,12],[12,13],[13,14],[14,15],[15,16],[16,17],[17,18],[18,19],[19,20],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10]],\"n\":21}"},"fields":{"title":"Degree Parity Needs Connected Nonisolated Support","stem":"Use the complete finite simple undirected graph on Fin(21)={0,…,20}, with canonical sorted edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7], [7, 8], [8, 9], [9, 10], [10, 11], [11, 12], [12, 13], [13, 14], [14, 15], [15, 16], [16, 17], [17, 18], [18, 19], [19, 20]] and zero-based edge indices in that order. The edge support contains exactly the nonisolated vertices. A trail may revisit vertices but may not repeat an edge; an Euler trail uses every edge once. Isolated vertices do not invalidate connected edge support, and no Hamiltonian conclusion follows. Give (degrees, odd_vertices, edge_support_connected, euler_type, criterion).","answer":{"kind":"expression","value":"([1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1], [0, 20], Yes, open_euler_trail, zero_or_two_odd_vertices_on_connected_edge_support)","explanation":"For a nonzero-edge graph, compute every degree, all odd vertices and edge-support connectivity. Connected support with zero odd vertices gives a circuit, with two gives an open Euler trail; otherwise none. Parity alone does not connect separated edge components. The exact result, retaining the requested order, is ([1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1], [0, 20], Yes, open_euler_trail, zero_or_two_odd_vertices_on_connected_edge_support)."},"skills":["Degree Parity Needs Connected Nonisolated Support"],"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-20261105-r04-eulerian-trail-0046","batch_id":"batch-20261014-advance-20261105-r04-eulerian-trail-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"474e3c52714c4cf6d24845fb327ae8ca44787bcb1fafc56bfab38bc4af43d6cf","family_id":"qf-mod36-euler-odd-degree-classification","seed":202611054046,"semantic_template":"qf-mod36-euler-odd-degree-classification/{\"edges\":[[0,1],[1,2],[10,11],[11,12],[12,13],[13,14],[14,15],[15,16],[16,17],[17,18],[18,19],[19,20],[2,3],[20,21],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10]],\"n\":22}"},"fields":{"title":"Degree Parity Needs Connected Nonisolated Support","stem":"Use the complete finite simple undirected graph on Fin(22)={0,…,21}, with canonical sorted edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7], [7, 8], [8, 9], [9, 10], [10, 11], [11, 12], [12, 13], [13, 14], [14, 15], [15, 16], [16, 17], [17, 18], [18, 19], [19, 20], [20, 21]] and zero-based edge indices in that order. The edge support contains exactly the nonisolated vertices. A trail may revisit vertices but may not repeat an edge; an Euler trail uses every edge once. Isolated vertices do not invalidate connected edge support, and no Hamiltonian conclusion follows. Give (degrees, odd_vertices, edge_support_connected, euler_type, criterion).","answer":{"kind":"expression","value":"([1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1], [0, 21], Yes, open_euler_trail, zero_or_two_odd_vertices_on_connected_edge_support)","explanation":"For a nonzero-edge graph, compute every degree, all odd vertices and edge-support connectivity. Connected support with zero odd vertices gives a circuit, with two gives an open Euler trail; otherwise none. Parity alone does not connect separated edge components. The exact result, retaining the requested order, is ([1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1], [0, 21], Yes, open_euler_trail, zero_or_two_odd_vertices_on_connected_edge_support)."},"skills":["Degree Parity Needs Connected Nonisolated Support"],"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-20261105-r04-eulerian-trail-0047","batch_id":"batch-20261014-advance-20261105-r04-eulerian-trail-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"87562e1179c64be821c1c22c32da67aee711b41330c072fee0213ef81449ad26","family_id":"qf-mod36-euler-odd-degree-classification","seed":202611054047,"semantic_template":"qf-mod36-euler-odd-degree-classification/{\"edges\":[[0,1],[1,2],[10,11],[11,12],[12,13],[13,14],[14,15],[15,16],[16,17],[17,18],[18,19],[19,20],[2,3],[20,21],[21,22],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10]],\"n\":23}"},"fields":{"title":"Degree Parity Needs Connected Nonisolated Support","stem":"Use the complete finite simple undirected graph on Fin(23)={0,…,22}, with canonical sorted edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7], [7, 8], [8, 9], [9, 10], [10, 11], [11, 12], [12, 13], [13, 14], [14, 15], [15, 16], [16, 17], [17, 18], [18, 19], [19, 20], [20, 21], [21, 22]] and zero-based edge indices in that order. The edge support contains exactly the nonisolated vertices. A trail may revisit vertices but may not repeat an edge; an Euler trail uses every edge once. Isolated vertices do not invalidate connected edge support, and no Hamiltonian conclusion follows. Give (degrees, odd_vertices, edge_support_connected, euler_type, criterion).","answer":{"kind":"expression","value":"([1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1], [0, 22], Yes, open_euler_trail, zero_or_two_odd_vertices_on_connected_edge_support)","explanation":"For a nonzero-edge graph, compute every degree, all odd vertices and edge-support connectivity. Connected support with zero odd vertices gives a circuit, with two gives an open Euler trail; otherwise none. Parity alone does not connect separated edge components. The exact result, retaining the requested order, is ([1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1], [0, 22], Yes, open_euler_trail, zero_or_two_odd_vertices_on_connected_edge_support)."},"skills":["Degree Parity Needs Connected Nonisolated Support"],"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-20261105-r04-eulerian-trail-0048","batch_id":"batch-20261014-advance-20261105-r04-eulerian-trail-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"60d6796ffb36fa7c525bfb92c797ceddc0a6e77371db578397491da575e2cfca","family_id":"qf-mod36-euler-odd-degree-classification","seed":202611054048,"semantic_template":"qf-mod36-euler-odd-degree-classification/{\"edges\":[[0,1],[1,2],[10,11],[11,12],[12,13],[13,14],[14,15],[15,16],[16,17],[17,18],[18,19],[19,20],[2,3],[20,21],[21,22],[22,23],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10]],\"n\":24}"},"fields":{"title":"Degree Parity Needs Connected Nonisolated Support","stem":"Use the complete finite simple undirected graph on Fin(24)={0,…,23}, with canonical sorted edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7], [7, 8], [8, 9], [9, 10], [10, 11], [11, 12], [12, 13], [13, 14], [14, 15], [15, 16], [16, 17], [17, 18], [18, 19], [19, 20], [20, 21], [21, 22], [22, 23]] and zero-based edge indices in that order. The edge support contains exactly the nonisolated vertices. A trail may revisit vertices but may not repeat an edge; an Euler trail uses every edge once. Isolated vertices do not invalidate connected edge support, and no Hamiltonian conclusion follows. Give (degrees, odd_vertices, edge_support_connected, euler_type, criterion).","answer":{"kind":"expression","value":"([1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1], [0, 23], Yes, open_euler_trail, zero_or_two_odd_vertices_on_connected_edge_support)","explanation":"For a nonzero-edge graph, compute every degree, all odd vertices and edge-support connectivity. Connected support with zero odd vertices gives a circuit, with two gives an open Euler trail; otherwise none. Parity alone does not connect separated edge components. The exact result, retaining the requested order, is ([1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1], [0, 23], Yes, open_euler_trail, zero_or_two_odd_vertices_on_connected_edge_support)."},"skills":["Degree Parity Needs Connected Nonisolated Support"],"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-20261105-r04-eulerian-trail-0049","batch_id":"batch-20261014-advance-20261105-r04-eulerian-trail-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7ba2fcf0b3d9dcc6fe9b13ee3edc266d088914fe641a89a25ea0c9a26a5d33ce","family_id":"qf-mod36-euler-odd-degree-classification","seed":202611054049,"semantic_template":"qf-mod36-euler-odd-degree-classification/{\"edges\":[[0,1],[1,2],[10,11],[11,12],[12,13],[13,14],[14,15],[15,16],[16,17],[17,18],[18,19],[19,20],[2,3],[20,21],[21,22],[22,23],[23,24],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10]],\"n\":25}"},"fields":{"title":"Degree Parity Needs Connected Nonisolated Support","stem":"Use the complete finite simple undirected graph on Fin(25)={0,…,24}, with canonical sorted edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7], [7, 8], [8, 9], [9, 10], [10, 11], [11, 12], [12, 13], [13, 14], [14, 15], [15, 16], [16, 17], [17, 18], [18, 19], [19, 20], [20, 21], [21, 22], [22, 23], [23, 24]] and zero-based edge indices in that order. The edge support contains exactly the nonisolated vertices. A trail may revisit vertices but may not repeat an edge; an Euler trail uses every edge once. Isolated vertices do not invalidate connected edge support, and no Hamiltonian conclusion follows. Give (degrees, odd_vertices, edge_support_connected, euler_type, criterion).","answer":{"kind":"expression","value":"([1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1], [0, 24], Yes, open_euler_trail, zero_or_two_odd_vertices_on_connected_edge_support)","explanation":"For a nonzero-edge graph, compute every degree, all odd vertices and edge-support connectivity. Connected support with zero odd vertices gives a circuit, with two gives an open Euler trail; otherwise none. Parity alone does not connect separated edge components. The exact result, retaining the requested order, is ([1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1], [0, 24], Yes, open_euler_trail, zero_or_two_odd_vertices_on_connected_edge_support)."},"skills":["Degree Parity Needs Connected Nonisolated Support"],"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-20261105-r04-eulerian-trail-0050","batch_id":"batch-20261014-advance-20261105-r04-eulerian-trail-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d1633bbb62bde13b5e05c40a8900b8c75aca84905e744236102274b5e882968a","family_id":"qf-mod36-euler-odd-degree-classification","seed":202611054050,"semantic_template":"qf-mod36-euler-odd-degree-classification/{\"edges\":[[0,1],[1,2],[10,11],[11,12],[12,13],[13,14],[14,15],[15,16],[16,17],[17,18],[18,19],[19,20],[2,3],[20,21],[21,22],[22,23],[23,24],[24,25],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10]],\"n\":26}"},"fields":{"title":"Degree Parity Needs Connected Nonisolated Support","stem":"Use the complete finite simple undirected graph on Fin(26)={0,…,25}, with canonical sorted edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7], [7, 8], [8, 9], [9, 10], [10, 11], [11, 12], [12, 13], [13, 14], [14, 15], [15, 16], [16, 17], [17, 18], [18, 19], [19, 20], [20, 21], [21, 22], [22, 23], [23, 24], [24, 25]] and zero-based edge indices in that order. The edge support contains exactly the nonisolated vertices. A trail may revisit vertices but may not repeat an edge; an Euler trail uses every edge once. Isolated vertices do not invalidate connected edge support, and no Hamiltonian conclusion follows. Give (degrees, odd_vertices, edge_support_connected, euler_type, criterion).","answer":{"kind":"expression","value":"([1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1], [0, 25], Yes, open_euler_trail, zero_or_two_odd_vertices_on_connected_edge_support)","explanation":"For a nonzero-edge graph, compute every degree, all odd vertices and edge-support connectivity. Connected support with zero odd vertices gives a circuit, with two gives an open Euler trail; otherwise none. Parity alone does not connect separated edge components. The exact result, retaining the requested order, is ([1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1], [0, 25], Yes, open_euler_trail, zero_or_two_odd_vertices_on_connected_edge_support)."},"skills":["Degree Parity Needs Connected Nonisolated Support"],"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-20261105-r04-eulerian-trail-0051","batch_id":"batch-20261014-advance-20261105-r04-eulerian-trail-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e5c157f10510023645330a8f620270178f9ca57c8843f098e1954cd9c832ff77","family_id":"qf-mod36-euler-support-connectivity","seed":202611054051,"semantic_template":"qf-mod36-euler-support-connectivity/{\"edges\":[[0,1]],\"n\":3}"},"fields":{"title":"Isolated Vertices Do Not Disconnect the Edge Support","stem":"Use the complete finite simple undirected graph on Fin(3)={0,…,2}, with canonical sorted edges=[[0, 1]] and zero-based edge indices in that order. The edge support contains exactly the nonisolated vertices. A trail may revisit vertices but may not repeat an edge; an Euler trail uses every edge once. Isolated vertices do not invalidate connected edge support, and no Hamiltonian conclusion follows. Give (nonisolated_vertices, edge_support_connected, whole_graph_connected, odd_vertices, euler_type, boundary).","answer":{"kind":"expression","value":"([0, 1], Yes, No, [0, 1], open_euler_trail, isolated_vertices_do_not_break_edge_support_connectivity)","explanation":"Return the complete nonisolated support and its connectivity separately from whole-carrier connectivity. Degree parity is applied to the connected edge support, not used to demand that every isolated vertex lie in the same graph component. This family requires at least one edge. The exact result, retaining the requested order, is ([0, 1], Yes, No, [0, 1], open_euler_trail, isolated_vertices_do_not_break_edge_support_connectivity)."},"skills":["Isolated Vertices Do Not Disconnect the Edge Support"],"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-20261105-r04-eulerian-trail-0052","batch_id":"batch-20261014-advance-20261105-r04-eulerian-trail-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5534b03c905bf6f4990f3390f6e63bd24b1f61599a8ad2cad55910ba2136d80f","family_id":"qf-mod36-euler-support-connectivity","seed":202611054052,"semantic_template":"qf-mod36-euler-support-connectivity/{\"edges\":[[0,1],[1,2]],\"n\":4}"},"fields":{"title":"Isolated Vertices Do Not Disconnect the Edge Support","stem":"Use the complete finite simple undirected graph on Fin(4)={0,…,3}, with canonical sorted edges=[[0, 1], [1, 2]] and zero-based edge indices in that order. The edge support contains exactly the nonisolated vertices. A trail may revisit vertices but may not repeat an edge; an Euler trail uses every edge once. Isolated vertices do not invalidate connected edge support, and no Hamiltonian conclusion follows. Give (nonisolated_vertices, edge_support_connected, whole_graph_connected, odd_vertices, euler_type, boundary).","answer":{"kind":"expression","value":"([0, 1, 2], Yes, No, [0, 2], open_euler_trail, isolated_vertices_do_not_break_edge_support_connectivity)","explanation":"Return the complete nonisolated support and its connectivity separately from whole-carrier connectivity. Degree parity is applied to the connected edge support, not used to demand that every isolated vertex lie in the same graph component. This family requires at least one edge. The exact result, retaining the requested order, is ([0, 1, 2], Yes, No, [0, 2], open_euler_trail, isolated_vertices_do_not_break_edge_support_connectivity)."},"skills":["Isolated Vertices Do Not Disconnect the Edge Support"],"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-20261105-r04-eulerian-trail-0053","batch_id":"batch-20261014-advance-20261105-r04-eulerian-trail-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"36f7a27fd42bc6028cc66199bee80f76649fff31b028b8fec51587562e156173","family_id":"qf-mod36-euler-support-connectivity","seed":202611054053,"semantic_template":"qf-mod36-euler-support-connectivity/{\"edges\":[[0,1],[1,2],[2,3]],\"n\":5}"},"fields":{"title":"Isolated Vertices Do Not Disconnect the Edge Support","stem":"Use the complete finite simple undirected graph on Fin(5)={0,…,4}, with canonical sorted edges=[[0, 1], [1, 2], [2, 3]] and zero-based edge indices in that order. The edge support contains exactly the nonisolated vertices. A trail may revisit vertices but may not repeat an edge; an Euler trail uses every edge once. Isolated vertices do not invalidate connected edge support, and no Hamiltonian conclusion follows. Give (nonisolated_vertices, edge_support_connected, whole_graph_connected, odd_vertices, euler_type, boundary).","answer":{"kind":"expression","value":"([0, 1, 2, 3], Yes, No, [0, 3], open_euler_trail, isolated_vertices_do_not_break_edge_support_connectivity)","explanation":"Return the complete nonisolated support and its connectivity separately from whole-carrier connectivity. Degree parity is applied to the connected edge support, not used to demand that every isolated vertex lie in the same graph component. This family requires at least one edge. The exact result, retaining the requested order, is ([0, 1, 2, 3], Yes, No, [0, 3], open_euler_trail, isolated_vertices_do_not_break_edge_support_connectivity)."},"skills":["Isolated Vertices Do Not Disconnect the Edge Support"],"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-20261105-r04-eulerian-trail-0054","batch_id":"batch-20261014-advance-20261105-r04-eulerian-trail-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"61b54b02a927c64906267eba941063e7c1f40d861234af3bee0ab2fe87c12eeb","family_id":"qf-mod36-euler-support-connectivity","seed":202611054054,"semantic_template":"qf-mod36-euler-support-connectivity/{\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"n\":6}"},"fields":{"title":"Isolated Vertices Do Not Disconnect the Edge Support","stem":"Use the complete finite simple undirected graph on Fin(6)={0,…,5}, with canonical sorted edges=[[0, 1], [1, 2], [2, 3], [3, 4]] and zero-based edge indices in that order. The edge support contains exactly the nonisolated vertices. A trail may revisit vertices but may not repeat an edge; an Euler trail uses every edge once. Isolated vertices do not invalidate connected edge support, and no Hamiltonian conclusion follows. Give (nonisolated_vertices, edge_support_connected, whole_graph_connected, odd_vertices, euler_type, boundary).","answer":{"kind":"expression","value":"([0, 1, 2, 3, 4], Yes, No, [0, 4], open_euler_trail, isolated_vertices_do_not_break_edge_support_connectivity)","explanation":"Return the complete nonisolated support and its connectivity separately from whole-carrier connectivity. Degree parity is applied to the connected edge support, not used to demand that every isolated vertex lie in the same graph component. This family requires at least one edge. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4], Yes, No, [0, 4], open_euler_trail, isolated_vertices_do_not_break_edge_support_connectivity)."},"skills":["Isolated Vertices Do Not Disconnect the Edge Support"],"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-20261105-r04-eulerian-trail-0055","batch_id":"batch-20261014-advance-20261105-r04-eulerian-trail-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"654f19db53b21cac29091f22bbaa911612c70deda3446ad92137e9c88e948f26","family_id":"qf-mod36-euler-support-connectivity","seed":202611054055,"semantic_template":"qf-mod36-euler-support-connectivity/{\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"n\":7}"},"fields":{"title":"Isolated Vertices Do Not Disconnect the Edge Support","stem":"Use the complete finite simple undirected graph on Fin(7)={0,…,6}, with canonical sorted edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5]] and zero-based edge indices in that order. The edge support contains exactly the nonisolated vertices. A trail may revisit vertices but may not repeat an edge; an Euler trail uses every edge once. Isolated vertices do not invalidate connected edge support, and no Hamiltonian conclusion follows. Give (nonisolated_vertices, edge_support_connected, whole_graph_connected, odd_vertices, euler_type, boundary).","answer":{"kind":"expression","value":"([0, 1, 2, 3, 4, 5], Yes, No, [0, 5], open_euler_trail, isolated_vertices_do_not_break_edge_support_connectivity)","explanation":"Return the complete nonisolated support and its connectivity separately from whole-carrier connectivity. Degree parity is applied to the connected edge support, not used to demand that every isolated vertex lie in the same graph component. This family requires at least one edge. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 5], Yes, No, [0, 5], open_euler_trail, isolated_vertices_do_not_break_edge_support_connectivity)."},"skills":["Isolated Vertices Do Not Disconnect the Edge Support"],"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-20261105-r04-eulerian-trail-0056","batch_id":"batch-20261014-advance-20261105-r04-eulerian-trail-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3a224c39778e3fd7e3884b5b7773c724901f648f43d6fb49d6d8546bbfecacb2","family_id":"qf-mod36-euler-support-connectivity","seed":202611054056,"semantic_template":"qf-mod36-euler-support-connectivity/{\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5],[5,6]],\"n\":8}"},"fields":{"title":"Isolated Vertices Do Not Disconnect the Edge Support","stem":"Use the complete finite simple undirected graph on Fin(8)={0,…,7}, with canonical sorted edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6]] and zero-based edge indices in that order. The edge support contains exactly the nonisolated vertices. A trail may revisit vertices but may not repeat an edge; an Euler trail uses every edge once. Isolated vertices do not invalidate connected edge support, and no Hamiltonian conclusion follows. Give (nonisolated_vertices, edge_support_connected, whole_graph_connected, odd_vertices, euler_type, boundary).","answer":{"kind":"expression","value":"([0, 1, 2, 3, 4, 5, 6], Yes, No, [0, 6], open_euler_trail, isolated_vertices_do_not_break_edge_support_connectivity)","explanation":"Return the complete nonisolated support and its connectivity separately from whole-carrier connectivity. Degree parity is applied to the connected edge support, not used to demand that every isolated vertex lie in the same graph component. This family requires at least one edge. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 5, 6], Yes, No, [0, 6], open_euler_trail, isolated_vertices_do_not_break_edge_support_connectivity)."},"skills":["Isolated Vertices Do Not Disconnect the Edge Support"],"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-20261105-r04-eulerian-trail-0057","batch_id":"batch-20261014-advance-20261105-r04-eulerian-trail-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bb1ccf3abd72cdc3af81f8d79354674af9815903cb17b7d319b9b63e388b77a3","family_id":"qf-mod36-euler-support-connectivity","seed":202611054057,"semantic_template":"qf-mod36-euler-support-connectivity/{\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5],[5,6],[6,7]],\"n\":9}"},"fields":{"title":"Isolated Vertices Do Not Disconnect the Edge Support","stem":"Use the complete finite simple undirected graph on Fin(9)={0,…,8}, with canonical sorted edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7]] and zero-based edge indices in that order. The edge support contains exactly the nonisolated vertices. A trail may revisit vertices but may not repeat an edge; an Euler trail uses every edge once. Isolated vertices do not invalidate connected edge support, and no Hamiltonian conclusion follows. Give (nonisolated_vertices, edge_support_connected, whole_graph_connected, odd_vertices, euler_type, boundary).","answer":{"kind":"expression","value":"([0, 1, 2, 3, 4, 5, 6, 7], Yes, No, [0, 7], open_euler_trail, isolated_vertices_do_not_break_edge_support_connectivity)","explanation":"Return the complete nonisolated support and its connectivity separately from whole-carrier connectivity. Degree parity is applied to the connected edge support, not used to demand that every isolated vertex lie in the same graph component. This family requires at least one edge. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 5, 6, 7], Yes, No, [0, 7], open_euler_trail, isolated_vertices_do_not_break_edge_support_connectivity)."},"skills":["Isolated Vertices Do Not Disconnect the Edge Support"],"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-20261105-r04-eulerian-trail-0058","batch_id":"batch-20261014-advance-20261105-r04-eulerian-trail-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"191d00b6cfa79590bb97a93e02be18f334e513b0af9c1d7e6a1072fd301d0a5b","family_id":"qf-mod36-euler-support-connectivity","seed":202611054058,"semantic_template":"qf-mod36-euler-support-connectivity/{\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8]],\"n\":10}"},"fields":{"title":"Isolated Vertices Do Not Disconnect the Edge Support","stem":"Use the complete finite simple undirected graph on Fin(10)={0,…,9}, with canonical sorted edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7], [7, 8]] and zero-based edge indices in that order. The edge support contains exactly the nonisolated vertices. A trail may revisit vertices but may not repeat an edge; an Euler trail uses every edge once. Isolated vertices do not invalidate connected edge support, and no Hamiltonian conclusion follows. Give (nonisolated_vertices, edge_support_connected, whole_graph_connected, odd_vertices, euler_type, boundary).","answer":{"kind":"expression","value":"([0, 1, 2, 3, 4, 5, 6, 7, 8], Yes, No, [0, 8], open_euler_trail, isolated_vertices_do_not_break_edge_support_connectivity)","explanation":"Return the complete nonisolated support and its connectivity separately from whole-carrier connectivity. Degree parity is applied to the connected edge support, not used to demand that every isolated vertex lie in the same graph component. This family requires at least one edge. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 5, 6, 7, 8], Yes, No, [0, 8], open_euler_trail, isolated_vertices_do_not_break_edge_support_connectivity)."},"skills":["Isolated Vertices Do Not Disconnect the Edge Support"],"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-20261105-r04-eulerian-trail-0059","batch_id":"batch-20261014-advance-20261105-r04-eulerian-trail-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6e1f416168978172078b958558e271920cd0567c064d8463db1a62ec9e77b204","family_id":"qf-mod36-euler-support-connectivity","seed":202611054059,"semantic_template":"qf-mod36-euler-support-connectivity/{\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9]],\"n\":11}"},"fields":{"title":"Isolated Vertices Do Not Disconnect the Edge Support","stem":"Use the complete finite simple undirected graph on Fin(11)={0,…,10}, with canonical sorted edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7], [7, 8], [8, 9]] and zero-based edge indices in that order. The edge support contains exactly the nonisolated vertices. A trail may revisit vertices but may not repeat an edge; an Euler trail uses every edge once. Isolated vertices do not invalidate connected edge support, and no Hamiltonian conclusion follows. Give (nonisolated_vertices, edge_support_connected, whole_graph_connected, odd_vertices, euler_type, boundary).","answer":{"kind":"expression","value":"([0, 1, 2, 3, 4, 5, 6, 7, 8, 9], Yes, No, [0, 9], open_euler_trail, isolated_vertices_do_not_break_edge_support_connectivity)","explanation":"Return the complete nonisolated support and its connectivity separately from whole-carrier connectivity. Degree parity is applied to the connected edge support, not used to demand that every isolated vertex lie in the same graph component. This family requires at least one edge. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 5, 6, 7, 8, 9], Yes, No, [0, 9], open_euler_trail, isolated_vertices_do_not_break_edge_support_connectivity)."},"skills":["Isolated Vertices Do Not Disconnect the Edge Support"],"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-20261105-r04-eulerian-trail-0060","batch_id":"batch-20261014-advance-20261105-r04-eulerian-trail-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c7df2f0070c789ee1b41dce64bedf50279e204d02685e2bbdd060290b1c042c5","family_id":"qf-mod36-euler-support-connectivity","seed":202611054060,"semantic_template":"qf-mod36-euler-support-connectivity/{\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10]],\"n\":12}"},"fields":{"title":"Isolated Vertices Do Not Disconnect the Edge Support","stem":"Use the complete finite simple undirected graph on Fin(12)={0,…,11}, with canonical sorted edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7], [7, 8], [8, 9], [9, 10]] and zero-based edge indices in that order. The edge support contains exactly the nonisolated vertices. A trail may revisit vertices but may not repeat an edge; an Euler trail uses every edge once. Isolated vertices do not invalidate connected edge support, and no Hamiltonian conclusion follows. Give (nonisolated_vertices, edge_support_connected, whole_graph_connected, odd_vertices, euler_type, boundary).","answer":{"kind":"expression","value":"([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], Yes, No, [0, 10], open_euler_trail, isolated_vertices_do_not_break_edge_support_connectivity)","explanation":"Return the complete nonisolated support and its connectivity separately from whole-carrier connectivity. Degree parity is applied to the connected edge support, not used to demand that every isolated vertex lie in the same graph component. This family requires at least one edge. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], Yes, No, [0, 10], open_euler_trail, isolated_vertices_do_not_break_edge_support_connectivity)."},"skills":["Isolated Vertices Do Not Disconnect the Edge Support"],"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-20261105-r04-eulerian-trail-0061","batch_id":"batch-20261014-advance-20261105-r04-eulerian-trail-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5864871ed984ac5325c42b53d48db09ec5f4bd24448bfba5fd5aac6af4f58406","family_id":"qf-mod36-euler-support-connectivity","seed":202611054061,"semantic_template":"qf-mod36-euler-support-connectivity/{\"edges\":[[0,1],[1,2],[10,11],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10]],\"n\":13}"},"fields":{"title":"Isolated Vertices Do Not Disconnect the Edge Support","stem":"Use the complete finite simple undirected graph on Fin(13)={0,…,12}, with canonical sorted edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7], [7, 8], [8, 9], [9, 10], [10, 11]] and zero-based edge indices in that order. The edge support contains exactly the nonisolated vertices. A trail may revisit vertices but may not repeat an edge; an Euler trail uses every edge once. Isolated vertices do not invalidate connected edge support, and no Hamiltonian conclusion follows. Give (nonisolated_vertices, edge_support_connected, whole_graph_connected, odd_vertices, euler_type, boundary).","answer":{"kind":"expression","value":"([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], Yes, No, [0, 11], open_euler_trail, isolated_vertices_do_not_break_edge_support_connectivity)","explanation":"Return the complete nonisolated support and its connectivity separately from whole-carrier connectivity. Degree parity is applied to the connected edge support, not used to demand that every isolated vertex lie in the same graph component. This family requires at least one edge. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], Yes, No, [0, 11], open_euler_trail, isolated_vertices_do_not_break_edge_support_connectivity)."},"skills":["Isolated Vertices Do Not Disconnect the Edge Support"],"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-20261105-r04-eulerian-trail-0062","batch_id":"batch-20261014-advance-20261105-r04-eulerian-trail-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5c615f19efd6c99e9c5015e3629ce6d8f5a027330ba0ed50c22693f62f053c3b","family_id":"qf-mod36-euler-support-connectivity","seed":202611054062,"semantic_template":"qf-mod36-euler-support-connectivity/{\"edges\":[[0,1],[1,2],[10,11],[11,12],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10]],\"n\":14}"},"fields":{"title":"Isolated Vertices Do Not Disconnect the Edge Support","stem":"Use the complete finite simple undirected graph on Fin(14)={0,…,13}, with canonical sorted edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7], [7, 8], [8, 9], [9, 10], [10, 11], [11, 12]] and zero-based edge indices in that order. The edge support contains exactly the nonisolated vertices. A trail may revisit vertices but may not repeat an edge; an Euler trail uses every edge once. Isolated vertices do not invalidate connected edge support, and no Hamiltonian conclusion follows. Give (nonisolated_vertices, edge_support_connected, whole_graph_connected, odd_vertices, euler_type, boundary).","answer":{"kind":"expression","value":"([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], Yes, No, [0, 12], open_euler_trail, isolated_vertices_do_not_break_edge_support_connectivity)","explanation":"Return the complete nonisolated support and its connectivity separately from whole-carrier connectivity. Degree parity is applied to the connected edge support, not used to demand that every isolated vertex lie in the same graph component. This family requires at least one edge. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], Yes, No, [0, 12], open_euler_trail, isolated_vertices_do_not_break_edge_support_connectivity)."},"skills":["Isolated Vertices Do Not Disconnect the Edge Support"],"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-20261105-r04-eulerian-trail-0063","batch_id":"batch-20261014-advance-20261105-r04-eulerian-trail-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0b80ef526db8f0fa9d8dc333fb76891f82f68457468d705f38acb523d8bd5792","family_id":"qf-mod36-euler-support-connectivity","seed":202611054063,"semantic_template":"qf-mod36-euler-support-connectivity/{\"edges\":[[0,1],[1,2],[10,11],[11,12],[12,13],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10]],\"n\":15}"},"fields":{"title":"Isolated Vertices Do Not Disconnect the Edge Support","stem":"Use the complete finite simple undirected graph on Fin(15)={0,…,14}, with canonical sorted edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7], [7, 8], [8, 9], [9, 10], [10, 11], [11, 12], [12, 13]] and zero-based edge indices in that order. The edge support contains exactly the nonisolated vertices. A trail may revisit vertices but may not repeat an edge; an Euler trail uses every edge once. Isolated vertices do not invalidate connected edge support, and no Hamiltonian conclusion follows. Give (nonisolated_vertices, edge_support_connected, whole_graph_connected, odd_vertices, euler_type, boundary).","answer":{"kind":"expression","value":"([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], Yes, No, [0, 13], open_euler_trail, isolated_vertices_do_not_break_edge_support_connectivity)","explanation":"Return the complete nonisolated support and its connectivity separately from whole-carrier connectivity. Degree parity is applied to the connected edge support, not used to demand that every isolated vertex lie in the same graph component. This family requires at least one edge. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], Yes, No, [0, 13], open_euler_trail, isolated_vertices_do_not_break_edge_support_connectivity)."},"skills":["Isolated Vertices Do Not Disconnect the Edge Support"],"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-20261105-r04-eulerian-trail-0064","batch_id":"batch-20261014-advance-20261105-r04-eulerian-trail-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a00a1a0486d4f306a2d8e69d14b7145c9f167395ae3a2c3acc772ccd629eb1a4","family_id":"qf-mod36-euler-support-connectivity","seed":202611054064,"semantic_template":"qf-mod36-euler-support-connectivity/{\"edges\":[[0,1],[1,2],[10,11],[11,12],[12,13],[13,14],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10]],\"n\":16}"},"fields":{"title":"Isolated Vertices Do Not Disconnect the Edge Support","stem":"Use the complete finite simple undirected graph on Fin(16)={0,…,15}, with canonical sorted edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7], [7, 8], [8, 9], [9, 10], [10, 11], [11, 12], [12, 13], [13, 14]] and zero-based edge indices in that order. The edge support contains exactly the nonisolated vertices. A trail may revisit vertices but may not repeat an edge; an Euler trail uses every edge once. Isolated vertices do not invalidate connected edge support, and no Hamiltonian conclusion follows. Give (nonisolated_vertices, edge_support_connected, whole_graph_connected, odd_vertices, euler_type, boundary).","answer":{"kind":"expression","value":"([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], Yes, No, [0, 14], open_euler_trail, isolated_vertices_do_not_break_edge_support_connectivity)","explanation":"Return the complete nonisolated support and its connectivity separately from whole-carrier connectivity. Degree parity is applied to the connected edge support, not used to demand that every isolated vertex lie in the same graph component. This family requires at least one edge. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], Yes, No, [0, 14], open_euler_trail, isolated_vertices_do_not_break_edge_support_connectivity)."},"skills":["Isolated Vertices Do Not Disconnect the Edge Support"],"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-20261105-r04-eulerian-trail-0065","batch_id":"batch-20261014-advance-20261105-r04-eulerian-trail-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"96407c4893d55d61e75f0fb81cacbab8bd44c1f60047bf91ffa8efe22c2f167c","family_id":"qf-mod36-euler-support-connectivity","seed":202611054065,"semantic_template":"qf-mod36-euler-support-connectivity/{\"edges\":[[0,1],[1,2],[10,11],[11,12],[12,13],[13,14],[14,15],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10]],\"n\":17}"},"fields":{"title":"Isolated Vertices Do Not Disconnect the Edge Support","stem":"Use the complete finite simple undirected graph on Fin(17)={0,…,16}, with canonical sorted edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7], [7, 8], [8, 9], [9, 10], [10, 11], [11, 12], [12, 13], [13, 14], [14, 15]] and zero-based edge indices in that order. The edge support contains exactly the nonisolated vertices. A trail may revisit vertices but may not repeat an edge; an Euler trail uses every edge once. Isolated vertices do not invalidate connected edge support, and no Hamiltonian conclusion follows. Give (nonisolated_vertices, edge_support_connected, whole_graph_connected, odd_vertices, euler_type, boundary).","answer":{"kind":"expression","value":"([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], Yes, No, [0, 15], open_euler_trail, isolated_vertices_do_not_break_edge_support_connectivity)","explanation":"Return the complete nonisolated support and its connectivity separately from whole-carrier connectivity. Degree parity is applied to the connected edge support, not used to demand that every isolated vertex lie in the same graph component. This family requires at least one edge. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], Yes, No, [0, 15], open_euler_trail, isolated_vertices_do_not_break_edge_support_connectivity)."},"skills":["Isolated Vertices Do Not Disconnect the Edge Support"],"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-20261105-r04-eulerian-trail-0066","batch_id":"batch-20261014-advance-20261105-r04-eulerian-trail-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a2e7db7608a6362aa7d4c6de3ec55c1ba0d3e4b173b6c7a5c670a80a1d62b3c0","family_id":"qf-mod36-euler-support-connectivity","seed":202611054066,"semantic_template":"qf-mod36-euler-support-connectivity/{\"edges\":[[0,1],[1,2],[10,11],[11,12],[12,13],[13,14],[14,15],[15,16],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10]],\"n\":18}"},"fields":{"title":"Isolated Vertices Do Not Disconnect the Edge Support","stem":"Use the complete finite simple undirected graph on Fin(18)={0,…,17}, with canonical sorted edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7], [7, 8], [8, 9], [9, 10], [10, 11], [11, 12], [12, 13], [13, 14], [14, 15], [15, 16]] and zero-based edge indices in that order. The edge support contains exactly the nonisolated vertices. A trail may revisit vertices but may not repeat an edge; an Euler trail uses every edge once. Isolated vertices do not invalidate connected edge support, and no Hamiltonian conclusion follows. Give (nonisolated_vertices, edge_support_connected, whole_graph_connected, odd_vertices, euler_type, boundary).","answer":{"kind":"expression","value":"([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], Yes, No, [0, 16], open_euler_trail, isolated_vertices_do_not_break_edge_support_connectivity)","explanation":"Return the complete nonisolated support and its connectivity separately from whole-carrier connectivity. Degree parity is applied to the connected edge support, not used to demand that every isolated vertex lie in the same graph component. This family requires at least one edge. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], Yes, No, [0, 16], open_euler_trail, isolated_vertices_do_not_break_edge_support_connectivity)."},"skills":["Isolated Vertices Do Not Disconnect the Edge Support"],"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-20261105-r04-eulerian-trail-0067","batch_id":"batch-20261014-advance-20261105-r04-eulerian-trail-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"45be08417f535ace02023adc67d6fad05ea1df1049192b5593b7a9a21ec403e5","family_id":"qf-mod36-euler-support-connectivity","seed":202611054067,"semantic_template":"qf-mod36-euler-support-connectivity/{\"edges\":[[0,1],[1,2],[10,11],[11,12],[12,13],[13,14],[14,15],[15,16],[16,17],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10]],\"n\":19}"},"fields":{"title":"Isolated Vertices Do Not Disconnect the Edge Support","stem":"Use the complete finite simple undirected graph on Fin(19)={0,…,18}, with canonical sorted edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7], [7, 8], [8, 9], [9, 10], [10, 11], [11, 12], [12, 13], [13, 14], [14, 15], [15, 16], [16, 17]] and zero-based edge indices in that order. The edge support contains exactly the nonisolated vertices. A trail may revisit vertices but may not repeat an edge; an Euler trail uses every edge once. Isolated vertices do not invalidate connected edge support, and no Hamiltonian conclusion follows. Give (nonisolated_vertices, edge_support_connected, whole_graph_connected, odd_vertices, euler_type, boundary).","answer":{"kind":"expression","value":"([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], Yes, No, [0, 17], open_euler_trail, isolated_vertices_do_not_break_edge_support_connectivity)","explanation":"Return the complete nonisolated support and its connectivity separately from whole-carrier connectivity. Degree parity is applied to the connected edge support, not used to demand that every isolated vertex lie in the same graph component. This family requires at least one edge. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], Yes, No, [0, 17], open_euler_trail, isolated_vertices_do_not_break_edge_support_connectivity)."},"skills":["Isolated Vertices Do Not Disconnect the Edge Support"],"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-20261105-r04-eulerian-trail-0068","batch_id":"batch-20261014-advance-20261105-r04-eulerian-trail-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"757b396a325632210e29ed84ad304590dc99a59130261a3cacbe3d88f1c21621","family_id":"qf-mod36-euler-support-connectivity","seed":202611054068,"semantic_template":"qf-mod36-euler-support-connectivity/{\"edges\":[[0,1],[1,2],[10,11],[11,12],[12,13],[13,14],[14,15],[15,16],[16,17],[17,18],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10]],\"n\":20}"},"fields":{"title":"Isolated Vertices Do Not Disconnect the Edge Support","stem":"Use the complete finite simple undirected graph on Fin(20)={0,…,19}, with canonical sorted edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7], [7, 8], [8, 9], [9, 10], [10, 11], [11, 12], [12, 13], [13, 14], [14, 15], [15, 16], [16, 17], [17, 18]] and zero-based edge indices in that order. The edge support contains exactly the nonisolated vertices. A trail may revisit vertices but may not repeat an edge; an Euler trail uses every edge once. Isolated vertices do not invalidate connected edge support, and no Hamiltonian conclusion follows. Give (nonisolated_vertices, edge_support_connected, whole_graph_connected, odd_vertices, euler_type, boundary).","answer":{"kind":"expression","value":"([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], Yes, No, [0, 18], open_euler_trail, isolated_vertices_do_not_break_edge_support_connectivity)","explanation":"Return the complete nonisolated support and its connectivity separately from whole-carrier connectivity. Degree parity is applied to the connected edge support, not used to demand that every isolated vertex lie in the same graph component. This family requires at least one edge. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], Yes, No, [0, 18], open_euler_trail, isolated_vertices_do_not_break_edge_support_connectivity)."},"skills":["Isolated Vertices Do Not Disconnect the Edge Support"],"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-20261105-r04-eulerian-trail-0069","batch_id":"batch-20261014-advance-20261105-r04-eulerian-trail-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bc768a933154ad4b081eb3ea418366de72784ff729dfa2db6efa75e5631e52d3","family_id":"qf-mod36-euler-support-connectivity","seed":202611054069,"semantic_template":"qf-mod36-euler-support-connectivity/{\"edges\":[[0,1],[1,2],[10,11],[11,12],[12,13],[13,14],[14,15],[15,16],[16,17],[17,18],[18,19],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10]],\"n\":21}"},"fields":{"title":"Isolated Vertices Do Not Disconnect the Edge Support","stem":"Use the complete finite simple undirected graph on Fin(21)={0,…,20}, with canonical sorted edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7], [7, 8], [8, 9], [9, 10], [10, 11], [11, 12], [12, 13], [13, 14], [14, 15], [15, 16], [16, 17], [17, 18], [18, 19]] and zero-based edge indices in that order. The edge support contains exactly the nonisolated vertices. A trail may revisit vertices but may not repeat an edge; an Euler trail uses every edge once. Isolated vertices do not invalidate connected edge support, and no Hamiltonian conclusion follows. Give (nonisolated_vertices, edge_support_connected, whole_graph_connected, odd_vertices, euler_type, boundary).","answer":{"kind":"expression","value":"([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], Yes, No, [0, 19], open_euler_trail, isolated_vertices_do_not_break_edge_support_connectivity)","explanation":"Return the complete nonisolated support and its connectivity separately from whole-carrier connectivity. Degree parity is applied to the connected edge support, not used to demand that every isolated vertex lie in the same graph component. This family requires at least one edge. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], Yes, No, [0, 19], open_euler_trail, isolated_vertices_do_not_break_edge_support_connectivity)."},"skills":["Isolated Vertices Do Not Disconnect the Edge Support"],"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-20261105-r04-eulerian-trail-0070","batch_id":"batch-20261014-advance-20261105-r04-eulerian-trail-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e440d7a7ea60d895a1ff3406594d69d2d42b5139446b4db5a402bd01ac8f6cef","family_id":"qf-mod36-euler-support-connectivity","seed":202611054070,"semantic_template":"qf-mod36-euler-support-connectivity/{\"edges\":[[0,1],[1,2],[10,11],[11,12],[12,13],[13,14],[14,15],[15,16],[16,17],[17,18],[18,19],[19,20],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10]],\"n\":22}"},"fields":{"title":"Isolated Vertices Do Not Disconnect the Edge Support","stem":"Use the complete finite simple undirected graph on Fin(22)={0,…,21}, with canonical sorted edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7], [7, 8], [8, 9], [9, 10], [10, 11], [11, 12], [12, 13], [13, 14], [14, 15], [15, 16], [16, 17], [17, 18], [18, 19], [19, 20]] and zero-based edge indices in that order. The edge support contains exactly the nonisolated vertices. A trail may revisit vertices but may not repeat an edge; an Euler trail uses every edge once. Isolated vertices do not invalidate connected edge support, and no Hamiltonian conclusion follows. Give (nonisolated_vertices, edge_support_connected, whole_graph_connected, odd_vertices, euler_type, boundary).","answer":{"kind":"expression","value":"([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], Yes, No, [0, 20], open_euler_trail, isolated_vertices_do_not_break_edge_support_connectivity)","explanation":"Return the complete nonisolated support and its connectivity separately from whole-carrier connectivity. Degree parity is applied to the connected edge support, not used to demand that every isolated vertex lie in the same graph component. This family requires at least one edge. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], Yes, No, [0, 20], open_euler_trail, isolated_vertices_do_not_break_edge_support_connectivity)."},"skills":["Isolated Vertices Do Not Disconnect the Edge Support"],"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-20261105-r04-eulerian-trail-0071","batch_id":"batch-20261014-advance-20261105-r04-eulerian-trail-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"97ea687be4de265dfa47f01af031fa04d115b48be4b2a060f574e9194be31764","family_id":"qf-mod36-euler-support-connectivity","seed":202611054071,"semantic_template":"qf-mod36-euler-support-connectivity/{\"edges\":[[0,1],[1,2],[10,11],[11,12],[12,13],[13,14],[14,15],[15,16],[16,17],[17,18],[18,19],[19,20],[2,3],[20,21],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10]],\"n\":23}"},"fields":{"title":"Isolated Vertices Do Not Disconnect the Edge Support","stem":"Use the complete finite simple undirected graph on Fin(23)={0,…,22}, with canonical sorted edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7], [7, 8], [8, 9], [9, 10], [10, 11], [11, 12], [12, 13], [13, 14], [14, 15], [15, 16], [16, 17], [17, 18], [18, 19], [19, 20], [20, 21]] and zero-based edge indices in that order. The edge support contains exactly the nonisolated vertices. A trail may revisit vertices but may not repeat an edge; an Euler trail uses every edge once. Isolated vertices do not invalidate connected edge support, and no Hamiltonian conclusion follows. Give (nonisolated_vertices, edge_support_connected, whole_graph_connected, odd_vertices, euler_type, boundary).","answer":{"kind":"expression","value":"([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], Yes, No, [0, 21], open_euler_trail, isolated_vertices_do_not_break_edge_support_connectivity)","explanation":"Return the complete nonisolated support and its connectivity separately from whole-carrier connectivity. Degree parity is applied to the connected edge support, not used to demand that every isolated vertex lie in the same graph component. This family requires at least one edge. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], Yes, No, [0, 21], open_euler_trail, isolated_vertices_do_not_break_edge_support_connectivity)."},"skills":["Isolated Vertices Do Not Disconnect the Edge Support"],"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-20261105-r04-eulerian-trail-0072","batch_id":"batch-20261014-advance-20261105-r04-eulerian-trail-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7458022faaee069215a59ec5b1b4ad6c65710cde375e8ff804ff512d16d105fb","family_id":"qf-mod36-euler-support-connectivity","seed":202611054072,"semantic_template":"qf-mod36-euler-support-connectivity/{\"edges\":[[0,1],[1,2],[10,11],[11,12],[12,13],[13,14],[14,15],[15,16],[16,17],[17,18],[18,19],[19,20],[2,3],[20,21],[21,22],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10]],\"n\":24}"},"fields":{"title":"Isolated Vertices Do Not Disconnect the Edge Support","stem":"Use the complete finite simple undirected graph on Fin(24)={0,…,23}, with canonical sorted edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7], [7, 8], [8, 9], [9, 10], [10, 11], [11, 12], [12, 13], [13, 14], [14, 15], [15, 16], [16, 17], [17, 18], [18, 19], [19, 20], [20, 21], [21, 22]] and zero-based edge indices in that order. The edge support contains exactly the nonisolated vertices. A trail may revisit vertices but may not repeat an edge; an Euler trail uses every edge once. Isolated vertices do not invalidate connected edge support, and no Hamiltonian conclusion follows. Give (nonisolated_vertices, edge_support_connected, whole_graph_connected, odd_vertices, euler_type, boundary).","answer":{"kind":"expression","value":"([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], Yes, No, [0, 22], open_euler_trail, isolated_vertices_do_not_break_edge_support_connectivity)","explanation":"Return the complete nonisolated support and its connectivity separately from whole-carrier connectivity. Degree parity is applied to the connected edge support, not used to demand that every isolated vertex lie in the same graph component. This family requires at least one edge. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], Yes, No, [0, 22], open_euler_trail, isolated_vertices_do_not_break_edge_support_connectivity)."},"skills":["Isolated Vertices Do Not Disconnect the Edge Support"],"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-20261105-r04-eulerian-trail-0073","batch_id":"batch-20261014-advance-20261105-r04-eulerian-trail-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d03555dc06e79e938bd232bf24aee6051411934e7b2e6b350b0e92fbd9c4de10","family_id":"qf-mod36-euler-support-connectivity","seed":202611054073,"semantic_template":"qf-mod36-euler-support-connectivity/{\"edges\":[[0,1],[1,2],[10,11],[11,12],[12,13],[13,14],[14,15],[15,16],[16,17],[17,18],[18,19],[19,20],[2,3],[20,21],[21,22],[22,23],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10]],\"n\":25}"},"fields":{"title":"Isolated Vertices Do Not Disconnect the Edge Support","stem":"Use the complete finite simple undirected graph on Fin(25)={0,…,24}, with canonical sorted edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7], [7, 8], [8, 9], [9, 10], [10, 11], [11, 12], [12, 13], [13, 14], [14, 15], [15, 16], [16, 17], [17, 18], [18, 19], [19, 20], [20, 21], [21, 22], [22, 23]] and zero-based edge indices in that order. The edge support contains exactly the nonisolated vertices. A trail may revisit vertices but may not repeat an edge; an Euler trail uses every edge once. Isolated vertices do not invalidate connected edge support, and no Hamiltonian conclusion follows. Give (nonisolated_vertices, edge_support_connected, whole_graph_connected, odd_vertices, euler_type, boundary).","answer":{"kind":"expression","value":"([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], Yes, No, [0, 23], open_euler_trail, isolated_vertices_do_not_break_edge_support_connectivity)","explanation":"Return the complete nonisolated support and its connectivity separately from whole-carrier connectivity. Degree parity is applied to the connected edge support, not used to demand that every isolated vertex lie in the same graph component. This family requires at least one edge. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], Yes, No, [0, 23], open_euler_trail, isolated_vertices_do_not_break_edge_support_connectivity)."},"skills":["Isolated Vertices Do Not Disconnect the Edge Support"],"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-20261105-r04-eulerian-trail-0074","batch_id":"batch-20261014-advance-20261105-r04-eulerian-trail-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6cc5caad9b133ae28f9aa235142f9719ac56ca94ea7de70b19dea671c5c3fd85","family_id":"qf-mod36-euler-support-connectivity","seed":202611054074,"semantic_template":"qf-mod36-euler-support-connectivity/{\"edges\":[[0,1],[1,2],[10,11],[11,12],[12,13],[13,14],[14,15],[15,16],[16,17],[17,18],[18,19],[19,20],[2,3],[20,21],[21,22],[22,23],[23,24],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10]],\"n\":26}"},"fields":{"title":"Isolated Vertices Do Not Disconnect the Edge Support","stem":"Use the complete finite simple undirected graph on Fin(26)={0,…,25}, with canonical sorted edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7], [7, 8], [8, 9], [9, 10], [10, 11], [11, 12], [12, 13], [13, 14], [14, 15], [15, 16], [16, 17], [17, 18], [18, 19], [19, 20], [20, 21], [21, 22], [22, 23], [23, 24]] and zero-based edge indices in that order. The edge support contains exactly the nonisolated vertices. A trail may revisit vertices but may not repeat an edge; an Euler trail uses every edge once. Isolated vertices do not invalidate connected edge support, and no Hamiltonian conclusion follows. Give (nonisolated_vertices, edge_support_connected, whole_graph_connected, odd_vertices, euler_type, boundary).","answer":{"kind":"expression","value":"([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24], Yes, No, [0, 24], open_euler_trail, isolated_vertices_do_not_break_edge_support_connectivity)","explanation":"Return the complete nonisolated support and its connectivity separately from whole-carrier connectivity. Degree parity is applied to the connected edge support, not used to demand that every isolated vertex lie in the same graph component. This family requires at least one edge. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24], Yes, No, [0, 24], open_euler_trail, isolated_vertices_do_not_break_edge_support_connectivity)."},"skills":["Isolated Vertices Do Not Disconnect the Edge Support"],"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-20261105-r04-eulerian-trail-0075","batch_id":"batch-20261014-advance-20261105-r04-eulerian-trail-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9e9fdf6f67c6f0ab0b65e2754edcc67ebad9f61ae044bf5eaa66f47a9305e423","family_id":"qf-mod36-euler-support-connectivity","seed":202611054075,"semantic_template":"qf-mod36-euler-support-connectivity/{\"edges\":[[0,1],[1,2],[10,11],[11,12],[12,13],[13,14],[14,15],[15,16],[16,17],[17,18],[18,19],[19,20],[2,3],[20,21],[21,22],[22,23],[23,24],[24,25],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10]],\"n\":27}"},"fields":{"title":"Isolated Vertices Do Not Disconnect the Edge Support","stem":"Use the complete finite simple undirected graph on Fin(27)={0,…,26}, with canonical sorted edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7], [7, 8], [8, 9], [9, 10], [10, 11], [11, 12], [12, 13], [13, 14], [14, 15], [15, 16], [16, 17], [17, 18], [18, 19], [19, 20], [20, 21], [21, 22], [22, 23], [23, 24], [24, 25]] and zero-based edge indices in that order. The edge support contains exactly the nonisolated vertices. A trail may revisit vertices but may not repeat an edge; an Euler trail uses every edge once. Isolated vertices do not invalidate connected edge support, and no Hamiltonian conclusion follows. Give (nonisolated_vertices, edge_support_connected, whole_graph_connected, odd_vertices, euler_type, boundary).","answer":{"kind":"expression","value":"([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], Yes, No, [0, 25], open_euler_trail, isolated_vertices_do_not_break_edge_support_connectivity)","explanation":"Return the complete nonisolated support and its connectivity separately from whole-carrier connectivity. Degree parity is applied to the connected edge support, not used to demand that every isolated vertex lie in the same graph component. This family requires at least one edge. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], Yes, No, [0, 25], open_euler_trail, isolated_vertices_do_not_break_edge_support_connectivity)."},"skills":["Isolated Vertices Do Not Disconnect the Edge Support"],"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-20261105-r04-eulerian-trail-0076","batch_id":"batch-20261014-advance-20261105-r04-eulerian-trail-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9dd23144290a35e33e11f8df2a83ff0617695c90ffb172ac10c921131a3e1b61","family_id":"qf-mod36-euler-zero-edge-convention","seed":202611054076,"semantic_template":"qf-mod36-euler-zero-edge-convention/{\"n\":1}"},"fields":{"title":"A Nonempty Vertex Set Admits the Declared Length-Zero Circuit","stem":"Use vertex carrier Fin(1)={0,…,0} and no edges, with n=1≥1. Give (vertex_count, edge_count, length_zero_circuit, convention, boundary).","answer":{"kind":"expression","value":"(1, 0, Yes, length_zero_circuit_on_nonempty_vertex_set, empty_vertex_graph_out_of_scope)","explanation":"Adopt a walk consisting of a single chosen vertex as a length-zero circuit using every edge vacuously. The empty-vertex graph is explicitly out of scope, and no Hamiltonian claim follows. The exact result, retaining the requested order, is (1, 0, Yes, length_zero_circuit_on_nonempty_vertex_set, empty_vertex_graph_out_of_scope)."},"skills":["A Nonempty Vertex Set Admits the Declared Length-Zero Circuit"],"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-20261105-r04-eulerian-trail-0077","batch_id":"batch-20261014-advance-20261105-r04-eulerian-trail-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2680c4e4d19f990f8abb98ac052eeabe6a7e2996f59cf970434753b3c9c962a7","family_id":"qf-mod36-euler-zero-edge-convention","seed":202611054077,"semantic_template":"qf-mod36-euler-zero-edge-convention/{\"n\":2}"},"fields":{"title":"A Nonempty Vertex Set Admits the Declared Length-Zero Circuit","stem":"Use vertex carrier Fin(2)={0,…,1} and no edges, with n=2≥1. Give (vertex_count, edge_count, length_zero_circuit, convention, boundary).","answer":{"kind":"expression","value":"(2, 0, Yes, length_zero_circuit_on_nonempty_vertex_set, empty_vertex_graph_out_of_scope)","explanation":"Adopt a walk consisting of a single chosen vertex as a length-zero circuit using every edge vacuously. The empty-vertex graph is explicitly out of scope, and no Hamiltonian claim follows. The exact result, retaining the requested order, is (2, 0, Yes, length_zero_circuit_on_nonempty_vertex_set, empty_vertex_graph_out_of_scope)."},"skills":["A Nonempty Vertex Set Admits the Declared Length-Zero Circuit"],"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-20261105-r04-eulerian-trail-0078","batch_id":"batch-20261014-advance-20261105-r04-eulerian-trail-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a6bc80b645df918492a660c478b58c53e6d916e11a9203b0803b1bfb35f99e98","family_id":"qf-mod36-euler-zero-edge-convention","seed":202611054078,"semantic_template":"qf-mod36-euler-zero-edge-convention/{\"n\":3}"},"fields":{"title":"A Nonempty Vertex Set Admits the Declared Length-Zero Circuit","stem":"Use vertex carrier Fin(3)={0,…,2} and no edges, with n=3≥1. Give (vertex_count, edge_count, length_zero_circuit, convention, boundary).","answer":{"kind":"expression","value":"(3, 0, Yes, length_zero_circuit_on_nonempty_vertex_set, empty_vertex_graph_out_of_scope)","explanation":"Adopt a walk consisting of a single chosen vertex as a length-zero circuit using every edge vacuously. The empty-vertex graph is explicitly out of scope, and no Hamiltonian claim follows. The exact result, retaining the requested order, is (3, 0, Yes, length_zero_circuit_on_nonempty_vertex_set, empty_vertex_graph_out_of_scope)."},"skills":["A Nonempty Vertex Set Admits the Declared Length-Zero Circuit"],"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-20261105-r04-eulerian-trail-0079","batch_id":"batch-20261014-advance-20261105-r04-eulerian-trail-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"83a70f7a51936f53dd450d5ca27e6a6ff26bee0089bdac0f25fab0cf1c4ff49a","family_id":"qf-mod36-euler-zero-edge-convention","seed":202611054079,"semantic_template":"qf-mod36-euler-zero-edge-convention/{\"n\":4}"},"fields":{"title":"A Nonempty Vertex Set Admits the Declared Length-Zero Circuit","stem":"Use vertex carrier Fin(4)={0,…,3} and no edges, with n=4≥1. Give (vertex_count, edge_count, length_zero_circuit, convention, boundary).","answer":{"kind":"expression","value":"(4, 0, Yes, length_zero_circuit_on_nonempty_vertex_set, empty_vertex_graph_out_of_scope)","explanation":"Adopt a walk consisting of a single chosen vertex as a length-zero circuit using every edge vacuously. The empty-vertex graph is explicitly out of scope, and no Hamiltonian claim follows. The exact result, retaining the requested order, is (4, 0, Yes, length_zero_circuit_on_nonempty_vertex_set, empty_vertex_graph_out_of_scope)."},"skills":["A Nonempty Vertex Set Admits the Declared Length-Zero Circuit"],"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-20261105-r04-eulerian-trail-0080","batch_id":"batch-20261014-advance-20261105-r04-eulerian-trail-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4fbc83e9923282fa2a687e9acfbc5c15df193b0def384ba5311b1aad87e67c32","family_id":"qf-mod36-euler-zero-edge-convention","seed":202611054080,"semantic_template":"qf-mod36-euler-zero-edge-convention/{\"n\":5}"},"fields":{"title":"A Nonempty Vertex Set Admits the Declared Length-Zero Circuit","stem":"Use vertex carrier Fin(5)={0,…,4} and no edges, with n=5≥1. Give (vertex_count, edge_count, length_zero_circuit, convention, boundary).","answer":{"kind":"expression","value":"(5, 0, Yes, length_zero_circuit_on_nonempty_vertex_set, empty_vertex_graph_out_of_scope)","explanation":"Adopt a walk consisting of a single chosen vertex as a length-zero circuit using every edge vacuously. The empty-vertex graph is explicitly out of scope, and no Hamiltonian claim follows. The exact result, retaining the requested order, is (5, 0, Yes, length_zero_circuit_on_nonempty_vertex_set, empty_vertex_graph_out_of_scope)."},"skills":["A Nonempty Vertex Set Admits the Declared Length-Zero Circuit"],"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-20261105-r04-eulerian-trail-0081","batch_id":"batch-20261014-advance-20261105-r04-eulerian-trail-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"83adf4224fb6a18eab7b88c057f0c76446c4b84037ba12de904feea81b221437","family_id":"qf-mod36-euler-zero-edge-convention","seed":202611054081,"semantic_template":"qf-mod36-euler-zero-edge-convention/{\"n\":6}"},"fields":{"title":"A Nonempty Vertex Set Admits the Declared Length-Zero Circuit","stem":"Use vertex carrier Fin(6)={0,…,5} and no edges, with n=6≥1. Give (vertex_count, edge_count, length_zero_circuit, convention, boundary).","answer":{"kind":"expression","value":"(6, 0, Yes, length_zero_circuit_on_nonempty_vertex_set, empty_vertex_graph_out_of_scope)","explanation":"Adopt a walk consisting of a single chosen vertex as a length-zero circuit using every edge vacuously. The empty-vertex graph is explicitly out of scope, and no Hamiltonian claim follows. The exact result, retaining the requested order, is (6, 0, Yes, length_zero_circuit_on_nonempty_vertex_set, empty_vertex_graph_out_of_scope)."},"skills":["A Nonempty Vertex Set Admits the Declared Length-Zero Circuit"],"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-20261105-r04-eulerian-trail-0082","batch_id":"batch-20261014-advance-20261105-r04-eulerian-trail-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0e9554fd1479ac5cc7402069215b18fef93a69d8b028215b4fa10d3bf5592927","family_id":"qf-mod36-euler-zero-edge-convention","seed":202611054082,"semantic_template":"qf-mod36-euler-zero-edge-convention/{\"n\":7}"},"fields":{"title":"A Nonempty Vertex Set Admits the Declared Length-Zero Circuit","stem":"Use vertex carrier Fin(7)={0,…,6} and no edges, with n=7≥1. Give (vertex_count, edge_count, length_zero_circuit, convention, boundary).","answer":{"kind":"expression","value":"(7, 0, Yes, length_zero_circuit_on_nonempty_vertex_set, empty_vertex_graph_out_of_scope)","explanation":"Adopt a walk consisting of a single chosen vertex as a length-zero circuit using every edge vacuously. The empty-vertex graph is explicitly out of scope, and no Hamiltonian claim follows. The exact result, retaining the requested order, is (7, 0, Yes, length_zero_circuit_on_nonempty_vertex_set, empty_vertex_graph_out_of_scope)."},"skills":["A Nonempty Vertex Set Admits the Declared Length-Zero Circuit"],"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-20261105-r04-eulerian-trail-0083","batch_id":"batch-20261014-advance-20261105-r04-eulerian-trail-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bb5b475c00843c0087e9a6be927f334a85a452be98f9a692328b9f538abe9ee0","family_id":"qf-mod36-euler-zero-edge-convention","seed":202611054083,"semantic_template":"qf-mod36-euler-zero-edge-convention/{\"n\":8}"},"fields":{"title":"A Nonempty Vertex Set Admits the Declared Length-Zero Circuit","stem":"Use vertex carrier Fin(8)={0,…,7} and no edges, with n=8≥1. Give (vertex_count, edge_count, length_zero_circuit, convention, boundary).","answer":{"kind":"expression","value":"(8, 0, Yes, length_zero_circuit_on_nonempty_vertex_set, empty_vertex_graph_out_of_scope)","explanation":"Adopt a walk consisting of a single chosen vertex as a length-zero circuit using every edge vacuously. The empty-vertex graph is explicitly out of scope, and no Hamiltonian claim follows. The exact result, retaining the requested order, is (8, 0, Yes, length_zero_circuit_on_nonempty_vertex_set, empty_vertex_graph_out_of_scope)."},"skills":["A Nonempty Vertex Set Admits the Declared Length-Zero Circuit"],"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-20261105-r04-eulerian-trail-0084","batch_id":"batch-20261014-advance-20261105-r04-eulerian-trail-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c3ff3cd7e1e003b90962dde58cf77d70ea8a6635d156f0bffa635c182ad91f61","family_id":"qf-mod36-euler-zero-edge-convention","seed":202611054084,"semantic_template":"qf-mod36-euler-zero-edge-convention/{\"n\":9}"},"fields":{"title":"A Nonempty Vertex Set Admits the Declared Length-Zero Circuit","stem":"Use vertex carrier Fin(9)={0,…,8} and no edges, with n=9≥1. Give (vertex_count, edge_count, length_zero_circuit, convention, boundary).","answer":{"kind":"expression","value":"(9, 0, Yes, length_zero_circuit_on_nonempty_vertex_set, empty_vertex_graph_out_of_scope)","explanation":"Adopt a walk consisting of a single chosen vertex as a length-zero circuit using every edge vacuously. The empty-vertex graph is explicitly out of scope, and no Hamiltonian claim follows. The exact result, retaining the requested order, is (9, 0, Yes, length_zero_circuit_on_nonempty_vertex_set, empty_vertex_graph_out_of_scope)."},"skills":["A Nonempty Vertex Set Admits the Declared Length-Zero Circuit"],"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-20261105-r04-eulerian-trail-0085","batch_id":"batch-20261014-advance-20261105-r04-eulerian-trail-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0cfe4b5ecefe6a26bea3782c259365706a6a88a205745d8a5f73927b05d3576a","family_id":"qf-mod36-euler-zero-edge-convention","seed":202611054085,"semantic_template":"qf-mod36-euler-zero-edge-convention/{\"n\":10}"},"fields":{"title":"A Nonempty Vertex Set Admits the Declared Length-Zero Circuit","stem":"Use vertex carrier Fin(10)={0,…,9} and no edges, with n=10≥1. Give (vertex_count, edge_count, length_zero_circuit, convention, boundary).","answer":{"kind":"expression","value":"(10, 0, Yes, length_zero_circuit_on_nonempty_vertex_set, empty_vertex_graph_out_of_scope)","explanation":"Adopt a walk consisting of a single chosen vertex as a length-zero circuit using every edge vacuously. The empty-vertex graph is explicitly out of scope, and no Hamiltonian claim follows. The exact result, retaining the requested order, is (10, 0, Yes, length_zero_circuit_on_nonempty_vertex_set, empty_vertex_graph_out_of_scope)."},"skills":["A Nonempty Vertex Set Admits the Declared Length-Zero Circuit"],"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-20261105-r04-eulerian-trail-0086","batch_id":"batch-20261014-advance-20261105-r04-eulerian-trail-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0890528aecb337f352383c6ba6525e53977689d3afa10bcc59a1c5ae7b10d1fb","family_id":"qf-mod36-euler-zero-edge-convention","seed":202611054086,"semantic_template":"qf-mod36-euler-zero-edge-convention/{\"n\":11}"},"fields":{"title":"A Nonempty Vertex Set Admits the Declared Length-Zero Circuit","stem":"Use vertex carrier Fin(11)={0,…,10} and no edges, with n=11≥1. Give (vertex_count, edge_count, length_zero_circuit, convention, boundary).","answer":{"kind":"expression","value":"(11, 0, Yes, length_zero_circuit_on_nonempty_vertex_set, empty_vertex_graph_out_of_scope)","explanation":"Adopt a walk consisting of a single chosen vertex as a length-zero circuit using every edge vacuously. The empty-vertex graph is explicitly out of scope, and no Hamiltonian claim follows. The exact result, retaining the requested order, is (11, 0, Yes, length_zero_circuit_on_nonempty_vertex_set, empty_vertex_graph_out_of_scope)."},"skills":["A Nonempty Vertex Set Admits the Declared Length-Zero Circuit"],"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-20261105-r04-eulerian-trail-0087","batch_id":"batch-20261014-advance-20261105-r04-eulerian-trail-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4a992d31926b8c8a1d2dd6e1ef7acf7bbf745d9e5517998ef46c9807814bb71e","family_id":"qf-mod36-euler-zero-edge-convention","seed":202611054087,"semantic_template":"qf-mod36-euler-zero-edge-convention/{\"n\":12}"},"fields":{"title":"A Nonempty Vertex Set Admits the Declared Length-Zero Circuit","stem":"Use vertex carrier Fin(12)={0,…,11} and no edges, with n=12≥1. Give (vertex_count, edge_count, length_zero_circuit, convention, boundary).","answer":{"kind":"expression","value":"(12, 0, Yes, length_zero_circuit_on_nonempty_vertex_set, empty_vertex_graph_out_of_scope)","explanation":"Adopt a walk consisting of a single chosen vertex as a length-zero circuit using every edge vacuously. The empty-vertex graph is explicitly out of scope, and no Hamiltonian claim follows. The exact result, retaining the requested order, is (12, 0, Yes, length_zero_circuit_on_nonempty_vertex_set, empty_vertex_graph_out_of_scope)."},"skills":["A Nonempty Vertex Set Admits the Declared Length-Zero Circuit"],"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-20261105-r04-eulerian-trail-0088","batch_id":"batch-20261014-advance-20261105-r04-eulerian-trail-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6a30cd525be80ad4b5266f87d444552e049b2a8883ac5f2d34c0b39ca8e11fe0","family_id":"qf-mod36-euler-zero-edge-convention","seed":202611054088,"semantic_template":"qf-mod36-euler-zero-edge-convention/{\"n\":13}"},"fields":{"title":"A Nonempty Vertex Set Admits the Declared Length-Zero Circuit","stem":"Use vertex carrier Fin(13)={0,…,12} and no edges, with n=13≥1. Give (vertex_count, edge_count, length_zero_circuit, convention, boundary).","answer":{"kind":"expression","value":"(13, 0, Yes, length_zero_circuit_on_nonempty_vertex_set, empty_vertex_graph_out_of_scope)","explanation":"Adopt a walk consisting of a single chosen vertex as a length-zero circuit using every edge vacuously. The empty-vertex graph is explicitly out of scope, and no Hamiltonian claim follows. The exact result, retaining the requested order, is (13, 0, Yes, length_zero_circuit_on_nonempty_vertex_set, empty_vertex_graph_out_of_scope)."},"skills":["A Nonempty Vertex Set Admits the Declared Length-Zero Circuit"],"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-20261105-r04-eulerian-trail-0089","batch_id":"batch-20261014-advance-20261105-r04-eulerian-trail-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a1ec46a8da11f1182cfda87a5e635ea9a0bef4402bc7cd472fd3c1fda9ec7f6b","family_id":"qf-mod36-euler-zero-edge-convention","seed":202611054089,"semantic_template":"qf-mod36-euler-zero-edge-convention/{\"n\":14}"},"fields":{"title":"A Nonempty Vertex Set Admits the Declared Length-Zero Circuit","stem":"Use vertex carrier Fin(14)={0,…,13} and no edges, with n=14≥1. Give (vertex_count, edge_count, length_zero_circuit, convention, boundary).","answer":{"kind":"expression","value":"(14, 0, Yes, length_zero_circuit_on_nonempty_vertex_set, empty_vertex_graph_out_of_scope)","explanation":"Adopt a walk consisting of a single chosen vertex as a length-zero circuit using every edge vacuously. The empty-vertex graph is explicitly out of scope, and no Hamiltonian claim follows. The exact result, retaining the requested order, is (14, 0, Yes, length_zero_circuit_on_nonempty_vertex_set, empty_vertex_graph_out_of_scope)."},"skills":["A Nonempty Vertex Set Admits the Declared Length-Zero Circuit"],"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-20261105-r04-eulerian-trail-0090","batch_id":"batch-20261014-advance-20261105-r04-eulerian-trail-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"20a67daf8dc3064dc55da608086c4e6c053bf8feb0292294250118d9833cd843","family_id":"qf-mod36-euler-zero-edge-convention","seed":202611054090,"semantic_template":"qf-mod36-euler-zero-edge-convention/{\"n\":15}"},"fields":{"title":"A Nonempty Vertex Set Admits the Declared Length-Zero Circuit","stem":"Use vertex carrier Fin(15)={0,…,14} and no edges, with n=15≥1. Give (vertex_count, edge_count, length_zero_circuit, convention, boundary).","answer":{"kind":"expression","value":"(15, 0, Yes, length_zero_circuit_on_nonempty_vertex_set, empty_vertex_graph_out_of_scope)","explanation":"Adopt a walk consisting of a single chosen vertex as a length-zero circuit using every edge vacuously. The empty-vertex graph is explicitly out of scope, and no Hamiltonian claim follows. The exact result, retaining the requested order, is (15, 0, Yes, length_zero_circuit_on_nonempty_vertex_set, empty_vertex_graph_out_of_scope)."},"skills":["A Nonempty Vertex Set Admits the Declared Length-Zero Circuit"],"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-20261105-r04-eulerian-trail-0091","batch_id":"batch-20261014-advance-20261105-r04-eulerian-trail-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"aac1614a7cb3c0079c5a936543fdc18e4dabbb227447f6103175c06b99ea852b","family_id":"qf-mod36-euler-zero-edge-convention","seed":202611054091,"semantic_template":"qf-mod36-euler-zero-edge-convention/{\"n\":16}"},"fields":{"title":"A Nonempty Vertex Set Admits the Declared Length-Zero Circuit","stem":"Use vertex carrier Fin(16)={0,…,15} and no edges, with n=16≥1. Give (vertex_count, edge_count, length_zero_circuit, convention, boundary).","answer":{"kind":"expression","value":"(16, 0, Yes, length_zero_circuit_on_nonempty_vertex_set, empty_vertex_graph_out_of_scope)","explanation":"Adopt a walk consisting of a single chosen vertex as a length-zero circuit using every edge vacuously. The empty-vertex graph is explicitly out of scope, and no Hamiltonian claim follows. The exact result, retaining the requested order, is (16, 0, Yes, length_zero_circuit_on_nonempty_vertex_set, empty_vertex_graph_out_of_scope)."},"skills":["A Nonempty Vertex Set Admits the Declared Length-Zero Circuit"],"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-20261105-r04-eulerian-trail-0092","batch_id":"batch-20261014-advance-20261105-r04-eulerian-trail-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bc203efcdcd87b0fb220dfcf95b287a018befeb259c301e83c82669a06c6bd9d","family_id":"qf-mod36-euler-zero-edge-convention","seed":202611054092,"semantic_template":"qf-mod36-euler-zero-edge-convention/{\"n\":17}"},"fields":{"title":"A Nonempty Vertex Set Admits the Declared Length-Zero Circuit","stem":"Use vertex carrier Fin(17)={0,…,16} and no edges, with n=17≥1. Give (vertex_count, edge_count, length_zero_circuit, convention, boundary).","answer":{"kind":"expression","value":"(17, 0, Yes, length_zero_circuit_on_nonempty_vertex_set, empty_vertex_graph_out_of_scope)","explanation":"Adopt a walk consisting of a single chosen vertex as a length-zero circuit using every edge vacuously. The empty-vertex graph is explicitly out of scope, and no Hamiltonian claim follows. The exact result, retaining the requested order, is (17, 0, Yes, length_zero_circuit_on_nonempty_vertex_set, empty_vertex_graph_out_of_scope)."},"skills":["A Nonempty Vertex Set Admits the Declared Length-Zero Circuit"],"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-20261105-r04-eulerian-trail-0093","batch_id":"batch-20261014-advance-20261105-r04-eulerian-trail-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7778ab0047382de91a1244c3f44388eda8104483539654e565afedc66d26ba8f","family_id":"qf-mod36-euler-zero-edge-convention","seed":202611054093,"semantic_template":"qf-mod36-euler-zero-edge-convention/{\"n\":18}"},"fields":{"title":"A Nonempty Vertex Set Admits the Declared Length-Zero Circuit","stem":"Use vertex carrier Fin(18)={0,…,17} and no edges, with n=18≥1. Give (vertex_count, edge_count, length_zero_circuit, convention, boundary).","answer":{"kind":"expression","value":"(18, 0, Yes, length_zero_circuit_on_nonempty_vertex_set, empty_vertex_graph_out_of_scope)","explanation":"Adopt a walk consisting of a single chosen vertex as a length-zero circuit using every edge vacuously. The empty-vertex graph is explicitly out of scope, and no Hamiltonian claim follows. The exact result, retaining the requested order, is (18, 0, Yes, length_zero_circuit_on_nonempty_vertex_set, empty_vertex_graph_out_of_scope)."},"skills":["A Nonempty Vertex Set Admits the Declared Length-Zero Circuit"],"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-20261105-r04-eulerian-trail-0094","batch_id":"batch-20261014-advance-20261105-r04-eulerian-trail-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2c7c9cf7446b82fbaaf9c7ef74c1fa2098638aa3bf5dae367cacc8fd54bcebdb","family_id":"qf-mod36-euler-zero-edge-convention","seed":202611054094,"semantic_template":"qf-mod36-euler-zero-edge-convention/{\"n\":19}"},"fields":{"title":"A Nonempty Vertex Set Admits the Declared Length-Zero Circuit","stem":"Use vertex carrier Fin(19)={0,…,18} and no edges, with n=19≥1. Give (vertex_count, edge_count, length_zero_circuit, convention, boundary).","answer":{"kind":"expression","value":"(19, 0, Yes, length_zero_circuit_on_nonempty_vertex_set, empty_vertex_graph_out_of_scope)","explanation":"Adopt a walk consisting of a single chosen vertex as a length-zero circuit using every edge vacuously. The empty-vertex graph is explicitly out of scope, and no Hamiltonian claim follows. The exact result, retaining the requested order, is (19, 0, Yes, length_zero_circuit_on_nonempty_vertex_set, empty_vertex_graph_out_of_scope)."},"skills":["A Nonempty Vertex Set Admits the Declared Length-Zero Circuit"],"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-20261105-r04-eulerian-trail-0095","batch_id":"batch-20261014-advance-20261105-r04-eulerian-trail-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c55186a9823bd11c412c40190d4ab1fe3ef3627dfd6e51033dfbe9a76e540d9e","family_id":"qf-mod36-euler-zero-edge-convention","seed":202611054095,"semantic_template":"qf-mod36-euler-zero-edge-convention/{\"n\":20}"},"fields":{"title":"A Nonempty Vertex Set Admits the Declared Length-Zero Circuit","stem":"Use vertex carrier Fin(20)={0,…,19} and no edges, with n=20≥1. Give (vertex_count, edge_count, length_zero_circuit, convention, boundary).","answer":{"kind":"expression","value":"(20, 0, Yes, length_zero_circuit_on_nonempty_vertex_set, empty_vertex_graph_out_of_scope)","explanation":"Adopt a walk consisting of a single chosen vertex as a length-zero circuit using every edge vacuously. The empty-vertex graph is explicitly out of scope, and no Hamiltonian claim follows. The exact result, retaining the requested order, is (20, 0, Yes, length_zero_circuit_on_nonempty_vertex_set, empty_vertex_graph_out_of_scope)."},"skills":["A Nonempty Vertex Set Admits the Declared Length-Zero Circuit"],"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-20261105-r04-eulerian-trail-0096","batch_id":"batch-20261014-advance-20261105-r04-eulerian-trail-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6dce2d2c046ebc2f0451068706a45debdc25fc7cceab8b63a9f7dd232cfc49e9","family_id":"qf-mod36-euler-zero-edge-convention","seed":202611054096,"semantic_template":"qf-mod36-euler-zero-edge-convention/{\"n\":21}"},"fields":{"title":"A Nonempty Vertex Set Admits the Declared Length-Zero Circuit","stem":"Use vertex carrier Fin(21)={0,…,20} and no edges, with n=21≥1. Give (vertex_count, edge_count, length_zero_circuit, convention, boundary).","answer":{"kind":"expression","value":"(21, 0, Yes, length_zero_circuit_on_nonempty_vertex_set, empty_vertex_graph_out_of_scope)","explanation":"Adopt a walk consisting of a single chosen vertex as a length-zero circuit using every edge vacuously. The empty-vertex graph is explicitly out of scope, and no Hamiltonian claim follows. The exact result, retaining the requested order, is (21, 0, Yes, length_zero_circuit_on_nonempty_vertex_set, empty_vertex_graph_out_of_scope)."},"skills":["A Nonempty Vertex Set Admits the Declared Length-Zero Circuit"],"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-20261105-r04-eulerian-trail-0097","batch_id":"batch-20261014-advance-20261105-r04-eulerian-trail-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5f352e5f4bda0956f1862e60177f51f22b8a6ccc009a72ac5739e5a7de8cf45c","family_id":"qf-mod36-euler-zero-edge-convention","seed":202611054097,"semantic_template":"qf-mod36-euler-zero-edge-convention/{\"n\":22}"},"fields":{"title":"A Nonempty Vertex Set Admits the Declared Length-Zero Circuit","stem":"Use vertex carrier Fin(22)={0,…,21} and no edges, with n=22≥1. Give (vertex_count, edge_count, length_zero_circuit, convention, boundary).","answer":{"kind":"expression","value":"(22, 0, Yes, length_zero_circuit_on_nonempty_vertex_set, empty_vertex_graph_out_of_scope)","explanation":"Adopt a walk consisting of a single chosen vertex as a length-zero circuit using every edge vacuously. The empty-vertex graph is explicitly out of scope, and no Hamiltonian claim follows. The exact result, retaining the requested order, is (22, 0, Yes, length_zero_circuit_on_nonempty_vertex_set, empty_vertex_graph_out_of_scope)."},"skills":["A Nonempty Vertex Set Admits the Declared Length-Zero Circuit"],"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-20261105-r04-eulerian-trail-0098","batch_id":"batch-20261014-advance-20261105-r04-eulerian-trail-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8c032f3fc13198fae03dd66552c699b3ab8ee3cf0a4dc27484abd8b276deb812","family_id":"qf-mod36-euler-zero-edge-convention","seed":202611054098,"semantic_template":"qf-mod36-euler-zero-edge-convention/{\"n\":23}"},"fields":{"title":"A Nonempty Vertex Set Admits the Declared Length-Zero Circuit","stem":"Use vertex carrier Fin(23)={0,…,22} and no edges, with n=23≥1. Give (vertex_count, edge_count, length_zero_circuit, convention, boundary).","answer":{"kind":"expression","value":"(23, 0, Yes, length_zero_circuit_on_nonempty_vertex_set, empty_vertex_graph_out_of_scope)","explanation":"Adopt a walk consisting of a single chosen vertex as a length-zero circuit using every edge vacuously. The empty-vertex graph is explicitly out of scope, and no Hamiltonian claim follows. The exact result, retaining the requested order, is (23, 0, Yes, length_zero_circuit_on_nonempty_vertex_set, empty_vertex_graph_out_of_scope)."},"skills":["A Nonempty Vertex Set Admits the Declared Length-Zero Circuit"],"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-20261105-r04-eulerian-trail-0099","batch_id":"batch-20261014-advance-20261105-r04-eulerian-trail-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5295e74f230a94836abc7f49ce163091404caa93a6df8ec1b21dec32d47bde2a","family_id":"qf-mod36-euler-zero-edge-convention","seed":202611054099,"semantic_template":"qf-mod36-euler-zero-edge-convention/{\"n\":24}"},"fields":{"title":"A Nonempty Vertex Set Admits the Declared Length-Zero Circuit","stem":"Use vertex carrier Fin(24)={0,…,23} and no edges, with n=24≥1. Give (vertex_count, edge_count, length_zero_circuit, convention, boundary).","answer":{"kind":"expression","value":"(24, 0, Yes, length_zero_circuit_on_nonempty_vertex_set, empty_vertex_graph_out_of_scope)","explanation":"Adopt a walk consisting of a single chosen vertex as a length-zero circuit using every edge vacuously. The empty-vertex graph is explicitly out of scope, and no Hamiltonian claim follows. The exact result, retaining the requested order, is (24, 0, Yes, length_zero_circuit_on_nonempty_vertex_set, empty_vertex_graph_out_of_scope)."},"skills":["A Nonempty Vertex Set Admits the Declared Length-Zero Circuit"],"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-20261105-r04-eulerian-trail-0100","batch_id":"batch-20261014-advance-20261105-r04-eulerian-trail-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"153b7844ee1bbcaae1c31c8ae095218043266b4586628b1e75bc97b4d2ce022c","family_id":"qf-mod36-euler-zero-edge-convention","seed":202611054100,"semantic_template":"qf-mod36-euler-zero-edge-convention/{\"n\":25}"},"fields":{"title":"A Nonempty Vertex Set Admits the Declared Length-Zero Circuit","stem":"Use vertex carrier Fin(25)={0,…,24} and no edges, with n=25≥1. Give (vertex_count, edge_count, length_zero_circuit, convention, boundary).","answer":{"kind":"expression","value":"(25, 0, Yes, length_zero_circuit_on_nonempty_vertex_set, empty_vertex_graph_out_of_scope)","explanation":"Adopt a walk consisting of a single chosen vertex as a length-zero circuit using every edge vacuously. The empty-vertex graph is explicitly out of scope, and no Hamiltonian claim follows. The exact result, retaining the requested order, is (25, 0, Yes, length_zero_circuit_on_nonempty_vertex_set, empty_vertex_graph_out_of_scope)."},"skills":["A Nonempty Vertex Set Admits the Declared Length-Zero Circuit"],"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."}}