{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r05-graph-0001","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e06cfc90fc9613da5566d9ed2c2b1d12da44100c6736449f306b88ffa2b9e191","family_id":"qf-density17-graph-adjacency-degrees","seed":202610175001,"semantic_template":"qf-density17-graph-adjacency-degrees/{\"edges\":[],\"n\":0}"},"fields":{"title":"Complete Adjacency Matrix, Degrees, and Handshake Counts","stem":"Use a complete simple undirected graph on Fin(0)=∅ with raw edges=[]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Give (n, normalized edges, Boolean adjacency matrix, degree list, degree sum, twice the edge count).","answer":{"kind":"expression","value":"(0, [], [], [], 0, 0)","explanation":"Read the complete symmetric adjacency relation and count each distinct undirected edge once; this is not a connectivity or tree test. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (0, [], [], [], 0, 0)."},"skills":["Complete Adjacency Matrix, Degrees, and Handshake Counts"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r05-graph-0002","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ccec157f81ece99465f7b78bf5f1c1a76eb2bad9fed525ead3aa552a959914f7","family_id":"qf-density17-graph-adjacency-degrees","seed":202610175002,"semantic_template":"qf-density17-graph-adjacency-degrees/{\"edges\":[],\"n\":1}"},"fields":{"title":"Complete Adjacency Matrix, Degrees, and Handshake Counts","stem":"Use a complete simple undirected graph on Fin(1)={0,…,0} with raw edges=[]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Give (n, normalized edges, Boolean adjacency matrix, degree list, degree sum, twice the edge count).","answer":{"kind":"expression","value":"(1, [], [[No]], [0], 0, 0)","explanation":"Read the complete symmetric adjacency relation and count each distinct undirected edge once; this is not a connectivity or tree test. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (1, [], [[No]], [0], 0, 0)."},"skills":["Complete Adjacency Matrix, Degrees, and Handshake Counts"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r05-graph-0003","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"376e50b249b29d39f5fcb03406606cf0e19e7691d6ddd5ed3d844983e07ad69a","family_id":"qf-density17-graph-adjacency-degrees","seed":202610175003,"semantic_template":"qf-density17-graph-adjacency-degrees/{\"edges\":[[1,2],[1,3],[2,3]],\"n\":4}"},"fields":{"title":"Complete Adjacency Matrix, Degrees, and Handshake Counts","stem":"Use a complete simple undirected graph on Fin(4)={0,…,3} with raw edges=[[1, 2], [1, 3], [2, 3]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Give (n, normalized edges, Boolean adjacency matrix, degree list, degree sum, twice the edge count).","answer":{"kind":"expression","value":"(4, [[1, 2], [1, 3], [2, 3]], [[No, No, No, No], [No, No, Yes, Yes], [No, Yes, No, Yes], [No, Yes, Yes, No]], [0, 2, 2, 2], 6, 6)","explanation":"Read the complete symmetric adjacency relation and count each distinct undirected edge once; this is not a connectivity or tree test. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [[1, 2], [1, 3], [2, 3]], [[No, No, No, No], [No, No, Yes, Yes], [No, Yes, No, Yes], [No, Yes, Yes, No]], [0, 2, 2, 2], 6, 6)."},"skills":["Complete Adjacency Matrix, Degrees, and Handshake Counts"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r05-graph-0004","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"443042d8412ed685d446738144ac78f240ca9b84415b1f701467d08a6ff96ac8","family_id":"qf-density17-graph-adjacency-degrees","seed":202610175004,"semantic_template":"qf-density17-graph-adjacency-degrees/{\"edges\":[[0,1],[2,3]],\"n\":4}"},"fields":{"title":"Complete Adjacency Matrix, Degrees, and Handshake Counts","stem":"Use a complete simple undirected graph on Fin(4)={0,…,3} with raw edges=[[0, 1], [2, 3]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Give (n, normalized edges, Boolean adjacency matrix, degree list, degree sum, twice the edge count).","answer":{"kind":"expression","value":"(4, [[0, 1], [2, 3]], [[No, Yes, No, No], [Yes, No, No, No], [No, No, No, Yes], [No, No, Yes, No]], [1, 1, 1, 1], 4, 4)","explanation":"Read the complete symmetric adjacency relation and count each distinct undirected edge once; this is not a connectivity or tree test. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [[0, 1], [2, 3]], [[No, Yes, No, No], [Yes, No, No, No], [No, No, No, Yes], [No, No, Yes, No]], [1, 1, 1, 1], 4, 4)."},"skills":["Complete Adjacency Matrix, Degrees, and Handshake Counts"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r05-graph-0005","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"867676b0c810764c5203a7303f739700f777d50cf402847cfd8719c1fd3f7e79","family_id":"qf-density17-graph-adjacency-degrees","seed":202610175005,"semantic_template":"qf-density17-graph-adjacency-degrees/{\"edges\":[[0,1],[1,2],[2,3]],\"n\":4}"},"fields":{"title":"Complete Adjacency Matrix, Degrees, and Handshake Counts","stem":"Use a complete simple undirected graph on Fin(4)={0,…,3} with raw edges=[[0, 1], [1, 2], [2, 3]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Give (n, normalized edges, Boolean adjacency matrix, degree list, degree sum, twice the edge count).","answer":{"kind":"expression","value":"(4, [[0, 1], [1, 2], [2, 3]], [[No, Yes, No, No], [Yes, No, Yes, No], [No, Yes, No, Yes], [No, No, Yes, No]], [1, 2, 2, 1], 6, 6)","explanation":"Read the complete symmetric adjacency relation and count each distinct undirected edge once; this is not a connectivity or tree test. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [[0, 1], [1, 2], [2, 3]], [[No, Yes, No, No], [Yes, No, Yes, No], [No, Yes, No, Yes], [No, No, Yes, No]], [1, 2, 2, 1], 6, 6)."},"skills":["Complete Adjacency Matrix, Degrees, and Handshake Counts"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r05-graph-0006","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"51da948cb228e121e03cb8f9affd75fdfecc52420bef2dcd031ad94791277590","family_id":"qf-density17-graph-adjacency-degrees","seed":202610175006,"semantic_template":"qf-density17-graph-adjacency-degrees/{\"edges\":[[0,1],[0,2],[0,3]],\"n\":4}"},"fields":{"title":"Complete Adjacency Matrix, Degrees, and Handshake Counts","stem":"Use a complete simple undirected graph on Fin(4)={0,…,3} with raw edges=[[0, 1], [0, 2], [0, 3]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Give (n, normalized edges, Boolean adjacency matrix, degree list, degree sum, twice the edge count).","answer":{"kind":"expression","value":"(4, [[0, 1], [0, 2], [0, 3]], [[No, Yes, Yes, Yes], [Yes, No, No, No], [Yes, No, No, No], [Yes, No, No, No]], [3, 1, 1, 1], 6, 6)","explanation":"Read the complete symmetric adjacency relation and count each distinct undirected edge once; this is not a connectivity or tree test. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [[0, 1], [0, 2], [0, 3]], [[No, Yes, Yes, Yes], [Yes, No, No, No], [Yes, No, No, No], [Yes, No, No, No]], [3, 1, 1, 1], 6, 6)."},"skills":["Complete Adjacency Matrix, Degrees, and Handshake Counts"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r05-graph-0007","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"abfea934946ba0eab6142ca1ecc8cee5763b41a466616e73b70fb2fdc158ff44","family_id":"qf-density17-graph-adjacency-degrees","seed":202610175007,"semantic_template":"qf-density17-graph-adjacency-degrees/{\"edges\":[[0,1]],\"n\":5}"},"fields":{"title":"Complete Adjacency Matrix, Degrees, and Handshake Counts","stem":"Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 1]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Give (n, normalized edges, Boolean adjacency matrix, degree list, degree sum, twice the edge count).","answer":{"kind":"expression","value":"(5, [[0, 1]], [[No, Yes, No, No, No], [Yes, No, No, No, No], [No, No, No, No, No], [No, No, No, No, No], [No, No, No, No, No]], [1, 1, 0, 0, 0], 2, 2)","explanation":"Read the complete symmetric adjacency relation and count each distinct undirected edge once; this is not a connectivity or tree test. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (5, [[0, 1]], [[No, Yes, No, No, No], [Yes, No, No, No, No], [No, No, No, No, No], [No, No, No, No, No], [No, No, No, No, No]], [1, 1, 0, 0, 0], 2, 2)."},"skills":["Complete Adjacency Matrix, Degrees, and Handshake Counts"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r05-graph-0008","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a5f8a2298e3c912256b0e73aa896d16ec6f5f8754d0cf8a05c7ef277e86265c8","family_id":"qf-density17-graph-adjacency-degrees","seed":202610175008,"semantic_template":"qf-density17-graph-adjacency-degrees/{\"edges\":[[0,2]],\"n\":5}"},"fields":{"title":"Complete Adjacency Matrix, Degrees, and Handshake Counts","stem":"Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 2]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Give (n, normalized edges, Boolean adjacency matrix, degree list, degree sum, twice the edge count).","answer":{"kind":"expression","value":"(5, [[0, 2]], [[No, No, Yes, No, No], [No, No, No, No, No], [Yes, No, No, No, No], [No, No, No, No, No], [No, No, No, No, No]], [1, 0, 1, 0, 0], 2, 2)","explanation":"Read the complete symmetric adjacency relation and count each distinct undirected edge once; this is not a connectivity or tree test. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (5, [[0, 2]], [[No, No, Yes, No, No], [No, No, No, No, No], [Yes, No, No, No, No], [No, No, No, No, No], [No, No, No, No, No]], [1, 0, 1, 0, 0], 2, 2)."},"skills":["Complete Adjacency Matrix, Degrees, and Handshake Counts"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r05-graph-0009","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"284c041ed959d9148e8739eb613cd079f61295f29b76601ef5e558a1fc1237b0","family_id":"qf-density17-graph-adjacency-degrees","seed":202610175009,"semantic_template":"qf-density17-graph-adjacency-degrees/{\"edges\":[[0,1],[0,2]],\"n\":5}"},"fields":{"title":"Complete Adjacency Matrix, Degrees, and Handshake Counts","stem":"Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 1], [0, 2]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Give (n, normalized edges, Boolean adjacency matrix, degree list, degree sum, twice the edge count).","answer":{"kind":"expression","value":"(5, [[0, 1], [0, 2]], [[No, Yes, Yes, No, No], [Yes, No, No, No, No], [Yes, No, No, No, No], [No, No, No, No, No], [No, No, No, No, No]], [2, 1, 1, 0, 0], 4, 4)","explanation":"Read the complete symmetric adjacency relation and count each distinct undirected edge once; this is not a connectivity or tree test. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (5, [[0, 1], [0, 2]], [[No, Yes, Yes, No, No], [Yes, No, No, No, No], [Yes, No, No, No, No], [No, No, No, No, No], [No, No, No, No, No]], [2, 1, 1, 0, 0], 4, 4)."},"skills":["Complete Adjacency Matrix, Degrees, and Handshake Counts"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r05-graph-0010","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6405db083616144e5d53cfb6e7661c025ca4881d74e0c07ad4312a02f06c11d6","family_id":"qf-density17-graph-adjacency-degrees","seed":202610175010,"semantic_template":"qf-density17-graph-adjacency-degrees/{\"edges\":[[0,3]],\"n\":5}"},"fields":{"title":"Complete Adjacency Matrix, Degrees, and Handshake Counts","stem":"Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 3]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Give (n, normalized edges, Boolean adjacency matrix, degree list, degree sum, twice the edge count).","answer":{"kind":"expression","value":"(5, [[0, 3]], [[No, No, No, Yes, No], [No, No, No, No, No], [No, No, No, No, No], [Yes, No, No, No, No], [No, No, No, No, No]], [1, 0, 0, 1, 0], 2, 2)","explanation":"Read the complete symmetric adjacency relation and count each distinct undirected edge once; this is not a connectivity or tree test. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (5, [[0, 3]], [[No, No, No, Yes, No], [No, No, No, No, No], [No, No, No, No, No], [Yes, No, No, No, No], [No, No, No, No, No]], [1, 0, 0, 1, 0], 2, 2)."},"skills":["Complete Adjacency Matrix, Degrees, and Handshake Counts"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r05-graph-0011","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d04d696f41b07e0a371e713798d8f87674e069a4c03d404fcc1efc99f247f70f","family_id":"qf-density17-graph-adjacency-degrees","seed":202610175011,"semantic_template":"qf-density17-graph-adjacency-degrees/{\"edges\":[[0,1],[0,3]],\"n\":5}"},"fields":{"title":"Complete Adjacency Matrix, Degrees, and Handshake Counts","stem":"Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 1], [0, 3]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Give (n, normalized edges, Boolean adjacency matrix, degree list, degree sum, twice the edge count).","answer":{"kind":"expression","value":"(5, [[0, 1], [0, 3]], [[No, Yes, No, Yes, No], [Yes, No, No, No, No], [No, No, No, No, No], [Yes, No, No, No, No], [No, No, No, No, No]], [2, 1, 0, 1, 0], 4, 4)","explanation":"Read the complete symmetric adjacency relation and count each distinct undirected edge once; this is not a connectivity or tree test. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (5, [[0, 1], [0, 3]], [[No, Yes, No, Yes, No], [Yes, No, No, No, No], [No, No, No, No, No], [Yes, No, No, No, No], [No, No, No, No, No]], [2, 1, 0, 1, 0], 4, 4)."},"skills":["Complete Adjacency Matrix, Degrees, and Handshake Counts"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r05-graph-0012","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c68053f626f730a9c53ce3008fdccb4b0d590c37b3216a0c735b8dc9b9e7f278","family_id":"qf-density17-graph-adjacency-degrees","seed":202610175012,"semantic_template":"qf-density17-graph-adjacency-degrees/{\"edges\":[[0,2],[0,3]],\"n\":5}"},"fields":{"title":"Complete Adjacency Matrix, Degrees, and Handshake Counts","stem":"Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 2], [0, 3]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Give (n, normalized edges, Boolean adjacency matrix, degree list, degree sum, twice the edge count).","answer":{"kind":"expression","value":"(5, [[0, 2], [0, 3]], [[No, No, Yes, Yes, No], [No, No, No, No, No], [Yes, No, No, No, No], [Yes, No, No, No, No], [No, No, No, No, No]], [2, 0, 1, 1, 0], 4, 4)","explanation":"Read the complete symmetric adjacency relation and count each distinct undirected edge once; this is not a connectivity or tree test. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (5, [[0, 2], [0, 3]], [[No, No, Yes, Yes, No], [No, No, No, No, No], [Yes, No, No, No, No], [Yes, No, No, No, No], [No, No, No, No, No]], [2, 0, 1, 1, 0], 4, 4)."},"skills":["Complete Adjacency Matrix, Degrees, and Handshake Counts"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r05-graph-0013","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b6538cbbb62728d496318f5908e661c448279dbf501a5290b29a972a750a8aca","family_id":"qf-density17-graph-adjacency-degrees","seed":202610175013,"semantic_template":"qf-density17-graph-adjacency-degrees/{\"edges\":[[0,1],[0,2],[0,3]],\"n\":5}"},"fields":{"title":"Complete Adjacency Matrix, Degrees, and Handshake Counts","stem":"Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 1], [0, 2], [0, 3]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Give (n, normalized edges, Boolean adjacency matrix, degree list, degree sum, twice the edge count).","answer":{"kind":"expression","value":"(5, [[0, 1], [0, 2], [0, 3]], [[No, Yes, Yes, Yes, No], [Yes, No, No, No, No], [Yes, No, No, No, No], [Yes, No, No, No, No], [No, No, No, No, No]], [3, 1, 1, 1, 0], 6, 6)","explanation":"Read the complete symmetric adjacency relation and count each distinct undirected edge once; this is not a connectivity or tree test. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (5, [[0, 1], [0, 2], [0, 3]], [[No, Yes, Yes, Yes, No], [Yes, No, No, No, No], [Yes, No, No, No, No], [Yes, No, No, No, No], [No, No, No, No, No]], [3, 1, 1, 1, 0], 6, 6)."},"skills":["Complete Adjacency Matrix, Degrees, and Handshake Counts"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r05-graph-0014","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d7b14ea615b75e3671002cef93f588e1428460b486e3d8b00933bf4ccbc9a05d","family_id":"qf-density17-graph-adjacency-degrees","seed":202610175014,"semantic_template":"qf-density17-graph-adjacency-degrees/{\"edges\":[[0,4]],\"n\":5}"},"fields":{"title":"Complete Adjacency Matrix, Degrees, and Handshake Counts","stem":"Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 4]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Give (n, normalized edges, Boolean adjacency matrix, degree list, degree sum, twice the edge count).","answer":{"kind":"expression","value":"(5, [[0, 4]], [[No, No, No, No, Yes], [No, No, No, No, No], [No, No, No, No, No], [No, No, No, No, No], [Yes, No, No, No, No]], [1, 0, 0, 0, 1], 2, 2)","explanation":"Read the complete symmetric adjacency relation and count each distinct undirected edge once; this is not a connectivity or tree test. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (5, [[0, 4]], [[No, No, No, No, Yes], [No, No, No, No, No], [No, No, No, No, No], [No, No, No, No, No], [Yes, No, No, No, No]], [1, 0, 0, 0, 1], 2, 2)."},"skills":["Complete Adjacency Matrix, Degrees, and Handshake Counts"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r05-graph-0015","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e4c4b761aad4bae3c08977d0ef6e331e6f0301a3136c66b832c465ea9c20791c","family_id":"qf-density17-graph-adjacency-degrees","seed":202610175015,"semantic_template":"qf-density17-graph-adjacency-degrees/{\"edges\":[[0,1],[0,4]],\"n\":5}"},"fields":{"title":"Complete Adjacency Matrix, Degrees, and Handshake Counts","stem":"Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 1], [0, 4]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Give (n, normalized edges, Boolean adjacency matrix, degree list, degree sum, twice the edge count).","answer":{"kind":"expression","value":"(5, [[0, 1], [0, 4]], [[No, Yes, No, No, Yes], [Yes, No, No, No, No], [No, No, No, No, No], [No, No, No, No, No], [Yes, No, No, No, No]], [2, 1, 0, 0, 1], 4, 4)","explanation":"Read the complete symmetric adjacency relation and count each distinct undirected edge once; this is not a connectivity or tree test. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (5, [[0, 1], [0, 4]], [[No, Yes, No, No, Yes], [Yes, No, No, No, No], [No, No, No, No, No], [No, No, No, No, No], [Yes, No, No, No, No]], [2, 1, 0, 0, 1], 4, 4)."},"skills":["Complete Adjacency Matrix, Degrees, and Handshake Counts"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r05-graph-0016","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cf8a71ffd9e68a2d90ffe186ea83b7f8f3a3c8580c64a4a6ac8a259e5142055f","family_id":"qf-density17-graph-adjacency-degrees","seed":202610175016,"semantic_template":"qf-density17-graph-adjacency-degrees/{\"edges\":[[0,2],[0,4]],\"n\":5}"},"fields":{"title":"Complete Adjacency Matrix, Degrees, and Handshake Counts","stem":"Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 2], [0, 4]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Give (n, normalized edges, Boolean adjacency matrix, degree list, degree sum, twice the edge count).","answer":{"kind":"expression","value":"(5, [[0, 2], [0, 4]], [[No, No, Yes, No, Yes], [No, No, No, No, No], [Yes, No, No, No, No], [No, No, No, No, No], [Yes, No, No, No, No]], [2, 0, 1, 0, 1], 4, 4)","explanation":"Read the complete symmetric adjacency relation and count each distinct undirected edge once; this is not a connectivity or tree test. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (5, [[0, 2], [0, 4]], [[No, No, Yes, No, Yes], [No, No, No, No, No], [Yes, No, No, No, No], [No, No, No, No, No], [Yes, No, No, No, No]], [2, 0, 1, 0, 1], 4, 4)."},"skills":["Complete Adjacency Matrix, Degrees, and Handshake Counts"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r05-graph-0017","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5597ead4e4240526b2d95eb4189071d64a6ba8c723a390fbcaefdf1e655fe0a5","family_id":"qf-density17-graph-adjacency-degrees","seed":202610175017,"semantic_template":"qf-density17-graph-adjacency-degrees/{\"edges\":[[0,1],[0,2],[0,4]],\"n\":5}"},"fields":{"title":"Complete Adjacency Matrix, Degrees, and Handshake Counts","stem":"Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 1], [0, 2], [0, 4]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Give (n, normalized edges, Boolean adjacency matrix, degree list, degree sum, twice the edge count).","answer":{"kind":"expression","value":"(5, [[0, 1], [0, 2], [0, 4]], [[No, Yes, Yes, No, Yes], [Yes, No, No, No, No], [Yes, No, No, No, No], [No, No, No, No, No], [Yes, No, No, No, No]], [3, 1, 1, 0, 1], 6, 6)","explanation":"Read the complete symmetric adjacency relation and count each distinct undirected edge once; this is not a connectivity or tree test. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (5, [[0, 1], [0, 2], [0, 4]], [[No, Yes, Yes, No, Yes], [Yes, No, No, No, No], [Yes, No, No, No, No], [No, No, No, No, No], [Yes, No, No, No, No]], [3, 1, 1, 0, 1], 6, 6)."},"skills":["Complete Adjacency Matrix, Degrees, and Handshake Counts"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r05-graph-0018","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1d7a4d1ff14ede9bfd6b28869cc3bf5a2e60b276e8819af927f7226671bce27f","family_id":"qf-density17-graph-adjacency-degrees","seed":202610175018,"semantic_template":"qf-density17-graph-adjacency-degrees/{\"edges\":[[0,3],[0,4]],\"n\":5}"},"fields":{"title":"Complete Adjacency Matrix, Degrees, and Handshake Counts","stem":"Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 3], [0, 4]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Give (n, normalized edges, Boolean adjacency matrix, degree list, degree sum, twice the edge count).","answer":{"kind":"expression","value":"(5, [[0, 3], [0, 4]], [[No, No, No, Yes, Yes], [No, No, No, No, No], [No, No, No, No, No], [Yes, No, No, No, No], [Yes, No, No, No, No]], [2, 0, 0, 1, 1], 4, 4)","explanation":"Read the complete symmetric adjacency relation and count each distinct undirected edge once; this is not a connectivity or tree test. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (5, [[0, 3], [0, 4]], [[No, No, No, Yes, Yes], [No, No, No, No, No], [No, No, No, No, No], [Yes, No, No, No, No], [Yes, No, No, No, No]], [2, 0, 0, 1, 1], 4, 4)."},"skills":["Complete Adjacency Matrix, Degrees, and Handshake Counts"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r05-graph-0019","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"71bfa61acf8ccf44d1fcd3d27a61d092f70e7911c459a880f9814ffbe7b67dc6","family_id":"qf-density17-graph-adjacency-degrees","seed":202610175019,"semantic_template":"qf-density17-graph-adjacency-degrees/{\"edges\":[[0,1],[0,3],[0,4]],\"n\":5}"},"fields":{"title":"Complete Adjacency Matrix, Degrees, and Handshake Counts","stem":"Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 1], [0, 3], [0, 4]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Give (n, normalized edges, Boolean adjacency matrix, degree list, degree sum, twice the edge count).","answer":{"kind":"expression","value":"(5, [[0, 1], [0, 3], [0, 4]], [[No, Yes, No, Yes, Yes], [Yes, No, No, No, No], [No, No, No, No, No], [Yes, No, No, No, No], [Yes, No, No, No, No]], [3, 1, 0, 1, 1], 6, 6)","explanation":"Read the complete symmetric adjacency relation and count each distinct undirected edge once; this is not a connectivity or tree test. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (5, [[0, 1], [0, 3], [0, 4]], [[No, Yes, No, Yes, Yes], [Yes, No, No, No, No], [No, No, No, No, No], [Yes, No, No, No, No], [Yes, No, No, No, No]], [3, 1, 0, 1, 1], 6, 6)."},"skills":["Complete Adjacency Matrix, Degrees, and Handshake Counts"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r05-graph-0020","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"166d84c384b4ce68d7182667a0e3463931d8effd30bf26f8830727f88e11824f","family_id":"qf-density17-graph-adjacency-degrees","seed":202610175020,"semantic_template":"qf-density17-graph-adjacency-degrees/{\"edges\":[[0,2],[0,3],[0,4]],\"n\":5}"},"fields":{"title":"Complete Adjacency Matrix, Degrees, and Handshake Counts","stem":"Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 2], [0, 3], [0, 4]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Give (n, normalized edges, Boolean adjacency matrix, degree list, degree sum, twice the edge count).","answer":{"kind":"expression","value":"(5, [[0, 2], [0, 3], [0, 4]], [[No, No, Yes, Yes, Yes], [No, No, No, No, No], [Yes, No, No, No, No], [Yes, No, No, No, No], [Yes, No, No, No, No]], [3, 0, 1, 1, 1], 6, 6)","explanation":"Read the complete symmetric adjacency relation and count each distinct undirected edge once; this is not a connectivity or tree test. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (5, [[0, 2], [0, 3], [0, 4]], [[No, No, Yes, Yes, Yes], [No, No, No, No, No], [Yes, No, No, No, No], [Yes, No, No, No, No], [Yes, No, No, No, No]], [3, 0, 1, 1, 1], 6, 6)."},"skills":["Complete Adjacency Matrix, Degrees, and Handshake Counts"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r05-graph-0021","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d41aa0084ca636afb442a84754b2dcb76d8e23d38102c8452c25d1272aa94763","family_id":"qf-density17-graph-adjacency-degrees","seed":202610175021,"semantic_template":"qf-density17-graph-adjacency-degrees/{\"edges\":[[0,1],[0,2],[0,3],[0,4]],\"n\":5}"},"fields":{"title":"Complete Adjacency Matrix, Degrees, and Handshake Counts","stem":"Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 1], [0, 2], [0, 3], [0, 4]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Give (n, normalized edges, Boolean adjacency matrix, degree list, degree sum, twice the edge count).","answer":{"kind":"expression","value":"(5, [[0, 1], [0, 2], [0, 3], [0, 4]], [[No, Yes, Yes, Yes, Yes], [Yes, No, No, No, No], [Yes, No, No, No, No], [Yes, No, No, No, No], [Yes, No, No, No, No]], [4, 1, 1, 1, 1], 8, 8)","explanation":"Read the complete symmetric adjacency relation and count each distinct undirected edge once; this is not a connectivity or tree test. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (5, [[0, 1], [0, 2], [0, 3], [0, 4]], [[No, Yes, Yes, Yes, Yes], [Yes, No, No, No, No], [Yes, No, No, No, No], [Yes, No, No, No, No], [Yes, No, No, No, No]], [4, 1, 1, 1, 1], 8, 8)."},"skills":["Complete Adjacency Matrix, Degrees, and Handshake Counts"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r05-graph-0022","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d40b7a6926887e1d39ad5a66e92de340d0325aec0d4c6836f73402f8ff6cee47","family_id":"qf-density17-graph-adjacency-degrees","seed":202610175022,"semantic_template":"qf-density17-graph-adjacency-degrees/{\"edges\":[[1,2]],\"n\":5}"},"fields":{"title":"Complete Adjacency Matrix, Degrees, and Handshake Counts","stem":"Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[1, 2]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Give (n, normalized edges, Boolean adjacency matrix, degree list, degree sum, twice the edge count).","answer":{"kind":"expression","value":"(5, [[1, 2]], [[No, No, No, No, No], [No, No, Yes, No, No], [No, Yes, No, No, No], [No, No, No, No, No], [No, No, No, No, No]], [0, 1, 1, 0, 0], 2, 2)","explanation":"Read the complete symmetric adjacency relation and count each distinct undirected edge once; this is not a connectivity or tree test. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (5, [[1, 2]], [[No, No, No, No, No], [No, No, Yes, No, No], [No, Yes, No, No, No], [No, No, No, No, No], [No, No, No, No, No]], [0, 1, 1, 0, 0], 2, 2)."},"skills":["Complete Adjacency Matrix, Degrees, and Handshake Counts"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r05-graph-0023","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a9f54a4aeb608916fb901af494b4bf2dd701286b2738b65a66dee337ce147eaa","family_id":"qf-density17-graph-adjacency-degrees","seed":202610175023,"semantic_template":"qf-density17-graph-adjacency-degrees/{\"edges\":[[0,1],[1,2]],\"n\":5}"},"fields":{"title":"Complete Adjacency Matrix, Degrees, and Handshake Counts","stem":"Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 1], [1, 2]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Give (n, normalized edges, Boolean adjacency matrix, degree list, degree sum, twice the edge count).","answer":{"kind":"expression","value":"(5, [[0, 1], [1, 2]], [[No, Yes, No, No, No], [Yes, No, Yes, No, No], [No, Yes, No, No, No], [No, No, No, No, No], [No, No, No, No, No]], [1, 2, 1, 0, 0], 4, 4)","explanation":"Read the complete symmetric adjacency relation and count each distinct undirected edge once; this is not a connectivity or tree test. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (5, [[0, 1], [1, 2]], [[No, Yes, No, No, No], [Yes, No, Yes, No, No], [No, Yes, No, No, No], [No, No, No, No, No], [No, No, No, No, No]], [1, 2, 1, 0, 0], 4, 4)."},"skills":["Complete Adjacency Matrix, Degrees, and Handshake Counts"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r05-graph-0024","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"feb9feec470e32f2c78990a055e4e5a5029b6b47d6fe88a0ae8010059eaa30ab","family_id":"qf-density17-graph-adjacency-degrees","seed":202610175024,"semantic_template":"qf-density17-graph-adjacency-degrees/{\"edges\":[[0,2],[1,2]],\"n\":5}"},"fields":{"title":"Complete Adjacency Matrix, Degrees, and Handshake Counts","stem":"Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 2], [1, 2]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Give (n, normalized edges, Boolean adjacency matrix, degree list, degree sum, twice the edge count).","answer":{"kind":"expression","value":"(5, [[0, 2], [1, 2]], [[No, No, Yes, No, No], [No, No, Yes, No, No], [Yes, Yes, No, No, No], [No, No, No, No, No], [No, No, No, No, No]], [1, 1, 2, 0, 0], 4, 4)","explanation":"Read the complete symmetric adjacency relation and count each distinct undirected edge once; this is not a connectivity or tree test. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (5, [[0, 2], [1, 2]], [[No, No, Yes, No, No], [No, No, Yes, No, No], [Yes, Yes, No, No, No], [No, No, No, No, No], [No, No, No, No, No]], [1, 1, 2, 0, 0], 4, 4)."},"skills":["Complete Adjacency Matrix, Degrees, and Handshake Counts"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r05-graph-0025","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1f023569d067375fce499c622979a5b7f2af1b8dba1d9e0cf33e664e53eaad46","family_id":"qf-density17-graph-adjacency-degrees","seed":202610175025,"semantic_template":"qf-density17-graph-adjacency-degrees/{\"edges\":[[0,1],[0,2],[1,2]],\"n\":5}"},"fields":{"title":"Complete Adjacency Matrix, Degrees, and Handshake Counts","stem":"Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 1], [0, 2], [1, 2]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Give (n, normalized edges, Boolean adjacency matrix, degree list, degree sum, twice the edge count).","answer":{"kind":"expression","value":"(5, [[0, 1], [0, 2], [1, 2]], [[No, Yes, Yes, No, No], [Yes, No, Yes, No, No], [Yes, Yes, No, No, No], [No, No, No, No, No], [No, No, No, No, No]], [2, 2, 2, 0, 0], 6, 6)","explanation":"Read the complete symmetric adjacency relation and count each distinct undirected edge once; this is not a connectivity or tree test. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (5, [[0, 1], [0, 2], [1, 2]], [[No, Yes, Yes, No, No], [Yes, No, Yes, No, No], [Yes, Yes, No, No, No], [No, No, No, No, No], [No, No, No, No, No]], [2, 2, 2, 0, 0], 6, 6)."},"skills":["Complete Adjacency Matrix, Degrees, and Handshake Counts"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r05-graph-0026","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ee62368407076b3d1e851789f4f51d43d0f513d59390a504e36bf1d7924647bf","family_id":"qf-density17-graph-complete-neighbor-partitions","seed":202610175026,"semantic_template":"qf-density17-graph-complete-neighbor-partitions/{\"edges\":[],\"n\":0}"},"fields":{"title":"Every Vertex Neighbor Partition and Isolated Vertices","stem":"Use a complete simple undirected graph on Fin(0)=∅ with raw edges=[]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Give (n, normalized edges, each vertex’s neighbors, each vertex’s distinct nonneighbors excluding itself, isolated vertices).","answer":{"kind":"expression","value":"(0, [], [], [], [])","explanation":"Partition the other vertices at each original label; isolated vertices have degree zero. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (0, [], [], [], [])."},"skills":["Every Vertex Neighbor Partition and Isolated Vertices"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r05-graph-0027","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2c0a6df4dcea8a37945ee7ce4161f7eb5953c64ee36ca408008d4e66d37d226f","family_id":"qf-density17-graph-complete-neighbor-partitions","seed":202610175027,"semantic_template":"qf-density17-graph-complete-neighbor-partitions/{\"edges\":[],\"n\":1}"},"fields":{"title":"Every Vertex Neighbor Partition and Isolated Vertices","stem":"Use a complete simple undirected graph on Fin(1)={0,…,0} with raw edges=[]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Give (n, normalized edges, each vertex’s neighbors, each vertex’s distinct nonneighbors excluding itself, isolated vertices).","answer":{"kind":"expression","value":"(1, [], [[]], [[]], [0])","explanation":"Partition the other vertices at each original label; isolated vertices have degree zero. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (1, [], [[]], [[]], [0])."},"skills":["Every Vertex Neighbor Partition and Isolated Vertices"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r05-graph-0028","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"45024e2cee19e8de0811acca9793a48a9ce1b3ea93c646a1c16502f30f715a32","family_id":"qf-density17-graph-complete-neighbor-partitions","seed":202610175028,"semantic_template":"qf-density17-graph-complete-neighbor-partitions/{\"edges\":[[1,2],[1,3],[2,3]],\"n\":4}"},"fields":{"title":"Every Vertex Neighbor Partition and Isolated Vertices","stem":"Use a complete simple undirected graph on Fin(4)={0,…,3} with raw edges=[[1, 2], [1, 3], [2, 3]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Give (n, normalized edges, each vertex’s neighbors, each vertex’s distinct nonneighbors excluding itself, isolated vertices).","answer":{"kind":"expression","value":"(4, [[1, 2], [1, 3], [2, 3]], [[], [2, 3], [1, 3], [1, 2]], [[1, 2, 3], [0], [0], [0]], [0])","explanation":"Partition the other vertices at each original label; isolated vertices have degree zero. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [[1, 2], [1, 3], [2, 3]], [[], [2, 3], [1, 3], [1, 2]], [[1, 2, 3], [0], [0], [0]], [0])."},"skills":["Every Vertex Neighbor Partition and Isolated Vertices"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r05-graph-0029","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"52e5d570f040345a507e552d0388d781abe849a5fdbc5f4e0b2c8f41513a218b","family_id":"qf-density17-graph-complete-neighbor-partitions","seed":202610175029,"semantic_template":"qf-density17-graph-complete-neighbor-partitions/{\"edges\":[[0,1],[2,3]],\"n\":4}"},"fields":{"title":"Every Vertex Neighbor Partition and Isolated Vertices","stem":"Use a complete simple undirected graph on Fin(4)={0,…,3} with raw edges=[[0, 1], [2, 3]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Give (n, normalized edges, each vertex’s neighbors, each vertex’s distinct nonneighbors excluding itself, isolated vertices).","answer":{"kind":"expression","value":"(4, [[0, 1], [2, 3]], [[1], [0], [3], [2]], [[2, 3], [2, 3], [0, 1], [0, 1]], [])","explanation":"Partition the other vertices at each original label; isolated vertices have degree zero. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [[0, 1], [2, 3]], [[1], [0], [3], [2]], [[2, 3], [2, 3], [0, 1], [0, 1]], [])."},"skills":["Every Vertex Neighbor Partition and Isolated Vertices"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r05-graph-0030","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"da571ac92de04cd1810bbf0c7c5107515d1eef3109e7e6cdffe77edd12400f38","family_id":"qf-density17-graph-complete-neighbor-partitions","seed":202610175030,"semantic_template":"qf-density17-graph-complete-neighbor-partitions/{\"edges\":[[0,1],[1,2],[2,3]],\"n\":4}"},"fields":{"title":"Every Vertex Neighbor Partition and Isolated Vertices","stem":"Use a complete simple undirected graph on Fin(4)={0,…,3} with raw edges=[[0, 1], [1, 2], [2, 3]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Give (n, normalized edges, each vertex’s neighbors, each vertex’s distinct nonneighbors excluding itself, isolated vertices).","answer":{"kind":"expression","value":"(4, [[0, 1], [1, 2], [2, 3]], [[1], [0, 2], [1, 3], [2]], [[2, 3], [3], [0], [0, 1]], [])","explanation":"Partition the other vertices at each original label; isolated vertices have degree zero. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [[0, 1], [1, 2], [2, 3]], [[1], [0, 2], [1, 3], [2]], [[2, 3], [3], [0], [0, 1]], [])."},"skills":["Every Vertex Neighbor Partition and Isolated Vertices"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r05-graph-0031","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6833cf5d12be7ebc3e148e0f9ba9764e2178624f9011aac0f5629e406d52b862","family_id":"qf-density17-graph-complete-neighbor-partitions","seed":202610175031,"semantic_template":"qf-density17-graph-complete-neighbor-partitions/{\"edges\":[[0,1],[0,2],[0,3]],\"n\":4}"},"fields":{"title":"Every Vertex Neighbor Partition and Isolated Vertices","stem":"Use a complete simple undirected graph on Fin(4)={0,…,3} with raw edges=[[0, 1], [0, 2], [0, 3]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Give (n, normalized edges, each vertex’s neighbors, each vertex’s distinct nonneighbors excluding itself, isolated vertices).","answer":{"kind":"expression","value":"(4, [[0, 1], [0, 2], [0, 3]], [[1, 2, 3], [0], [0], [0]], [[], [2, 3], [1, 3], [1, 2]], [])","explanation":"Partition the other vertices at each original label; isolated vertices have degree zero. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [[0, 1], [0, 2], [0, 3]], [[1, 2, 3], [0], [0], [0]], [[], [2, 3], [1, 3], [1, 2]], [])."},"skills":["Every Vertex Neighbor Partition and Isolated Vertices"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r05-graph-0032","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4f99c656b41ce1fcea44d2c5e9fe0caf7746a235e50fa136a5f0822f0ee26945","family_id":"qf-density17-graph-complete-neighbor-partitions","seed":202610175032,"semantic_template":"qf-density17-graph-complete-neighbor-partitions/{\"edges\":[[0,1]],\"n\":5}"},"fields":{"title":"Every Vertex Neighbor Partition and Isolated Vertices","stem":"Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 1]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Give (n, normalized edges, each vertex’s neighbors, each vertex’s distinct nonneighbors excluding itself, isolated vertices).","answer":{"kind":"expression","value":"(5, [[0, 1]], [[1], [0], [], [], []], [[2, 3, 4], [2, 3, 4], [0, 1, 3, 4], [0, 1, 2, 4], [0, 1, 2, 3]], [2, 3, 4])","explanation":"Partition the other vertices at each original label; isolated vertices have degree zero. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (5, [[0, 1]], [[1], [0], [], [], []], [[2, 3, 4], [2, 3, 4], [0, 1, 3, 4], [0, 1, 2, 4], [0, 1, 2, 3]], [2, 3, 4])."},"skills":["Every Vertex Neighbor Partition and Isolated Vertices"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r05-graph-0033","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ec9b2c74324f62cd3c04c8e6113af89f56846a9f80a4179de329f1e55348e2e3","family_id":"qf-density17-graph-complete-neighbor-partitions","seed":202610175033,"semantic_template":"qf-density17-graph-complete-neighbor-partitions/{\"edges\":[[0,2]],\"n\":5}"},"fields":{"title":"Every Vertex Neighbor Partition and Isolated Vertices","stem":"Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 2]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Give (n, normalized edges, each vertex’s neighbors, each vertex’s distinct nonneighbors excluding itself, isolated vertices).","answer":{"kind":"expression","value":"(5, [[0, 2]], [[2], [], [0], [], []], [[1, 3, 4], [0, 2, 3, 4], [1, 3, 4], [0, 1, 2, 4], [0, 1, 2, 3]], [1, 3, 4])","explanation":"Partition the other vertices at each original label; isolated vertices have degree zero. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (5, [[0, 2]], [[2], [], [0], [], []], [[1, 3, 4], [0, 2, 3, 4], [1, 3, 4], [0, 1, 2, 4], [0, 1, 2, 3]], [1, 3, 4])."},"skills":["Every Vertex Neighbor Partition and Isolated Vertices"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r05-graph-0034","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bce879d14c9a28bf84f3976ae6c607aa30afc3bf2136b44dbf79dfa5a2edff69","family_id":"qf-density17-graph-complete-neighbor-partitions","seed":202610175034,"semantic_template":"qf-density17-graph-complete-neighbor-partitions/{\"edges\":[[0,1],[0,2]],\"n\":5}"},"fields":{"title":"Every Vertex Neighbor Partition and Isolated Vertices","stem":"Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 1], [0, 2]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Give (n, normalized edges, each vertex’s neighbors, each vertex’s distinct nonneighbors excluding itself, isolated vertices).","answer":{"kind":"expression","value":"(5, [[0, 1], [0, 2]], [[1, 2], [0], [0], [], []], [[3, 4], [2, 3, 4], [1, 3, 4], [0, 1, 2, 4], [0, 1, 2, 3]], [3, 4])","explanation":"Partition the other vertices at each original label; isolated vertices have degree zero. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (5, [[0, 1], [0, 2]], [[1, 2], [0], [0], [], []], [[3, 4], [2, 3, 4], [1, 3, 4], [0, 1, 2, 4], [0, 1, 2, 3]], [3, 4])."},"skills":["Every Vertex Neighbor Partition and Isolated Vertices"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r05-graph-0035","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d24ac094aaf3220aa1c2e4d47dad0f66d3c69794ff7c909e4d34bafbef2b8098","family_id":"qf-density17-graph-complete-neighbor-partitions","seed":202610175035,"semantic_template":"qf-density17-graph-complete-neighbor-partitions/{\"edges\":[[0,3]],\"n\":5}"},"fields":{"title":"Every Vertex Neighbor Partition and Isolated Vertices","stem":"Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 3]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Give (n, normalized edges, each vertex’s neighbors, each vertex’s distinct nonneighbors excluding itself, isolated vertices).","answer":{"kind":"expression","value":"(5, [[0, 3]], [[3], [], [], [0], []], [[1, 2, 4], [0, 2, 3, 4], [0, 1, 3, 4], [1, 2, 4], [0, 1, 2, 3]], [1, 2, 4])","explanation":"Partition the other vertices at each original label; isolated vertices have degree zero. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (5, [[0, 3]], [[3], [], [], [0], []], [[1, 2, 4], [0, 2, 3, 4], [0, 1, 3, 4], [1, 2, 4], [0, 1, 2, 3]], [1, 2, 4])."},"skills":["Every Vertex Neighbor Partition and Isolated Vertices"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r05-graph-0036","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f65f860d5992b6981989b3a352555f21ea2d62689042536dbf617d3578450989","family_id":"qf-density17-graph-complete-neighbor-partitions","seed":202610175036,"semantic_template":"qf-density17-graph-complete-neighbor-partitions/{\"edges\":[[0,1],[0,3]],\"n\":5}"},"fields":{"title":"Every Vertex Neighbor Partition and Isolated Vertices","stem":"Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 1], [0, 3]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Give (n, normalized edges, each vertex’s neighbors, each vertex’s distinct nonneighbors excluding itself, isolated vertices).","answer":{"kind":"expression","value":"(5, [[0, 1], [0, 3]], [[1, 3], [0], [], [0], []], [[2, 4], [2, 3, 4], [0, 1, 3, 4], [1, 2, 4], [0, 1, 2, 3]], [2, 4])","explanation":"Partition the other vertices at each original label; isolated vertices have degree zero. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (5, [[0, 1], [0, 3]], [[1, 3], [0], [], [0], []], [[2, 4], [2, 3, 4], [0, 1, 3, 4], [1, 2, 4], [0, 1, 2, 3]], [2, 4])."},"skills":["Every Vertex Neighbor Partition and Isolated Vertices"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r05-graph-0037","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"779bda9767d32ecf626df7c387e5e5d9b4d02114139964b5209ca7b5480b974a","family_id":"qf-density17-graph-complete-neighbor-partitions","seed":202610175037,"semantic_template":"qf-density17-graph-complete-neighbor-partitions/{\"edges\":[[0,2],[0,3]],\"n\":5}"},"fields":{"title":"Every Vertex Neighbor Partition and Isolated Vertices","stem":"Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 2], [0, 3]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Give (n, normalized edges, each vertex’s neighbors, each vertex’s distinct nonneighbors excluding itself, isolated vertices).","answer":{"kind":"expression","value":"(5, [[0, 2], [0, 3]], [[2, 3], [], [0], [0], []], [[1, 4], [0, 2, 3, 4], [1, 3, 4], [1, 2, 4], [0, 1, 2, 3]], [1, 4])","explanation":"Partition the other vertices at each original label; isolated vertices have degree zero. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (5, [[0, 2], [0, 3]], [[2, 3], [], [0], [0], []], [[1, 4], [0, 2, 3, 4], [1, 3, 4], [1, 2, 4], [0, 1, 2, 3]], [1, 4])."},"skills":["Every Vertex Neighbor Partition and Isolated Vertices"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r05-graph-0038","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"38de83f7e38bca402d8b2a18df037ff2ba0dd541e7fd2032c0ac71280813a2e2","family_id":"qf-density17-graph-complete-neighbor-partitions","seed":202610175038,"semantic_template":"qf-density17-graph-complete-neighbor-partitions/{\"edges\":[[0,1],[0,2],[0,3]],\"n\":5}"},"fields":{"title":"Every Vertex Neighbor Partition and Isolated Vertices","stem":"Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 1], [0, 2], [0, 3]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Give (n, normalized edges, each vertex’s neighbors, each vertex’s distinct nonneighbors excluding itself, isolated vertices).","answer":{"kind":"expression","value":"(5, [[0, 1], [0, 2], [0, 3]], [[1, 2, 3], [0], [0], [0], []], [[4], [2, 3, 4], [1, 3, 4], [1, 2, 4], [0, 1, 2, 3]], [4])","explanation":"Partition the other vertices at each original label; isolated vertices have degree zero. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (5, [[0, 1], [0, 2], [0, 3]], [[1, 2, 3], [0], [0], [0], []], [[4], [2, 3, 4], [1, 3, 4], [1, 2, 4], [0, 1, 2, 3]], [4])."},"skills":["Every Vertex Neighbor Partition and Isolated Vertices"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r05-graph-0039","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8d2ce8fc59613e958058bd7d613b6f3c9ba27ab77ddc223c4acd388ee24f57b8","family_id":"qf-density17-graph-complete-neighbor-partitions","seed":202610175039,"semantic_template":"qf-density17-graph-complete-neighbor-partitions/{\"edges\":[[0,4]],\"n\":5}"},"fields":{"title":"Every Vertex Neighbor Partition and Isolated Vertices","stem":"Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 4]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Give (n, normalized edges, each vertex’s neighbors, each vertex’s distinct nonneighbors excluding itself, isolated vertices).","answer":{"kind":"expression","value":"(5, [[0, 4]], [[4], [], [], [], [0]], [[1, 2, 3], [0, 2, 3, 4], [0, 1, 3, 4], [0, 1, 2, 4], [1, 2, 3]], [1, 2, 3])","explanation":"Partition the other vertices at each original label; isolated vertices have degree zero. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (5, [[0, 4]], [[4], [], [], [], [0]], [[1, 2, 3], [0, 2, 3, 4], [0, 1, 3, 4], [0, 1, 2, 4], [1, 2, 3]], [1, 2, 3])."},"skills":["Every Vertex Neighbor Partition and Isolated Vertices"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r05-graph-0040","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a83378c68dd1b96a21f53b097d9bbd5d1d6dc46bedbeb750549f88dd352b6010","family_id":"qf-density17-graph-complete-neighbor-partitions","seed":202610175040,"semantic_template":"qf-density17-graph-complete-neighbor-partitions/{\"edges\":[[0,1],[0,4]],\"n\":5}"},"fields":{"title":"Every Vertex Neighbor Partition and Isolated Vertices","stem":"Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 1], [0, 4]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Give (n, normalized edges, each vertex’s neighbors, each vertex’s distinct nonneighbors excluding itself, isolated vertices).","answer":{"kind":"expression","value":"(5, [[0, 1], [0, 4]], [[1, 4], [0], [], [], [0]], [[2, 3], [2, 3, 4], [0, 1, 3, 4], [0, 1, 2, 4], [1, 2, 3]], [2, 3])","explanation":"Partition the other vertices at each original label; isolated vertices have degree zero. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (5, [[0, 1], [0, 4]], [[1, 4], [0], [], [], [0]], [[2, 3], [2, 3, 4], [0, 1, 3, 4], [0, 1, 2, 4], [1, 2, 3]], [2, 3])."},"skills":["Every Vertex Neighbor Partition and Isolated Vertices"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r05-graph-0041","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5068f1aa44a96d2c6bd5ef5796bd821043c436a131826b2d0d40b66afd9d0a1c","family_id":"qf-density17-graph-complete-neighbor-partitions","seed":202610175041,"semantic_template":"qf-density17-graph-complete-neighbor-partitions/{\"edges\":[[0,2],[0,4]],\"n\":5}"},"fields":{"title":"Every Vertex Neighbor Partition and Isolated Vertices","stem":"Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 2], [0, 4]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Give (n, normalized edges, each vertex’s neighbors, each vertex’s distinct nonneighbors excluding itself, isolated vertices).","answer":{"kind":"expression","value":"(5, [[0, 2], [0, 4]], [[2, 4], [], [0], [], [0]], [[1, 3], [0, 2, 3, 4], [1, 3, 4], [0, 1, 2, 4], [1, 2, 3]], [1, 3])","explanation":"Partition the other vertices at each original label; isolated vertices have degree zero. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (5, [[0, 2], [0, 4]], [[2, 4], [], [0], [], [0]], [[1, 3], [0, 2, 3, 4], [1, 3, 4], [0, 1, 2, 4], [1, 2, 3]], [1, 3])."},"skills":["Every Vertex Neighbor Partition and Isolated Vertices"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r05-graph-0042","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5566667531d6724654c5291aad2a6356a5b53543604499da3e050de2d5ed3b0c","family_id":"qf-density17-graph-complete-neighbor-partitions","seed":202610175042,"semantic_template":"qf-density17-graph-complete-neighbor-partitions/{\"edges\":[[0,1],[0,2],[0,4]],\"n\":5}"},"fields":{"title":"Every Vertex Neighbor Partition and Isolated Vertices","stem":"Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 1], [0, 2], [0, 4]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Give (n, normalized edges, each vertex’s neighbors, each vertex’s distinct nonneighbors excluding itself, isolated vertices).","answer":{"kind":"expression","value":"(5, [[0, 1], [0, 2], [0, 4]], [[1, 2, 4], [0], [0], [], [0]], [[3], [2, 3, 4], [1, 3, 4], [0, 1, 2, 4], [1, 2, 3]], [3])","explanation":"Partition the other vertices at each original label; isolated vertices have degree zero. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (5, [[0, 1], [0, 2], [0, 4]], [[1, 2, 4], [0], [0], [], [0]], [[3], [2, 3, 4], [1, 3, 4], [0, 1, 2, 4], [1, 2, 3]], [3])."},"skills":["Every Vertex Neighbor Partition and Isolated Vertices"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r05-graph-0043","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cf255e4e5f67291495e6c279fd688ed21684f5a06111e77db65bb05fad7420c0","family_id":"qf-density17-graph-complete-neighbor-partitions","seed":202610175043,"semantic_template":"qf-density17-graph-complete-neighbor-partitions/{\"edges\":[[0,3],[0,4]],\"n\":5}"},"fields":{"title":"Every Vertex Neighbor Partition and Isolated Vertices","stem":"Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 3], [0, 4]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Give (n, normalized edges, each vertex’s neighbors, each vertex’s distinct nonneighbors excluding itself, isolated vertices).","answer":{"kind":"expression","value":"(5, [[0, 3], [0, 4]], [[3, 4], [], [], [0], [0]], [[1, 2], [0, 2, 3, 4], [0, 1, 3, 4], [1, 2, 4], [1, 2, 3]], [1, 2])","explanation":"Partition the other vertices at each original label; isolated vertices have degree zero. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (5, [[0, 3], [0, 4]], [[3, 4], [], [], [0], [0]], [[1, 2], [0, 2, 3, 4], [0, 1, 3, 4], [1, 2, 4], [1, 2, 3]], [1, 2])."},"skills":["Every Vertex Neighbor Partition and Isolated Vertices"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r05-graph-0044","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ef7924e0f68dcaa122f7e8d8404f7e0836e6da030e7d86927d138409f93cd368","family_id":"qf-density17-graph-complete-neighbor-partitions","seed":202610175044,"semantic_template":"qf-density17-graph-complete-neighbor-partitions/{\"edges\":[[0,1],[0,3],[0,4]],\"n\":5}"},"fields":{"title":"Every Vertex Neighbor Partition and Isolated Vertices","stem":"Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 1], [0, 3], [0, 4]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Give (n, normalized edges, each vertex’s neighbors, each vertex’s distinct nonneighbors excluding itself, isolated vertices).","answer":{"kind":"expression","value":"(5, [[0, 1], [0, 3], [0, 4]], [[1, 3, 4], [0], [], [0], [0]], [[2], [2, 3, 4], [0, 1, 3, 4], [1, 2, 4], [1, 2, 3]], [2])","explanation":"Partition the other vertices at each original label; isolated vertices have degree zero. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (5, [[0, 1], [0, 3], [0, 4]], [[1, 3, 4], [0], [], [0], [0]], [[2], [2, 3, 4], [0, 1, 3, 4], [1, 2, 4], [1, 2, 3]], [2])."},"skills":["Every Vertex Neighbor Partition and Isolated Vertices"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r05-graph-0045","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"59f75fdb4c68e6343d725af07bba0c341924b5a5a2f368de735b8daa591ba2b6","family_id":"qf-density17-graph-complete-neighbor-partitions","seed":202610175045,"semantic_template":"qf-density17-graph-complete-neighbor-partitions/{\"edges\":[[0,2],[0,3],[0,4]],\"n\":5}"},"fields":{"title":"Every Vertex Neighbor Partition and Isolated Vertices","stem":"Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 2], [0, 3], [0, 4]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Give (n, normalized edges, each vertex’s neighbors, each vertex’s distinct nonneighbors excluding itself, isolated vertices).","answer":{"kind":"expression","value":"(5, [[0, 2], [0, 3], [0, 4]], [[2, 3, 4], [], [0], [0], [0]], [[1], [0, 2, 3, 4], [1, 3, 4], [1, 2, 4], [1, 2, 3]], [1])","explanation":"Partition the other vertices at each original label; isolated vertices have degree zero. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (5, [[0, 2], [0, 3], [0, 4]], [[2, 3, 4], [], [0], [0], [0]], [[1], [0, 2, 3, 4], [1, 3, 4], [1, 2, 4], [1, 2, 3]], [1])."},"skills":["Every Vertex Neighbor Partition and Isolated Vertices"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r05-graph-0046","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d9238b36d4b6a4b5a51fd8e758440264e0700e87d7dc7433a0222fca6091224b","family_id":"qf-density17-graph-complete-neighbor-partitions","seed":202610175046,"semantic_template":"qf-density17-graph-complete-neighbor-partitions/{\"edges\":[[0,1],[0,2],[0,3],[0,4]],\"n\":5}"},"fields":{"title":"Every Vertex Neighbor Partition and Isolated Vertices","stem":"Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 1], [0, 2], [0, 3], [0, 4]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Give (n, normalized edges, each vertex’s neighbors, each vertex’s distinct nonneighbors excluding itself, isolated vertices).","answer":{"kind":"expression","value":"(5, [[0, 1], [0, 2], [0, 3], [0, 4]], [[1, 2, 3, 4], [0], [0], [0], [0]], [[], [2, 3, 4], [1, 3, 4], [1, 2, 4], [1, 2, 3]], [])","explanation":"Partition the other vertices at each original label; isolated vertices have degree zero. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (5, [[0, 1], [0, 2], [0, 3], [0, 4]], [[1, 2, 3, 4], [0], [0], [0], [0]], [[], [2, 3, 4], [1, 3, 4], [1, 2, 4], [1, 2, 3]], [])."},"skills":["Every Vertex Neighbor Partition and Isolated Vertices"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r05-graph-0047","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"252e53d920174d14daffc2aa574f4760188fa65fa5f8fb63e6e8b5dffd9a91f7","family_id":"qf-density17-graph-complete-neighbor-partitions","seed":202610175047,"semantic_template":"qf-density17-graph-complete-neighbor-partitions/{\"edges\":[[1,2]],\"n\":5}"},"fields":{"title":"Every Vertex Neighbor Partition and Isolated Vertices","stem":"Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[1, 2]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Give (n, normalized edges, each vertex’s neighbors, each vertex’s distinct nonneighbors excluding itself, isolated vertices).","answer":{"kind":"expression","value":"(5, [[1, 2]], [[], [2], [1], [], []], [[1, 2, 3, 4], [0, 3, 4], [0, 3, 4], [0, 1, 2, 4], [0, 1, 2, 3]], [0, 3, 4])","explanation":"Partition the other vertices at each original label; isolated vertices have degree zero. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (5, [[1, 2]], [[], [2], [1], [], []], [[1, 2, 3, 4], [0, 3, 4], [0, 3, 4], [0, 1, 2, 4], [0, 1, 2, 3]], [0, 3, 4])."},"skills":["Every Vertex Neighbor Partition and Isolated Vertices"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r05-graph-0048","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"14dcaaa378398fb76eaeab916eb8d3ea1c4c8a6b057a100c6a02696b23b16d1c","family_id":"qf-density17-graph-complete-neighbor-partitions","seed":202610175048,"semantic_template":"qf-density17-graph-complete-neighbor-partitions/{\"edges\":[[0,1],[1,2]],\"n\":5}"},"fields":{"title":"Every Vertex Neighbor Partition and Isolated Vertices","stem":"Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 1], [1, 2]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Give (n, normalized edges, each vertex’s neighbors, each vertex’s distinct nonneighbors excluding itself, isolated vertices).","answer":{"kind":"expression","value":"(5, [[0, 1], [1, 2]], [[1], [0, 2], [1], [], []], [[2, 3, 4], [3, 4], [0, 3, 4], [0, 1, 2, 4], [0, 1, 2, 3]], [3, 4])","explanation":"Partition the other vertices at each original label; isolated vertices have degree zero. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (5, [[0, 1], [1, 2]], [[1], [0, 2], [1], [], []], [[2, 3, 4], [3, 4], [0, 3, 4], [0, 1, 2, 4], [0, 1, 2, 3]], [3, 4])."},"skills":["Every Vertex Neighbor Partition and Isolated Vertices"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r05-graph-0049","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c261dee30c2f90331a0f9ae8722a034337667975f1dcc74e4709f72bcc84f3a8","family_id":"qf-density17-graph-complete-neighbor-partitions","seed":202610175049,"semantic_template":"qf-density17-graph-complete-neighbor-partitions/{\"edges\":[[0,2],[1,2]],\"n\":5}"},"fields":{"title":"Every Vertex Neighbor Partition and Isolated Vertices","stem":"Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 2], [1, 2]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Give (n, normalized edges, each vertex’s neighbors, each vertex’s distinct nonneighbors excluding itself, isolated vertices).","answer":{"kind":"expression","value":"(5, [[0, 2], [1, 2]], [[2], [2], [0, 1], [], []], [[1, 3, 4], [0, 3, 4], [3, 4], [0, 1, 2, 4], [0, 1, 2, 3]], [3, 4])","explanation":"Partition the other vertices at each original label; isolated vertices have degree zero. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (5, [[0, 2], [1, 2]], [[2], [2], [0, 1], [], []], [[1, 3, 4], [0, 3, 4], [3, 4], [0, 1, 2, 4], [0, 1, 2, 3]], [3, 4])."},"skills":["Every Vertex Neighbor Partition and Isolated Vertices"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r05-graph-0050","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6d5cb9ec667203a150654435d26a5c636b3138e202167d4e289e0f0ba9d7f1ff","family_id":"qf-density17-graph-complete-neighbor-partitions","seed":202610175050,"semantic_template":"qf-density17-graph-complete-neighbor-partitions/{\"edges\":[[0,1],[0,2],[1,2]],\"n\":5}"},"fields":{"title":"Every Vertex Neighbor Partition and Isolated Vertices","stem":"Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 1], [0, 2], [1, 2]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Give (n, normalized edges, each vertex’s neighbors, each vertex’s distinct nonneighbors excluding itself, isolated vertices).","answer":{"kind":"expression","value":"(5, [[0, 1], [0, 2], [1, 2]], [[1, 2], [0, 2], [0, 1], [], []], [[3, 4], [3, 4], [3, 4], [0, 1, 2, 4], [0, 1, 2, 3]], [3, 4])","explanation":"Partition the other vertices at each original label; isolated vertices have degree zero. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (5, [[0, 1], [0, 2], [1, 2]], [[1, 2], [0, 2], [0, 1], [], []], [[3, 4], [3, 4], [3, 4], [0, 1, 2, 4], [0, 1, 2, 3]], [3, 4])."},"skills":["Every Vertex Neighbor Partition and Isolated Vertices"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r05-graph-0051","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b619efbe7fa084833fb39c3efb07b7f3476e32215281d765caa3638d8067b1bd","family_id":"qf-density17-graph-edge-toggle-delta","seed":202610175051,"semantic_template":"qf-density17-graph-edge-toggle-delta/{\"edge\":[0,1],\"edges\":[[0,1]],\"n\":2}"},"fields":{"title":"Toggle a Typed Nonloop Edge and Compute Degree Changes","stem":"Use a complete simple undirected graph on Fin(2)={0,…,1} with raw edges=[[0, 1]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Toggle edge=[1, 0]. Give (original normalized edges, normalized toggled edge, removed/added action, new edges, old degrees, new degrees, degree differences, degree-sum difference).","answer":{"kind":"expression","value":"([[0, 1]], [0, 1], removed, [], [1, 1], [0, 0], [-1, -1], -2)","explanation":"Remove an existing edge or insert an absent one, then independently count both degree lists; no loop or invalid endpoint is admitted. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, 1]], [0, 1], removed, [], [1, 1], [0, 0], [-1, -1], -2)."},"skills":["Toggle a Typed Nonloop Edge and Compute Degree Changes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r05-graph-0052","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a14a54dc4eeb157bc4616e271357d044c72bfc7ac8ecf7bd9484e3854c1678ce","family_id":"qf-density17-graph-edge-toggle-delta","seed":202610175052,"semantic_template":"qf-density17-graph-edge-toggle-delta/{\"edge\":[0,1],\"edges\":[],\"n\":2}"},"fields":{"title":"Toggle a Typed Nonloop Edge and Compute Degree Changes","stem":"Use a complete simple undirected graph on Fin(2)={0,…,1} with raw edges=[]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Toggle edge=[0, 1]. Give (original normalized edges, normalized toggled edge, removed/added action, new edges, old degrees, new degrees, degree differences, degree-sum difference).","answer":{"kind":"expression","value":"([], [0, 1], added, [[0, 1]], [0, 0], [1, 1], [1, 1], 2)","explanation":"Remove an existing edge or insert an absent one, then independently count both degree lists; no loop or invalid endpoint is admitted. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([], [0, 1], added, [[0, 1]], [0, 0], [1, 1], [1, 1], 2)."},"skills":["Toggle a Typed Nonloop Edge and Compute Degree Changes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r05-graph-0053","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"17b44e46dbb8d415d13f67c4cc05702560907792de6a9ebb3d8dfd92f7315f05","family_id":"qf-density17-graph-edge-toggle-delta","seed":202610175053,"semantic_template":"qf-density17-graph-edge-toggle-delta/{\"edge\":[0,2],\"edges\":[[0,1],[1,2]],\"n\":3}"},"fields":{"title":"Toggle a Typed Nonloop Edge and Compute Degree Changes","stem":"Use a complete simple undirected graph on Fin(3)={0,…,2} with raw edges=[[0, 1], [1, 2]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Toggle edge=[2, 0]. Give (original normalized edges, normalized toggled edge, removed/added action, new edges, old degrees, new degrees, degree differences, degree-sum difference).","answer":{"kind":"expression","value":"([[0, 1], [1, 2]], [0, 2], added, [[0, 1], [0, 2], [1, 2]], [1, 2, 1], [2, 2, 2], [1, 0, 1], 2)","explanation":"Remove an existing edge or insert an absent one, then independently count both degree lists; no loop or invalid endpoint is admitted. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, 1], [1, 2]], [0, 2], added, [[0, 1], [0, 2], [1, 2]], [1, 2, 1], [2, 2, 2], [1, 0, 1], 2)."},"skills":["Toggle a Typed Nonloop Edge and Compute Degree Changes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r05-graph-0054","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"63a234e53d797313329dafee8af5c07cbf78e7ad938838e1939cdc5ef7e050bf","family_id":"qf-density17-graph-edge-toggle-delta","seed":202610175054,"semantic_template":"qf-density17-graph-edge-toggle-delta/{\"edge\":[0,1],\"edges\":[[0,1],[0,2]],\"n\":3}"},"fields":{"title":"Toggle a Typed Nonloop Edge and Compute Degree Changes","stem":"Use a complete simple undirected graph on Fin(3)={0,…,2} with raw edges=[[0, 1], [0, 2]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Toggle edge=[0, 1]. Give (original normalized edges, normalized toggled edge, removed/added action, new edges, old degrees, new degrees, degree differences, degree-sum difference).","answer":{"kind":"expression","value":"([[0, 1], [0, 2]], [0, 1], removed, [[0, 2]], [2, 1, 1], [1, 0, 1], [-1, -1, 0], -2)","explanation":"Remove an existing edge or insert an absent one, then independently count both degree lists; no loop or invalid endpoint is admitted. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, 1], [0, 2]], [0, 1], removed, [[0, 2]], [2, 1, 1], [1, 0, 1], [-1, -1, 0], -2)."},"skills":["Toggle a Typed Nonloop Edge and Compute Degree Changes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r05-graph-0055","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"21ffb81bed5f5b37503ec3ba019a166a5d018af449507b3f54b761daed8dae84","family_id":"qf-density17-graph-edge-toggle-delta","seed":202610175055,"semantic_template":"qf-density17-graph-edge-toggle-delta/{\"edge\":[0,4],\"edges\":[[0,1]],\"n\":5}"},"fields":{"title":"Toggle a Typed Nonloop Edge and Compute Degree Changes","stem":"Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 1]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Toggle edge=[0, 4]. Give (original normalized edges, normalized toggled edge, removed/added action, new edges, old degrees, new degrees, degree differences, degree-sum difference).","answer":{"kind":"expression","value":"([[0, 1]], [0, 4], added, [[0, 1], [0, 4]], [1, 1, 0, 0, 0], [2, 1, 0, 0, 1], [1, 0, 0, 0, 1], 2)","explanation":"Remove an existing edge or insert an absent one, then independently count both degree lists; no loop or invalid endpoint is admitted. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, 1]], [0, 4], added, [[0, 1], [0, 4]], [1, 1, 0, 0, 0], [2, 1, 0, 0, 1], [1, 0, 0, 0, 1], 2)."},"skills":["Toggle a Typed Nonloop Edge and Compute Degree Changes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r05-graph-0056","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fb51e135185ab39e73cdf3afab7650d60e92d7be04ae9103377bf31df152f735","family_id":"qf-density17-graph-edge-toggle-delta","seed":202610175056,"semantic_template":"qf-density17-graph-edge-toggle-delta/{\"edge\":[0,4],\"edges\":[[0,2]],\"n\":5}"},"fields":{"title":"Toggle a Typed Nonloop Edge and Compute Degree Changes","stem":"Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 2]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Toggle edge=[0, 4]. Give (original normalized edges, normalized toggled edge, removed/added action, new edges, old degrees, new degrees, degree differences, degree-sum difference).","answer":{"kind":"expression","value":"([[0, 2]], [0, 4], added, [[0, 2], [0, 4]], [1, 0, 1, 0, 0], [2, 0, 1, 0, 1], [1, 0, 0, 0, 1], 2)","explanation":"Remove an existing edge or insert an absent one, then independently count both degree lists; no loop or invalid endpoint is admitted. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, 2]], [0, 4], added, [[0, 2], [0, 4]], [1, 0, 1, 0, 0], [2, 0, 1, 0, 1], [1, 0, 0, 0, 1], 2)."},"skills":["Toggle a Typed Nonloop Edge and Compute Degree Changes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r05-graph-0057","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"944d5620bf8f51826c9288a10dc83fb0efcb1f099566dd18af480b0b195dc71a","family_id":"qf-density17-graph-edge-toggle-delta","seed":202610175057,"semantic_template":"qf-density17-graph-edge-toggle-delta/{\"edge\":[0,4],\"edges\":[[0,1],[0,2]],\"n\":5}"},"fields":{"title":"Toggle a Typed Nonloop Edge and Compute Degree Changes","stem":"Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 1], [0, 2]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Toggle edge=[0, 4]. Give (original normalized edges, normalized toggled edge, removed/added action, new edges, old degrees, new degrees, degree differences, degree-sum difference).","answer":{"kind":"expression","value":"([[0, 1], [0, 2]], [0, 4], added, [[0, 1], [0, 2], [0, 4]], [2, 1, 1, 0, 0], [3, 1, 1, 0, 1], [1, 0, 0, 0, 1], 2)","explanation":"Remove an existing edge or insert an absent one, then independently count both degree lists; no loop or invalid endpoint is admitted. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, 1], [0, 2]], [0, 4], added, [[0, 1], [0, 2], [0, 4]], [2, 1, 1, 0, 0], [3, 1, 1, 0, 1], [1, 0, 0, 0, 1], 2)."},"skills":["Toggle a Typed Nonloop Edge and Compute Degree Changes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r05-graph-0058","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"aeeef07a1ea1ce13d220070bba9d8820de970f6a6448f2d9cb4c11c29435e50a","family_id":"qf-density17-graph-edge-toggle-delta","seed":202610175058,"semantic_template":"qf-density17-graph-edge-toggle-delta/{\"edge\":[0,4],\"edges\":[[0,3]],\"n\":5}"},"fields":{"title":"Toggle a Typed Nonloop Edge and Compute Degree Changes","stem":"Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 3]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Toggle edge=[0, 4]. Give (original normalized edges, normalized toggled edge, removed/added action, new edges, old degrees, new degrees, degree differences, degree-sum difference).","answer":{"kind":"expression","value":"([[0, 3]], [0, 4], added, [[0, 3], [0, 4]], [1, 0, 0, 1, 0], [2, 0, 0, 1, 1], [1, 0, 0, 0, 1], 2)","explanation":"Remove an existing edge or insert an absent one, then independently count both degree lists; no loop or invalid endpoint is admitted. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, 3]], [0, 4], added, [[0, 3], [0, 4]], [1, 0, 0, 1, 0], [2, 0, 0, 1, 1], [1, 0, 0, 0, 1], 2)."},"skills":["Toggle a Typed Nonloop Edge and Compute Degree Changes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r05-graph-0059","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bd37f4eea6d9d1e3e731a0fb898be561ec4ed905ace33676773f8672ff2ba575","family_id":"qf-density17-graph-edge-toggle-delta","seed":202610175059,"semantic_template":"qf-density17-graph-edge-toggle-delta/{\"edge\":[0,4],\"edges\":[[0,1],[0,3]],\"n\":5}"},"fields":{"title":"Toggle a Typed Nonloop Edge and Compute Degree Changes","stem":"Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 1], [0, 3]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Toggle edge=[0, 4]. Give (original normalized edges, normalized toggled edge, removed/added action, new edges, old degrees, new degrees, degree differences, degree-sum difference).","answer":{"kind":"expression","value":"([[0, 1], [0, 3]], [0, 4], added, [[0, 1], [0, 3], [0, 4]], [2, 1, 0, 1, 0], [3, 1, 0, 1, 1], [1, 0, 0, 0, 1], 2)","explanation":"Remove an existing edge or insert an absent one, then independently count both degree lists; no loop or invalid endpoint is admitted. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, 1], [0, 3]], [0, 4], added, [[0, 1], [0, 3], [0, 4]], [2, 1, 0, 1, 0], [3, 1, 0, 1, 1], [1, 0, 0, 0, 1], 2)."},"skills":["Toggle a Typed Nonloop Edge and Compute Degree Changes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r05-graph-0060","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"271c5bd76364e51c94bbde720cbbda52aeea49baf3963432c9b6db852ed9c283","family_id":"qf-density17-graph-edge-toggle-delta","seed":202610175060,"semantic_template":"qf-density17-graph-edge-toggle-delta/{\"edge\":[0,4],\"edges\":[[0,2],[0,3]],\"n\":5}"},"fields":{"title":"Toggle a Typed Nonloop Edge and Compute Degree Changes","stem":"Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 2], [0, 3]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Toggle edge=[0, 4]. Give (original normalized edges, normalized toggled edge, removed/added action, new edges, old degrees, new degrees, degree differences, degree-sum difference).","answer":{"kind":"expression","value":"([[0, 2], [0, 3]], [0, 4], added, [[0, 2], [0, 3], [0, 4]], [2, 0, 1, 1, 0], [3, 0, 1, 1, 1], [1, 0, 0, 0, 1], 2)","explanation":"Remove an existing edge or insert an absent one, then independently count both degree lists; no loop or invalid endpoint is admitted. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, 2], [0, 3]], [0, 4], added, [[0, 2], [0, 3], [0, 4]], [2, 0, 1, 1, 0], [3, 0, 1, 1, 1], [1, 0, 0, 0, 1], 2)."},"skills":["Toggle a Typed Nonloop Edge and Compute Degree Changes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r05-graph-0061","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e5501ee080f2c7ea6ceb9c126409666db6678d18e7cca35fc867b947fcb34cde","family_id":"qf-density17-graph-edge-toggle-delta","seed":202610175061,"semantic_template":"qf-density17-graph-edge-toggle-delta/{\"edge\":[0,4],\"edges\":[[0,1],[0,2],[0,3]],\"n\":5}"},"fields":{"title":"Toggle a Typed Nonloop Edge and Compute Degree Changes","stem":"Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 1], [0, 2], [0, 3]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Toggle edge=[0, 4]. Give (original normalized edges, normalized toggled edge, removed/added action, new edges, old degrees, new degrees, degree differences, degree-sum difference).","answer":{"kind":"expression","value":"([[0, 1], [0, 2], [0, 3]], [0, 4], added, [[0, 1], [0, 2], [0, 3], [0, 4]], [3, 1, 1, 1, 0], [4, 1, 1, 1, 1], [1, 0, 0, 0, 1], 2)","explanation":"Remove an existing edge or insert an absent one, then independently count both degree lists; no loop or invalid endpoint is admitted. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, 1], [0, 2], [0, 3]], [0, 4], added, [[0, 1], [0, 2], [0, 3], [0, 4]], [3, 1, 1, 1, 0], [4, 1, 1, 1, 1], [1, 0, 0, 0, 1], 2)."},"skills":["Toggle a Typed Nonloop Edge and Compute Degree Changes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r05-graph-0062","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7d951148ff5cff384a10f91d1ac08b89dc2a446690686a539007deff92a2e6fa","family_id":"qf-density17-graph-edge-toggle-delta","seed":202610175062,"semantic_template":"qf-density17-graph-edge-toggle-delta/{\"edge\":[0,4],\"edges\":[[0,4]],\"n\":5}"},"fields":{"title":"Toggle a Typed Nonloop Edge and Compute Degree Changes","stem":"Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 4]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Toggle edge=[0, 4]. Give (original normalized edges, normalized toggled edge, removed/added action, new edges, old degrees, new degrees, degree differences, degree-sum difference).","answer":{"kind":"expression","value":"([[0, 4]], [0, 4], removed, [], [1, 0, 0, 0, 1], [0, 0, 0, 0, 0], [-1, 0, 0, 0, -1], -2)","explanation":"Remove an existing edge or insert an absent one, then independently count both degree lists; no loop or invalid endpoint is admitted. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, 4]], [0, 4], removed, [], [1, 0, 0, 0, 1], [0, 0, 0, 0, 0], [-1, 0, 0, 0, -1], -2)."},"skills":["Toggle a Typed Nonloop Edge and Compute Degree Changes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r05-graph-0063","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"72db29795b3bcead876d3585e1f107737144ddedc21ca16b21731903eeb74197","family_id":"qf-density17-graph-edge-toggle-delta","seed":202610175063,"semantic_template":"qf-density17-graph-edge-toggle-delta/{\"edge\":[0,4],\"edges\":[[0,1],[0,4]],\"n\":5}"},"fields":{"title":"Toggle a Typed Nonloop Edge and Compute Degree Changes","stem":"Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 1], [0, 4]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Toggle edge=[0, 4]. Give (original normalized edges, normalized toggled edge, removed/added action, new edges, old degrees, new degrees, degree differences, degree-sum difference).","answer":{"kind":"expression","value":"([[0, 1], [0, 4]], [0, 4], removed, [[0, 1]], [2, 1, 0, 0, 1], [1, 1, 0, 0, 0], [-1, 0, 0, 0, -1], -2)","explanation":"Remove an existing edge or insert an absent one, then independently count both degree lists; no loop or invalid endpoint is admitted. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, 1], [0, 4]], [0, 4], removed, [[0, 1]], [2, 1, 0, 0, 1], [1, 1, 0, 0, 0], [-1, 0, 0, 0, -1], -2)."},"skills":["Toggle a Typed Nonloop Edge and Compute Degree Changes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r05-graph-0064","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e96c3303994a5f5d6b293fb7608eacf3c69fda16ad372eda8eaaf6ce537a70f3","family_id":"qf-density17-graph-edge-toggle-delta","seed":202610175064,"semantic_template":"qf-density17-graph-edge-toggle-delta/{\"edge\":[0,4],\"edges\":[[0,2],[0,4]],\"n\":5}"},"fields":{"title":"Toggle a Typed Nonloop Edge and Compute Degree Changes","stem":"Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 2], [0, 4]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Toggle edge=[0, 4]. Give (original normalized edges, normalized toggled edge, removed/added action, new edges, old degrees, new degrees, degree differences, degree-sum difference).","answer":{"kind":"expression","value":"([[0, 2], [0, 4]], [0, 4], removed, [[0, 2]], [2, 0, 1, 0, 1], [1, 0, 1, 0, 0], [-1, 0, 0, 0, -1], -2)","explanation":"Remove an existing edge or insert an absent one, then independently count both degree lists; no loop or invalid endpoint is admitted. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, 2], [0, 4]], [0, 4], removed, [[0, 2]], [2, 0, 1, 0, 1], [1, 0, 1, 0, 0], [-1, 0, 0, 0, -1], -2)."},"skills":["Toggle a Typed Nonloop Edge and Compute Degree Changes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r05-graph-0065","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2797db49ca3d3413d03f5eef9d12dfbf4de7e9ff7d62ae18334e6d73119009eb","family_id":"qf-density17-graph-edge-toggle-delta","seed":202610175065,"semantic_template":"qf-density17-graph-edge-toggle-delta/{\"edge\":[0,4],\"edges\":[[0,1],[0,2],[0,4]],\"n\":5}"},"fields":{"title":"Toggle a Typed Nonloop Edge and Compute Degree Changes","stem":"Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 1], [0, 2], [0, 4]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Toggle edge=[0, 4]. Give (original normalized edges, normalized toggled edge, removed/added action, new edges, old degrees, new degrees, degree differences, degree-sum difference).","answer":{"kind":"expression","value":"([[0, 1], [0, 2], [0, 4]], [0, 4], removed, [[0, 1], [0, 2]], [3, 1, 1, 0, 1], [2, 1, 1, 0, 0], [-1, 0, 0, 0, -1], -2)","explanation":"Remove an existing edge or insert an absent one, then independently count both degree lists; no loop or invalid endpoint is admitted. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, 1], [0, 2], [0, 4]], [0, 4], removed, [[0, 1], [0, 2]], [3, 1, 1, 0, 1], [2, 1, 1, 0, 0], [-1, 0, 0, 0, -1], -2)."},"skills":["Toggle a Typed Nonloop Edge and Compute Degree Changes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r05-graph-0066","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0a6478e654ec08bd3934987cc14804c4b9645a24fc7b3b99b7227592c63227c5","family_id":"qf-density17-graph-edge-toggle-delta","seed":202610175066,"semantic_template":"qf-density17-graph-edge-toggle-delta/{\"edge\":[0,4],\"edges\":[[0,3],[0,4]],\"n\":5}"},"fields":{"title":"Toggle a Typed Nonloop Edge and Compute Degree Changes","stem":"Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 3], [0, 4]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Toggle edge=[0, 4]. Give (original normalized edges, normalized toggled edge, removed/added action, new edges, old degrees, new degrees, degree differences, degree-sum difference).","answer":{"kind":"expression","value":"([[0, 3], [0, 4]], [0, 4], removed, [[0, 3]], [2, 0, 0, 1, 1], [1, 0, 0, 1, 0], [-1, 0, 0, 0, -1], -2)","explanation":"Remove an existing edge or insert an absent one, then independently count both degree lists; no loop or invalid endpoint is admitted. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, 3], [0, 4]], [0, 4], removed, [[0, 3]], [2, 0, 0, 1, 1], [1, 0, 0, 1, 0], [-1, 0, 0, 0, -1], -2)."},"skills":["Toggle a Typed Nonloop Edge and Compute Degree Changes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r05-graph-0067","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"019496250152f9b1625a4393abe2450197b95e9c316dcada188e8897e7dc365e","family_id":"qf-density17-graph-edge-toggle-delta","seed":202610175067,"semantic_template":"qf-density17-graph-edge-toggle-delta/{\"edge\":[0,4],\"edges\":[[0,1],[0,3],[0,4]],\"n\":5}"},"fields":{"title":"Toggle a Typed Nonloop Edge and Compute Degree Changes","stem":"Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 1], [0, 3], [0, 4]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Toggle edge=[0, 4]. Give (original normalized edges, normalized toggled edge, removed/added action, new edges, old degrees, new degrees, degree differences, degree-sum difference).","answer":{"kind":"expression","value":"([[0, 1], [0, 3], [0, 4]], [0, 4], removed, [[0, 1], [0, 3]], [3, 1, 0, 1, 1], [2, 1, 0, 1, 0], [-1, 0, 0, 0, -1], -2)","explanation":"Remove an existing edge or insert an absent one, then independently count both degree lists; no loop or invalid endpoint is admitted. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, 1], [0, 3], [0, 4]], [0, 4], removed, [[0, 1], [0, 3]], [3, 1, 0, 1, 1], [2, 1, 0, 1, 0], [-1, 0, 0, 0, -1], -2)."},"skills":["Toggle a Typed Nonloop Edge and Compute Degree Changes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r05-graph-0068","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"39025384039b0a7b6123f0fb6002f7d2068f9ef11aa04b5d5b96aaddab1538d7","family_id":"qf-density17-graph-edge-toggle-delta","seed":202610175068,"semantic_template":"qf-density17-graph-edge-toggle-delta/{\"edge\":[0,4],\"edges\":[[0,2],[0,3],[0,4]],\"n\":5}"},"fields":{"title":"Toggle a Typed Nonloop Edge and Compute Degree Changes","stem":"Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 2], [0, 3], [0, 4]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Toggle edge=[0, 4]. Give (original normalized edges, normalized toggled edge, removed/added action, new edges, old degrees, new degrees, degree differences, degree-sum difference).","answer":{"kind":"expression","value":"([[0, 2], [0, 3], [0, 4]], [0, 4], removed, [[0, 2], [0, 3]], [3, 0, 1, 1, 1], [2, 0, 1, 1, 0], [-1, 0, 0, 0, -1], -2)","explanation":"Remove an existing edge or insert an absent one, then independently count both degree lists; no loop or invalid endpoint is admitted. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, 2], [0, 3], [0, 4]], [0, 4], removed, [[0, 2], [0, 3]], [3, 0, 1, 1, 1], [2, 0, 1, 1, 0], [-1, 0, 0, 0, -1], -2)."},"skills":["Toggle a Typed Nonloop Edge and Compute Degree Changes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r05-graph-0069","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3f4ce94555d4988a895cfe377d4c6cb372ec1a1b3317e7e4de0da24468750b9c","family_id":"qf-density17-graph-edge-toggle-delta","seed":202610175069,"semantic_template":"qf-density17-graph-edge-toggle-delta/{\"edge\":[0,4],\"edges\":[[0,1],[0,2],[0,3],[0,4]],\"n\":5}"},"fields":{"title":"Toggle a Typed Nonloop Edge and Compute Degree Changes","stem":"Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 1], [0, 2], [0, 3], [0, 4]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Toggle edge=[0, 4]. Give (original normalized edges, normalized toggled edge, removed/added action, new edges, old degrees, new degrees, degree differences, degree-sum difference).","answer":{"kind":"expression","value":"([[0, 1], [0, 2], [0, 3], [0, 4]], [0, 4], removed, [[0, 1], [0, 2], [0, 3]], [4, 1, 1, 1, 1], [3, 1, 1, 1, 0], [-1, 0, 0, 0, -1], -2)","explanation":"Remove an existing edge or insert an absent one, then independently count both degree lists; no loop or invalid endpoint is admitted. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, 1], [0, 2], [0, 3], [0, 4]], [0, 4], removed, [[0, 1], [0, 2], [0, 3]], [4, 1, 1, 1, 1], [3, 1, 1, 1, 0], [-1, 0, 0, 0, -1], -2)."},"skills":["Toggle a Typed Nonloop Edge and Compute Degree Changes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r05-graph-0070","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1ffd115bc0b6c958c6cd9ef97ef4899fb876b6b2401ddd03c5e784cfd285dbc7","family_id":"qf-density17-graph-edge-toggle-delta","seed":202610175070,"semantic_template":"qf-density17-graph-edge-toggle-delta/{\"edge\":[0,4],\"edges\":[[1,2]],\"n\":5}"},"fields":{"title":"Toggle a Typed Nonloop Edge and Compute Degree Changes","stem":"Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[1, 2]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Toggle edge=[0, 4]. Give (original normalized edges, normalized toggled edge, removed/added action, new edges, old degrees, new degrees, degree differences, degree-sum difference).","answer":{"kind":"expression","value":"([[1, 2]], [0, 4], added, [[0, 4], [1, 2]], [0, 1, 1, 0, 0], [1, 1, 1, 0, 1], [1, 0, 0, 0, 1], 2)","explanation":"Remove an existing edge or insert an absent one, then independently count both degree lists; no loop or invalid endpoint is admitted. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[1, 2]], [0, 4], added, [[0, 4], [1, 2]], [0, 1, 1, 0, 0], [1, 1, 1, 0, 1], [1, 0, 0, 0, 1], 2)."},"skills":["Toggle a Typed Nonloop Edge and Compute Degree Changes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r05-graph-0071","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"313389f1bc228bb5c8314eb21fc186ef0f252f5ec8d2771fced1804f45741494","family_id":"qf-density17-graph-edge-toggle-delta","seed":202610175071,"semantic_template":"qf-density17-graph-edge-toggle-delta/{\"edge\":[0,4],\"edges\":[[0,1],[1,2]],\"n\":5}"},"fields":{"title":"Toggle a Typed Nonloop Edge and Compute Degree Changes","stem":"Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 1], [1, 2]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Toggle edge=[0, 4]. Give (original normalized edges, normalized toggled edge, removed/added action, new edges, old degrees, new degrees, degree differences, degree-sum difference).","answer":{"kind":"expression","value":"([[0, 1], [1, 2]], [0, 4], added, [[0, 1], [0, 4], [1, 2]], [1, 2, 1, 0, 0], [2, 2, 1, 0, 1], [1, 0, 0, 0, 1], 2)","explanation":"Remove an existing edge or insert an absent one, then independently count both degree lists; no loop or invalid endpoint is admitted. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, 1], [1, 2]], [0, 4], added, [[0, 1], [0, 4], [1, 2]], [1, 2, 1, 0, 0], [2, 2, 1, 0, 1], [1, 0, 0, 0, 1], 2)."},"skills":["Toggle a Typed Nonloop Edge and Compute Degree Changes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r05-graph-0072","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c8ab120570daa1b4afbf0d002fab20e6c0a41bb7644fbd73b67c776c81e8404d","family_id":"qf-density17-graph-edge-toggle-delta","seed":202610175072,"semantic_template":"qf-density17-graph-edge-toggle-delta/{\"edge\":[0,4],\"edges\":[[0,2],[1,2]],\"n\":5}"},"fields":{"title":"Toggle a Typed Nonloop Edge and Compute Degree Changes","stem":"Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 2], [1, 2]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Toggle edge=[0, 4]. Give (original normalized edges, normalized toggled edge, removed/added action, new edges, old degrees, new degrees, degree differences, degree-sum difference).","answer":{"kind":"expression","value":"([[0, 2], [1, 2]], [0, 4], added, [[0, 2], [0, 4], [1, 2]], [1, 1, 2, 0, 0], [2, 1, 2, 0, 1], [1, 0, 0, 0, 1], 2)","explanation":"Remove an existing edge or insert an absent one, then independently count both degree lists; no loop or invalid endpoint is admitted. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, 2], [1, 2]], [0, 4], added, [[0, 2], [0, 4], [1, 2]], [1, 1, 2, 0, 0], [2, 1, 2, 0, 1], [1, 0, 0, 0, 1], 2)."},"skills":["Toggle a Typed Nonloop Edge and Compute Degree Changes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r05-graph-0073","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"77bc58f13db4c9eabe0797ace46e334df6b2d7677dfa3d47fbc1881d9d7127e5","family_id":"qf-density17-graph-edge-toggle-delta","seed":202610175073,"semantic_template":"qf-density17-graph-edge-toggle-delta/{\"edge\":[0,4],\"edges\":[[0,1],[0,2],[1,2]],\"n\":5}"},"fields":{"title":"Toggle a Typed Nonloop Edge and Compute Degree Changes","stem":"Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 1], [0, 2], [1, 2]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Toggle edge=[0, 4]. Give (original normalized edges, normalized toggled edge, removed/added action, new edges, old degrees, new degrees, degree differences, degree-sum difference).","answer":{"kind":"expression","value":"([[0, 1], [0, 2], [1, 2]], [0, 4], added, [[0, 1], [0, 2], [0, 4], [1, 2]], [2, 2, 2, 0, 0], [3, 2, 2, 0, 1], [1, 0, 0, 0, 1], 2)","explanation":"Remove an existing edge or insert an absent one, then independently count both degree lists; no loop or invalid endpoint is admitted. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, 1], [0, 2], [1, 2]], [0, 4], added, [[0, 1], [0, 2], [0, 4], [1, 2]], [2, 2, 2, 0, 0], [3, 2, 2, 0, 1], [1, 0, 0, 0, 1], 2)."},"skills":["Toggle a Typed Nonloop Edge and Compute Degree Changes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r05-graph-0074","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"94eba69e3f77deaf38625d0ce8fcb037fa7d2824119c58d9375208a278c695f1","family_id":"qf-density17-graph-edge-toggle-delta","seed":202610175074,"semantic_template":"qf-density17-graph-edge-toggle-delta/{\"edge\":[0,4],\"edges\":[[0,3],[1,2]],\"n\":5}"},"fields":{"title":"Toggle a Typed Nonloop Edge and Compute Degree Changes","stem":"Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 3], [1, 2]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Toggle edge=[0, 4]. Give (original normalized edges, normalized toggled edge, removed/added action, new edges, old degrees, new degrees, degree differences, degree-sum difference).","answer":{"kind":"expression","value":"([[0, 3], [1, 2]], [0, 4], added, [[0, 3], [0, 4], [1, 2]], [1, 1, 1, 1, 0], [2, 1, 1, 1, 1], [1, 0, 0, 0, 1], 2)","explanation":"Remove an existing edge or insert an absent one, then independently count both degree lists; no loop or invalid endpoint is admitted. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, 3], [1, 2]], [0, 4], added, [[0, 3], [0, 4], [1, 2]], [1, 1, 1, 1, 0], [2, 1, 1, 1, 1], [1, 0, 0, 0, 1], 2)."},"skills":["Toggle a Typed Nonloop Edge and Compute Degree Changes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r05-graph-0075","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"28b3934967a2eda7a72590bb3dea8544f46839637f49adad662f5c7cedf17bfb","family_id":"qf-density17-graph-edge-toggle-delta","seed":202610175075,"semantic_template":"qf-density17-graph-edge-toggle-delta/{\"edge\":[0,4],\"edges\":[[0,1],[0,3],[1,2]],\"n\":5}"},"fields":{"title":"Toggle a Typed Nonloop Edge and Compute Degree Changes","stem":"Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 1], [0, 3], [1, 2]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Toggle edge=[0, 4]. Give (original normalized edges, normalized toggled edge, removed/added action, new edges, old degrees, new degrees, degree differences, degree-sum difference).","answer":{"kind":"expression","value":"([[0, 1], [0, 3], [1, 2]], [0, 4], added, [[0, 1], [0, 3], [0, 4], [1, 2]], [2, 2, 1, 1, 0], [3, 2, 1, 1, 1], [1, 0, 0, 0, 1], 2)","explanation":"Remove an existing edge or insert an absent one, then independently count both degree lists; no loop or invalid endpoint is admitted. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, 1], [0, 3], [1, 2]], [0, 4], added, [[0, 1], [0, 3], [0, 4], [1, 2]], [2, 2, 1, 1, 0], [3, 2, 1, 1, 1], [1, 0, 0, 0, 1], 2)."},"skills":["Toggle a Typed Nonloop Edge and Compute Degree Changes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r05-graph-0076","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d27806bc80c92286a2a8b73b5b29d809e0d2118e295c619ff21b8b523459c66b","family_id":"qf-density17-graph-induced-subset","seed":202610175076,"semantic_template":"qf-density17-graph-induced-subset/{\"edges\":[],\"n\":0,\"subset\":[]}"},"fields":{"title":"Induced Graph with Original Vertex Labels","stem":"Use a complete simple undirected graph on Fin(0)=∅ with raw edges=[]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Retain source subset=[], normalized as a set. Give (old normalized edges, sorted retained labels, induced edges, retained-label degrees, retained-label neighbor lists).","answer":{"kind":"expression","value":"([], [], [], [], [])","explanation":"Keep exactly edges with both endpoints retained, and do not relabel the induced vertices. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([], [], [], [], [])."},"skills":["Induced Graph with Original Vertex Labels"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r05-graph-0077","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"095d9c1eb9d0c1d93fcf3ef5af548f61994564d2be6661c32ddd10d3fa06f9c9","family_id":"qf-density17-graph-induced-subset","seed":202610175077,"semantic_template":"qf-density17-graph-induced-subset/{\"edges\":[[0,1],[1,2]],\"n\":3,\"subset\":[]}"},"fields":{"title":"Induced Graph with Original Vertex Labels","stem":"Use a complete simple undirected graph on Fin(3)={0,…,2} with raw edges=[[0, 1], [1, 2]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Retain source subset=[], normalized as a set. Give (old normalized edges, sorted retained labels, induced edges, retained-label degrees, retained-label neighbor lists).","answer":{"kind":"expression","value":"([[0, 1], [1, 2]], [], [], [], [])","explanation":"Keep exactly edges with both endpoints retained, and do not relabel the induced vertices. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, 1], [1, 2]], [], [], [], [])."},"skills":["Induced Graph with Original Vertex Labels"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r05-graph-0078","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3b58c16b6efbee528b6c4155ad0e521c74f4bb751d9196f76e61b3d96aa35b45","family_id":"qf-density17-graph-induced-subset","seed":202610175078,"semantic_template":"qf-density17-graph-induced-subset/{\"edges\":[[0,1],[1,2]],\"n\":3,\"subset\":[0,2]}"},"fields":{"title":"Induced Graph with Original Vertex Labels","stem":"Use a complete simple undirected graph on Fin(3)={0,…,2} with raw edges=[[0, 1], [1, 2]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Retain source subset=[0, 2], normalized as a set. Give (old normalized edges, sorted retained labels, induced edges, retained-label degrees, retained-label neighbor lists).","answer":{"kind":"expression","value":"([[0, 1], [1, 2]], [0, 2], [], [0, 0], [[], []])","explanation":"Keep exactly edges with both endpoints retained, and do not relabel the induced vertices. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, 1], [1, 2]], [0, 2], [], [0, 0], [[], []])."},"skills":["Induced Graph with Original Vertex Labels"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r05-graph-0079","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d92f4340b627576ae79acf71ce36860c9d7e04107d789aa6064f4f2090085261","family_id":"qf-density17-graph-induced-subset","seed":202610175079,"semantic_template":"qf-density17-graph-induced-subset/{\"edges\":[[0,1],[1,2]],\"n\":3,\"subset\":[1,2]}"},"fields":{"title":"Induced Graph with Original Vertex Labels","stem":"Use a complete simple undirected graph on Fin(3)={0,…,2} with raw edges=[[0, 1], [1, 2]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Retain source subset=[2, 1], normalized as a set. Give (old normalized edges, sorted retained labels, induced edges, retained-label degrees, retained-label neighbor lists).","answer":{"kind":"expression","value":"([[0, 1], [1, 2]], [1, 2], [[1, 2]], [1, 1], [[2], [1]])","explanation":"Keep exactly edges with both endpoints retained, and do not relabel the induced vertices. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, 1], [1, 2]], [1, 2], [[1, 2]], [1, 1], [[2], [1]])."},"skills":["Induced Graph with Original Vertex Labels"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r05-graph-0080","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0a107ac2b48dab9dfa79a1e155485247922c5d4881ebe3dd31c590d3831fd432","family_id":"qf-density17-graph-induced-subset","seed":202610175080,"semantic_template":"qf-density17-graph-induced-subset/{\"edges\":[[0,1],[2,3]],\"n\":4,\"subset\":[0,2,3]}"},"fields":{"title":"Induced Graph with Original Vertex Labels","stem":"Use a complete simple undirected graph on Fin(4)={0,…,3} with raw edges=[[0, 1], [2, 3]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Retain source subset=[3, 2, 0], normalized as a set. Give (old normalized edges, sorted retained labels, induced edges, retained-label degrees, retained-label neighbor lists).","answer":{"kind":"expression","value":"([[0, 1], [2, 3]], [0, 2, 3], [[2, 3]], [0, 1, 1], [[], [3], [2]])","explanation":"Keep exactly edges with both endpoints retained, and do not relabel the induced vertices. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, 1], [2, 3]], [0, 2, 3], [[2, 3]], [0, 1, 1], [[], [3], [2]])."},"skills":["Induced Graph with Original Vertex Labels"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r05-graph-0081","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"66cdfc532037a6a6c362ca694bc25eba0f4effca00b56eae281244a5a86022fc","family_id":"qf-density17-graph-induced-subset","seed":202610175081,"semantic_template":"qf-density17-graph-induced-subset/{\"edges\":[[0,1]],\"n\":5,\"subset\":[0,2,4]}"},"fields":{"title":"Induced Graph with Original Vertex Labels","stem":"Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 1]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Retain source subset=[4, 2, 0], normalized as a set. Give (old normalized edges, sorted retained labels, induced edges, retained-label degrees, retained-label neighbor lists).","answer":{"kind":"expression","value":"([[0, 1]], [0, 2, 4], [], [0, 0, 0], [[], [], []])","explanation":"Keep exactly edges with both endpoints retained, and do not relabel the induced vertices. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, 1]], [0, 2, 4], [], [0, 0, 0], [[], [], []])."},"skills":["Induced Graph with Original Vertex Labels"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r05-graph-0082","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6ac397d45697fb6dae9a8c13c62909f0071ed7e8dc4ebfcdce7c791188cc9dbe","family_id":"qf-density17-graph-induced-subset","seed":202610175082,"semantic_template":"qf-density17-graph-induced-subset/{\"edges\":[[0,2]],\"n\":5,\"subset\":[0,2,4]}"},"fields":{"title":"Induced Graph with Original Vertex Labels","stem":"Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 2]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Retain source subset=[4, 2, 0], normalized as a set. Give (old normalized edges, sorted retained labels, induced edges, retained-label degrees, retained-label neighbor lists).","answer":{"kind":"expression","value":"([[0, 2]], [0, 2, 4], [[0, 2]], [1, 1, 0], [[2], [0], []])","explanation":"Keep exactly edges with both endpoints retained, and do not relabel the induced vertices. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, 2]], [0, 2, 4], [[0, 2]], [1, 1, 0], [[2], [0], []])."},"skills":["Induced Graph with Original Vertex Labels"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r05-graph-0083","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0c3cd1f7bcff418e846d3e0cedfa0cfbb5bbdc39899e5adeb7e00a3bcc6e7b59","family_id":"qf-density17-graph-induced-subset","seed":202610175083,"semantic_template":"qf-density17-graph-induced-subset/{\"edges\":[[0,1],[0,2]],\"n\":5,\"subset\":[0,2,4]}"},"fields":{"title":"Induced Graph with Original Vertex Labels","stem":"Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 1], [0, 2]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Retain source subset=[4, 2, 0], normalized as a set. Give (old normalized edges, sorted retained labels, induced edges, retained-label degrees, retained-label neighbor lists).","answer":{"kind":"expression","value":"([[0, 1], [0, 2]], [0, 2, 4], [[0, 2]], [1, 1, 0], [[2], [0], []])","explanation":"Keep exactly edges with both endpoints retained, and do not relabel the induced vertices. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, 1], [0, 2]], [0, 2, 4], [[0, 2]], [1, 1, 0], [[2], [0], []])."},"skills":["Induced Graph with Original Vertex Labels"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r05-graph-0084","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a041757109a437a42c0296913b7bffe699deb3f741171cc41f83c1b7d81fcec4","family_id":"qf-density17-graph-induced-subset","seed":202610175084,"semantic_template":"qf-density17-graph-induced-subset/{\"edges\":[[0,3]],\"n\":5,\"subset\":[0,2,4]}"},"fields":{"title":"Induced Graph with Original Vertex Labels","stem":"Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 3]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Retain source subset=[4, 2, 0], normalized as a set. Give (old normalized edges, sorted retained labels, induced edges, retained-label degrees, retained-label neighbor lists).","answer":{"kind":"expression","value":"([[0, 3]], [0, 2, 4], [], [0, 0, 0], [[], [], []])","explanation":"Keep exactly edges with both endpoints retained, and do not relabel the induced vertices. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, 3]], [0, 2, 4], [], [0, 0, 0], [[], [], []])."},"skills":["Induced Graph with Original Vertex Labels"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r05-graph-0085","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7da123de5286c1bf4e204c5deefa93bcbbe7bb7cc3a9293c3042f0dd6199e8f5","family_id":"qf-density17-graph-induced-subset","seed":202610175085,"semantic_template":"qf-density17-graph-induced-subset/{\"edges\":[[0,1],[0,3]],\"n\":5,\"subset\":[0,2,4]}"},"fields":{"title":"Induced Graph with Original Vertex Labels","stem":"Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 1], [0, 3]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Retain source subset=[4, 2, 0], normalized as a set. Give (old normalized edges, sorted retained labels, induced edges, retained-label degrees, retained-label neighbor lists).","answer":{"kind":"expression","value":"([[0, 1], [0, 3]], [0, 2, 4], [], [0, 0, 0], [[], [], []])","explanation":"Keep exactly edges with both endpoints retained, and do not relabel the induced vertices. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, 1], [0, 3]], [0, 2, 4], [], [0, 0, 0], [[], [], []])."},"skills":["Induced Graph with Original Vertex Labels"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r05-graph-0086","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"527886e86b35a65a9950b058f05e470ac0989fd471993c127b0fe25428a3627a","family_id":"qf-density17-graph-induced-subset","seed":202610175086,"semantic_template":"qf-density17-graph-induced-subset/{\"edges\":[[0,2],[0,3]],\"n\":5,\"subset\":[0,2,4]}"},"fields":{"title":"Induced Graph with Original Vertex Labels","stem":"Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 2], [0, 3]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Retain source subset=[4, 2, 0], normalized as a set. Give (old normalized edges, sorted retained labels, induced edges, retained-label degrees, retained-label neighbor lists).","answer":{"kind":"expression","value":"([[0, 2], [0, 3]], [0, 2, 4], [[0, 2]], [1, 1, 0], [[2], [0], []])","explanation":"Keep exactly edges with both endpoints retained, and do not relabel the induced vertices. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, 2], [0, 3]], [0, 2, 4], [[0, 2]], [1, 1, 0], [[2], [0], []])."},"skills":["Induced Graph with Original Vertex Labels"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r05-graph-0087","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"72704e04d0cfb7ea97bf13343fda986e914873cbc16193d077397d9d6fc16dfd","family_id":"qf-density17-graph-induced-subset","seed":202610175087,"semantic_template":"qf-density17-graph-induced-subset/{\"edges\":[[0,1],[0,2],[0,3]],\"n\":5,\"subset\":[0,2,4]}"},"fields":{"title":"Induced Graph with Original Vertex Labels","stem":"Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 1], [0, 2], [0, 3]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Retain source subset=[4, 2, 0], normalized as a set. Give (old normalized edges, sorted retained labels, induced edges, retained-label degrees, retained-label neighbor lists).","answer":{"kind":"expression","value":"([[0, 1], [0, 2], [0, 3]], [0, 2, 4], [[0, 2]], [1, 1, 0], [[2], [0], []])","explanation":"Keep exactly edges with both endpoints retained, and do not relabel the induced vertices. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, 1], [0, 2], [0, 3]], [0, 2, 4], [[0, 2]], [1, 1, 0], [[2], [0], []])."},"skills":["Induced Graph with Original Vertex Labels"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r05-graph-0088","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a688ed7b15d136794c9ac4c8def03981229a85995b87d198ce32ea7922b84d34","family_id":"qf-density17-graph-induced-subset","seed":202610175088,"semantic_template":"qf-density17-graph-induced-subset/{\"edges\":[[0,4]],\"n\":5,\"subset\":[0,2,4]}"},"fields":{"title":"Induced Graph with Original Vertex Labels","stem":"Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 4]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Retain source subset=[4, 2, 0], normalized as a set. Give (old normalized edges, sorted retained labels, induced edges, retained-label degrees, retained-label neighbor lists).","answer":{"kind":"expression","value":"([[0, 4]], [0, 2, 4], [[0, 4]], [1, 0, 1], [[4], [], [0]])","explanation":"Keep exactly edges with both endpoints retained, and do not relabel the induced vertices. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, 4]], [0, 2, 4], [[0, 4]], [1, 0, 1], [[4], [], [0]])."},"skills":["Induced Graph with Original Vertex Labels"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r05-graph-0089","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c85d225e37379cd6dad4327eb711655733289620db6c881d992e3261c86c4042","family_id":"qf-density17-graph-induced-subset","seed":202610175089,"semantic_template":"qf-density17-graph-induced-subset/{\"edges\":[[0,1],[0,4]],\"n\":5,\"subset\":[0,2,4]}"},"fields":{"title":"Induced Graph with Original Vertex Labels","stem":"Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 1], [0, 4]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Retain source subset=[4, 2, 0], normalized as a set. Give (old normalized edges, sorted retained labels, induced edges, retained-label degrees, retained-label neighbor lists).","answer":{"kind":"expression","value":"([[0, 1], [0, 4]], [0, 2, 4], [[0, 4]], [1, 0, 1], [[4], [], [0]])","explanation":"Keep exactly edges with both endpoints retained, and do not relabel the induced vertices. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, 1], [0, 4]], [0, 2, 4], [[0, 4]], [1, 0, 1], [[4], [], [0]])."},"skills":["Induced Graph with Original Vertex Labels"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r05-graph-0090","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"480b8641bc56722ca119519fdbf7900a5ac8611de915599fe77f15ba7d8b7c5b","family_id":"qf-density17-graph-induced-subset","seed":202610175090,"semantic_template":"qf-density17-graph-induced-subset/{\"edges\":[[0,2],[0,4]],\"n\":5,\"subset\":[0,2,4]}"},"fields":{"title":"Induced Graph with Original Vertex Labels","stem":"Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 2], [0, 4]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Retain source subset=[4, 2, 0], normalized as a set. Give (old normalized edges, sorted retained labels, induced edges, retained-label degrees, retained-label neighbor lists).","answer":{"kind":"expression","value":"([[0, 2], [0, 4]], [0, 2, 4], [[0, 2], [0, 4]], [2, 1, 1], [[2, 4], [0], [0]])","explanation":"Keep exactly edges with both endpoints retained, and do not relabel the induced vertices. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, 2], [0, 4]], [0, 2, 4], [[0, 2], [0, 4]], [2, 1, 1], [[2, 4], [0], [0]])."},"skills":["Induced Graph with Original Vertex Labels"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r05-graph-0091","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9031f59766daa53e981ac0b8a36a4b5b35c5a8567ffb7cd4a6653c79e749cbd9","family_id":"qf-density17-graph-induced-subset","seed":202610175091,"semantic_template":"qf-density17-graph-induced-subset/{\"edges\":[[0,1],[0,2],[0,4]],\"n\":5,\"subset\":[0,2,4]}"},"fields":{"title":"Induced Graph with Original Vertex Labels","stem":"Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 1], [0, 2], [0, 4]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Retain source subset=[4, 2, 0], normalized as a set. Give (old normalized edges, sorted retained labels, induced edges, retained-label degrees, retained-label neighbor lists).","answer":{"kind":"expression","value":"([[0, 1], [0, 2], [0, 4]], [0, 2, 4], [[0, 2], [0, 4]], [2, 1, 1], [[2, 4], [0], [0]])","explanation":"Keep exactly edges with both endpoints retained, and do not relabel the induced vertices. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, 1], [0, 2], [0, 4]], [0, 2, 4], [[0, 2], [0, 4]], [2, 1, 1], [[2, 4], [0], [0]])."},"skills":["Induced Graph with Original Vertex Labels"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r05-graph-0092","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0e8e5e409870dae8fa1a8f786c77ef5c921edcfd73f89e6f0e020a798ec785c5","family_id":"qf-density17-graph-induced-subset","seed":202610175092,"semantic_template":"qf-density17-graph-induced-subset/{\"edges\":[[0,3],[0,4]],\"n\":5,\"subset\":[0,2,4]}"},"fields":{"title":"Induced Graph with Original Vertex Labels","stem":"Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 3], [0, 4]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Retain source subset=[4, 2, 0], normalized as a set. Give (old normalized edges, sorted retained labels, induced edges, retained-label degrees, retained-label neighbor lists).","answer":{"kind":"expression","value":"([[0, 3], [0, 4]], [0, 2, 4], [[0, 4]], [1, 0, 1], [[4], [], [0]])","explanation":"Keep exactly edges with both endpoints retained, and do not relabel the induced vertices. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, 3], [0, 4]], [0, 2, 4], [[0, 4]], [1, 0, 1], [[4], [], [0]])."},"skills":["Induced Graph with Original Vertex Labels"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r05-graph-0093","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"566798bc632844d0c95f25d4b3e679f7ce29549f8a074047e919b5f2c1c66d93","family_id":"qf-density17-graph-induced-subset","seed":202610175093,"semantic_template":"qf-density17-graph-induced-subset/{\"edges\":[[0,1],[0,3],[0,4]],\"n\":5,\"subset\":[0,2,4]}"},"fields":{"title":"Induced Graph with Original Vertex Labels","stem":"Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 1], [0, 3], [0, 4]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Retain source subset=[4, 2, 0], normalized as a set. Give (old normalized edges, sorted retained labels, induced edges, retained-label degrees, retained-label neighbor lists).","answer":{"kind":"expression","value":"([[0, 1], [0, 3], [0, 4]], [0, 2, 4], [[0, 4]], [1, 0, 1], [[4], [], [0]])","explanation":"Keep exactly edges with both endpoints retained, and do not relabel the induced vertices. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, 1], [0, 3], [0, 4]], [0, 2, 4], [[0, 4]], [1, 0, 1], [[4], [], [0]])."},"skills":["Induced Graph with Original Vertex Labels"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r05-graph-0094","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f9d0ae2b1cee56973aa6ccad2749d7b148dc9398e03c0abbd15bc2260d4cae9e","family_id":"qf-density17-graph-induced-subset","seed":202610175094,"semantic_template":"qf-density17-graph-induced-subset/{\"edges\":[[0,2],[0,3],[0,4]],\"n\":5,\"subset\":[0,2,4]}"},"fields":{"title":"Induced Graph with Original Vertex Labels","stem":"Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 2], [0, 3], [0, 4]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Retain source subset=[4, 2, 0], normalized as a set. Give (old normalized edges, sorted retained labels, induced edges, retained-label degrees, retained-label neighbor lists).","answer":{"kind":"expression","value":"([[0, 2], [0, 3], [0, 4]], [0, 2, 4], [[0, 2], [0, 4]], [2, 1, 1], [[2, 4], [0], [0]])","explanation":"Keep exactly edges with both endpoints retained, and do not relabel the induced vertices. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, 2], [0, 3], [0, 4]], [0, 2, 4], [[0, 2], [0, 4]], [2, 1, 1], [[2, 4], [0], [0]])."},"skills":["Induced Graph with Original Vertex Labels"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r05-graph-0095","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"24afddc902c7ed9900db08e5850092edf6c3515c2b8dc848ec2f846799e0858d","family_id":"qf-density17-graph-induced-subset","seed":202610175095,"semantic_template":"qf-density17-graph-induced-subset/{\"edges\":[[0,1],[0,2],[0,3],[0,4]],\"n\":5,\"subset\":[0,2,4]}"},"fields":{"title":"Induced Graph with Original Vertex Labels","stem":"Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 1], [0, 2], [0, 3], [0, 4]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Retain source subset=[4, 2, 0], normalized as a set. Give (old normalized edges, sorted retained labels, induced edges, retained-label degrees, retained-label neighbor lists).","answer":{"kind":"expression","value":"([[0, 1], [0, 2], [0, 3], [0, 4]], [0, 2, 4], [[0, 2], [0, 4]], [2, 1, 1], [[2, 4], [0], [0]])","explanation":"Keep exactly edges with both endpoints retained, and do not relabel the induced vertices. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, 1], [0, 2], [0, 3], [0, 4]], [0, 2, 4], [[0, 2], [0, 4]], [2, 1, 1], [[2, 4], [0], [0]])."},"skills":["Induced Graph with Original Vertex Labels"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r05-graph-0096","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"15d2884a7f6a370114027ae432987b14b4161f584b67baa00dcebaaa0228d7f7","family_id":"qf-density17-graph-induced-subset","seed":202610175096,"semantic_template":"qf-density17-graph-induced-subset/{\"edges\":[[1,2]],\"n\":5,\"subset\":[0,2,4]}"},"fields":{"title":"Induced Graph with Original Vertex Labels","stem":"Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[1, 2]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Retain source subset=[4, 2, 0], normalized as a set. Give (old normalized edges, sorted retained labels, induced edges, retained-label degrees, retained-label neighbor lists).","answer":{"kind":"expression","value":"([[1, 2]], [0, 2, 4], [], [0, 0, 0], [[], [], []])","explanation":"Keep exactly edges with both endpoints retained, and do not relabel the induced vertices. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[1, 2]], [0, 2, 4], [], [0, 0, 0], [[], [], []])."},"skills":["Induced Graph with Original Vertex Labels"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r05-graph-0097","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2b1c8d0974502ef2b4c7728874e6e24261c3a7cd2d45b13aad02524c4471a21b","family_id":"qf-density17-graph-induced-subset","seed":202610175097,"semantic_template":"qf-density17-graph-induced-subset/{\"edges\":[[0,1],[1,2]],\"n\":5,\"subset\":[0,2,4]}"},"fields":{"title":"Induced Graph with Original Vertex Labels","stem":"Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 1], [1, 2]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Retain source subset=[4, 2, 0], normalized as a set. Give (old normalized edges, sorted retained labels, induced edges, retained-label degrees, retained-label neighbor lists).","answer":{"kind":"expression","value":"([[0, 1], [1, 2]], [0, 2, 4], [], [0, 0, 0], [[], [], []])","explanation":"Keep exactly edges with both endpoints retained, and do not relabel the induced vertices. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, 1], [1, 2]], [0, 2, 4], [], [0, 0, 0], [[], [], []])."},"skills":["Induced Graph with Original Vertex Labels"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r05-graph-0098","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"78374329f1b273f8426d02114c3c802a6e872c84ee648035186262d63d4bde2c","family_id":"qf-density17-graph-induced-subset","seed":202610175098,"semantic_template":"qf-density17-graph-induced-subset/{\"edges\":[[0,2],[1,2]],\"n\":5,\"subset\":[0,2,4]}"},"fields":{"title":"Induced Graph with Original Vertex Labels","stem":"Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 2], [1, 2]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Retain source subset=[4, 2, 0], normalized as a set. Give (old normalized edges, sorted retained labels, induced edges, retained-label degrees, retained-label neighbor lists).","answer":{"kind":"expression","value":"([[0, 2], [1, 2]], [0, 2, 4], [[0, 2]], [1, 1, 0], [[2], [0], []])","explanation":"Keep exactly edges with both endpoints retained, and do not relabel the induced vertices. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, 2], [1, 2]], [0, 2, 4], [[0, 2]], [1, 1, 0], [[2], [0], []])."},"skills":["Induced Graph with Original Vertex Labels"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r05-graph-0099","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"022b214a74865533686cd3262b0dcf0e4331824a56e7e31aef41fe7e863a9390","family_id":"qf-density17-graph-induced-subset","seed":202610175099,"semantic_template":"qf-density17-graph-induced-subset/{\"edges\":[[0,1],[0,2],[1,2]],\"n\":5,\"subset\":[0,2,4]}"},"fields":{"title":"Induced Graph with Original Vertex Labels","stem":"Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 1], [0, 2], [1, 2]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Retain source subset=[4, 2, 0], normalized as a set. Give (old normalized edges, sorted retained labels, induced edges, retained-label degrees, retained-label neighbor lists).","answer":{"kind":"expression","value":"([[0, 1], [0, 2], [1, 2]], [0, 2, 4], [[0, 2]], [1, 1, 0], [[2], [0], []])","explanation":"Keep exactly edges with both endpoints retained, and do not relabel the induced vertices. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, 1], [0, 2], [1, 2]], [0, 2, 4], [[0, 2]], [1, 1, 0], [[2], [0], []])."},"skills":["Induced Graph with Original Vertex Labels"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r05-graph-0100","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d8587e6a329e82469cebc3171a2ba5f3cc64fea83e517789359dd294f84412a1","family_id":"qf-density17-graph-induced-subset","seed":202610175100,"semantic_template":"qf-density17-graph-induced-subset/{\"edges\":[[0,3],[1,2]],\"n\":5,\"subset\":[0,2,4]}"},"fields":{"title":"Induced Graph with Original Vertex Labels","stem":"Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 3], [1, 2]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Retain source subset=[4, 2, 0], normalized as a set. Give (old normalized edges, sorted retained labels, induced edges, retained-label degrees, retained-label neighbor lists).","answer":{"kind":"expression","value":"([[0, 3], [1, 2]], [0, 2, 4], [], [0, 0, 0], [[], [], []])","explanation":"Keep exactly edges with both endpoints retained, and do not relabel the induced vertices. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, 3], [1, 2]], [0, 2, 4], [], [0, 0, 0], [[], [], []])."},"skills":["Induced Graph with Original Vertex Labels"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}