{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r06-spanning-tree-0001","batch_id":"batch-20261014-advance-20261104-r06-spanning-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b69819c0b56757418d70f5e58dfe85bdab5e60e15b771c456b39fa97edda7d22","family_id":"qf-mod35-spantree-audit","seed":202611046001,"semantic_template":"qf-mod35-spantree-audit/{\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"n\":5,\"selected\":[0,1,2,3]}"},"fields":{"title":"A Spanning Tree Uses the Entire Parent Vertex Set","stem":"Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [1, 2], [2, 3], [3, 4]], and distinct parent-edge selection=[0, 1, 2, 3], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (selected_indices, parent_connected, selected_connected, selected_acyclic, spanning_tree).","answer":{"kind":"expression","value":"([0, 1, 2, 3], Yes, Yes, Yes, Yes)","explanation":"Read the selected parent edges and report all connectivity and acyclicity obligations separately. A disconnected candidate parent has no spanning tree; a failed selection does not refute existence for another selection in a connected parent. The exact result, retaining the requested order, is ([0, 1, 2, 3], Yes, Yes, Yes, Yes)."},"skills":["A Spanning Tree Uses the Entire Parent Vertex Set"],"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-20261104-r06-spanning-tree-0002","batch_id":"batch-20261014-advance-20261104-r06-spanning-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5b49a7ec4c4727fdc3a9e7b3b886dfb3afaf918cb28311aa958d96f58e028be2","family_id":"qf-mod35-spantree-audit","seed":202611046002,"semantic_template":"qf-mod35-spantree-audit/{\"edges\":[[0,1],[0,2],[1,2],[2,3],[3,4]],\"n\":5,\"selected\":[0,2,3,4]}"},"fields":{"title":"A Spanning Tree Uses the Entire Parent Vertex Set","stem":"Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [0, 2], [1, 2], [2, 3], [3, 4]], and distinct parent-edge selection=[0, 2, 3, 4], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (selected_indices, parent_connected, selected_connected, selected_acyclic, spanning_tree).","answer":{"kind":"expression","value":"([0, 2, 3, 4], Yes, Yes, Yes, Yes)","explanation":"Read the selected parent edges and report all connectivity and acyclicity obligations separately. A disconnected candidate parent has no spanning tree; a failed selection does not refute existence for another selection in a connected parent. The exact result, retaining the requested order, is ([0, 2, 3, 4], Yes, Yes, Yes, Yes)."},"skills":["A Spanning Tree Uses the Entire Parent Vertex Set"],"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-20261104-r06-spanning-tree-0003","batch_id":"batch-20261014-advance-20261104-r06-spanning-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7ec7fe5d57372f6e7558d2b72f6ce8090716b09ef3e1618d1bd6aca213de1bd4","family_id":"qf-mod35-spantree-audit","seed":202611046003,"semantic_template":"qf-mod35-spantree-audit/{\"edges\":[[0,1],[0,3],[1,2],[2,3],[3,4]],\"n\":5,\"selected\":[0,2,3,4]}"},"fields":{"title":"A Spanning Tree Uses the Entire Parent Vertex Set","stem":"Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [0, 3], [1, 2], [2, 3], [3, 4]], and distinct parent-edge selection=[0, 2, 3, 4], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (selected_indices, parent_connected, selected_connected, selected_acyclic, spanning_tree).","answer":{"kind":"expression","value":"([0, 2, 3, 4], Yes, Yes, Yes, Yes)","explanation":"Read the selected parent edges and report all connectivity and acyclicity obligations separately. A disconnected candidate parent has no spanning tree; a failed selection does not refute existence for another selection in a connected parent. The exact result, retaining the requested order, is ([0, 2, 3, 4], Yes, Yes, Yes, Yes)."},"skills":["A Spanning Tree Uses the Entire Parent Vertex Set"],"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-20261104-r06-spanning-tree-0004","batch_id":"batch-20261014-advance-20261104-r06-spanning-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"95bcfec225a2a95f9543552bfa77077a9fa5c08ae3f15c5940328ab410fcfac6","family_id":"qf-mod35-spantree-audit","seed":202611046004,"semantic_template":"qf-mod35-spantree-audit/{\"edges\":[[0,1],[0,2],[0,3],[1,2],[2,3],[3,4]],\"n\":5,\"selected\":[0,3,4,5]}"},"fields":{"title":"A Spanning Tree Uses the Entire Parent Vertex Set","stem":"Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [0, 2], [0, 3], [1, 2], [2, 3], [3, 4]], and distinct parent-edge selection=[0, 3, 4, 5], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (selected_indices, parent_connected, selected_connected, selected_acyclic, spanning_tree).","answer":{"kind":"expression","value":"([0, 3, 4, 5], Yes, Yes, Yes, Yes)","explanation":"Read the selected parent edges and report all connectivity and acyclicity obligations separately. A disconnected candidate parent has no spanning tree; a failed selection does not refute existence for another selection in a connected parent. The exact result, retaining the requested order, is ([0, 3, 4, 5], Yes, Yes, Yes, Yes)."},"skills":["A Spanning Tree Uses the Entire Parent Vertex Set"],"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-20261104-r06-spanning-tree-0005","batch_id":"batch-20261014-advance-20261104-r06-spanning-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"04ddb9e19ffd970cb31dcd39b4d303fa4b4c7baa96cec77679a1105b0b848053","family_id":"qf-mod35-spantree-audit","seed":202611046005,"semantic_template":"qf-mod35-spantree-audit/{\"edges\":[[0,1],[1,2],[1,3],[2,3],[3,4]],\"n\":5,\"selected\":[0,1,3,4]}"},"fields":{"title":"A Spanning Tree Uses the Entire Parent Vertex Set","stem":"Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [1, 2], [1, 3], [2, 3], [3, 4]], and distinct parent-edge selection=[0, 1, 3, 4], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (selected_indices, parent_connected, selected_connected, selected_acyclic, spanning_tree).","answer":{"kind":"expression","value":"([0, 1, 3, 4], Yes, Yes, Yes, Yes)","explanation":"Read the selected parent edges and report all connectivity and acyclicity obligations separately. A disconnected candidate parent has no spanning tree; a failed selection does not refute existence for another selection in a connected parent. The exact result, retaining the requested order, is ([0, 1, 3, 4], Yes, Yes, Yes, Yes)."},"skills":["A Spanning Tree Uses the Entire Parent Vertex Set"],"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-20261104-r06-spanning-tree-0006","batch_id":"batch-20261014-advance-20261104-r06-spanning-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d38a68acaf9a3b98b7dbae4744e252e4b81e436cb450497d6d280c48c2cdcc2e","family_id":"qf-mod35-spantree-audit","seed":202611046006,"semantic_template":"qf-mod35-spantree-audit/{\"edges\":[[0,1],[0,2],[1,2],[1,3],[2,3],[3,4]],\"n\":5,\"selected\":[0,2,4,5]}"},"fields":{"title":"A Spanning Tree Uses the Entire Parent Vertex Set","stem":"Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [0, 2], [1, 2], [1, 3], [2, 3], [3, 4]], and distinct parent-edge selection=[0, 2, 4, 5], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (selected_indices, parent_connected, selected_connected, selected_acyclic, spanning_tree).","answer":{"kind":"expression","value":"([0, 2, 4, 5], Yes, Yes, Yes, Yes)","explanation":"Read the selected parent edges and report all connectivity and acyclicity obligations separately. A disconnected candidate parent has no spanning tree; a failed selection does not refute existence for another selection in a connected parent. The exact result, retaining the requested order, is ([0, 2, 4, 5], Yes, Yes, Yes, Yes)."},"skills":["A Spanning Tree Uses the Entire Parent Vertex Set"],"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-20261104-r06-spanning-tree-0007","batch_id":"batch-20261014-advance-20261104-r06-spanning-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c27c6f89b3cc664a22ea83cfac7a1479e2395cd5a4143d8b81e4c1a1a4e565a7","family_id":"qf-mod35-spantree-audit","seed":202611046007,"semantic_template":"qf-mod35-spantree-audit/{\"edges\":[[0,1],[0,3],[1,2],[1,3],[2,3],[3,4]],\"n\":5,\"selected\":[0,2,4,5]}"},"fields":{"title":"A Spanning Tree Uses the Entire Parent Vertex Set","stem":"Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [0, 3], [1, 2], [1, 3], [2, 3], [3, 4]], and distinct parent-edge selection=[0, 2, 4, 5], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (selected_indices, parent_connected, selected_connected, selected_acyclic, spanning_tree).","answer":{"kind":"expression","value":"([0, 2, 4, 5], Yes, Yes, Yes, Yes)","explanation":"Read the selected parent edges and report all connectivity and acyclicity obligations separately. A disconnected candidate parent has no spanning tree; a failed selection does not refute existence for another selection in a connected parent. The exact result, retaining the requested order, is ([0, 2, 4, 5], Yes, Yes, Yes, Yes)."},"skills":["A Spanning Tree Uses the Entire Parent Vertex Set"],"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-20261104-r06-spanning-tree-0008","batch_id":"batch-20261014-advance-20261104-r06-spanning-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3d868edcbb4a21b08a9beab074103f2f74b1b2006c074aaa2be31290cd74c243","family_id":"qf-mod35-spantree-audit","seed":202611046008,"semantic_template":"qf-mod35-spantree-audit/{\"edges\":[[0,1],[0,2],[0,3],[1,2],[1,3],[2,3],[3,4]],\"n\":5,\"selected\":[0,3,5,6]}"},"fields":{"title":"A Spanning Tree Uses the Entire Parent Vertex Set","stem":"Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [0, 2], [0, 3], [1, 2], [1, 3], [2, 3], [3, 4]], and distinct parent-edge selection=[0, 3, 5, 6], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (selected_indices, parent_connected, selected_connected, selected_acyclic, spanning_tree).","answer":{"kind":"expression","value":"([0, 3, 5, 6], Yes, Yes, Yes, Yes)","explanation":"Read the selected parent edges and report all connectivity and acyclicity obligations separately. A disconnected candidate parent has no spanning tree; a failed selection does not refute existence for another selection in a connected parent. The exact result, retaining the requested order, is ([0, 3, 5, 6], Yes, Yes, Yes, Yes)."},"skills":["A Spanning Tree Uses the Entire Parent Vertex Set"],"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-20261104-r06-spanning-tree-0009","batch_id":"batch-20261014-advance-20261104-r06-spanning-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f584d4f4801e584f78698e941dd0f8c4236d0a202351ee42d76207ebc09e6507","family_id":"qf-mod35-spantree-audit","seed":202611046009,"semantic_template":"qf-mod35-spantree-audit/{\"edges\":[[0,1],[1,2],[1,4],[2,3],[3,4]],\"n\":5,\"selected\":[0,1,3,4]}"},"fields":{"title":"A Spanning Tree Uses the Entire Parent Vertex Set","stem":"Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [1, 2], [1, 4], [2, 3], [3, 4]], and distinct parent-edge selection=[0, 1, 3, 4], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (selected_indices, parent_connected, selected_connected, selected_acyclic, spanning_tree).","answer":{"kind":"expression","value":"([0, 1, 3, 4], Yes, Yes, Yes, Yes)","explanation":"Read the selected parent edges and report all connectivity and acyclicity obligations separately. A disconnected candidate parent has no spanning tree; a failed selection does not refute existence for another selection in a connected parent. The exact result, retaining the requested order, is ([0, 1, 3, 4], Yes, Yes, Yes, Yes)."},"skills":["A Spanning Tree Uses the Entire Parent Vertex Set"],"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-20261104-r06-spanning-tree-0010","batch_id":"batch-20261014-advance-20261104-r06-spanning-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"47b783f2f46f8dba6dd2eccb158936cf920dc8fd835defff951b6e549d67e00e","family_id":"qf-mod35-spantree-audit","seed":202611046010,"semantic_template":"qf-mod35-spantree-audit/{\"edges\":[[0,1],[0,2],[1,2],[1,4],[2,3],[3,4]],\"n\":5,\"selected\":[0,2,4,5]}"},"fields":{"title":"A Spanning Tree Uses the Entire Parent Vertex Set","stem":"Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [0, 2], [1, 2], [1, 4], [2, 3], [3, 4]], and distinct parent-edge selection=[0, 2, 4, 5], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (selected_indices, parent_connected, selected_connected, selected_acyclic, spanning_tree).","answer":{"kind":"expression","value":"([0, 2, 4, 5], Yes, Yes, Yes, Yes)","explanation":"Read the selected parent edges and report all connectivity and acyclicity obligations separately. A disconnected candidate parent has no spanning tree; a failed selection does not refute existence for another selection in a connected parent. The exact result, retaining the requested order, is ([0, 2, 4, 5], Yes, Yes, Yes, Yes)."},"skills":["A Spanning Tree Uses the Entire Parent Vertex Set"],"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-20261104-r06-spanning-tree-0011","batch_id":"batch-20261014-advance-20261104-r06-spanning-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"59d4e4ea8a9010ce6dc014b5d70b75b0024df1d1ea1220bc442cdbfbbd2a2f9c","family_id":"qf-mod35-spantree-audit","seed":202611046011,"semantic_template":"qf-mod35-spantree-audit/{\"edges\":[[0,1],[0,3],[1,2],[1,4],[2,3],[3,4]],\"n\":5,\"selected\":[0,2,4,5]}"},"fields":{"title":"A Spanning Tree Uses the Entire Parent Vertex Set","stem":"Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [0, 3], [1, 2], [1, 4], [2, 3], [3, 4]], and distinct parent-edge selection=[0, 2, 4, 5], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (selected_indices, parent_connected, selected_connected, selected_acyclic, spanning_tree).","answer":{"kind":"expression","value":"([0, 2, 4, 5], Yes, Yes, Yes, Yes)","explanation":"Read the selected parent edges and report all connectivity and acyclicity obligations separately. A disconnected candidate parent has no spanning tree; a failed selection does not refute existence for another selection in a connected parent. The exact result, retaining the requested order, is ([0, 2, 4, 5], Yes, Yes, Yes, Yes)."},"skills":["A Spanning Tree Uses the Entire Parent Vertex Set"],"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-20261104-r06-spanning-tree-0012","batch_id":"batch-20261014-advance-20261104-r06-spanning-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f1c496d7fca91fc97f1966f3bfcd50e4c96da38aa57135149d046f16fd082892","family_id":"qf-mod35-spantree-audit","seed":202611046012,"semantic_template":"qf-mod35-spantree-audit/{\"edges\":[[0,1],[0,2],[0,3],[1,2],[1,4],[2,3],[3,4]],\"n\":5,\"selected\":[0,3,5,6]}"},"fields":{"title":"A Spanning Tree Uses the Entire Parent Vertex Set","stem":"Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [0, 2], [0, 3], [1, 2], [1, 4], [2, 3], [3, 4]], and distinct parent-edge selection=[0, 3, 5, 6], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (selected_indices, parent_connected, selected_connected, selected_acyclic, spanning_tree).","answer":{"kind":"expression","value":"([0, 3, 5, 6], Yes, Yes, Yes, Yes)","explanation":"Read the selected parent edges and report all connectivity and acyclicity obligations separately. A disconnected candidate parent has no spanning tree; a failed selection does not refute existence for another selection in a connected parent. The exact result, retaining the requested order, is ([0, 3, 5, 6], Yes, Yes, Yes, Yes)."},"skills":["A Spanning Tree Uses the Entire Parent Vertex Set"],"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-20261104-r06-spanning-tree-0013","batch_id":"batch-20261014-advance-20261104-r06-spanning-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e3e843f668025868f892cd23862df6407eb4fc2157b4a48370c784e18c277f3a","family_id":"qf-mod35-spantree-audit","seed":202611046013,"semantic_template":"qf-mod35-spantree-audit/{\"edges\":[[0,1],[1,2],[1,3],[1,4],[2,3],[3,4]],\"n\":5,\"selected\":[0,1,4,5]}"},"fields":{"title":"A Spanning Tree Uses the Entire Parent Vertex Set","stem":"Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [1, 2], [1, 3], [1, 4], [2, 3], [3, 4]], and distinct parent-edge selection=[0, 1, 4, 5], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (selected_indices, parent_connected, selected_connected, selected_acyclic, spanning_tree).","answer":{"kind":"expression","value":"([0, 1, 4, 5], Yes, Yes, Yes, Yes)","explanation":"Read the selected parent edges and report all connectivity and acyclicity obligations separately. A disconnected candidate parent has no spanning tree; a failed selection does not refute existence for another selection in a connected parent. The exact result, retaining the requested order, is ([0, 1, 4, 5], Yes, Yes, Yes, Yes)."},"skills":["A Spanning Tree Uses the Entire Parent Vertex Set"],"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-20261104-r06-spanning-tree-0014","batch_id":"batch-20261014-advance-20261104-r06-spanning-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c782f5c78a1170e7fa5eafedee42346521b9b305292a7bbf01edb3d5662d52d2","family_id":"qf-mod35-spantree-audit","seed":202611046014,"semantic_template":"qf-mod35-spantree-audit/{\"edges\":[[0,1],[0,2],[1,2],[1,3],[1,4],[2,3],[3,4]],\"n\":5,\"selected\":[0,2,5,6]}"},"fields":{"title":"A Spanning Tree Uses the Entire Parent Vertex Set","stem":"Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [0, 2], [1, 2], [1, 3], [1, 4], [2, 3], [3, 4]], and distinct parent-edge selection=[0, 2, 5, 6], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (selected_indices, parent_connected, selected_connected, selected_acyclic, spanning_tree).","answer":{"kind":"expression","value":"([0, 2, 5, 6], Yes, Yes, Yes, Yes)","explanation":"Read the selected parent edges and report all connectivity and acyclicity obligations separately. A disconnected candidate parent has no spanning tree; a failed selection does not refute existence for another selection in a connected parent. The exact result, retaining the requested order, is ([0, 2, 5, 6], Yes, Yes, Yes, Yes)."},"skills":["A Spanning Tree Uses the Entire Parent Vertex Set"],"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-20261104-r06-spanning-tree-0015","batch_id":"batch-20261014-advance-20261104-r06-spanning-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2aa44925f6a9725e3d5328fd1a8fe04acd8b1846fe5e6fc1d6a2198e1bc61bad","family_id":"qf-mod35-spantree-audit","seed":202611046015,"semantic_template":"qf-mod35-spantree-audit/{\"edges\":[[0,1],[0,3],[1,2],[1,3],[1,4],[2,3],[3,4]],\"n\":5,\"selected\":[0,2,5,6]}"},"fields":{"title":"A Spanning Tree Uses the Entire Parent Vertex Set","stem":"Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [0, 3], [1, 2], [1, 3], [1, 4], [2, 3], [3, 4]], and distinct parent-edge selection=[0, 2, 5, 6], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (selected_indices, parent_connected, selected_connected, selected_acyclic, spanning_tree).","answer":{"kind":"expression","value":"([0, 2, 5, 6], Yes, Yes, Yes, Yes)","explanation":"Read the selected parent edges and report all connectivity and acyclicity obligations separately. A disconnected candidate parent has no spanning tree; a failed selection does not refute existence for another selection in a connected parent. The exact result, retaining the requested order, is ([0, 2, 5, 6], Yes, Yes, Yes, Yes)."},"skills":["A Spanning Tree Uses the Entire Parent Vertex Set"],"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-20261104-r06-spanning-tree-0016","batch_id":"batch-20261014-advance-20261104-r06-spanning-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bee39c1095d2123be095dd2eae8c2511d4ab0bd5e6062e3ed57a2de5ffc7fb24","family_id":"qf-mod35-spantree-audit","seed":202611046016,"semantic_template":"qf-mod35-spantree-audit/{\"edges\":[[0,1],[0,2],[0,3],[1,2],[1,3],[1,4],[2,3],[3,4]],\"n\":5,\"selected\":[0,3,6,7]}"},"fields":{"title":"A Spanning Tree Uses the Entire Parent Vertex Set","stem":"Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [0, 2], [0, 3], [1, 2], [1, 3], [1, 4], [2, 3], [3, 4]], and distinct parent-edge selection=[0, 3, 6, 7], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (selected_indices, parent_connected, selected_connected, selected_acyclic, spanning_tree).","answer":{"kind":"expression","value":"([0, 3, 6, 7], Yes, Yes, Yes, Yes)","explanation":"Read the selected parent edges and report all connectivity and acyclicity obligations separately. A disconnected candidate parent has no spanning tree; a failed selection does not refute existence for another selection in a connected parent. The exact result, retaining the requested order, is ([0, 3, 6, 7], Yes, Yes, Yes, Yes)."},"skills":["A Spanning Tree Uses the Entire Parent Vertex Set"],"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-20261104-r06-spanning-tree-0017","batch_id":"batch-20261014-advance-20261104-r06-spanning-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"be3401f6dcf63b73b3b8a057312ebb8bd8bd2d6bc10160a243d2933624240b63","family_id":"qf-mod35-spantree-audit","seed":202611046017,"semantic_template":"qf-mod35-spantree-audit/{\"edges\":[[0,1],[1,2],[2,3],[2,4],[3,4]],\"n\":5,\"selected\":[0,1,2,4]}"},"fields":{"title":"A Spanning Tree Uses the Entire Parent Vertex Set","stem":"Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [1, 2], [2, 3], [2, 4], [3, 4]], and distinct parent-edge selection=[0, 1, 2, 4], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (selected_indices, parent_connected, selected_connected, selected_acyclic, spanning_tree).","answer":{"kind":"expression","value":"([0, 1, 2, 4], Yes, Yes, Yes, Yes)","explanation":"Read the selected parent edges and report all connectivity and acyclicity obligations separately. A disconnected candidate parent has no spanning tree; a failed selection does not refute existence for another selection in a connected parent. The exact result, retaining the requested order, is ([0, 1, 2, 4], Yes, Yes, Yes, Yes)."},"skills":["A Spanning Tree Uses the Entire Parent Vertex Set"],"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-20261104-r06-spanning-tree-0018","batch_id":"batch-20261014-advance-20261104-r06-spanning-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"79d21b4593371f44d351b847387dd59fa90ac7a3ac66029b22b12daf0a986336","family_id":"qf-mod35-spantree-audit","seed":202611046018,"semantic_template":"qf-mod35-spantree-audit/{\"edges\":[[0,1],[0,2],[1,2],[2,3],[2,4],[3,4]],\"n\":5,\"selected\":[0,2,3,5]}"},"fields":{"title":"A Spanning Tree Uses the Entire Parent Vertex Set","stem":"Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [0, 2], [1, 2], [2, 3], [2, 4], [3, 4]], and distinct parent-edge selection=[0, 2, 3, 5], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (selected_indices, parent_connected, selected_connected, selected_acyclic, spanning_tree).","answer":{"kind":"expression","value":"([0, 2, 3, 5], Yes, Yes, Yes, Yes)","explanation":"Read the selected parent edges and report all connectivity and acyclicity obligations separately. A disconnected candidate parent has no spanning tree; a failed selection does not refute existence for another selection in a connected parent. The exact result, retaining the requested order, is ([0, 2, 3, 5], Yes, Yes, Yes, Yes)."},"skills":["A Spanning Tree Uses the Entire Parent Vertex Set"],"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-20261104-r06-spanning-tree-0019","batch_id":"batch-20261014-advance-20261104-r06-spanning-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b74b986b06bddd1861c36725024c87d34d6144039bd9011b6fc6a8e4ceb264b1","family_id":"qf-mod35-spantree-audit","seed":202611046019,"semantic_template":"qf-mod35-spantree-audit/{\"edges\":[[0,1],[0,3],[1,2],[2,3],[2,4],[3,4]],\"n\":5,\"selected\":[0,2,3,5]}"},"fields":{"title":"A Spanning Tree Uses the Entire Parent Vertex Set","stem":"Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [0, 3], [1, 2], [2, 3], [2, 4], [3, 4]], and distinct parent-edge selection=[0, 2, 3, 5], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (selected_indices, parent_connected, selected_connected, selected_acyclic, spanning_tree).","answer":{"kind":"expression","value":"([0, 2, 3, 5], Yes, Yes, Yes, Yes)","explanation":"Read the selected parent edges and report all connectivity and acyclicity obligations separately. A disconnected candidate parent has no spanning tree; a failed selection does not refute existence for another selection in a connected parent. The exact result, retaining the requested order, is ([0, 2, 3, 5], Yes, Yes, Yes, Yes)."},"skills":["A Spanning Tree Uses the Entire Parent Vertex Set"],"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-20261104-r06-spanning-tree-0020","batch_id":"batch-20261014-advance-20261104-r06-spanning-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"21c7f8016613e0609b3a0a21ae397e0c567820e18f667e091ce8466f9af227d1","family_id":"qf-mod35-spantree-audit","seed":202611046020,"semantic_template":"qf-mod35-spantree-audit/{\"edges\":[[0,1],[0,2],[0,3],[1,2],[2,3],[2,4],[3,4]],\"n\":5,\"selected\":[0,3,4,6]}"},"fields":{"title":"A Spanning Tree Uses the Entire Parent Vertex Set","stem":"Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [0, 2], [0, 3], [1, 2], [2, 3], [2, 4], [3, 4]], and distinct parent-edge selection=[0, 3, 4, 6], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (selected_indices, parent_connected, selected_connected, selected_acyclic, spanning_tree).","answer":{"kind":"expression","value":"([0, 3, 4, 6], Yes, Yes, Yes, Yes)","explanation":"Read the selected parent edges and report all connectivity and acyclicity obligations separately. A disconnected candidate parent has no spanning tree; a failed selection does not refute existence for another selection in a connected parent. The exact result, retaining the requested order, is ([0, 3, 4, 6], Yes, Yes, Yes, Yes)."},"skills":["A Spanning Tree Uses the Entire Parent Vertex Set"],"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-20261104-r06-spanning-tree-0021","batch_id":"batch-20261014-advance-20261104-r06-spanning-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"06260d2828ccf5e067a03c630a82572449eb929826c2f801a0e13460a1e3c397","family_id":"qf-mod35-spantree-audit","seed":202611046021,"semantic_template":"qf-mod35-spantree-audit/{\"edges\":[[0,1],[1,2],[1,3],[2,3],[2,4],[3,4]],\"n\":5,\"selected\":[0,1,3,5]}"},"fields":{"title":"A Spanning Tree Uses the Entire Parent Vertex Set","stem":"Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [1, 2], [1, 3], [2, 3], [2, 4], [3, 4]], and distinct parent-edge selection=[0, 1, 3, 5], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (selected_indices, parent_connected, selected_connected, selected_acyclic, spanning_tree).","answer":{"kind":"expression","value":"([0, 1, 3, 5], Yes, Yes, Yes, Yes)","explanation":"Read the selected parent edges and report all connectivity and acyclicity obligations separately. A disconnected candidate parent has no spanning tree; a failed selection does not refute existence for another selection in a connected parent. The exact result, retaining the requested order, is ([0, 1, 3, 5], Yes, Yes, Yes, Yes)."},"skills":["A Spanning Tree Uses the Entire Parent Vertex Set"],"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-20261104-r06-spanning-tree-0022","batch_id":"batch-20261014-advance-20261104-r06-spanning-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b681061b66f5d5bc01700ecc933dd67d717c53fdb39d6e9f7a190a4ea23034b9","family_id":"qf-mod35-spantree-audit","seed":202611046022,"semantic_template":"qf-mod35-spantree-audit/{\"edges\":[[0,1],[0,2],[1,2],[1,3],[2,3],[2,4],[3,4]],\"n\":5,\"selected\":[0,2,4,6]}"},"fields":{"title":"A Spanning Tree Uses the Entire Parent Vertex Set","stem":"Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [0, 2], [1, 2], [1, 3], [2, 3], [2, 4], [3, 4]], and distinct parent-edge selection=[0, 2, 4, 6], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (selected_indices, parent_connected, selected_connected, selected_acyclic, spanning_tree).","answer":{"kind":"expression","value":"([0, 2, 4, 6], Yes, Yes, Yes, Yes)","explanation":"Read the selected parent edges and report all connectivity and acyclicity obligations separately. A disconnected candidate parent has no spanning tree; a failed selection does not refute existence for another selection in a connected parent. The exact result, retaining the requested order, is ([0, 2, 4, 6], Yes, Yes, Yes, Yes)."},"skills":["A Spanning Tree Uses the Entire Parent Vertex Set"],"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-20261104-r06-spanning-tree-0023","batch_id":"batch-20261014-advance-20261104-r06-spanning-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b87235c164f362fd49d6c61e3a9b5f62b137aba3fcbdc6b6142aeecdf513c509","family_id":"qf-mod35-spantree-audit","seed":202611046023,"semantic_template":"qf-mod35-spantree-audit/{\"edges\":[[0,1],[0,3],[1,2],[1,3],[2,3],[2,4],[3,4]],\"n\":5,\"selected\":[0,2,4,6]}"},"fields":{"title":"A Spanning Tree Uses the Entire Parent Vertex Set","stem":"Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [0, 3], [1, 2], [1, 3], [2, 3], [2, 4], [3, 4]], and distinct parent-edge selection=[0, 2, 4, 6], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (selected_indices, parent_connected, selected_connected, selected_acyclic, spanning_tree).","answer":{"kind":"expression","value":"([0, 2, 4, 6], Yes, Yes, Yes, Yes)","explanation":"Read the selected parent edges and report all connectivity and acyclicity obligations separately. A disconnected candidate parent has no spanning tree; a failed selection does not refute existence for another selection in a connected parent. The exact result, retaining the requested order, is ([0, 2, 4, 6], Yes, Yes, Yes, Yes)."},"skills":["A Spanning Tree Uses the Entire Parent Vertex Set"],"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-20261104-r06-spanning-tree-0024","batch_id":"batch-20261014-advance-20261104-r06-spanning-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a871ef01f0329b6c6769399aca234812b849a99354c12eed8ac4bb50bbf5c409","family_id":"qf-mod35-spantree-audit","seed":202611046024,"semantic_template":"qf-mod35-spantree-audit/{\"edges\":[[0,1],[0,2],[0,3],[1,2],[1,3],[2,3],[2,4],[3,4]],\"n\":5,\"selected\":[0,3,5,7]}"},"fields":{"title":"A Spanning Tree Uses the Entire Parent Vertex Set","stem":"Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [0, 2], [0, 3], [1, 2], [1, 3], [2, 3], [2, 4], [3, 4]], and distinct parent-edge selection=[0, 3, 5, 7], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (selected_indices, parent_connected, selected_connected, selected_acyclic, spanning_tree).","answer":{"kind":"expression","value":"([0, 3, 5, 7], Yes, Yes, Yes, Yes)","explanation":"Read the selected parent edges and report all connectivity and acyclicity obligations separately. A disconnected candidate parent has no spanning tree; a failed selection does not refute existence for another selection in a connected parent. The exact result, retaining the requested order, is ([0, 3, 5, 7], Yes, Yes, Yes, Yes)."},"skills":["A Spanning Tree Uses the Entire Parent Vertex Set"],"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-20261104-r06-spanning-tree-0025","batch_id":"batch-20261014-advance-20261104-r06-spanning-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"47ff0e57d7d5c04c3c7b4869a3c3b313d0d422e96c4d4441d3d64bf372d3ce5d","family_id":"qf-mod35-spantree-audit","seed":202611046025,"semantic_template":"qf-mod35-spantree-audit/{\"edges\":[[0,1],[1,2],[1,4],[2,3],[2,4],[3,4]],\"n\":5,\"selected\":[0,1,3,5]}"},"fields":{"title":"A Spanning Tree Uses the Entire Parent Vertex Set","stem":"Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [1, 2], [1, 4], [2, 3], [2, 4], [3, 4]], and distinct parent-edge selection=[0, 1, 3, 5], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (selected_indices, parent_connected, selected_connected, selected_acyclic, spanning_tree).","answer":{"kind":"expression","value":"([0, 1, 3, 5], Yes, Yes, Yes, Yes)","explanation":"Read the selected parent edges and report all connectivity and acyclicity obligations separately. A disconnected candidate parent has no spanning tree; a failed selection does not refute existence for another selection in a connected parent. The exact result, retaining the requested order, is ([0, 1, 3, 5], Yes, Yes, Yes, Yes)."},"skills":["A Spanning Tree Uses the Entire Parent Vertex Set"],"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-20261104-r06-spanning-tree-0026","batch_id":"batch-20261014-advance-20261104-r06-spanning-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b8e27be1cc67842179ce45c5d465b3a87f2a4a7448f3497c913dccfdf2645b95","family_id":"qf-mod35-spantree-n-minus-one-certificate","seed":202611046026,"semantic_template":"qf-mod35-spantree-n-minus-one-certificate/{\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"n\":5,\"selected\":[0,1,2,3]}"},"fields":{"title":"Counting n−1 Edges Alone Does Not Establish a Tree","stem":"Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [1, 2], [2, 3], [3, 4]], and distinct parent-edge selection=[0, 1, 2, 3], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (vertex_count, selected_edge_count, connected, acyclic, edge_count_is_n_minus_one, spanning_tree, boundary).","answer":{"kind":"expression","value":"(5, 4, Yes, Yes, Yes, Yes, n_minus_one_alone_insufficient)","explanation":"Compare the selected edge count with n−1 and retain both connectivity and acyclicity checks. A cycle on some vertices with an isolated vertex can have n−1 edges and still fail to span as a tree. The exact result, retaining the requested order, is (5, 4, Yes, Yes, Yes, Yes, n_minus_one_alone_insufficient)."},"skills":["Counting n−1 Edges Alone Does Not Establish a Tree"],"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-20261104-r06-spanning-tree-0027","batch_id":"batch-20261014-advance-20261104-r06-spanning-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"aea0ec6d559a43334741f3db6ecd7d82855c0a4814d30c01dc247e1459e6050a","family_id":"qf-mod35-spantree-n-minus-one-certificate","seed":202611046027,"semantic_template":"qf-mod35-spantree-n-minus-one-certificate/{\"edges\":[[0,1],[0,2],[1,2],[2,3],[3,4]],\"n\":5,\"selected\":[0,2,3,4]}"},"fields":{"title":"Counting n−1 Edges Alone Does Not Establish a Tree","stem":"Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [0, 2], [1, 2], [2, 3], [3, 4]], and distinct parent-edge selection=[0, 2, 3, 4], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (vertex_count, selected_edge_count, connected, acyclic, edge_count_is_n_minus_one, spanning_tree, boundary).","answer":{"kind":"expression","value":"(5, 4, Yes, Yes, Yes, Yes, n_minus_one_alone_insufficient)","explanation":"Compare the selected edge count with n−1 and retain both connectivity and acyclicity checks. A cycle on some vertices with an isolated vertex can have n−1 edges and still fail to span as a tree. The exact result, retaining the requested order, is (5, 4, Yes, Yes, Yes, Yes, n_minus_one_alone_insufficient)."},"skills":["Counting n−1 Edges Alone Does Not Establish a Tree"],"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-20261104-r06-spanning-tree-0028","batch_id":"batch-20261014-advance-20261104-r06-spanning-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2a69e26dde70ef7c499e30197aaceaeb8876142a6d53966bf1cddcc5ff091b94","family_id":"qf-mod35-spantree-n-minus-one-certificate","seed":202611046028,"semantic_template":"qf-mod35-spantree-n-minus-one-certificate/{\"edges\":[[0,1],[0,3],[1,2],[2,3],[3,4]],\"n\":5,\"selected\":[0,2,3,4]}"},"fields":{"title":"Counting n−1 Edges Alone Does Not Establish a Tree","stem":"Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [0, 3], [1, 2], [2, 3], [3, 4]], and distinct parent-edge selection=[0, 2, 3, 4], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (vertex_count, selected_edge_count, connected, acyclic, edge_count_is_n_minus_one, spanning_tree, boundary).","answer":{"kind":"expression","value":"(5, 4, Yes, Yes, Yes, Yes, n_minus_one_alone_insufficient)","explanation":"Compare the selected edge count with n−1 and retain both connectivity and acyclicity checks. A cycle on some vertices with an isolated vertex can have n−1 edges and still fail to span as a tree. The exact result, retaining the requested order, is (5, 4, Yes, Yes, Yes, Yes, n_minus_one_alone_insufficient)."},"skills":["Counting n−1 Edges Alone Does Not Establish a Tree"],"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-20261104-r06-spanning-tree-0029","batch_id":"batch-20261014-advance-20261104-r06-spanning-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"02c34975cadc7b940c291f509de4b4df9fdd0ba2d4cdbccd557373a3c712ccf0","family_id":"qf-mod35-spantree-n-minus-one-certificate","seed":202611046029,"semantic_template":"qf-mod35-spantree-n-minus-one-certificate/{\"edges\":[[0,1],[0,2],[0,3],[1,2],[2,3],[3,4]],\"n\":5,\"selected\":[0,3,4,5]}"},"fields":{"title":"Counting n−1 Edges Alone Does Not Establish a Tree","stem":"Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [0, 2], [0, 3], [1, 2], [2, 3], [3, 4]], and distinct parent-edge selection=[0, 3, 4, 5], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (vertex_count, selected_edge_count, connected, acyclic, edge_count_is_n_minus_one, spanning_tree, boundary).","answer":{"kind":"expression","value":"(5, 4, Yes, Yes, Yes, Yes, n_minus_one_alone_insufficient)","explanation":"Compare the selected edge count with n−1 and retain both connectivity and acyclicity checks. A cycle on some vertices with an isolated vertex can have n−1 edges and still fail to span as a tree. The exact result, retaining the requested order, is (5, 4, Yes, Yes, Yes, Yes, n_minus_one_alone_insufficient)."},"skills":["Counting n−1 Edges Alone Does Not Establish a Tree"],"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-20261104-r06-spanning-tree-0030","batch_id":"batch-20261014-advance-20261104-r06-spanning-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9009f3143b4e6c7ed79c8e619ed79d67869cd6c8c1aaa56bd3865e4b2311daa5","family_id":"qf-mod35-spantree-n-minus-one-certificate","seed":202611046030,"semantic_template":"qf-mod35-spantree-n-minus-one-certificate/{\"edges\":[[0,1],[1,2],[1,3],[2,3],[3,4]],\"n\":5,\"selected\":[0,1,3,4]}"},"fields":{"title":"Counting n−1 Edges Alone Does Not Establish a Tree","stem":"Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [1, 2], [1, 3], [2, 3], [3, 4]], and distinct parent-edge selection=[0, 1, 3, 4], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (vertex_count, selected_edge_count, connected, acyclic, edge_count_is_n_minus_one, spanning_tree, boundary).","answer":{"kind":"expression","value":"(5, 4, Yes, Yes, Yes, Yes, n_minus_one_alone_insufficient)","explanation":"Compare the selected edge count with n−1 and retain both connectivity and acyclicity checks. A cycle on some vertices with an isolated vertex can have n−1 edges and still fail to span as a tree. The exact result, retaining the requested order, is (5, 4, Yes, Yes, Yes, Yes, n_minus_one_alone_insufficient)."},"skills":["Counting n−1 Edges Alone Does Not Establish a Tree"],"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-20261104-r06-spanning-tree-0031","batch_id":"batch-20261014-advance-20261104-r06-spanning-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"60175edd5dd87e8811cd3d9fa572ad268ece14ec39ba06914dd460e4b360feba","family_id":"qf-mod35-spantree-n-minus-one-certificate","seed":202611046031,"semantic_template":"qf-mod35-spantree-n-minus-one-certificate/{\"edges\":[[0,1],[0,2],[1,2],[1,3],[2,3],[3,4]],\"n\":5,\"selected\":[0,2,4,5]}"},"fields":{"title":"Counting n−1 Edges Alone Does Not Establish a Tree","stem":"Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [0, 2], [1, 2], [1, 3], [2, 3], [3, 4]], and distinct parent-edge selection=[0, 2, 4, 5], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (vertex_count, selected_edge_count, connected, acyclic, edge_count_is_n_minus_one, spanning_tree, boundary).","answer":{"kind":"expression","value":"(5, 4, Yes, Yes, Yes, Yes, n_minus_one_alone_insufficient)","explanation":"Compare the selected edge count with n−1 and retain both connectivity and acyclicity checks. A cycle on some vertices with an isolated vertex can have n−1 edges and still fail to span as a tree. The exact result, retaining the requested order, is (5, 4, Yes, Yes, Yes, Yes, n_minus_one_alone_insufficient)."},"skills":["Counting n−1 Edges Alone Does Not Establish a Tree"],"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-20261104-r06-spanning-tree-0032","batch_id":"batch-20261014-advance-20261104-r06-spanning-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f8d65971c272b155c11e3b156d0acb49847f9a0a095ae89c4f0f34a89998d4f2","family_id":"qf-mod35-spantree-n-minus-one-certificate","seed":202611046032,"semantic_template":"qf-mod35-spantree-n-minus-one-certificate/{\"edges\":[[0,1],[0,3],[1,2],[1,3],[2,3],[3,4]],\"n\":5,\"selected\":[0,2,4,5]}"},"fields":{"title":"Counting n−1 Edges Alone Does Not Establish a Tree","stem":"Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [0, 3], [1, 2], [1, 3], [2, 3], [3, 4]], and distinct parent-edge selection=[0, 2, 4, 5], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (vertex_count, selected_edge_count, connected, acyclic, edge_count_is_n_minus_one, spanning_tree, boundary).","answer":{"kind":"expression","value":"(5, 4, Yes, Yes, Yes, Yes, n_minus_one_alone_insufficient)","explanation":"Compare the selected edge count with n−1 and retain both connectivity and acyclicity checks. A cycle on some vertices with an isolated vertex can have n−1 edges and still fail to span as a tree. The exact result, retaining the requested order, is (5, 4, Yes, Yes, Yes, Yes, n_minus_one_alone_insufficient)."},"skills":["Counting n−1 Edges Alone Does Not Establish a Tree"],"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-20261104-r06-spanning-tree-0033","batch_id":"batch-20261014-advance-20261104-r06-spanning-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"042f877edae400153b9c9305261af116a0df2e545692813ad30f0b877d70067d","family_id":"qf-mod35-spantree-n-minus-one-certificate","seed":202611046033,"semantic_template":"qf-mod35-spantree-n-minus-one-certificate/{\"edges\":[[0,1],[0,2],[0,3],[1,2],[1,3],[2,3],[3,4]],\"n\":5,\"selected\":[0,3,5,6]}"},"fields":{"title":"Counting n−1 Edges Alone Does Not Establish a Tree","stem":"Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [0, 2], [0, 3], [1, 2], [1, 3], [2, 3], [3, 4]], and distinct parent-edge selection=[0, 3, 5, 6], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (vertex_count, selected_edge_count, connected, acyclic, edge_count_is_n_minus_one, spanning_tree, boundary).","answer":{"kind":"expression","value":"(5, 4, Yes, Yes, Yes, Yes, n_minus_one_alone_insufficient)","explanation":"Compare the selected edge count with n−1 and retain both connectivity and acyclicity checks. A cycle on some vertices with an isolated vertex can have n−1 edges and still fail to span as a tree. The exact result, retaining the requested order, is (5, 4, Yes, Yes, Yes, Yes, n_minus_one_alone_insufficient)."},"skills":["Counting n−1 Edges Alone Does Not Establish a Tree"],"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-20261104-r06-spanning-tree-0034","batch_id":"batch-20261014-advance-20261104-r06-spanning-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"35940e36490f497855a81ab69cc72616302ac6afe2068e3a865f33bca7f5d6ca","family_id":"qf-mod35-spantree-n-minus-one-certificate","seed":202611046034,"semantic_template":"qf-mod35-spantree-n-minus-one-certificate/{\"edges\":[[0,1],[1,2],[1,4],[2,3],[3,4]],\"n\":5,\"selected\":[0,1,3,4]}"},"fields":{"title":"Counting n−1 Edges Alone Does Not Establish a Tree","stem":"Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [1, 2], [1, 4], [2, 3], [3, 4]], and distinct parent-edge selection=[0, 1, 3, 4], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (vertex_count, selected_edge_count, connected, acyclic, edge_count_is_n_minus_one, spanning_tree, boundary).","answer":{"kind":"expression","value":"(5, 4, Yes, Yes, Yes, Yes, n_minus_one_alone_insufficient)","explanation":"Compare the selected edge count with n−1 and retain both connectivity and acyclicity checks. A cycle on some vertices with an isolated vertex can have n−1 edges and still fail to span as a tree. The exact result, retaining the requested order, is (5, 4, Yes, Yes, Yes, Yes, n_minus_one_alone_insufficient)."},"skills":["Counting n−1 Edges Alone Does Not Establish a Tree"],"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-20261104-r06-spanning-tree-0035","batch_id":"batch-20261014-advance-20261104-r06-spanning-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ba09a581cee5aae30462010403aa0042ead4a3739641283eca91fd51dcb2ab2a","family_id":"qf-mod35-spantree-n-minus-one-certificate","seed":202611046035,"semantic_template":"qf-mod35-spantree-n-minus-one-certificate/{\"edges\":[[0,1],[0,2],[1,2],[1,4],[2,3],[3,4]],\"n\":5,\"selected\":[0,2,4,5]}"},"fields":{"title":"Counting n−1 Edges Alone Does Not Establish a Tree","stem":"Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [0, 2], [1, 2], [1, 4], [2, 3], [3, 4]], and distinct parent-edge selection=[0, 2, 4, 5], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (vertex_count, selected_edge_count, connected, acyclic, edge_count_is_n_minus_one, spanning_tree, boundary).","answer":{"kind":"expression","value":"(5, 4, Yes, Yes, Yes, Yes, n_minus_one_alone_insufficient)","explanation":"Compare the selected edge count with n−1 and retain both connectivity and acyclicity checks. A cycle on some vertices with an isolated vertex can have n−1 edges and still fail to span as a tree. The exact result, retaining the requested order, is (5, 4, Yes, Yes, Yes, Yes, n_minus_one_alone_insufficient)."},"skills":["Counting n−1 Edges Alone Does Not Establish a Tree"],"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-20261104-r06-spanning-tree-0036","batch_id":"batch-20261014-advance-20261104-r06-spanning-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"30ff16dd0535e020912afbc561635738ce33ae570df7716b0acb80693dc5c95e","family_id":"qf-mod35-spantree-n-minus-one-certificate","seed":202611046036,"semantic_template":"qf-mod35-spantree-n-minus-one-certificate/{\"edges\":[[0,1],[0,3],[1,2],[1,4],[2,3],[3,4]],\"n\":5,\"selected\":[0,2,4,5]}"},"fields":{"title":"Counting n−1 Edges Alone Does Not Establish a Tree","stem":"Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [0, 3], [1, 2], [1, 4], [2, 3], [3, 4]], and distinct parent-edge selection=[0, 2, 4, 5], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (vertex_count, selected_edge_count, connected, acyclic, edge_count_is_n_minus_one, spanning_tree, boundary).","answer":{"kind":"expression","value":"(5, 4, Yes, Yes, Yes, Yes, n_minus_one_alone_insufficient)","explanation":"Compare the selected edge count with n−1 and retain both connectivity and acyclicity checks. A cycle on some vertices with an isolated vertex can have n−1 edges and still fail to span as a tree. The exact result, retaining the requested order, is (5, 4, Yes, Yes, Yes, Yes, n_minus_one_alone_insufficient)."},"skills":["Counting n−1 Edges Alone Does Not Establish a Tree"],"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-20261104-r06-spanning-tree-0037","batch_id":"batch-20261014-advance-20261104-r06-spanning-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"57bf35a28c01948eaeaf283d0a523bb3b36f77a6e2a9fed74924b961058e9548","family_id":"qf-mod35-spantree-n-minus-one-certificate","seed":202611046037,"semantic_template":"qf-mod35-spantree-n-minus-one-certificate/{\"edges\":[[0,1],[0,2],[0,3],[1,2],[1,4],[2,3],[3,4]],\"n\":5,\"selected\":[0,3,5,6]}"},"fields":{"title":"Counting n−1 Edges Alone Does Not Establish a Tree","stem":"Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [0, 2], [0, 3], [1, 2], [1, 4], [2, 3], [3, 4]], and distinct parent-edge selection=[0, 3, 5, 6], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (vertex_count, selected_edge_count, connected, acyclic, edge_count_is_n_minus_one, spanning_tree, boundary).","answer":{"kind":"expression","value":"(5, 4, Yes, Yes, Yes, Yes, n_minus_one_alone_insufficient)","explanation":"Compare the selected edge count with n−1 and retain both connectivity and acyclicity checks. A cycle on some vertices with an isolated vertex can have n−1 edges and still fail to span as a tree. The exact result, retaining the requested order, is (5, 4, Yes, Yes, Yes, Yes, n_minus_one_alone_insufficient)."},"skills":["Counting n−1 Edges Alone Does Not Establish a Tree"],"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-20261104-r06-spanning-tree-0038","batch_id":"batch-20261014-advance-20261104-r06-spanning-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c83e0ee2a8a956289cdc71917c2d6d45b208ca718e9fabc4eb80a09d55e6b9dc","family_id":"qf-mod35-spantree-n-minus-one-certificate","seed":202611046038,"semantic_template":"qf-mod35-spantree-n-minus-one-certificate/{\"edges\":[[0,1],[1,2],[1,3],[1,4],[2,3],[3,4]],\"n\":5,\"selected\":[0,1,4,5]}"},"fields":{"title":"Counting n−1 Edges Alone Does Not Establish a Tree","stem":"Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [1, 2], [1, 3], [1, 4], [2, 3], [3, 4]], and distinct parent-edge selection=[0, 1, 4, 5], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (vertex_count, selected_edge_count, connected, acyclic, edge_count_is_n_minus_one, spanning_tree, boundary).","answer":{"kind":"expression","value":"(5, 4, Yes, Yes, Yes, Yes, n_minus_one_alone_insufficient)","explanation":"Compare the selected edge count with n−1 and retain both connectivity and acyclicity checks. A cycle on some vertices with an isolated vertex can have n−1 edges and still fail to span as a tree. The exact result, retaining the requested order, is (5, 4, Yes, Yes, Yes, Yes, n_minus_one_alone_insufficient)."},"skills":["Counting n−1 Edges Alone Does Not Establish a Tree"],"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-20261104-r06-spanning-tree-0039","batch_id":"batch-20261014-advance-20261104-r06-spanning-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1eff1d3d0da335eed9cea11ffd85779bed41390dc61ff3c989d5fb4a4780f8bd","family_id":"qf-mod35-spantree-n-minus-one-certificate","seed":202611046039,"semantic_template":"qf-mod35-spantree-n-minus-one-certificate/{\"edges\":[[0,1],[0,2],[1,2],[1,3],[1,4],[2,3],[3,4]],\"n\":5,\"selected\":[0,2,5,6]}"},"fields":{"title":"Counting n−1 Edges Alone Does Not Establish a Tree","stem":"Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [0, 2], [1, 2], [1, 3], [1, 4], [2, 3], [3, 4]], and distinct parent-edge selection=[0, 2, 5, 6], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (vertex_count, selected_edge_count, connected, acyclic, edge_count_is_n_minus_one, spanning_tree, boundary).","answer":{"kind":"expression","value":"(5, 4, Yes, Yes, Yes, Yes, n_minus_one_alone_insufficient)","explanation":"Compare the selected edge count with n−1 and retain both connectivity and acyclicity checks. A cycle on some vertices with an isolated vertex can have n−1 edges and still fail to span as a tree. The exact result, retaining the requested order, is (5, 4, Yes, Yes, Yes, Yes, n_minus_one_alone_insufficient)."},"skills":["Counting n−1 Edges Alone Does Not Establish a Tree"],"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-20261104-r06-spanning-tree-0040","batch_id":"batch-20261014-advance-20261104-r06-spanning-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f8c98c4a85da5f3f624ab295ef06aeb4c4341cbf5725515271df034ad533f391","family_id":"qf-mod35-spantree-n-minus-one-certificate","seed":202611046040,"semantic_template":"qf-mod35-spantree-n-minus-one-certificate/{\"edges\":[[0,1],[0,3],[1,2],[1,3],[1,4],[2,3],[3,4]],\"n\":5,\"selected\":[0,2,5,6]}"},"fields":{"title":"Counting n−1 Edges Alone Does Not Establish a Tree","stem":"Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [0, 3], [1, 2], [1, 3], [1, 4], [2, 3], [3, 4]], and distinct parent-edge selection=[0, 2, 5, 6], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (vertex_count, selected_edge_count, connected, acyclic, edge_count_is_n_minus_one, spanning_tree, boundary).","answer":{"kind":"expression","value":"(5, 4, Yes, Yes, Yes, Yes, n_minus_one_alone_insufficient)","explanation":"Compare the selected edge count with n−1 and retain both connectivity and acyclicity checks. A cycle on some vertices with an isolated vertex can have n−1 edges and still fail to span as a tree. The exact result, retaining the requested order, is (5, 4, Yes, Yes, Yes, Yes, n_minus_one_alone_insufficient)."},"skills":["Counting n−1 Edges Alone Does Not Establish a Tree"],"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-20261104-r06-spanning-tree-0041","batch_id":"batch-20261014-advance-20261104-r06-spanning-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1a1e30a742d501d4e30b55ca928c9ff89725ec0ffdc48fc154ef3a2a3f3aea03","family_id":"qf-mod35-spantree-n-minus-one-certificate","seed":202611046041,"semantic_template":"qf-mod35-spantree-n-minus-one-certificate/{\"edges\":[[0,1],[0,2],[0,3],[1,2],[1,3],[1,4],[2,3],[3,4]],\"n\":5,\"selected\":[0,3,6,7]}"},"fields":{"title":"Counting n−1 Edges Alone Does Not Establish a Tree","stem":"Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [0, 2], [0, 3], [1, 2], [1, 3], [1, 4], [2, 3], [3, 4]], and distinct parent-edge selection=[0, 3, 6, 7], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (vertex_count, selected_edge_count, connected, acyclic, edge_count_is_n_minus_one, spanning_tree, boundary).","answer":{"kind":"expression","value":"(5, 4, Yes, Yes, Yes, Yes, n_minus_one_alone_insufficient)","explanation":"Compare the selected edge count with n−1 and retain both connectivity and acyclicity checks. A cycle on some vertices with an isolated vertex can have n−1 edges and still fail to span as a tree. The exact result, retaining the requested order, is (5, 4, Yes, Yes, Yes, Yes, n_minus_one_alone_insufficient)."},"skills":["Counting n−1 Edges Alone Does Not Establish a Tree"],"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-20261104-r06-spanning-tree-0042","batch_id":"batch-20261014-advance-20261104-r06-spanning-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"12d62f1d497832d9e52d627c6604c42a02588778bcd9026fc6f30e30e2d4f7a2","family_id":"qf-mod35-spantree-n-minus-one-certificate","seed":202611046042,"semantic_template":"qf-mod35-spantree-n-minus-one-certificate/{\"edges\":[[0,1],[1,2],[2,3],[2,4],[3,4]],\"n\":5,\"selected\":[0,1,2,4]}"},"fields":{"title":"Counting n−1 Edges Alone Does Not Establish a Tree","stem":"Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [1, 2], [2, 3], [2, 4], [3, 4]], and distinct parent-edge selection=[0, 1, 2, 4], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (vertex_count, selected_edge_count, connected, acyclic, edge_count_is_n_minus_one, spanning_tree, boundary).","answer":{"kind":"expression","value":"(5, 4, Yes, Yes, Yes, Yes, n_minus_one_alone_insufficient)","explanation":"Compare the selected edge count with n−1 and retain both connectivity and acyclicity checks. A cycle on some vertices with an isolated vertex can have n−1 edges and still fail to span as a tree. The exact result, retaining the requested order, is (5, 4, Yes, Yes, Yes, Yes, n_minus_one_alone_insufficient)."},"skills":["Counting n−1 Edges Alone Does Not Establish a Tree"],"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-20261104-r06-spanning-tree-0043","batch_id":"batch-20261014-advance-20261104-r06-spanning-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"670c2adaadaa1ba1a79a5dfe24b8a252838c3267f34309c96e663312dac22f38","family_id":"qf-mod35-spantree-n-minus-one-certificate","seed":202611046043,"semantic_template":"qf-mod35-spantree-n-minus-one-certificate/{\"edges\":[[0,1],[0,2],[1,2],[2,3],[2,4],[3,4]],\"n\":5,\"selected\":[0,2,3,5]}"},"fields":{"title":"Counting n−1 Edges Alone Does Not Establish a Tree","stem":"Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [0, 2], [1, 2], [2, 3], [2, 4], [3, 4]], and distinct parent-edge selection=[0, 2, 3, 5], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (vertex_count, selected_edge_count, connected, acyclic, edge_count_is_n_minus_one, spanning_tree, boundary).","answer":{"kind":"expression","value":"(5, 4, Yes, Yes, Yes, Yes, n_minus_one_alone_insufficient)","explanation":"Compare the selected edge count with n−1 and retain both connectivity and acyclicity checks. A cycle on some vertices with an isolated vertex can have n−1 edges and still fail to span as a tree. The exact result, retaining the requested order, is (5, 4, Yes, Yes, Yes, Yes, n_minus_one_alone_insufficient)."},"skills":["Counting n−1 Edges Alone Does Not Establish a Tree"],"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-20261104-r06-spanning-tree-0044","batch_id":"batch-20261014-advance-20261104-r06-spanning-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3143bf8672fb3c1f047cfc2965869d13d71b9945a579418f42eb0fd4bd9a7d59","family_id":"qf-mod35-spantree-n-minus-one-certificate","seed":202611046044,"semantic_template":"qf-mod35-spantree-n-minus-one-certificate/{\"edges\":[[0,1],[0,3],[1,2],[2,3],[2,4],[3,4]],\"n\":5,\"selected\":[0,2,3,5]}"},"fields":{"title":"Counting n−1 Edges Alone Does Not Establish a Tree","stem":"Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [0, 3], [1, 2], [2, 3], [2, 4], [3, 4]], and distinct parent-edge selection=[0, 2, 3, 5], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (vertex_count, selected_edge_count, connected, acyclic, edge_count_is_n_minus_one, spanning_tree, boundary).","answer":{"kind":"expression","value":"(5, 4, Yes, Yes, Yes, Yes, n_minus_one_alone_insufficient)","explanation":"Compare the selected edge count with n−1 and retain both connectivity and acyclicity checks. A cycle on some vertices with an isolated vertex can have n−1 edges and still fail to span as a tree. The exact result, retaining the requested order, is (5, 4, Yes, Yes, Yes, Yes, n_minus_one_alone_insufficient)."},"skills":["Counting n−1 Edges Alone Does Not Establish a Tree"],"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-20261104-r06-spanning-tree-0045","batch_id":"batch-20261014-advance-20261104-r06-spanning-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cc9a3357df760d807260d93f09cfe3eecbed683970edc2aa339369fd8df869a3","family_id":"qf-mod35-spantree-n-minus-one-certificate","seed":202611046045,"semantic_template":"qf-mod35-spantree-n-minus-one-certificate/{\"edges\":[[0,1],[0,2],[0,3],[1,2],[2,3],[2,4],[3,4]],\"n\":5,\"selected\":[0,3,4,6]}"},"fields":{"title":"Counting n−1 Edges Alone Does Not Establish a Tree","stem":"Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [0, 2], [0, 3], [1, 2], [2, 3], [2, 4], [3, 4]], and distinct parent-edge selection=[0, 3, 4, 6], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (vertex_count, selected_edge_count, connected, acyclic, edge_count_is_n_minus_one, spanning_tree, boundary).","answer":{"kind":"expression","value":"(5, 4, Yes, Yes, Yes, Yes, n_minus_one_alone_insufficient)","explanation":"Compare the selected edge count with n−1 and retain both connectivity and acyclicity checks. A cycle on some vertices with an isolated vertex can have n−1 edges and still fail to span as a tree. The exact result, retaining the requested order, is (5, 4, Yes, Yes, Yes, Yes, n_minus_one_alone_insufficient)."},"skills":["Counting n−1 Edges Alone Does Not Establish a Tree"],"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-20261104-r06-spanning-tree-0046","batch_id":"batch-20261014-advance-20261104-r06-spanning-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f05d956343a16939a0665e77ed449ae0d4633d3c7de37346dfb73fb2b1a695d8","family_id":"qf-mod35-spantree-n-minus-one-certificate","seed":202611046046,"semantic_template":"qf-mod35-spantree-n-minus-one-certificate/{\"edges\":[[0,1],[1,2],[1,3],[2,3],[2,4],[3,4]],\"n\":5,\"selected\":[0,1,3,5]}"},"fields":{"title":"Counting n−1 Edges Alone Does Not Establish a Tree","stem":"Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [1, 2], [1, 3], [2, 3], [2, 4], [3, 4]], and distinct parent-edge selection=[0, 1, 3, 5], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (vertex_count, selected_edge_count, connected, acyclic, edge_count_is_n_minus_one, spanning_tree, boundary).","answer":{"kind":"expression","value":"(5, 4, Yes, Yes, Yes, Yes, n_minus_one_alone_insufficient)","explanation":"Compare the selected edge count with n−1 and retain both connectivity and acyclicity checks. A cycle on some vertices with an isolated vertex can have n−1 edges and still fail to span as a tree. The exact result, retaining the requested order, is (5, 4, Yes, Yes, Yes, Yes, n_minus_one_alone_insufficient)."},"skills":["Counting n−1 Edges Alone Does Not Establish a Tree"],"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-20261104-r06-spanning-tree-0047","batch_id":"batch-20261014-advance-20261104-r06-spanning-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"79fa976fbf0c3ad302d5d93387414bb5360aa746e719a32173875ac71a9d340f","family_id":"qf-mod35-spantree-n-minus-one-certificate","seed":202611046047,"semantic_template":"qf-mod35-spantree-n-minus-one-certificate/{\"edges\":[[0,1],[0,2],[1,2],[1,3],[2,3],[2,4],[3,4]],\"n\":5,\"selected\":[0,2,4,6]}"},"fields":{"title":"Counting n−1 Edges Alone Does Not Establish a Tree","stem":"Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [0, 2], [1, 2], [1, 3], [2, 3], [2, 4], [3, 4]], and distinct parent-edge selection=[0, 2, 4, 6], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (vertex_count, selected_edge_count, connected, acyclic, edge_count_is_n_minus_one, spanning_tree, boundary).","answer":{"kind":"expression","value":"(5, 4, Yes, Yes, Yes, Yes, n_minus_one_alone_insufficient)","explanation":"Compare the selected edge count with n−1 and retain both connectivity and acyclicity checks. A cycle on some vertices with an isolated vertex can have n−1 edges and still fail to span as a tree. The exact result, retaining the requested order, is (5, 4, Yes, Yes, Yes, Yes, n_minus_one_alone_insufficient)."},"skills":["Counting n−1 Edges Alone Does Not Establish a Tree"],"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-20261104-r06-spanning-tree-0048","batch_id":"batch-20261014-advance-20261104-r06-spanning-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"265a2c612d833d215f9cb0fcb28bf92844322e2f4d962b7bd004708728ddb5b4","family_id":"qf-mod35-spantree-n-minus-one-certificate","seed":202611046048,"semantic_template":"qf-mod35-spantree-n-minus-one-certificate/{\"edges\":[[0,1],[0,3],[1,2],[1,3],[2,3],[2,4],[3,4]],\"n\":5,\"selected\":[0,2,4,6]}"},"fields":{"title":"Counting n−1 Edges Alone Does Not Establish a Tree","stem":"Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [0, 3], [1, 2], [1, 3], [2, 3], [2, 4], [3, 4]], and distinct parent-edge selection=[0, 2, 4, 6], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (vertex_count, selected_edge_count, connected, acyclic, edge_count_is_n_minus_one, spanning_tree, boundary).","answer":{"kind":"expression","value":"(5, 4, Yes, Yes, Yes, Yes, n_minus_one_alone_insufficient)","explanation":"Compare the selected edge count with n−1 and retain both connectivity and acyclicity checks. A cycle on some vertices with an isolated vertex can have n−1 edges and still fail to span as a tree. The exact result, retaining the requested order, is (5, 4, Yes, Yes, Yes, Yes, n_minus_one_alone_insufficient)."},"skills":["Counting n−1 Edges Alone Does Not Establish a Tree"],"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-20261104-r06-spanning-tree-0049","batch_id":"batch-20261014-advance-20261104-r06-spanning-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b32bb4d07b421d9b4fcee38fe99f141e01e77f1cf3de1f40dc847de6b8905f17","family_id":"qf-mod35-spantree-n-minus-one-certificate","seed":202611046049,"semantic_template":"qf-mod35-spantree-n-minus-one-certificate/{\"edges\":[[0,1],[0,2],[0,3],[1,2],[1,3],[2,3],[2,4],[3,4]],\"n\":5,\"selected\":[0,3,5,7]}"},"fields":{"title":"Counting n−1 Edges Alone Does Not Establish a Tree","stem":"Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [0, 2], [0, 3], [1, 2], [1, 3], [2, 3], [2, 4], [3, 4]], and distinct parent-edge selection=[0, 3, 5, 7], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (vertex_count, selected_edge_count, connected, acyclic, edge_count_is_n_minus_one, spanning_tree, boundary).","answer":{"kind":"expression","value":"(5, 4, Yes, Yes, Yes, Yes, n_minus_one_alone_insufficient)","explanation":"Compare the selected edge count with n−1 and retain both connectivity and acyclicity checks. A cycle on some vertices with an isolated vertex can have n−1 edges and still fail to span as a tree. The exact result, retaining the requested order, is (5, 4, Yes, Yes, Yes, Yes, n_minus_one_alone_insufficient)."},"skills":["Counting n−1 Edges Alone Does Not Establish a Tree"],"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-20261104-r06-spanning-tree-0050","batch_id":"batch-20261014-advance-20261104-r06-spanning-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8ab95f2d587347e8a93e898630c8f28e0e4002150cd59e19c6eb5ec90dec3d4a","family_id":"qf-mod35-spantree-n-minus-one-certificate","seed":202611046050,"semantic_template":"qf-mod35-spantree-n-minus-one-certificate/{\"edges\":[[0,1],[1,2],[1,4],[2,3],[2,4],[3,4]],\"n\":5,\"selected\":[0,1,3,5]}"},"fields":{"title":"Counting n−1 Edges Alone Does Not Establish a Tree","stem":"Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [1, 2], [1, 4], [2, 3], [2, 4], [3, 4]], and distinct parent-edge selection=[0, 1, 3, 5], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (vertex_count, selected_edge_count, connected, acyclic, edge_count_is_n_minus_one, spanning_tree, boundary).","answer":{"kind":"expression","value":"(5, 4, Yes, Yes, Yes, Yes, n_minus_one_alone_insufficient)","explanation":"Compare the selected edge count with n−1 and retain both connectivity and acyclicity checks. A cycle on some vertices with an isolated vertex can have n−1 edges and still fail to span as a tree. The exact result, retaining the requested order, is (5, 4, Yes, Yes, Yes, Yes, n_minus_one_alone_insufficient)."},"skills":["Counting n−1 Edges Alone Does Not Establish a Tree"],"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-20261104-r06-spanning-tree-0051","batch_id":"batch-20261014-advance-20261104-r06-spanning-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c93712cafd793bf24b41337f83a522777b97ea1358984ddb6edeb2acbfbb2bf3","family_id":"qf-mod35-spantree-one-vertex","seed":202611046051,"semantic_template":"qf-mod35-spantree-one-vertex/{\"edges\":[],\"n\":1,\"selected\":[]}"},"fields":{"title":"Recognize the Single-Vertex Empty-Edge Tree without Assuming It","stem":"Use the complete finite undirected loop-free simple parent graph on Fin(1)={0,…,0}, canonical sorted edges=[], and distinct parent-edge selection=[], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (vertex_count, selected_indices, spanning_tree, one_vertex_empty_selection).","answer":{"kind":"expression","value":"(1, [], Yes, Yes)","explanation":"The final flag checks whether the actual carrier has one vertex and the selection is empty. That case is a valid spanning tree. For a larger carrier the flag is No even when a nonempty selection forms a spanning tree; it is not a mislabeled single-vertex theorem. The exact result, retaining the requested order, is (1, [], Yes, Yes)."},"skills":["Recognize the Single-Vertex Empty-Edge Tree without Assuming It"],"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-20261104-r06-spanning-tree-0052","batch_id":"batch-20261014-advance-20261104-r06-spanning-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"91a415c07765d8cb99033ff66a0908153dfcdf8b03a02a82e321bf2108729f04","family_id":"qf-mod35-spantree-one-vertex","seed":202611046052,"semantic_template":"qf-mod35-spantree-one-vertex/{\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"n\":5,\"selected\":[0,1,2,3]}"},"fields":{"title":"Recognize the Single-Vertex Empty-Edge Tree without Assuming It","stem":"Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [1, 2], [2, 3], [3, 4]], and distinct parent-edge selection=[0, 1, 2, 3], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (vertex_count, selected_indices, spanning_tree, one_vertex_empty_selection).","answer":{"kind":"expression","value":"(5, [0, 1, 2, 3], Yes, No)","explanation":"The final flag checks whether the actual carrier has one vertex and the selection is empty. That case is a valid spanning tree. For a larger carrier the flag is No even when a nonempty selection forms a spanning tree; it is not a mislabeled single-vertex theorem. The exact result, retaining the requested order, is (5, [0, 1, 2, 3], Yes, No)."},"skills":["Recognize the Single-Vertex Empty-Edge Tree without Assuming It"],"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-20261104-r06-spanning-tree-0053","batch_id":"batch-20261014-advance-20261104-r06-spanning-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"32f93b59d49591f727900e98fc25cf3183cdc0e7f4f4145d1e13bd6c1f927931","family_id":"qf-mod35-spantree-one-vertex","seed":202611046053,"semantic_template":"qf-mod35-spantree-one-vertex/{\"edges\":[[0,1],[0,2],[1,2],[2,3],[3,4]],\"n\":5,\"selected\":[0,2,3,4]}"},"fields":{"title":"Recognize the Single-Vertex Empty-Edge Tree without Assuming It","stem":"Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [0, 2], [1, 2], [2, 3], [3, 4]], and distinct parent-edge selection=[0, 2, 3, 4], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (vertex_count, selected_indices, spanning_tree, one_vertex_empty_selection).","answer":{"kind":"expression","value":"(5, [0, 2, 3, 4], Yes, No)","explanation":"The final flag checks whether the actual carrier has one vertex and the selection is empty. That case is a valid spanning tree. For a larger carrier the flag is No even when a nonempty selection forms a spanning tree; it is not a mislabeled single-vertex theorem. The exact result, retaining the requested order, is (5, [0, 2, 3, 4], Yes, No)."},"skills":["Recognize the Single-Vertex Empty-Edge Tree without Assuming It"],"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-20261104-r06-spanning-tree-0054","batch_id":"batch-20261014-advance-20261104-r06-spanning-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c6a8844fbe6e80affa497c22786cf35da677e40a833ee22ca169a0e1dca71bc7","family_id":"qf-mod35-spantree-one-vertex","seed":202611046054,"semantic_template":"qf-mod35-spantree-one-vertex/{\"edges\":[[0,1],[0,3],[1,2],[2,3],[3,4]],\"n\":5,\"selected\":[0,2,3,4]}"},"fields":{"title":"Recognize the Single-Vertex Empty-Edge Tree without Assuming It","stem":"Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [0, 3], [1, 2], [2, 3], [3, 4]], and distinct parent-edge selection=[0, 2, 3, 4], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (vertex_count, selected_indices, spanning_tree, one_vertex_empty_selection).","answer":{"kind":"expression","value":"(5, [0, 2, 3, 4], Yes, No)","explanation":"The final flag checks whether the actual carrier has one vertex and the selection is empty. That case is a valid spanning tree. For a larger carrier the flag is No even when a nonempty selection forms a spanning tree; it is not a mislabeled single-vertex theorem. The exact result, retaining the requested order, is (5, [0, 2, 3, 4], Yes, No)."},"skills":["Recognize the Single-Vertex Empty-Edge Tree without Assuming It"],"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-20261104-r06-spanning-tree-0055","batch_id":"batch-20261014-advance-20261104-r06-spanning-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0e0d5bc7f43703c1081d8dfcd4dbd777b31d64c3abdb2867ca70e697e75fba22","family_id":"qf-mod35-spantree-one-vertex","seed":202611046055,"semantic_template":"qf-mod35-spantree-one-vertex/{\"edges\":[[0,1],[0,2],[0,3],[1,2],[2,3],[3,4]],\"n\":5,\"selected\":[0,3,4,5]}"},"fields":{"title":"Recognize the Single-Vertex Empty-Edge Tree without Assuming It","stem":"Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [0, 2], [0, 3], [1, 2], [2, 3], [3, 4]], and distinct parent-edge selection=[0, 3, 4, 5], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (vertex_count, selected_indices, spanning_tree, one_vertex_empty_selection).","answer":{"kind":"expression","value":"(5, [0, 3, 4, 5], Yes, No)","explanation":"The final flag checks whether the actual carrier has one vertex and the selection is empty. That case is a valid spanning tree. For a larger carrier the flag is No even when a nonempty selection forms a spanning tree; it is not a mislabeled single-vertex theorem. The exact result, retaining the requested order, is (5, [0, 3, 4, 5], Yes, No)."},"skills":["Recognize the Single-Vertex Empty-Edge Tree without Assuming It"],"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-20261104-r06-spanning-tree-0056","batch_id":"batch-20261014-advance-20261104-r06-spanning-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"135c9fa861a247c1648852a4cc9ecae84b2a5513ec1e2f3eccd37a761d726bf9","family_id":"qf-mod35-spantree-one-vertex","seed":202611046056,"semantic_template":"qf-mod35-spantree-one-vertex/{\"edges\":[[0,1],[1,2],[1,3],[2,3],[3,4]],\"n\":5,\"selected\":[0,1,3,4]}"},"fields":{"title":"Recognize the Single-Vertex Empty-Edge Tree without Assuming It","stem":"Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [1, 2], [1, 3], [2, 3], [3, 4]], and distinct parent-edge selection=[0, 1, 3, 4], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (vertex_count, selected_indices, spanning_tree, one_vertex_empty_selection).","answer":{"kind":"expression","value":"(5, [0, 1, 3, 4], Yes, No)","explanation":"The final flag checks whether the actual carrier has one vertex and the selection is empty. That case is a valid spanning tree. For a larger carrier the flag is No even when a nonempty selection forms a spanning tree; it is not a mislabeled single-vertex theorem. The exact result, retaining the requested order, is (5, [0, 1, 3, 4], Yes, No)."},"skills":["Recognize the Single-Vertex Empty-Edge Tree without Assuming It"],"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-20261104-r06-spanning-tree-0057","batch_id":"batch-20261014-advance-20261104-r06-spanning-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"54b836782cfa5658b74b8ee6d0031d37e3a0cfe1bca7c319417ffab767346ac8","family_id":"qf-mod35-spantree-one-vertex","seed":202611046057,"semantic_template":"qf-mod35-spantree-one-vertex/{\"edges\":[[0,1],[0,2],[1,2],[1,3],[2,3],[3,4]],\"n\":5,\"selected\":[0,2,4,5]}"},"fields":{"title":"Recognize the Single-Vertex Empty-Edge Tree without Assuming It","stem":"Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [0, 2], [1, 2], [1, 3], [2, 3], [3, 4]], and distinct parent-edge selection=[0, 2, 4, 5], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (vertex_count, selected_indices, spanning_tree, one_vertex_empty_selection).","answer":{"kind":"expression","value":"(5, [0, 2, 4, 5], Yes, No)","explanation":"The final flag checks whether the actual carrier has one vertex and the selection is empty. That case is a valid spanning tree. For a larger carrier the flag is No even when a nonempty selection forms a spanning tree; it is not a mislabeled single-vertex theorem. The exact result, retaining the requested order, is (5, [0, 2, 4, 5], Yes, No)."},"skills":["Recognize the Single-Vertex Empty-Edge Tree without Assuming It"],"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-20261104-r06-spanning-tree-0058","batch_id":"batch-20261014-advance-20261104-r06-spanning-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"196915c47111d82cd40e340bb7969be65b52c1d6bbe7bb49f2f11170bc2637ab","family_id":"qf-mod35-spantree-one-vertex","seed":202611046058,"semantic_template":"qf-mod35-spantree-one-vertex/{\"edges\":[[0,1],[0,3],[1,2],[1,3],[2,3],[3,4]],\"n\":5,\"selected\":[0,2,4,5]}"},"fields":{"title":"Recognize the Single-Vertex Empty-Edge Tree without Assuming It","stem":"Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [0, 3], [1, 2], [1, 3], [2, 3], [3, 4]], and distinct parent-edge selection=[0, 2, 4, 5], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (vertex_count, selected_indices, spanning_tree, one_vertex_empty_selection).","answer":{"kind":"expression","value":"(5, [0, 2, 4, 5], Yes, No)","explanation":"The final flag checks whether the actual carrier has one vertex and the selection is empty. That case is a valid spanning tree. For a larger carrier the flag is No even when a nonempty selection forms a spanning tree; it is not a mislabeled single-vertex theorem. The exact result, retaining the requested order, is (5, [0, 2, 4, 5], Yes, No)."},"skills":["Recognize the Single-Vertex Empty-Edge Tree without Assuming It"],"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-20261104-r06-spanning-tree-0059","batch_id":"batch-20261014-advance-20261104-r06-spanning-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5cd4297170a3a5c4b6d3496c7556aa24c4def0b789ab4b2c687b91969b8eaece","family_id":"qf-mod35-spantree-one-vertex","seed":202611046059,"semantic_template":"qf-mod35-spantree-one-vertex/{\"edges\":[[0,1],[0,2],[0,3],[1,2],[1,3],[2,3],[3,4]],\"n\":5,\"selected\":[0,3,5,6]}"},"fields":{"title":"Recognize the Single-Vertex Empty-Edge Tree without Assuming It","stem":"Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [0, 2], [0, 3], [1, 2], [1, 3], [2, 3], [3, 4]], and distinct parent-edge selection=[0, 3, 5, 6], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (vertex_count, selected_indices, spanning_tree, one_vertex_empty_selection).","answer":{"kind":"expression","value":"(5, [0, 3, 5, 6], Yes, No)","explanation":"The final flag checks whether the actual carrier has one vertex and the selection is empty. That case is a valid spanning tree. For a larger carrier the flag is No even when a nonempty selection forms a spanning tree; it is not a mislabeled single-vertex theorem. The exact result, retaining the requested order, is (5, [0, 3, 5, 6], Yes, No)."},"skills":["Recognize the Single-Vertex Empty-Edge Tree without Assuming It"],"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-20261104-r06-spanning-tree-0060","batch_id":"batch-20261014-advance-20261104-r06-spanning-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fc61f9ceca9be586151fac2ec6675f4cff535aa88af691a395c3b5976bedcb09","family_id":"qf-mod35-spantree-one-vertex","seed":202611046060,"semantic_template":"qf-mod35-spantree-one-vertex/{\"edges\":[[0,1],[1,2],[1,4],[2,3],[3,4]],\"n\":5,\"selected\":[0,1,3,4]}"},"fields":{"title":"Recognize the Single-Vertex Empty-Edge Tree without Assuming It","stem":"Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [1, 2], [1, 4], [2, 3], [3, 4]], and distinct parent-edge selection=[0, 1, 3, 4], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (vertex_count, selected_indices, spanning_tree, one_vertex_empty_selection).","answer":{"kind":"expression","value":"(5, [0, 1, 3, 4], Yes, No)","explanation":"The final flag checks whether the actual carrier has one vertex and the selection is empty. That case is a valid spanning tree. For a larger carrier the flag is No even when a nonempty selection forms a spanning tree; it is not a mislabeled single-vertex theorem. The exact result, retaining the requested order, is (5, [0, 1, 3, 4], Yes, No)."},"skills":["Recognize the Single-Vertex Empty-Edge Tree without Assuming It"],"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-20261104-r06-spanning-tree-0061","batch_id":"batch-20261014-advance-20261104-r06-spanning-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fc32b754bb6821e21cc1ba994827ff7d206f1c2566ed49762b8d194c87717f00","family_id":"qf-mod35-spantree-one-vertex","seed":202611046061,"semantic_template":"qf-mod35-spantree-one-vertex/{\"edges\":[[0,1],[0,2],[1,2],[1,4],[2,3],[3,4]],\"n\":5,\"selected\":[0,2,4,5]}"},"fields":{"title":"Recognize the Single-Vertex Empty-Edge Tree without Assuming It","stem":"Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [0, 2], [1, 2], [1, 4], [2, 3], [3, 4]], and distinct parent-edge selection=[0, 2, 4, 5], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (vertex_count, selected_indices, spanning_tree, one_vertex_empty_selection).","answer":{"kind":"expression","value":"(5, [0, 2, 4, 5], Yes, No)","explanation":"The final flag checks whether the actual carrier has one vertex and the selection is empty. That case is a valid spanning tree. For a larger carrier the flag is No even when a nonempty selection forms a spanning tree; it is not a mislabeled single-vertex theorem. The exact result, retaining the requested order, is (5, [0, 2, 4, 5], Yes, No)."},"skills":["Recognize the Single-Vertex Empty-Edge Tree without Assuming It"],"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-20261104-r06-spanning-tree-0062","batch_id":"batch-20261014-advance-20261104-r06-spanning-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"70eb4e1e7a2e979e86618bca77ecccb41229aebd4959cfc6da97a6ad684f4874","family_id":"qf-mod35-spantree-one-vertex","seed":202611046062,"semantic_template":"qf-mod35-spantree-one-vertex/{\"edges\":[[0,1],[0,3],[1,2],[1,4],[2,3],[3,4]],\"n\":5,\"selected\":[0,2,4,5]}"},"fields":{"title":"Recognize the Single-Vertex Empty-Edge Tree without Assuming It","stem":"Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [0, 3], [1, 2], [1, 4], [2, 3], [3, 4]], and distinct parent-edge selection=[0, 2, 4, 5], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (vertex_count, selected_indices, spanning_tree, one_vertex_empty_selection).","answer":{"kind":"expression","value":"(5, [0, 2, 4, 5], Yes, No)","explanation":"The final flag checks whether the actual carrier has one vertex and the selection is empty. That case is a valid spanning tree. For a larger carrier the flag is No even when a nonempty selection forms a spanning tree; it is not a mislabeled single-vertex theorem. The exact result, retaining the requested order, is (5, [0, 2, 4, 5], Yes, No)."},"skills":["Recognize the Single-Vertex Empty-Edge Tree without Assuming It"],"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-20261104-r06-spanning-tree-0063","batch_id":"batch-20261014-advance-20261104-r06-spanning-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0c6338d92217a91bc8c696ac84d354f8f4aa983da72ce8c0587bfc7f6922c8ed","family_id":"qf-mod35-spantree-one-vertex","seed":202611046063,"semantic_template":"qf-mod35-spantree-one-vertex/{\"edges\":[[0,1],[0,2],[0,3],[1,2],[1,4],[2,3],[3,4]],\"n\":5,\"selected\":[0,3,5,6]}"},"fields":{"title":"Recognize the Single-Vertex Empty-Edge Tree without Assuming It","stem":"Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [0, 2], [0, 3], [1, 2], [1, 4], [2, 3], [3, 4]], and distinct parent-edge selection=[0, 3, 5, 6], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (vertex_count, selected_indices, spanning_tree, one_vertex_empty_selection).","answer":{"kind":"expression","value":"(5, [0, 3, 5, 6], Yes, No)","explanation":"The final flag checks whether the actual carrier has one vertex and the selection is empty. That case is a valid spanning tree. For a larger carrier the flag is No even when a nonempty selection forms a spanning tree; it is not a mislabeled single-vertex theorem. The exact result, retaining the requested order, is (5, [0, 3, 5, 6], Yes, No)."},"skills":["Recognize the Single-Vertex Empty-Edge Tree without Assuming It"],"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-20261104-r06-spanning-tree-0064","batch_id":"batch-20261014-advance-20261104-r06-spanning-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"780d95a677d10091fcb0d7770ec608a28663490c74d35ba9d2e6cd5aec5eb550","family_id":"qf-mod35-spantree-one-vertex","seed":202611046064,"semantic_template":"qf-mod35-spantree-one-vertex/{\"edges\":[[0,1],[1,2],[1,3],[1,4],[2,3],[3,4]],\"n\":5,\"selected\":[0,1,4,5]}"},"fields":{"title":"Recognize the Single-Vertex Empty-Edge Tree without Assuming It","stem":"Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [1, 2], [1, 3], [1, 4], [2, 3], [3, 4]], and distinct parent-edge selection=[0, 1, 4, 5], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (vertex_count, selected_indices, spanning_tree, one_vertex_empty_selection).","answer":{"kind":"expression","value":"(5, [0, 1, 4, 5], Yes, No)","explanation":"The final flag checks whether the actual carrier has one vertex and the selection is empty. That case is a valid spanning tree. For a larger carrier the flag is No even when a nonempty selection forms a spanning tree; it is not a mislabeled single-vertex theorem. The exact result, retaining the requested order, is (5, [0, 1, 4, 5], Yes, No)."},"skills":["Recognize the Single-Vertex Empty-Edge Tree without Assuming It"],"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-20261104-r06-spanning-tree-0065","batch_id":"batch-20261014-advance-20261104-r06-spanning-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2e08bfab280f9097d0ca5c26d6c0dccf975ebc69f413d3f8f2e07d66fe3e0aba","family_id":"qf-mod35-spantree-one-vertex","seed":202611046065,"semantic_template":"qf-mod35-spantree-one-vertex/{\"edges\":[[0,1],[0,2],[1,2],[1,3],[1,4],[2,3],[3,4]],\"n\":5,\"selected\":[0,2,5,6]}"},"fields":{"title":"Recognize the Single-Vertex Empty-Edge Tree without Assuming It","stem":"Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [0, 2], [1, 2], [1, 3], [1, 4], [2, 3], [3, 4]], and distinct parent-edge selection=[0, 2, 5, 6], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (vertex_count, selected_indices, spanning_tree, one_vertex_empty_selection).","answer":{"kind":"expression","value":"(5, [0, 2, 5, 6], Yes, No)","explanation":"The final flag checks whether the actual carrier has one vertex and the selection is empty. That case is a valid spanning tree. For a larger carrier the flag is No even when a nonempty selection forms a spanning tree; it is not a mislabeled single-vertex theorem. The exact result, retaining the requested order, is (5, [0, 2, 5, 6], Yes, No)."},"skills":["Recognize the Single-Vertex Empty-Edge Tree without Assuming It"],"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-20261104-r06-spanning-tree-0066","batch_id":"batch-20261014-advance-20261104-r06-spanning-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8e5d5ce3c174550a7fe248479cd61f2520f5d46b2d677c244da289b0006d13d0","family_id":"qf-mod35-spantree-one-vertex","seed":202611046066,"semantic_template":"qf-mod35-spantree-one-vertex/{\"edges\":[[0,1],[0,3],[1,2],[1,3],[1,4],[2,3],[3,4]],\"n\":5,\"selected\":[0,2,5,6]}"},"fields":{"title":"Recognize the Single-Vertex Empty-Edge Tree without Assuming It","stem":"Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [0, 3], [1, 2], [1, 3], [1, 4], [2, 3], [3, 4]], and distinct parent-edge selection=[0, 2, 5, 6], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (vertex_count, selected_indices, spanning_tree, one_vertex_empty_selection).","answer":{"kind":"expression","value":"(5, [0, 2, 5, 6], Yes, No)","explanation":"The final flag checks whether the actual carrier has one vertex and the selection is empty. That case is a valid spanning tree. For a larger carrier the flag is No even when a nonempty selection forms a spanning tree; it is not a mislabeled single-vertex theorem. The exact result, retaining the requested order, is (5, [0, 2, 5, 6], Yes, No)."},"skills":["Recognize the Single-Vertex Empty-Edge Tree without Assuming It"],"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-20261104-r06-spanning-tree-0067","batch_id":"batch-20261014-advance-20261104-r06-spanning-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4d16988a0f04326a3639fcfa8009132d9525cdf7d88c594ecf679e861c17ad39","family_id":"qf-mod35-spantree-one-vertex","seed":202611046067,"semantic_template":"qf-mod35-spantree-one-vertex/{\"edges\":[[0,1],[0,2],[0,3],[1,2],[1,3],[1,4],[2,3],[3,4]],\"n\":5,\"selected\":[0,3,6,7]}"},"fields":{"title":"Recognize the Single-Vertex Empty-Edge Tree without Assuming It","stem":"Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [0, 2], [0, 3], [1, 2], [1, 3], [1, 4], [2, 3], [3, 4]], and distinct parent-edge selection=[0, 3, 6, 7], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (vertex_count, selected_indices, spanning_tree, one_vertex_empty_selection).","answer":{"kind":"expression","value":"(5, [0, 3, 6, 7], Yes, No)","explanation":"The final flag checks whether the actual carrier has one vertex and the selection is empty. That case is a valid spanning tree. For a larger carrier the flag is No even when a nonempty selection forms a spanning tree; it is not a mislabeled single-vertex theorem. The exact result, retaining the requested order, is (5, [0, 3, 6, 7], Yes, No)."},"skills":["Recognize the Single-Vertex Empty-Edge Tree without Assuming It"],"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-20261104-r06-spanning-tree-0068","batch_id":"batch-20261014-advance-20261104-r06-spanning-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5b0d271985d97b9b1040398b35a986c97ec4392291c855c3cb1599b05ed8d102","family_id":"qf-mod35-spantree-one-vertex","seed":202611046068,"semantic_template":"qf-mod35-spantree-one-vertex/{\"edges\":[[0,1],[1,2],[2,3],[2,4],[3,4]],\"n\":5,\"selected\":[0,1,2,4]}"},"fields":{"title":"Recognize the Single-Vertex Empty-Edge Tree without Assuming It","stem":"Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [1, 2], [2, 3], [2, 4], [3, 4]], and distinct parent-edge selection=[0, 1, 2, 4], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (vertex_count, selected_indices, spanning_tree, one_vertex_empty_selection).","answer":{"kind":"expression","value":"(5, [0, 1, 2, 4], Yes, No)","explanation":"The final flag checks whether the actual carrier has one vertex and the selection is empty. That case is a valid spanning tree. For a larger carrier the flag is No even when a nonempty selection forms a spanning tree; it is not a mislabeled single-vertex theorem. The exact result, retaining the requested order, is (5, [0, 1, 2, 4], Yes, No)."},"skills":["Recognize the Single-Vertex Empty-Edge Tree without Assuming It"],"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-20261104-r06-spanning-tree-0069","batch_id":"batch-20261014-advance-20261104-r06-spanning-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d8bd01d30ccfdc68cd7cd8e886f003c58da56cd2bcd92d5b4188b5a323a81b27","family_id":"qf-mod35-spantree-one-vertex","seed":202611046069,"semantic_template":"qf-mod35-spantree-one-vertex/{\"edges\":[[0,1],[0,2],[1,2],[2,3],[2,4],[3,4]],\"n\":5,\"selected\":[0,2,3,5]}"},"fields":{"title":"Recognize the Single-Vertex Empty-Edge Tree without Assuming It","stem":"Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [0, 2], [1, 2], [2, 3], [2, 4], [3, 4]], and distinct parent-edge selection=[0, 2, 3, 5], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (vertex_count, selected_indices, spanning_tree, one_vertex_empty_selection).","answer":{"kind":"expression","value":"(5, [0, 2, 3, 5], Yes, No)","explanation":"The final flag checks whether the actual carrier has one vertex and the selection is empty. That case is a valid spanning tree. For a larger carrier the flag is No even when a nonempty selection forms a spanning tree; it is not a mislabeled single-vertex theorem. The exact result, retaining the requested order, is (5, [0, 2, 3, 5], Yes, No)."},"skills":["Recognize the Single-Vertex Empty-Edge Tree without Assuming It"],"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-20261104-r06-spanning-tree-0070","batch_id":"batch-20261014-advance-20261104-r06-spanning-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f1e2ee13333bca006f9f6ae694e107b1bf2348f3dafa535f4165bbc39492a73d","family_id":"qf-mod35-spantree-one-vertex","seed":202611046070,"semantic_template":"qf-mod35-spantree-one-vertex/{\"edges\":[[0,1],[0,3],[1,2],[2,3],[2,4],[3,4]],\"n\":5,\"selected\":[0,2,3,5]}"},"fields":{"title":"Recognize the Single-Vertex Empty-Edge Tree without Assuming It","stem":"Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [0, 3], [1, 2], [2, 3], [2, 4], [3, 4]], and distinct parent-edge selection=[0, 2, 3, 5], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (vertex_count, selected_indices, spanning_tree, one_vertex_empty_selection).","answer":{"kind":"expression","value":"(5, [0, 2, 3, 5], Yes, No)","explanation":"The final flag checks whether the actual carrier has one vertex and the selection is empty. That case is a valid spanning tree. For a larger carrier the flag is No even when a nonempty selection forms a spanning tree; it is not a mislabeled single-vertex theorem. The exact result, retaining the requested order, is (5, [0, 2, 3, 5], Yes, No)."},"skills":["Recognize the Single-Vertex Empty-Edge Tree without Assuming It"],"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-20261104-r06-spanning-tree-0071","batch_id":"batch-20261014-advance-20261104-r06-spanning-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3ef90ac2d5addc0ef8ea0a10f2d954034970bd17dabf2e789f4ec3e064194020","family_id":"qf-mod35-spantree-one-vertex","seed":202611046071,"semantic_template":"qf-mod35-spantree-one-vertex/{\"edges\":[[0,1],[0,2],[0,3],[1,2],[2,3],[2,4],[3,4]],\"n\":5,\"selected\":[0,3,4,6]}"},"fields":{"title":"Recognize the Single-Vertex Empty-Edge Tree without Assuming It","stem":"Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [0, 2], [0, 3], [1, 2], [2, 3], [2, 4], [3, 4]], and distinct parent-edge selection=[0, 3, 4, 6], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (vertex_count, selected_indices, spanning_tree, one_vertex_empty_selection).","answer":{"kind":"expression","value":"(5, [0, 3, 4, 6], Yes, No)","explanation":"The final flag checks whether the actual carrier has one vertex and the selection is empty. That case is a valid spanning tree. For a larger carrier the flag is No even when a nonempty selection forms a spanning tree; it is not a mislabeled single-vertex theorem. The exact result, retaining the requested order, is (5, [0, 3, 4, 6], Yes, No)."},"skills":["Recognize the Single-Vertex Empty-Edge Tree without Assuming It"],"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-20261104-r06-spanning-tree-0072","batch_id":"batch-20261014-advance-20261104-r06-spanning-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"debb865be2bd367dd234aacc96cac86fb4721b5e48eb8d583f1d1e1d48c83217","family_id":"qf-mod35-spantree-one-vertex","seed":202611046072,"semantic_template":"qf-mod35-spantree-one-vertex/{\"edges\":[[0,1],[1,2],[1,3],[2,3],[2,4],[3,4]],\"n\":5,\"selected\":[0,1,3,5]}"},"fields":{"title":"Recognize the Single-Vertex Empty-Edge Tree without Assuming It","stem":"Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [1, 2], [1, 3], [2, 3], [2, 4], [3, 4]], and distinct parent-edge selection=[0, 1, 3, 5], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (vertex_count, selected_indices, spanning_tree, one_vertex_empty_selection).","answer":{"kind":"expression","value":"(5, [0, 1, 3, 5], Yes, No)","explanation":"The final flag checks whether the actual carrier has one vertex and the selection is empty. That case is a valid spanning tree. For a larger carrier the flag is No even when a nonempty selection forms a spanning tree; it is not a mislabeled single-vertex theorem. The exact result, retaining the requested order, is (5, [0, 1, 3, 5], Yes, No)."},"skills":["Recognize the Single-Vertex Empty-Edge Tree without Assuming It"],"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-20261104-r06-spanning-tree-0073","batch_id":"batch-20261014-advance-20261104-r06-spanning-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ffb589df5da4cc645058211b0a85e623bac0c557c256d3ce82b0e295eb1c6232","family_id":"qf-mod35-spantree-one-vertex","seed":202611046073,"semantic_template":"qf-mod35-spantree-one-vertex/{\"edges\":[[0,1],[0,2],[1,2],[1,3],[2,3],[2,4],[3,4]],\"n\":5,\"selected\":[0,2,4,6]}"},"fields":{"title":"Recognize the Single-Vertex Empty-Edge Tree without Assuming It","stem":"Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [0, 2], [1, 2], [1, 3], [2, 3], [2, 4], [3, 4]], and distinct parent-edge selection=[0, 2, 4, 6], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (vertex_count, selected_indices, spanning_tree, one_vertex_empty_selection).","answer":{"kind":"expression","value":"(5, [0, 2, 4, 6], Yes, No)","explanation":"The final flag checks whether the actual carrier has one vertex and the selection is empty. That case is a valid spanning tree. For a larger carrier the flag is No even when a nonempty selection forms a spanning tree; it is not a mislabeled single-vertex theorem. The exact result, retaining the requested order, is (5, [0, 2, 4, 6], Yes, No)."},"skills":["Recognize the Single-Vertex Empty-Edge Tree without Assuming It"],"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-20261104-r06-spanning-tree-0074","batch_id":"batch-20261014-advance-20261104-r06-spanning-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3cdc099fc58b6e0a3d359398dc28c79790d0fb64236506ac21558b742e920349","family_id":"qf-mod35-spantree-one-vertex","seed":202611046074,"semantic_template":"qf-mod35-spantree-one-vertex/{\"edges\":[[0,1],[0,3],[1,2],[1,3],[2,3],[2,4],[3,4]],\"n\":5,\"selected\":[0,2,4,6]}"},"fields":{"title":"Recognize the Single-Vertex Empty-Edge Tree without Assuming It","stem":"Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [0, 3], [1, 2], [1, 3], [2, 3], [2, 4], [3, 4]], and distinct parent-edge selection=[0, 2, 4, 6], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (vertex_count, selected_indices, spanning_tree, one_vertex_empty_selection).","answer":{"kind":"expression","value":"(5, [0, 2, 4, 6], Yes, No)","explanation":"The final flag checks whether the actual carrier has one vertex and the selection is empty. That case is a valid spanning tree. For a larger carrier the flag is No even when a nonempty selection forms a spanning tree; it is not a mislabeled single-vertex theorem. The exact result, retaining the requested order, is (5, [0, 2, 4, 6], Yes, No)."},"skills":["Recognize the Single-Vertex Empty-Edge Tree without Assuming It"],"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-20261104-r06-spanning-tree-0075","batch_id":"batch-20261014-advance-20261104-r06-spanning-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6aceacb2177f654033e55e9fc27dd2e16c349b7ab06eda3956e1a1868dee9eae","family_id":"qf-mod35-spantree-one-vertex","seed":202611046075,"semantic_template":"qf-mod35-spantree-one-vertex/{\"edges\":[[0,1],[0,2],[0,3],[1,2],[1,3],[2,3],[2,4],[3,4]],\"n\":5,\"selected\":[0,3,5,7]}"},"fields":{"title":"Recognize the Single-Vertex Empty-Edge Tree without Assuming It","stem":"Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [0, 2], [0, 3], [1, 2], [1, 3], [2, 3], [2, 4], [3, 4]], and distinct parent-edge selection=[0, 3, 5, 7], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (vertex_count, selected_indices, spanning_tree, one_vertex_empty_selection).","answer":{"kind":"expression","value":"(5, [0, 3, 5, 7], Yes, No)","explanation":"The final flag checks whether the actual carrier has one vertex and the selection is empty. That case is a valid spanning tree. For a larger carrier the flag is No even when a nonempty selection forms a spanning tree; it is not a mislabeled single-vertex theorem. The exact result, retaining the requested order, is (5, [0, 3, 5, 7], Yes, No)."},"skills":["Recognize the Single-Vertex Empty-Edge Tree without Assuming It"],"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-20261104-r06-spanning-tree-0076","batch_id":"batch-20261014-advance-20261104-r06-spanning-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"417b6969902f5f0d923278cdff7736963fb39b10ecc1d4a205cf8e8121ef7d20","family_id":"qf-mod35-spantree-non-mst-boundary","seed":202611046076,"semantic_template":"qf-mod35-spantree-non-mst-boundary/{\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"n\":5,\"selected\":[0,1,2,3]}"},"fields":{"title":"An Unweighted Tree Audit Withholds a Weighted-Minimum Claim","stem":"Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [1, 2], [2, 3], [3, 4]], and distinct parent-edge selection=[0, 1, 2, 3], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (spanning_tree, weighted_minimum_claim, boundary).","answer":{"kind":"expression","value":"(Yes, No, unweighted_spanning_tree_not_minimum_spanning_tree_claim)","explanation":"Report actual spanning-tree validity, then No for the absent weighted-minimum claim. This is not a statement that the tree cannot be minimum under some separately supplied weights. No weighted optimization problem was specified. The exact result, retaining the requested order, is (Yes, No, unweighted_spanning_tree_not_minimum_spanning_tree_claim)."},"skills":["An Unweighted Tree Audit Withholds a Weighted-Minimum Claim"],"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-20261104-r06-spanning-tree-0077","batch_id":"batch-20261014-advance-20261104-r06-spanning-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7e9e06df6764f33bd9a4ea910cc76dca5fd69a4b1b1bd003aa7c9f86b460a4eb","family_id":"qf-mod35-spantree-non-mst-boundary","seed":202611046077,"semantic_template":"qf-mod35-spantree-non-mst-boundary/{\"edges\":[[0,1],[0,2],[1,2],[2,3],[3,4]],\"n\":5,\"selected\":[0,2,3,4]}"},"fields":{"title":"An Unweighted Tree Audit Withholds a Weighted-Minimum Claim","stem":"Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [0, 2], [1, 2], [2, 3], [3, 4]], and distinct parent-edge selection=[0, 2, 3, 4], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (spanning_tree, weighted_minimum_claim, boundary).","answer":{"kind":"expression","value":"(Yes, No, unweighted_spanning_tree_not_minimum_spanning_tree_claim)","explanation":"Report actual spanning-tree validity, then No for the absent weighted-minimum claim. This is not a statement that the tree cannot be minimum under some separately supplied weights. No weighted optimization problem was specified. The exact result, retaining the requested order, is (Yes, No, unweighted_spanning_tree_not_minimum_spanning_tree_claim)."},"skills":["An Unweighted Tree Audit Withholds a Weighted-Minimum Claim"],"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-20261104-r06-spanning-tree-0078","batch_id":"batch-20261014-advance-20261104-r06-spanning-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9aeff91b2f3a81d58904b25bc89bff589a3e573662f98923c038e5671c9d7bfe","family_id":"qf-mod35-spantree-non-mst-boundary","seed":202611046078,"semantic_template":"qf-mod35-spantree-non-mst-boundary/{\"edges\":[[0,1],[0,3],[1,2],[2,3],[3,4]],\"n\":5,\"selected\":[0,2,3,4]}"},"fields":{"title":"An Unweighted Tree Audit Withholds a Weighted-Minimum Claim","stem":"Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [0, 3], [1, 2], [2, 3], [3, 4]], and distinct parent-edge selection=[0, 2, 3, 4], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (spanning_tree, weighted_minimum_claim, boundary).","answer":{"kind":"expression","value":"(Yes, No, unweighted_spanning_tree_not_minimum_spanning_tree_claim)","explanation":"Report actual spanning-tree validity, then No for the absent weighted-minimum claim. This is not a statement that the tree cannot be minimum under some separately supplied weights. No weighted optimization problem was specified. The exact result, retaining the requested order, is (Yes, No, unweighted_spanning_tree_not_minimum_spanning_tree_claim)."},"skills":["An Unweighted Tree Audit Withholds a Weighted-Minimum Claim"],"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-20261104-r06-spanning-tree-0079","batch_id":"batch-20261014-advance-20261104-r06-spanning-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b8fb244de84afe550be8cdf6bb1328832c0f64d1efeefc693272280a50e9473b","family_id":"qf-mod35-spantree-non-mst-boundary","seed":202611046079,"semantic_template":"qf-mod35-spantree-non-mst-boundary/{\"edges\":[[0,1],[0,2],[0,3],[1,2],[2,3],[3,4]],\"n\":5,\"selected\":[0,3,4,5]}"},"fields":{"title":"An Unweighted Tree Audit Withholds a Weighted-Minimum Claim","stem":"Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [0, 2], [0, 3], [1, 2], [2, 3], [3, 4]], and distinct parent-edge selection=[0, 3, 4, 5], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (spanning_tree, weighted_minimum_claim, boundary).","answer":{"kind":"expression","value":"(Yes, No, unweighted_spanning_tree_not_minimum_spanning_tree_claim)","explanation":"Report actual spanning-tree validity, then No for the absent weighted-minimum claim. This is not a statement that the tree cannot be minimum under some separately supplied weights. No weighted optimization problem was specified. The exact result, retaining the requested order, is (Yes, No, unweighted_spanning_tree_not_minimum_spanning_tree_claim)."},"skills":["An Unweighted Tree Audit Withholds a Weighted-Minimum Claim"],"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-20261104-r06-spanning-tree-0080","batch_id":"batch-20261014-advance-20261104-r06-spanning-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9bbd59bec2c9df72eb6462c27dc4dd5facff135d8a63319fd476c5549bd7d02f","family_id":"qf-mod35-spantree-non-mst-boundary","seed":202611046080,"semantic_template":"qf-mod35-spantree-non-mst-boundary/{\"edges\":[[0,1],[1,2],[1,3],[2,3],[3,4]],\"n\":5,\"selected\":[0,1,3,4]}"},"fields":{"title":"An Unweighted Tree Audit Withholds a Weighted-Minimum Claim","stem":"Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [1, 2], [1, 3], [2, 3], [3, 4]], and distinct parent-edge selection=[0, 1, 3, 4], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (spanning_tree, weighted_minimum_claim, boundary).","answer":{"kind":"expression","value":"(Yes, No, unweighted_spanning_tree_not_minimum_spanning_tree_claim)","explanation":"Report actual spanning-tree validity, then No for the absent weighted-minimum claim. This is not a statement that the tree cannot be minimum under some separately supplied weights. No weighted optimization problem was specified. The exact result, retaining the requested order, is (Yes, No, unweighted_spanning_tree_not_minimum_spanning_tree_claim)."},"skills":["An Unweighted Tree Audit Withholds a Weighted-Minimum Claim"],"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-20261104-r06-spanning-tree-0081","batch_id":"batch-20261014-advance-20261104-r06-spanning-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ed51d516070ce911f31919dd76a6102d75ca395726ea09653d9e029817c8b5eb","family_id":"qf-mod35-spantree-non-mst-boundary","seed":202611046081,"semantic_template":"qf-mod35-spantree-non-mst-boundary/{\"edges\":[[0,1],[0,2],[1,2],[1,3],[2,3],[3,4]],\"n\":5,\"selected\":[0,2,4,5]}"},"fields":{"title":"An Unweighted Tree Audit Withholds a Weighted-Minimum Claim","stem":"Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [0, 2], [1, 2], [1, 3], [2, 3], [3, 4]], and distinct parent-edge selection=[0, 2, 4, 5], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (spanning_tree, weighted_minimum_claim, boundary).","answer":{"kind":"expression","value":"(Yes, No, unweighted_spanning_tree_not_minimum_spanning_tree_claim)","explanation":"Report actual spanning-tree validity, then No for the absent weighted-minimum claim. This is not a statement that the tree cannot be minimum under some separately supplied weights. No weighted optimization problem was specified. The exact result, retaining the requested order, is (Yes, No, unweighted_spanning_tree_not_minimum_spanning_tree_claim)."},"skills":["An Unweighted Tree Audit Withholds a Weighted-Minimum Claim"],"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-20261104-r06-spanning-tree-0082","batch_id":"batch-20261014-advance-20261104-r06-spanning-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"72c981fecbb83e27dbf26f7fcff54a109f0d7abd54ca7dda70d2884885dc9aa6","family_id":"qf-mod35-spantree-non-mst-boundary","seed":202611046082,"semantic_template":"qf-mod35-spantree-non-mst-boundary/{\"edges\":[[0,1],[0,3],[1,2],[1,3],[2,3],[3,4]],\"n\":5,\"selected\":[0,2,4,5]}"},"fields":{"title":"An Unweighted Tree Audit Withholds a Weighted-Minimum Claim","stem":"Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [0, 3], [1, 2], [1, 3], [2, 3], [3, 4]], and distinct parent-edge selection=[0, 2, 4, 5], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (spanning_tree, weighted_minimum_claim, boundary).","answer":{"kind":"expression","value":"(Yes, No, unweighted_spanning_tree_not_minimum_spanning_tree_claim)","explanation":"Report actual spanning-tree validity, then No for the absent weighted-minimum claim. This is not a statement that the tree cannot be minimum under some separately supplied weights. No weighted optimization problem was specified. The exact result, retaining the requested order, is (Yes, No, unweighted_spanning_tree_not_minimum_spanning_tree_claim)."},"skills":["An Unweighted Tree Audit Withholds a Weighted-Minimum Claim"],"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-20261104-r06-spanning-tree-0083","batch_id":"batch-20261014-advance-20261104-r06-spanning-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"87b123a06f9f4454284487b266482d8399684ab95cf58d1f2c9c93bdf049d0a2","family_id":"qf-mod35-spantree-non-mst-boundary","seed":202611046083,"semantic_template":"qf-mod35-spantree-non-mst-boundary/{\"edges\":[[0,1],[0,2],[0,3],[1,2],[1,3],[2,3],[3,4]],\"n\":5,\"selected\":[0,3,5,6]}"},"fields":{"title":"An Unweighted Tree Audit Withholds a Weighted-Minimum Claim","stem":"Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [0, 2], [0, 3], [1, 2], [1, 3], [2, 3], [3, 4]], and distinct parent-edge selection=[0, 3, 5, 6], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (spanning_tree, weighted_minimum_claim, boundary).","answer":{"kind":"expression","value":"(Yes, No, unweighted_spanning_tree_not_minimum_spanning_tree_claim)","explanation":"Report actual spanning-tree validity, then No for the absent weighted-minimum claim. This is not a statement that the tree cannot be minimum under some separately supplied weights. No weighted optimization problem was specified. The exact result, retaining the requested order, is (Yes, No, unweighted_spanning_tree_not_minimum_spanning_tree_claim)."},"skills":["An Unweighted Tree Audit Withholds a Weighted-Minimum Claim"],"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-20261104-r06-spanning-tree-0084","batch_id":"batch-20261014-advance-20261104-r06-spanning-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"429893c46078f53fa3e8bd818ef40b9edf19d8ba5628733df03f8613224c6482","family_id":"qf-mod35-spantree-non-mst-boundary","seed":202611046084,"semantic_template":"qf-mod35-spantree-non-mst-boundary/{\"edges\":[[0,1],[1,2],[1,4],[2,3],[3,4]],\"n\":5,\"selected\":[0,1,3,4]}"},"fields":{"title":"An Unweighted Tree Audit Withholds a Weighted-Minimum Claim","stem":"Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [1, 2], [1, 4], [2, 3], [3, 4]], and distinct parent-edge selection=[0, 1, 3, 4], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (spanning_tree, weighted_minimum_claim, boundary).","answer":{"kind":"expression","value":"(Yes, No, unweighted_spanning_tree_not_minimum_spanning_tree_claim)","explanation":"Report actual spanning-tree validity, then No for the absent weighted-minimum claim. This is not a statement that the tree cannot be minimum under some separately supplied weights. No weighted optimization problem was specified. The exact result, retaining the requested order, is (Yes, No, unweighted_spanning_tree_not_minimum_spanning_tree_claim)."},"skills":["An Unweighted Tree Audit Withholds a Weighted-Minimum Claim"],"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-20261104-r06-spanning-tree-0085","batch_id":"batch-20261014-advance-20261104-r06-spanning-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e951849db19e048a1f2602d18f821033526547182ac2893f4248f227e6c8b474","family_id":"qf-mod35-spantree-non-mst-boundary","seed":202611046085,"semantic_template":"qf-mod35-spantree-non-mst-boundary/{\"edges\":[[0,1],[0,2],[1,2],[1,4],[2,3],[3,4]],\"n\":5,\"selected\":[0,2,4,5]}"},"fields":{"title":"An Unweighted Tree Audit Withholds a Weighted-Minimum Claim","stem":"Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [0, 2], [1, 2], [1, 4], [2, 3], [3, 4]], and distinct parent-edge selection=[0, 2, 4, 5], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (spanning_tree, weighted_minimum_claim, boundary).","answer":{"kind":"expression","value":"(Yes, No, unweighted_spanning_tree_not_minimum_spanning_tree_claim)","explanation":"Report actual spanning-tree validity, then No for the absent weighted-minimum claim. This is not a statement that the tree cannot be minimum under some separately supplied weights. No weighted optimization problem was specified. The exact result, retaining the requested order, is (Yes, No, unweighted_spanning_tree_not_minimum_spanning_tree_claim)."},"skills":["An Unweighted Tree Audit Withholds a Weighted-Minimum Claim"],"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-20261104-r06-spanning-tree-0086","batch_id":"batch-20261014-advance-20261104-r06-spanning-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e87b93781639c6b38a93b80f94a4da10a47404d2070c88689c2f42015f85b671","family_id":"qf-mod35-spantree-non-mst-boundary","seed":202611046086,"semantic_template":"qf-mod35-spantree-non-mst-boundary/{\"edges\":[[0,1],[0,3],[1,2],[1,4],[2,3],[3,4]],\"n\":5,\"selected\":[0,2,4,5]}"},"fields":{"title":"An Unweighted Tree Audit Withholds a Weighted-Minimum Claim","stem":"Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [0, 3], [1, 2], [1, 4], [2, 3], [3, 4]], and distinct parent-edge selection=[0, 2, 4, 5], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (spanning_tree, weighted_minimum_claim, boundary).","answer":{"kind":"expression","value":"(Yes, No, unweighted_spanning_tree_not_minimum_spanning_tree_claim)","explanation":"Report actual spanning-tree validity, then No for the absent weighted-minimum claim. This is not a statement that the tree cannot be minimum under some separately supplied weights. No weighted optimization problem was specified. The exact result, retaining the requested order, is (Yes, No, unweighted_spanning_tree_not_minimum_spanning_tree_claim)."},"skills":["An Unweighted Tree Audit Withholds a Weighted-Minimum Claim"],"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-20261104-r06-spanning-tree-0087","batch_id":"batch-20261014-advance-20261104-r06-spanning-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f693a21ced97c2f3b4c593677983d47cd60f092172bf6b2c8bdb3bb63b163a3d","family_id":"qf-mod35-spantree-non-mst-boundary","seed":202611046087,"semantic_template":"qf-mod35-spantree-non-mst-boundary/{\"edges\":[[0,1],[0,2],[0,3],[1,2],[1,4],[2,3],[3,4]],\"n\":5,\"selected\":[0,3,5,6]}"},"fields":{"title":"An Unweighted Tree Audit Withholds a Weighted-Minimum Claim","stem":"Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [0, 2], [0, 3], [1, 2], [1, 4], [2, 3], [3, 4]], and distinct parent-edge selection=[0, 3, 5, 6], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (spanning_tree, weighted_minimum_claim, boundary).","answer":{"kind":"expression","value":"(Yes, No, unweighted_spanning_tree_not_minimum_spanning_tree_claim)","explanation":"Report actual spanning-tree validity, then No for the absent weighted-minimum claim. This is not a statement that the tree cannot be minimum under some separately supplied weights. No weighted optimization problem was specified. The exact result, retaining the requested order, is (Yes, No, unweighted_spanning_tree_not_minimum_spanning_tree_claim)."},"skills":["An Unweighted Tree Audit Withholds a Weighted-Minimum Claim"],"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-20261104-r06-spanning-tree-0088","batch_id":"batch-20261014-advance-20261104-r06-spanning-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"39b546226e4df0e64ff5381745efcc048b4e5f580b1bb2d0b5aa0e465c897787","family_id":"qf-mod35-spantree-non-mst-boundary","seed":202611046088,"semantic_template":"qf-mod35-spantree-non-mst-boundary/{\"edges\":[[0,1],[1,2],[1,3],[1,4],[2,3],[3,4]],\"n\":5,\"selected\":[0,1,4,5]}"},"fields":{"title":"An Unweighted Tree Audit Withholds a Weighted-Minimum Claim","stem":"Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [1, 2], [1, 3], [1, 4], [2, 3], [3, 4]], and distinct parent-edge selection=[0, 1, 4, 5], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (spanning_tree, weighted_minimum_claim, boundary).","answer":{"kind":"expression","value":"(Yes, No, unweighted_spanning_tree_not_minimum_spanning_tree_claim)","explanation":"Report actual spanning-tree validity, then No for the absent weighted-minimum claim. This is not a statement that the tree cannot be minimum under some separately supplied weights. No weighted optimization problem was specified. The exact result, retaining the requested order, is (Yes, No, unweighted_spanning_tree_not_minimum_spanning_tree_claim)."},"skills":["An Unweighted Tree Audit Withholds a Weighted-Minimum Claim"],"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-20261104-r06-spanning-tree-0089","batch_id":"batch-20261014-advance-20261104-r06-spanning-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8dc081e5fc58ee0198fb0f03bb457ff316cf5f1debc24228eed4c9263edfd2e7","family_id":"qf-mod35-spantree-non-mst-boundary","seed":202611046089,"semantic_template":"qf-mod35-spantree-non-mst-boundary/{\"edges\":[[0,1],[0,2],[1,2],[1,3],[1,4],[2,3],[3,4]],\"n\":5,\"selected\":[0,2,5,6]}"},"fields":{"title":"An Unweighted Tree Audit Withholds a Weighted-Minimum Claim","stem":"Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [0, 2], [1, 2], [1, 3], [1, 4], [2, 3], [3, 4]], and distinct parent-edge selection=[0, 2, 5, 6], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (spanning_tree, weighted_minimum_claim, boundary).","answer":{"kind":"expression","value":"(Yes, No, unweighted_spanning_tree_not_minimum_spanning_tree_claim)","explanation":"Report actual spanning-tree validity, then No for the absent weighted-minimum claim. This is not a statement that the tree cannot be minimum under some separately supplied weights. No weighted optimization problem was specified. The exact result, retaining the requested order, is (Yes, No, unweighted_spanning_tree_not_minimum_spanning_tree_claim)."},"skills":["An Unweighted Tree Audit Withholds a Weighted-Minimum Claim"],"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-20261104-r06-spanning-tree-0090","batch_id":"batch-20261014-advance-20261104-r06-spanning-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"958834824aba4cbb37af28845d7e90c2051921fce1a6ec88cf270a76100c67cc","family_id":"qf-mod35-spantree-non-mst-boundary","seed":202611046090,"semantic_template":"qf-mod35-spantree-non-mst-boundary/{\"edges\":[[0,1],[0,3],[1,2],[1,3],[1,4],[2,3],[3,4]],\"n\":5,\"selected\":[0,2,5,6]}"},"fields":{"title":"An Unweighted Tree Audit Withholds a Weighted-Minimum Claim","stem":"Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [0, 3], [1, 2], [1, 3], [1, 4], [2, 3], [3, 4]], and distinct parent-edge selection=[0, 2, 5, 6], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (spanning_tree, weighted_minimum_claim, boundary).","answer":{"kind":"expression","value":"(Yes, No, unweighted_spanning_tree_not_minimum_spanning_tree_claim)","explanation":"Report actual spanning-tree validity, then No for the absent weighted-minimum claim. This is not a statement that the tree cannot be minimum under some separately supplied weights. No weighted optimization problem was specified. The exact result, retaining the requested order, is (Yes, No, unweighted_spanning_tree_not_minimum_spanning_tree_claim)."},"skills":["An Unweighted Tree Audit Withholds a Weighted-Minimum Claim"],"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-20261104-r06-spanning-tree-0091","batch_id":"batch-20261014-advance-20261104-r06-spanning-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c347c27b95c936748eb13b3b1ca4eee7c2503d0051ec816c72d07dfe3334d60e","family_id":"qf-mod35-spantree-non-mst-boundary","seed":202611046091,"semantic_template":"qf-mod35-spantree-non-mst-boundary/{\"edges\":[[0,1],[0,2],[0,3],[1,2],[1,3],[1,4],[2,3],[3,4]],\"n\":5,\"selected\":[0,3,6,7]}"},"fields":{"title":"An Unweighted Tree Audit Withholds a Weighted-Minimum Claim","stem":"Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [0, 2], [0, 3], [1, 2], [1, 3], [1, 4], [2, 3], [3, 4]], and distinct parent-edge selection=[0, 3, 6, 7], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (spanning_tree, weighted_minimum_claim, boundary).","answer":{"kind":"expression","value":"(Yes, No, unweighted_spanning_tree_not_minimum_spanning_tree_claim)","explanation":"Report actual spanning-tree validity, then No for the absent weighted-minimum claim. This is not a statement that the tree cannot be minimum under some separately supplied weights. No weighted optimization problem was specified. The exact result, retaining the requested order, is (Yes, No, unweighted_spanning_tree_not_minimum_spanning_tree_claim)."},"skills":["An Unweighted Tree Audit Withholds a Weighted-Minimum Claim"],"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-20261104-r06-spanning-tree-0092","batch_id":"batch-20261014-advance-20261104-r06-spanning-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2d832225bb39977546f83f4f4382e0f039584ed015ebb35310118410b247746d","family_id":"qf-mod35-spantree-non-mst-boundary","seed":202611046092,"semantic_template":"qf-mod35-spantree-non-mst-boundary/{\"edges\":[[0,1],[1,2],[2,3],[2,4],[3,4]],\"n\":5,\"selected\":[0,1,2,4]}"},"fields":{"title":"An Unweighted Tree Audit Withholds a Weighted-Minimum Claim","stem":"Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [1, 2], [2, 3], [2, 4], [3, 4]], and distinct parent-edge selection=[0, 1, 2, 4], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (spanning_tree, weighted_minimum_claim, boundary).","answer":{"kind":"expression","value":"(Yes, No, unweighted_spanning_tree_not_minimum_spanning_tree_claim)","explanation":"Report actual spanning-tree validity, then No for the absent weighted-minimum claim. This is not a statement that the tree cannot be minimum under some separately supplied weights. No weighted optimization problem was specified. The exact result, retaining the requested order, is (Yes, No, unweighted_spanning_tree_not_minimum_spanning_tree_claim)."},"skills":["An Unweighted Tree Audit Withholds a Weighted-Minimum Claim"],"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-20261104-r06-spanning-tree-0093","batch_id":"batch-20261014-advance-20261104-r06-spanning-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"54ab6a3a93841a24d4e67edbf3205b571d8dbb6bbcb5ebdaa5c3736cd80d77cd","family_id":"qf-mod35-spantree-non-mst-boundary","seed":202611046093,"semantic_template":"qf-mod35-spantree-non-mst-boundary/{\"edges\":[[0,1],[0,2],[1,2],[2,3],[2,4],[3,4]],\"n\":5,\"selected\":[0,2,3,5]}"},"fields":{"title":"An Unweighted Tree Audit Withholds a Weighted-Minimum Claim","stem":"Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [0, 2], [1, 2], [2, 3], [2, 4], [3, 4]], and distinct parent-edge selection=[0, 2, 3, 5], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (spanning_tree, weighted_minimum_claim, boundary).","answer":{"kind":"expression","value":"(Yes, No, unweighted_spanning_tree_not_minimum_spanning_tree_claim)","explanation":"Report actual spanning-tree validity, then No for the absent weighted-minimum claim. This is not a statement that the tree cannot be minimum under some separately supplied weights. No weighted optimization problem was specified. The exact result, retaining the requested order, is (Yes, No, unweighted_spanning_tree_not_minimum_spanning_tree_claim)."},"skills":["An Unweighted Tree Audit Withholds a Weighted-Minimum Claim"],"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-20261104-r06-spanning-tree-0094","batch_id":"batch-20261014-advance-20261104-r06-spanning-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2970fb6964498ce29aa1efb9644972eb1ef1390db8182ee54d330cc8bfafd6ec","family_id":"qf-mod35-spantree-non-mst-boundary","seed":202611046094,"semantic_template":"qf-mod35-spantree-non-mst-boundary/{\"edges\":[[0,1],[0,3],[1,2],[2,3],[2,4],[3,4]],\"n\":5,\"selected\":[0,2,3,5]}"},"fields":{"title":"An Unweighted Tree Audit Withholds a Weighted-Minimum Claim","stem":"Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [0, 3], [1, 2], [2, 3], [2, 4], [3, 4]], and distinct parent-edge selection=[0, 2, 3, 5], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (spanning_tree, weighted_minimum_claim, boundary).","answer":{"kind":"expression","value":"(Yes, No, unweighted_spanning_tree_not_minimum_spanning_tree_claim)","explanation":"Report actual spanning-tree validity, then No for the absent weighted-minimum claim. This is not a statement that the tree cannot be minimum under some separately supplied weights. No weighted optimization problem was specified. The exact result, retaining the requested order, is (Yes, No, unweighted_spanning_tree_not_minimum_spanning_tree_claim)."},"skills":["An Unweighted Tree Audit Withholds a Weighted-Minimum Claim"],"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-20261104-r06-spanning-tree-0095","batch_id":"batch-20261014-advance-20261104-r06-spanning-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6fad60546484dc1694860443f70ba41ed3a60a5325a5caf9bf458826af1c1c44","family_id":"qf-mod35-spantree-non-mst-boundary","seed":202611046095,"semantic_template":"qf-mod35-spantree-non-mst-boundary/{\"edges\":[[0,1],[0,2],[0,3],[1,2],[2,3],[2,4],[3,4]],\"n\":5,\"selected\":[0,3,4,6]}"},"fields":{"title":"An Unweighted Tree Audit Withholds a Weighted-Minimum Claim","stem":"Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [0, 2], [0, 3], [1, 2], [2, 3], [2, 4], [3, 4]], and distinct parent-edge selection=[0, 3, 4, 6], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (spanning_tree, weighted_minimum_claim, boundary).","answer":{"kind":"expression","value":"(Yes, No, unweighted_spanning_tree_not_minimum_spanning_tree_claim)","explanation":"Report actual spanning-tree validity, then No for the absent weighted-minimum claim. This is not a statement that the tree cannot be minimum under some separately supplied weights. No weighted optimization problem was specified. The exact result, retaining the requested order, is (Yes, No, unweighted_spanning_tree_not_minimum_spanning_tree_claim)."},"skills":["An Unweighted Tree Audit Withholds a Weighted-Minimum Claim"],"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-20261104-r06-spanning-tree-0096","batch_id":"batch-20261014-advance-20261104-r06-spanning-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d7d4fa7ce00c662a967eb3a59c9d51933804ffd87bd1c7ced5f8225846a8188f","family_id":"qf-mod35-spantree-non-mst-boundary","seed":202611046096,"semantic_template":"qf-mod35-spantree-non-mst-boundary/{\"edges\":[[0,1],[1,2],[1,3],[2,3],[2,4],[3,4]],\"n\":5,\"selected\":[0,1,3,5]}"},"fields":{"title":"An Unweighted Tree Audit Withholds a Weighted-Minimum Claim","stem":"Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [1, 2], [1, 3], [2, 3], [2, 4], [3, 4]], and distinct parent-edge selection=[0, 1, 3, 5], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (spanning_tree, weighted_minimum_claim, boundary).","answer":{"kind":"expression","value":"(Yes, No, unweighted_spanning_tree_not_minimum_spanning_tree_claim)","explanation":"Report actual spanning-tree validity, then No for the absent weighted-minimum claim. This is not a statement that the tree cannot be minimum under some separately supplied weights. No weighted optimization problem was specified. The exact result, retaining the requested order, is (Yes, No, unweighted_spanning_tree_not_minimum_spanning_tree_claim)."},"skills":["An Unweighted Tree Audit Withholds a Weighted-Minimum Claim"],"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-20261104-r06-spanning-tree-0097","batch_id":"batch-20261014-advance-20261104-r06-spanning-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e28b5c5fd757759fd6b0554934a52aa5b51d2cc265c39d35605ecce763cdf3c1","family_id":"qf-mod35-spantree-non-mst-boundary","seed":202611046097,"semantic_template":"qf-mod35-spantree-non-mst-boundary/{\"edges\":[[0,1],[0,2],[1,2],[1,3],[2,3],[2,4],[3,4]],\"n\":5,\"selected\":[0,2,4,6]}"},"fields":{"title":"An Unweighted Tree Audit Withholds a Weighted-Minimum Claim","stem":"Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [0, 2], [1, 2], [1, 3], [2, 3], [2, 4], [3, 4]], and distinct parent-edge selection=[0, 2, 4, 6], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (spanning_tree, weighted_minimum_claim, boundary).","answer":{"kind":"expression","value":"(Yes, No, unweighted_spanning_tree_not_minimum_spanning_tree_claim)","explanation":"Report actual spanning-tree validity, then No for the absent weighted-minimum claim. This is not a statement that the tree cannot be minimum under some separately supplied weights. No weighted optimization problem was specified. The exact result, retaining the requested order, is (Yes, No, unweighted_spanning_tree_not_minimum_spanning_tree_claim)."},"skills":["An Unweighted Tree Audit Withholds a Weighted-Minimum Claim"],"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-20261104-r06-spanning-tree-0098","batch_id":"batch-20261014-advance-20261104-r06-spanning-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c25ffb2bc47e00a3b0479d8b9b459f7cef6fb3feb298c09c943a808e6790cec6","family_id":"qf-mod35-spantree-non-mst-boundary","seed":202611046098,"semantic_template":"qf-mod35-spantree-non-mst-boundary/{\"edges\":[[0,1],[0,3],[1,2],[1,3],[2,3],[2,4],[3,4]],\"n\":5,\"selected\":[0,2,4,6]}"},"fields":{"title":"An Unweighted Tree Audit Withholds a Weighted-Minimum Claim","stem":"Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [0, 3], [1, 2], [1, 3], [2, 3], [2, 4], [3, 4]], and distinct parent-edge selection=[0, 2, 4, 6], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (spanning_tree, weighted_minimum_claim, boundary).","answer":{"kind":"expression","value":"(Yes, No, unweighted_spanning_tree_not_minimum_spanning_tree_claim)","explanation":"Report actual spanning-tree validity, then No for the absent weighted-minimum claim. This is not a statement that the tree cannot be minimum under some separately supplied weights. No weighted optimization problem was specified. The exact result, retaining the requested order, is (Yes, No, unweighted_spanning_tree_not_minimum_spanning_tree_claim)."},"skills":["An Unweighted Tree Audit Withholds a Weighted-Minimum Claim"],"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-20261104-r06-spanning-tree-0099","batch_id":"batch-20261014-advance-20261104-r06-spanning-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b91c957c90f46147d63513a5b6501ad4f2162df5adc1cfc9495d47155e3e4fb2","family_id":"qf-mod35-spantree-non-mst-boundary","seed":202611046099,"semantic_template":"qf-mod35-spantree-non-mst-boundary/{\"edges\":[[0,1],[0,2],[0,3],[1,2],[1,3],[2,3],[2,4],[3,4]],\"n\":5,\"selected\":[0,3,5,7]}"},"fields":{"title":"An Unweighted Tree Audit Withholds a Weighted-Minimum Claim","stem":"Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [0, 2], [0, 3], [1, 2], [1, 3], [2, 3], [2, 4], [3, 4]], and distinct parent-edge selection=[0, 3, 5, 7], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (spanning_tree, weighted_minimum_claim, boundary).","answer":{"kind":"expression","value":"(Yes, No, unweighted_spanning_tree_not_minimum_spanning_tree_claim)","explanation":"Report actual spanning-tree validity, then No for the absent weighted-minimum claim. This is not a statement that the tree cannot be minimum under some separately supplied weights. No weighted optimization problem was specified. The exact result, retaining the requested order, is (Yes, No, unweighted_spanning_tree_not_minimum_spanning_tree_claim)."},"skills":["An Unweighted Tree Audit Withholds a Weighted-Minimum Claim"],"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-20261104-r06-spanning-tree-0100","batch_id":"batch-20261014-advance-20261104-r06-spanning-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"434ab9a5934fdc2e6501775723dee4f447922be5af7827d3a55a5b122a5d7849","family_id":"qf-mod35-spantree-non-mst-boundary","seed":202611046100,"semantic_template":"qf-mod35-spantree-non-mst-boundary/{\"edges\":[[0,1],[1,2],[1,4],[2,3],[2,4],[3,4]],\"n\":5,\"selected\":[0,1,3,5]}"},"fields":{"title":"An Unweighted Tree Audit Withholds a Weighted-Minimum Claim","stem":"Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [1, 2], [1, 4], [2, 3], [2, 4], [3, 4]], and distinct parent-edge selection=[0, 1, 3, 5], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (spanning_tree, weighted_minimum_claim, boundary).","answer":{"kind":"expression","value":"(Yes, No, unweighted_spanning_tree_not_minimum_spanning_tree_claim)","explanation":"Report actual spanning-tree validity, then No for the absent weighted-minimum claim. This is not a statement that the tree cannot be minimum under some separately supplied weights. No weighted optimization problem was specified. The exact result, retaining the requested order, is (Yes, No, unweighted_spanning_tree_not_minimum_spanning_tree_claim)."},"skills":["An Unweighted Tree Audit Withholds a Weighted-Minimum Claim"],"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."}}