{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r02-bipartite-graph-0001","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e560b8e841390b3a23ff9e8e9bdcdcae78ca335a2149c61f3225bb737df0e076","family_id":"qf-mod31-bipartite-partition-audit","seed":202610312001,"semantic_template":"qf-mod31-bipartite-partition-audit/{\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"n\":6,\"parts\":[0,0,0,0,0,0]}"},"fields":{"title":"Audit This Bipartition without Misclassifying the Entire Graph","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 4]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. The total part assignment=[0, 0, 0, 0, 0, 0] uses labels 0 or 1 for every vertex. Its two inverse-image parts are disjoint and cover the whole carrier. Every edge must cross parts; empty parts are allowed in an edgeless graph. Failure of this assignment alone does not prove the graph nonbipartite. Give (parts,conflicting_edges,proper?,classification). List all same-part edges in normalized edge order. Classification is bipartition_certificate or this_partition_fails_not_nonbipartite_proof.","answer":{"kind":"expression","value":"([0, 0, 0, 0, 0, 0], [[0, 1], [1, 2], [2, 3], [3, 4]], No, this_partition_fails_not_nonbipartite_proof)","explanation":"Check both endpoints of every complete edge. A passing assignment is a bipartition certificate. A failing assignment may be replaced by a different valid assignment; an odd-cycle witness is a separate obstruction. The exact result, retaining the requested order, is ([0, 0, 0, 0, 0, 0], [[0, 1], [1, 2], [2, 3], [3, 4]], No, this_partition_fails_not_nonbipartite_proof)."},"skills":["Audit This Bipartition without Misclassifying the Entire Graph"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r02-bipartite-graph-0002","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7cd6e84643a0a1772aa39eaae36b89f911da00bbe296ec88f496243baa791783","family_id":"qf-mod31-bipartite-partition-audit","seed":202610312002,"semantic_template":"qf-mod31-bipartite-partition-audit/{\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"n\":6,\"parts\":[1,0,0,0,0,0]}"},"fields":{"title":"Audit This Bipartition without Misclassifying the Entire Graph","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 4]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. The total part assignment=[1, 0, 0, 0, 0, 0] uses labels 0 or 1 for every vertex. Its two inverse-image parts are disjoint and cover the whole carrier. Every edge must cross parts; empty parts are allowed in an edgeless graph. Failure of this assignment alone does not prove the graph nonbipartite. Give (parts,conflicting_edges,proper?,classification). List all same-part edges in normalized edge order. Classification is bipartition_certificate or this_partition_fails_not_nonbipartite_proof.","answer":{"kind":"expression","value":"([1, 0, 0, 0, 0, 0], [[1, 2], [2, 3], [3, 4]], No, this_partition_fails_not_nonbipartite_proof)","explanation":"Check both endpoints of every complete edge. A passing assignment is a bipartition certificate. A failing assignment may be replaced by a different valid assignment; an odd-cycle witness is a separate obstruction. The exact result, retaining the requested order, is ([1, 0, 0, 0, 0, 0], [[1, 2], [2, 3], [3, 4]], No, this_partition_fails_not_nonbipartite_proof)."},"skills":["Audit This Bipartition without Misclassifying the Entire Graph"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r02-bipartite-graph-0003","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1bd4f621d95b3552a81d48360574d267f58abe6276db50f5de7c405d70b3c6db","family_id":"qf-mod31-bipartite-partition-audit","seed":202610312003,"semantic_template":"qf-mod31-bipartite-partition-audit/{\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"n\":6,\"parts\":[0,1,0,0,0,0]}"},"fields":{"title":"Audit This Bipartition without Misclassifying the Entire Graph","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 4]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. The total part assignment=[0, 1, 0, 0, 0, 0] uses labels 0 or 1 for every vertex. Its two inverse-image parts are disjoint and cover the whole carrier. Every edge must cross parts; empty parts are allowed in an edgeless graph. Failure of this assignment alone does not prove the graph nonbipartite. Give (parts,conflicting_edges,proper?,classification). List all same-part edges in normalized edge order. Classification is bipartition_certificate or this_partition_fails_not_nonbipartite_proof.","answer":{"kind":"expression","value":"([0, 1, 0, 0, 0, 0], [[2, 3], [3, 4]], No, this_partition_fails_not_nonbipartite_proof)","explanation":"Check both endpoints of every complete edge. A passing assignment is a bipartition certificate. A failing assignment may be replaced by a different valid assignment; an odd-cycle witness is a separate obstruction. The exact result, retaining the requested order, is ([0, 1, 0, 0, 0, 0], [[2, 3], [3, 4]], No, this_partition_fails_not_nonbipartite_proof)."},"skills":["Audit This Bipartition without Misclassifying the Entire Graph"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r02-bipartite-graph-0004","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c3f659f9be30120fe2cb1cdb4cc93fc8464ca68070457a15b3444b6d04ab9c92","family_id":"qf-mod31-bipartite-partition-audit","seed":202610312004,"semantic_template":"qf-mod31-bipartite-partition-audit/{\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"n\":6,\"parts\":[1,1,0,0,0,0]}"},"fields":{"title":"Audit This Bipartition without Misclassifying the Entire Graph","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 4]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. The total part assignment=[1, 1, 0, 0, 0, 0] uses labels 0 or 1 for every vertex. Its two inverse-image parts are disjoint and cover the whole carrier. Every edge must cross parts; empty parts are allowed in an edgeless graph. Failure of this assignment alone does not prove the graph nonbipartite. Give (parts,conflicting_edges,proper?,classification). List all same-part edges in normalized edge order. Classification is bipartition_certificate or this_partition_fails_not_nonbipartite_proof.","answer":{"kind":"expression","value":"([1, 1, 0, 0, 0, 0], [[0, 1], [2, 3], [3, 4]], No, this_partition_fails_not_nonbipartite_proof)","explanation":"Check both endpoints of every complete edge. A passing assignment is a bipartition certificate. A failing assignment may be replaced by a different valid assignment; an odd-cycle witness is a separate obstruction. The exact result, retaining the requested order, is ([1, 1, 0, 0, 0, 0], [[0, 1], [2, 3], [3, 4]], No, this_partition_fails_not_nonbipartite_proof)."},"skills":["Audit This Bipartition without Misclassifying the Entire Graph"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r02-bipartite-graph-0005","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"088b9d6c3b87d8dc91ab56ec1997255ee714738f405c8f94c8c2dd9e5defb606","family_id":"qf-mod31-bipartite-partition-audit","seed":202610312005,"semantic_template":"qf-mod31-bipartite-partition-audit/{\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"n\":6,\"parts\":[0,0,1,0,0,0]}"},"fields":{"title":"Audit This Bipartition without Misclassifying the Entire Graph","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 4]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. The total part assignment=[0, 0, 1, 0, 0, 0] uses labels 0 or 1 for every vertex. Its two inverse-image parts are disjoint and cover the whole carrier. Every edge must cross parts; empty parts are allowed in an edgeless graph. Failure of this assignment alone does not prove the graph nonbipartite. Give (parts,conflicting_edges,proper?,classification). List all same-part edges in normalized edge order. Classification is bipartition_certificate or this_partition_fails_not_nonbipartite_proof.","answer":{"kind":"expression","value":"([0, 0, 1, 0, 0, 0], [[0, 1], [3, 4]], No, this_partition_fails_not_nonbipartite_proof)","explanation":"Check both endpoints of every complete edge. A passing assignment is a bipartition certificate. A failing assignment may be replaced by a different valid assignment; an odd-cycle witness is a separate obstruction. The exact result, retaining the requested order, is ([0, 0, 1, 0, 0, 0], [[0, 1], [3, 4]], No, this_partition_fails_not_nonbipartite_proof)."},"skills":["Audit This Bipartition without Misclassifying the Entire Graph"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r02-bipartite-graph-0006","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4a6558a6bb66083c75ef15d46d700333a02220b203b24acc9aeead8aae2b7c89","family_id":"qf-mod31-bipartite-partition-audit","seed":202610312006,"semantic_template":"qf-mod31-bipartite-partition-audit/{\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"n\":6,\"parts\":[1,0,1,0,0,0]}"},"fields":{"title":"Audit This Bipartition without Misclassifying the Entire Graph","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 4]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. The total part assignment=[1, 0, 1, 0, 0, 0] uses labels 0 or 1 for every vertex. Its two inverse-image parts are disjoint and cover the whole carrier. Every edge must cross parts; empty parts are allowed in an edgeless graph. Failure of this assignment alone does not prove the graph nonbipartite. Give (parts,conflicting_edges,proper?,classification). List all same-part edges in normalized edge order. Classification is bipartition_certificate or this_partition_fails_not_nonbipartite_proof.","answer":{"kind":"expression","value":"([1, 0, 1, 0, 0, 0], [[3, 4]], No, this_partition_fails_not_nonbipartite_proof)","explanation":"Check both endpoints of every complete edge. A passing assignment is a bipartition certificate. A failing assignment may be replaced by a different valid assignment; an odd-cycle witness is a separate obstruction. The exact result, retaining the requested order, is ([1, 0, 1, 0, 0, 0], [[3, 4]], No, this_partition_fails_not_nonbipartite_proof)."},"skills":["Audit This Bipartition without Misclassifying the Entire Graph"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r02-bipartite-graph-0007","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"55bbe17de4211898281383c9f9dfe9df25268814c7321f13b4790fc7ec861707","family_id":"qf-mod31-bipartite-partition-audit","seed":202610312007,"semantic_template":"qf-mod31-bipartite-partition-audit/{\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"n\":6,\"parts\":[0,1,1,0,0,0]}"},"fields":{"title":"Audit This Bipartition without Misclassifying the Entire Graph","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 4]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. The total part assignment=[0, 1, 1, 0, 0, 0] uses labels 0 or 1 for every vertex. Its two inverse-image parts are disjoint and cover the whole carrier. Every edge must cross parts; empty parts are allowed in an edgeless graph. Failure of this assignment alone does not prove the graph nonbipartite. Give (parts,conflicting_edges,proper?,classification). List all same-part edges in normalized edge order. Classification is bipartition_certificate or this_partition_fails_not_nonbipartite_proof.","answer":{"kind":"expression","value":"([0, 1, 1, 0, 0, 0], [[1, 2], [3, 4]], No, this_partition_fails_not_nonbipartite_proof)","explanation":"Check both endpoints of every complete edge. A passing assignment is a bipartition certificate. A failing assignment may be replaced by a different valid assignment; an odd-cycle witness is a separate obstruction. The exact result, retaining the requested order, is ([0, 1, 1, 0, 0, 0], [[1, 2], [3, 4]], No, this_partition_fails_not_nonbipartite_proof)."},"skills":["Audit This Bipartition without Misclassifying the Entire Graph"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r02-bipartite-graph-0008","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"51d14ef9b5b91c5edd15f0934a483d385c3f9e6a83291c68a256b9478714df6d","family_id":"qf-mod31-bipartite-partition-audit","seed":202610312008,"semantic_template":"qf-mod31-bipartite-partition-audit/{\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"n\":6,\"parts\":[1,1,1,0,0,0]}"},"fields":{"title":"Audit This Bipartition without Misclassifying the Entire Graph","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 4]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. The total part assignment=[1, 1, 1, 0, 0, 0] uses labels 0 or 1 for every vertex. Its two inverse-image parts are disjoint and cover the whole carrier. Every edge must cross parts; empty parts are allowed in an edgeless graph. Failure of this assignment alone does not prove the graph nonbipartite. Give (parts,conflicting_edges,proper?,classification). List all same-part edges in normalized edge order. Classification is bipartition_certificate or this_partition_fails_not_nonbipartite_proof.","answer":{"kind":"expression","value":"([1, 1, 1, 0, 0, 0], [[0, 1], [1, 2], [3, 4]], No, this_partition_fails_not_nonbipartite_proof)","explanation":"Check both endpoints of every complete edge. A passing assignment is a bipartition certificate. A failing assignment may be replaced by a different valid assignment; an odd-cycle witness is a separate obstruction. The exact result, retaining the requested order, is ([1, 1, 1, 0, 0, 0], [[0, 1], [1, 2], [3, 4]], No, this_partition_fails_not_nonbipartite_proof)."},"skills":["Audit This Bipartition without Misclassifying the Entire Graph"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r02-bipartite-graph-0009","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"19f91cadd5b110946cc9d3f9cb1f3af6cb2fdc04fbeafc0f56404e8009c9fc46","family_id":"qf-mod31-bipartite-partition-audit","seed":202610312009,"semantic_template":"qf-mod31-bipartite-partition-audit/{\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"n\":6,\"parts\":[0,0,0,1,0,0]}"},"fields":{"title":"Audit This Bipartition without Misclassifying the Entire Graph","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 4]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. The total part assignment=[0, 0, 0, 1, 0, 0] uses labels 0 or 1 for every vertex. Its two inverse-image parts are disjoint and cover the whole carrier. Every edge must cross parts; empty parts are allowed in an edgeless graph. Failure of this assignment alone does not prove the graph nonbipartite. Give (parts,conflicting_edges,proper?,classification). List all same-part edges in normalized edge order. Classification is bipartition_certificate or this_partition_fails_not_nonbipartite_proof.","answer":{"kind":"expression","value":"([0, 0, 0, 1, 0, 0], [[0, 1], [1, 2]], No, this_partition_fails_not_nonbipartite_proof)","explanation":"Check both endpoints of every complete edge. A passing assignment is a bipartition certificate. A failing assignment may be replaced by a different valid assignment; an odd-cycle witness is a separate obstruction. The exact result, retaining the requested order, is ([0, 0, 0, 1, 0, 0], [[0, 1], [1, 2]], No, this_partition_fails_not_nonbipartite_proof)."},"skills":["Audit This Bipartition without Misclassifying the Entire Graph"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r02-bipartite-graph-0010","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3d22f4b3876c01b46919c33cdf80e7462425fdd56766a8ed0ee9dbf29955d4de","family_id":"qf-mod31-bipartite-partition-audit","seed":202610312010,"semantic_template":"qf-mod31-bipartite-partition-audit/{\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"n\":6,\"parts\":[1,0,0,1,0,0]}"},"fields":{"title":"Audit This Bipartition without Misclassifying the Entire Graph","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 4]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. The total part assignment=[1, 0, 0, 1, 0, 0] uses labels 0 or 1 for every vertex. Its two inverse-image parts are disjoint and cover the whole carrier. Every edge must cross parts; empty parts are allowed in an edgeless graph. Failure of this assignment alone does not prove the graph nonbipartite. Give (parts,conflicting_edges,proper?,classification). List all same-part edges in normalized edge order. Classification is bipartition_certificate or this_partition_fails_not_nonbipartite_proof.","answer":{"kind":"expression","value":"([1, 0, 0, 1, 0, 0], [[1, 2]], No, this_partition_fails_not_nonbipartite_proof)","explanation":"Check both endpoints of every complete edge. A passing assignment is a bipartition certificate. A failing assignment may be replaced by a different valid assignment; an odd-cycle witness is a separate obstruction. The exact result, retaining the requested order, is ([1, 0, 0, 1, 0, 0], [[1, 2]], No, this_partition_fails_not_nonbipartite_proof)."},"skills":["Audit This Bipartition without Misclassifying the Entire Graph"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r02-bipartite-graph-0011","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"88b9288ed074bbc694affe1efb04c58f302730095c3840d8ab188fdf86a05b1f","family_id":"qf-mod31-bipartite-partition-audit","seed":202610312011,"semantic_template":"qf-mod31-bipartite-partition-audit/{\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"n\":6,\"parts\":[0,1,0,1,0,0]}"},"fields":{"title":"Audit This Bipartition without Misclassifying the Entire Graph","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 4]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. The total part assignment=[0, 1, 0, 1, 0, 0] uses labels 0 or 1 for every vertex. Its two inverse-image parts are disjoint and cover the whole carrier. Every edge must cross parts; empty parts are allowed in an edgeless graph. Failure of this assignment alone does not prove the graph nonbipartite. Give (parts,conflicting_edges,proper?,classification). List all same-part edges in normalized edge order. Classification is bipartition_certificate or this_partition_fails_not_nonbipartite_proof.","answer":{"kind":"expression","value":"([0, 1, 0, 1, 0, 0], [], Yes, bipartition_certificate)","explanation":"Check both endpoints of every complete edge. A passing assignment is a bipartition certificate. A failing assignment may be replaced by a different valid assignment; an odd-cycle witness is a separate obstruction. The exact result, retaining the requested order, is ([0, 1, 0, 1, 0, 0], [], Yes, bipartition_certificate)."},"skills":["Audit This Bipartition without Misclassifying the Entire Graph"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r02-bipartite-graph-0012","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0617af15d506714d6744859b1f1b7989d6958a7e09f56f1ef684d6c298066dcf","family_id":"qf-mod31-bipartite-partition-audit","seed":202610312012,"semantic_template":"qf-mod31-bipartite-partition-audit/{\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"n\":6,\"parts\":[1,1,0,1,0,0]}"},"fields":{"title":"Audit This Bipartition without Misclassifying the Entire Graph","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 4]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. The total part assignment=[1, 1, 0, 1, 0, 0] uses labels 0 or 1 for every vertex. Its two inverse-image parts are disjoint and cover the whole carrier. Every edge must cross parts; empty parts are allowed in an edgeless graph. Failure of this assignment alone does not prove the graph nonbipartite. Give (parts,conflicting_edges,proper?,classification). List all same-part edges in normalized edge order. Classification is bipartition_certificate or this_partition_fails_not_nonbipartite_proof.","answer":{"kind":"expression","value":"([1, 1, 0, 1, 0, 0], [[0, 1]], No, this_partition_fails_not_nonbipartite_proof)","explanation":"Check both endpoints of every complete edge. A passing assignment is a bipartition certificate. A failing assignment may be replaced by a different valid assignment; an odd-cycle witness is a separate obstruction. The exact result, retaining the requested order, is ([1, 1, 0, 1, 0, 0], [[0, 1]], No, this_partition_fails_not_nonbipartite_proof)."},"skills":["Audit This Bipartition without Misclassifying the Entire Graph"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r02-bipartite-graph-0013","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7cb2edaff5cfdcd451e7be7dd0f0958ef258d3d8a4a0ef6b40af7bd2789b4a1b","family_id":"qf-mod31-bipartite-partition-audit","seed":202610312013,"semantic_template":"qf-mod31-bipartite-partition-audit/{\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"n\":6,\"parts\":[0,0,1,1,0,0]}"},"fields":{"title":"Audit This Bipartition without Misclassifying the Entire Graph","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 4]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. The total part assignment=[0, 0, 1, 1, 0, 0] uses labels 0 or 1 for every vertex. Its two inverse-image parts are disjoint and cover the whole carrier. Every edge must cross parts; empty parts are allowed in an edgeless graph. Failure of this assignment alone does not prove the graph nonbipartite. Give (parts,conflicting_edges,proper?,classification). List all same-part edges in normalized edge order. Classification is bipartition_certificate or this_partition_fails_not_nonbipartite_proof.","answer":{"kind":"expression","value":"([0, 0, 1, 1, 0, 0], [[0, 1], [2, 3]], No, this_partition_fails_not_nonbipartite_proof)","explanation":"Check both endpoints of every complete edge. A passing assignment is a bipartition certificate. A failing assignment may be replaced by a different valid assignment; an odd-cycle witness is a separate obstruction. The exact result, retaining the requested order, is ([0, 0, 1, 1, 0, 0], [[0, 1], [2, 3]], No, this_partition_fails_not_nonbipartite_proof)."},"skills":["Audit This Bipartition without Misclassifying the Entire Graph"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r02-bipartite-graph-0014","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"219e73ed835fd4a01992ef16634218ef3a6b7bdc1ad6c1bb9ff54af42c787f92","family_id":"qf-mod31-bipartite-partition-audit","seed":202610312014,"semantic_template":"qf-mod31-bipartite-partition-audit/{\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"n\":6,\"parts\":[1,0,1,1,0,0]}"},"fields":{"title":"Audit This Bipartition without Misclassifying the Entire Graph","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 4]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. The total part assignment=[1, 0, 1, 1, 0, 0] uses labels 0 or 1 for every vertex. Its two inverse-image parts are disjoint and cover the whole carrier. Every edge must cross parts; empty parts are allowed in an edgeless graph. Failure of this assignment alone does not prove the graph nonbipartite. Give (parts,conflicting_edges,proper?,classification). List all same-part edges in normalized edge order. Classification is bipartition_certificate or this_partition_fails_not_nonbipartite_proof.","answer":{"kind":"expression","value":"([1, 0, 1, 1, 0, 0], [[2, 3]], No, this_partition_fails_not_nonbipartite_proof)","explanation":"Check both endpoints of every complete edge. A passing assignment is a bipartition certificate. A failing assignment may be replaced by a different valid assignment; an odd-cycle witness is a separate obstruction. The exact result, retaining the requested order, is ([1, 0, 1, 1, 0, 0], [[2, 3]], No, this_partition_fails_not_nonbipartite_proof)."},"skills":["Audit This Bipartition without Misclassifying the Entire Graph"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r02-bipartite-graph-0015","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1a421afdd96d5ed82848aa9992d4986bf215244ccda7020904df9f235342466c","family_id":"qf-mod31-bipartite-partition-audit","seed":202610312015,"semantic_template":"qf-mod31-bipartite-partition-audit/{\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"n\":6,\"parts\":[0,1,1,1,0,0]}"},"fields":{"title":"Audit This Bipartition without Misclassifying the Entire Graph","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 4]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. The total part assignment=[0, 1, 1, 1, 0, 0] uses labels 0 or 1 for every vertex. Its two inverse-image parts are disjoint and cover the whole carrier. Every edge must cross parts; empty parts are allowed in an edgeless graph. Failure of this assignment alone does not prove the graph nonbipartite. Give (parts,conflicting_edges,proper?,classification). List all same-part edges in normalized edge order. Classification is bipartition_certificate or this_partition_fails_not_nonbipartite_proof.","answer":{"kind":"expression","value":"([0, 1, 1, 1, 0, 0], [[1, 2], [2, 3]], No, this_partition_fails_not_nonbipartite_proof)","explanation":"Check both endpoints of every complete edge. A passing assignment is a bipartition certificate. A failing assignment may be replaced by a different valid assignment; an odd-cycle witness is a separate obstruction. The exact result, retaining the requested order, is ([0, 1, 1, 1, 0, 0], [[1, 2], [2, 3]], No, this_partition_fails_not_nonbipartite_proof)."},"skills":["Audit This Bipartition without Misclassifying the Entire Graph"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r02-bipartite-graph-0016","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7a5260a42aa76aa52959439a34ab400d86d388493a9913b61a299533deb0de0c","family_id":"qf-mod31-bipartite-partition-audit","seed":202610312016,"semantic_template":"qf-mod31-bipartite-partition-audit/{\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"n\":6,\"parts\":[1,1,1,1,0,0]}"},"fields":{"title":"Audit This Bipartition without Misclassifying the Entire Graph","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 4]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. The total part assignment=[1, 1, 1, 1, 0, 0] uses labels 0 or 1 for every vertex. Its two inverse-image parts are disjoint and cover the whole carrier. Every edge must cross parts; empty parts are allowed in an edgeless graph. Failure of this assignment alone does not prove the graph nonbipartite. Give (parts,conflicting_edges,proper?,classification). List all same-part edges in normalized edge order. Classification is bipartition_certificate or this_partition_fails_not_nonbipartite_proof.","answer":{"kind":"expression","value":"([1, 1, 1, 1, 0, 0], [[0, 1], [1, 2], [2, 3]], No, this_partition_fails_not_nonbipartite_proof)","explanation":"Check both endpoints of every complete edge. A passing assignment is a bipartition certificate. A failing assignment may be replaced by a different valid assignment; an odd-cycle witness is a separate obstruction. The exact result, retaining the requested order, is ([1, 1, 1, 1, 0, 0], [[0, 1], [1, 2], [2, 3]], No, this_partition_fails_not_nonbipartite_proof)."},"skills":["Audit This Bipartition without Misclassifying the Entire Graph"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r02-bipartite-graph-0017","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fe301b423650ab3c58df5a52e2ed50f57df331678a355bd0e003e1f36276ec98","family_id":"qf-mod31-bipartite-partition-audit","seed":202610312017,"semantic_template":"qf-mod31-bipartite-partition-audit/{\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"n\":6,\"parts\":[0,0,0,0,1,0]}"},"fields":{"title":"Audit This Bipartition without Misclassifying the Entire Graph","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 4]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. The total part assignment=[0, 0, 0, 0, 1, 0] uses labels 0 or 1 for every vertex. Its two inverse-image parts are disjoint and cover the whole carrier. Every edge must cross parts; empty parts are allowed in an edgeless graph. Failure of this assignment alone does not prove the graph nonbipartite. Give (parts,conflicting_edges,proper?,classification). List all same-part edges in normalized edge order. Classification is bipartition_certificate or this_partition_fails_not_nonbipartite_proof.","answer":{"kind":"expression","value":"([0, 0, 0, 0, 1, 0], [[0, 1], [1, 2], [2, 3]], No, this_partition_fails_not_nonbipartite_proof)","explanation":"Check both endpoints of every complete edge. A passing assignment is a bipartition certificate. A failing assignment may be replaced by a different valid assignment; an odd-cycle witness is a separate obstruction. The exact result, retaining the requested order, is ([0, 0, 0, 0, 1, 0], [[0, 1], [1, 2], [2, 3]], No, this_partition_fails_not_nonbipartite_proof)."},"skills":["Audit This Bipartition without Misclassifying the Entire Graph"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r02-bipartite-graph-0018","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4847c9008a73d6ea21be804e4dd2b6b50b740ab5398bee34537ef98386d0f9f6","family_id":"qf-mod31-bipartite-partition-audit","seed":202610312018,"semantic_template":"qf-mod31-bipartite-partition-audit/{\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"n\":6,\"parts\":[1,0,0,0,1,0]}"},"fields":{"title":"Audit This Bipartition without Misclassifying the Entire Graph","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 4]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. The total part assignment=[1, 0, 0, 0, 1, 0] uses labels 0 or 1 for every vertex. Its two inverse-image parts are disjoint and cover the whole carrier. Every edge must cross parts; empty parts are allowed in an edgeless graph. Failure of this assignment alone does not prove the graph nonbipartite. Give (parts,conflicting_edges,proper?,classification). List all same-part edges in normalized edge order. Classification is bipartition_certificate or this_partition_fails_not_nonbipartite_proof.","answer":{"kind":"expression","value":"([1, 0, 0, 0, 1, 0], [[1, 2], [2, 3]], No, this_partition_fails_not_nonbipartite_proof)","explanation":"Check both endpoints of every complete edge. A passing assignment is a bipartition certificate. A failing assignment may be replaced by a different valid assignment; an odd-cycle witness is a separate obstruction. The exact result, retaining the requested order, is ([1, 0, 0, 0, 1, 0], [[1, 2], [2, 3]], No, this_partition_fails_not_nonbipartite_proof)."},"skills":["Audit This Bipartition without Misclassifying the Entire Graph"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r02-bipartite-graph-0019","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"79623f27d3ed9f9fc988e2f0d42d956b221be8cfeed4f25303185537934a4346","family_id":"qf-mod31-bipartite-partition-audit","seed":202610312019,"semantic_template":"qf-mod31-bipartite-partition-audit/{\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"n\":6,\"parts\":[0,1,0,0,1,0]}"},"fields":{"title":"Audit This Bipartition without Misclassifying the Entire Graph","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 4]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. The total part assignment=[0, 1, 0, 0, 1, 0] uses labels 0 or 1 for every vertex. Its two inverse-image parts are disjoint and cover the whole carrier. Every edge must cross parts; empty parts are allowed in an edgeless graph. Failure of this assignment alone does not prove the graph nonbipartite. Give (parts,conflicting_edges,proper?,classification). List all same-part edges in normalized edge order. Classification is bipartition_certificate or this_partition_fails_not_nonbipartite_proof.","answer":{"kind":"expression","value":"([0, 1, 0, 0, 1, 0], [[2, 3]], No, this_partition_fails_not_nonbipartite_proof)","explanation":"Check both endpoints of every complete edge. A passing assignment is a bipartition certificate. A failing assignment may be replaced by a different valid assignment; an odd-cycle witness is a separate obstruction. The exact result, retaining the requested order, is ([0, 1, 0, 0, 1, 0], [[2, 3]], No, this_partition_fails_not_nonbipartite_proof)."},"skills":["Audit This Bipartition without Misclassifying the Entire Graph"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r02-bipartite-graph-0020","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d3191696378b870f5ab8ca0ad2b884141b4610111ec14daa9adbede583f9eed7","family_id":"qf-mod31-bipartite-partition-audit","seed":202610312020,"semantic_template":"qf-mod31-bipartite-partition-audit/{\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"n\":6,\"parts\":[1,1,0,0,1,0]}"},"fields":{"title":"Audit This Bipartition without Misclassifying the Entire Graph","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 4]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. The total part assignment=[1, 1, 0, 0, 1, 0] uses labels 0 or 1 for every vertex. Its two inverse-image parts are disjoint and cover the whole carrier. Every edge must cross parts; empty parts are allowed in an edgeless graph. Failure of this assignment alone does not prove the graph nonbipartite. Give (parts,conflicting_edges,proper?,classification). List all same-part edges in normalized edge order. Classification is bipartition_certificate or this_partition_fails_not_nonbipartite_proof.","answer":{"kind":"expression","value":"([1, 1, 0, 0, 1, 0], [[0, 1], [2, 3]], No, this_partition_fails_not_nonbipartite_proof)","explanation":"Check both endpoints of every complete edge. A passing assignment is a bipartition certificate. A failing assignment may be replaced by a different valid assignment; an odd-cycle witness is a separate obstruction. The exact result, retaining the requested order, is ([1, 1, 0, 0, 1, 0], [[0, 1], [2, 3]], No, this_partition_fails_not_nonbipartite_proof)."},"skills":["Audit This Bipartition without Misclassifying the Entire Graph"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r02-bipartite-graph-0021","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"02066bb2ed8f03d039421cd65edd44611894cef39105cb5a682a784dad28264c","family_id":"qf-mod31-bipartite-partition-audit","seed":202610312021,"semantic_template":"qf-mod31-bipartite-partition-audit/{\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"n\":6,\"parts\":[0,0,1,0,1,0]}"},"fields":{"title":"Audit This Bipartition without Misclassifying the Entire Graph","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 4]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. The total part assignment=[0, 0, 1, 0, 1, 0] uses labels 0 or 1 for every vertex. Its two inverse-image parts are disjoint and cover the whole carrier. Every edge must cross parts; empty parts are allowed in an edgeless graph. Failure of this assignment alone does not prove the graph nonbipartite. Give (parts,conflicting_edges,proper?,classification). List all same-part edges in normalized edge order. Classification is bipartition_certificate or this_partition_fails_not_nonbipartite_proof.","answer":{"kind":"expression","value":"([0, 0, 1, 0, 1, 0], [[0, 1]], No, this_partition_fails_not_nonbipartite_proof)","explanation":"Check both endpoints of every complete edge. A passing assignment is a bipartition certificate. A failing assignment may be replaced by a different valid assignment; an odd-cycle witness is a separate obstruction. The exact result, retaining the requested order, is ([0, 0, 1, 0, 1, 0], [[0, 1]], No, this_partition_fails_not_nonbipartite_proof)."},"skills":["Audit This Bipartition without Misclassifying the Entire Graph"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r02-bipartite-graph-0022","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a6584ffceb5b931523d7942dcbb676f6e1ff5bed1f6792e94e7b668b361de7cc","family_id":"qf-mod31-bipartite-partition-audit","seed":202610312022,"semantic_template":"qf-mod31-bipartite-partition-audit/{\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"n\":6,\"parts\":[1,0,1,0,1,0]}"},"fields":{"title":"Audit This Bipartition without Misclassifying the Entire Graph","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 4]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. The total part assignment=[1, 0, 1, 0, 1, 0] uses labels 0 or 1 for every vertex. Its two inverse-image parts are disjoint and cover the whole carrier. Every edge must cross parts; empty parts are allowed in an edgeless graph. Failure of this assignment alone does not prove the graph nonbipartite. Give (parts,conflicting_edges,proper?,classification). List all same-part edges in normalized edge order. Classification is bipartition_certificate or this_partition_fails_not_nonbipartite_proof.","answer":{"kind":"expression","value":"([1, 0, 1, 0, 1, 0], [], Yes, bipartition_certificate)","explanation":"Check both endpoints of every complete edge. A passing assignment is a bipartition certificate. A failing assignment may be replaced by a different valid assignment; an odd-cycle witness is a separate obstruction. The exact result, retaining the requested order, is ([1, 0, 1, 0, 1, 0], [], Yes, bipartition_certificate)."},"skills":["Audit This Bipartition without Misclassifying the Entire Graph"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r02-bipartite-graph-0023","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d8a8a6c34291496be7f29d919701b12cf19f565d1545255faba457f51c6ec91d","family_id":"qf-mod31-bipartite-partition-audit","seed":202610312023,"semantic_template":"qf-mod31-bipartite-partition-audit/{\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"n\":6,\"parts\":[0,1,1,0,1,0]}"},"fields":{"title":"Audit This Bipartition without Misclassifying the Entire Graph","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 4]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. The total part assignment=[0, 1, 1, 0, 1, 0] uses labels 0 or 1 for every vertex. Its two inverse-image parts are disjoint and cover the whole carrier. Every edge must cross parts; empty parts are allowed in an edgeless graph. Failure of this assignment alone does not prove the graph nonbipartite. Give (parts,conflicting_edges,proper?,classification). List all same-part edges in normalized edge order. Classification is bipartition_certificate or this_partition_fails_not_nonbipartite_proof.","answer":{"kind":"expression","value":"([0, 1, 1, 0, 1, 0], [[1, 2]], No, this_partition_fails_not_nonbipartite_proof)","explanation":"Check both endpoints of every complete edge. A passing assignment is a bipartition certificate. A failing assignment may be replaced by a different valid assignment; an odd-cycle witness is a separate obstruction. The exact result, retaining the requested order, is ([0, 1, 1, 0, 1, 0], [[1, 2]], No, this_partition_fails_not_nonbipartite_proof)."},"skills":["Audit This Bipartition without Misclassifying the Entire Graph"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r02-bipartite-graph-0024","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d88b7ea04296732e702c293c59d687a06bcbf41836adc93c5152f4c3f4a2e652","family_id":"qf-mod31-bipartite-partition-audit","seed":202610312024,"semantic_template":"qf-mod31-bipartite-partition-audit/{\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"n\":6,\"parts\":[1,1,1,0,1,0]}"},"fields":{"title":"Audit This Bipartition without Misclassifying the Entire Graph","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 4]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. The total part assignment=[1, 1, 1, 0, 1, 0] uses labels 0 or 1 for every vertex. Its two inverse-image parts are disjoint and cover the whole carrier. Every edge must cross parts; empty parts are allowed in an edgeless graph. Failure of this assignment alone does not prove the graph nonbipartite. Give (parts,conflicting_edges,proper?,classification). List all same-part edges in normalized edge order. Classification is bipartition_certificate or this_partition_fails_not_nonbipartite_proof.","answer":{"kind":"expression","value":"([1, 1, 1, 0, 1, 0], [[0, 1], [1, 2]], No, this_partition_fails_not_nonbipartite_proof)","explanation":"Check both endpoints of every complete edge. A passing assignment is a bipartition certificate. A failing assignment may be replaced by a different valid assignment; an odd-cycle witness is a separate obstruction. The exact result, retaining the requested order, is ([1, 1, 1, 0, 1, 0], [[0, 1], [1, 2]], No, this_partition_fails_not_nonbipartite_proof)."},"skills":["Audit This Bipartition without Misclassifying the Entire Graph"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r02-bipartite-graph-0025","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a22e683f903ff0d1668f9d23ff23950df3e866f0376a03dc6c04a93f7c64d845","family_id":"qf-mod31-bipartite-partition-audit","seed":202610312025,"semantic_template":"qf-mod31-bipartite-partition-audit/{\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"n\":6,\"parts\":[0,0,0,1,1,0]}"},"fields":{"title":"Audit This Bipartition without Misclassifying the Entire Graph","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 4]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. The total part assignment=[0, 0, 0, 1, 1, 0] uses labels 0 or 1 for every vertex. Its two inverse-image parts are disjoint and cover the whole carrier. Every edge must cross parts; empty parts are allowed in an edgeless graph. Failure of this assignment alone does not prove the graph nonbipartite. Give (parts,conflicting_edges,proper?,classification). List all same-part edges in normalized edge order. Classification is bipartition_certificate or this_partition_fails_not_nonbipartite_proof.","answer":{"kind":"expression","value":"([0, 0, 0, 1, 1, 0], [[0, 1], [1, 2], [3, 4]], No, this_partition_fails_not_nonbipartite_proof)","explanation":"Check both endpoints of every complete edge. A passing assignment is a bipartition certificate. A failing assignment may be replaced by a different valid assignment; an odd-cycle witness is a separate obstruction. The exact result, retaining the requested order, is ([0, 0, 0, 1, 1, 0], [[0, 1], [1, 2], [3, 4]], No, this_partition_fails_not_nonbipartite_proof)."},"skills":["Audit This Bipartition without Misclassifying the Entire Graph"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r02-bipartite-graph-0026","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3cc771a1498d098f2ecb79d3209f3911bd262b82c1e275a1665504190d3e05c1","family_id":"qf-mod31-bipartite-edge-crossing","seed":202610312026,"semantic_template":"qf-mod31-bipartite-edge-crossing/{\"edge_index\":0,\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"n\":6,\"parts\":[0,0,0,0,0,0]}"},"fields":{"title":"One Edge Crossing Is Separate from the Whole Partition Audit","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 4]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. The total part assignment=[0, 0, 0, 0, 0, 0] uses labels 0 or 1 for every vertex. Its two inverse-image parts are disjoint and cover the whole carrier. Every edge must cross parts; empty parts are allowed in an edgeless graph. Failure of this assignment alone does not prove the graph nonbipartite. edge_index=0 selects the normalized sorted edge list, not the raw input order. Give (edge_index,edge,left_part,right_part,crosses?,whole_partition_proper?).","answer":{"kind":"expression","value":"(0, [0, 1], 0, 0, No, No)","explanation":"Read the selected normalized edge and report its two part labels, then retain the independent all-edge properness flag. One crossing edge cannot certify all remaining edges. The exact result, retaining the requested order, is (0, [0, 1], 0, 0, No, No)."},"skills":["One Edge Crossing Is Separate from the Whole Partition Audit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r02-bipartite-graph-0027","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c7739eff6eb7df47bf8050bfdbdd19cd82b047f43af2101460f97ef21864b977","family_id":"qf-mod31-bipartite-edge-crossing","seed":202610312027,"semantic_template":"qf-mod31-bipartite-edge-crossing/{\"edge_index\":1,\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"n\":6,\"parts\":[1,0,0,0,0,0]}"},"fields":{"title":"One Edge Crossing Is Separate from the Whole Partition Audit","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 4]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. The total part assignment=[1, 0, 0, 0, 0, 0] uses labels 0 or 1 for every vertex. Its two inverse-image parts are disjoint and cover the whole carrier. Every edge must cross parts; empty parts are allowed in an edgeless graph. Failure of this assignment alone does not prove the graph nonbipartite. edge_index=1 selects the normalized sorted edge list, not the raw input order. Give (edge_index,edge,left_part,right_part,crosses?,whole_partition_proper?).","answer":{"kind":"expression","value":"(1, [1, 2], 0, 0, No, No)","explanation":"Read the selected normalized edge and report its two part labels, then retain the independent all-edge properness flag. One crossing edge cannot certify all remaining edges. The exact result, retaining the requested order, is (1, [1, 2], 0, 0, No, No)."},"skills":["One Edge Crossing Is Separate from the Whole Partition Audit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r02-bipartite-graph-0028","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9e065a61e0ed510903bfaa5d970124ccd91554e621f8ffe59e6c0fff069f65e2","family_id":"qf-mod31-bipartite-edge-crossing","seed":202610312028,"semantic_template":"qf-mod31-bipartite-edge-crossing/{\"edge_index\":2,\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"n\":6,\"parts\":[0,1,0,0,0,0]}"},"fields":{"title":"One Edge Crossing Is Separate from the Whole Partition Audit","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 4]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. The total part assignment=[0, 1, 0, 0, 0, 0] uses labels 0 or 1 for every vertex. Its two inverse-image parts are disjoint and cover the whole carrier. Every edge must cross parts; empty parts are allowed in an edgeless graph. Failure of this assignment alone does not prove the graph nonbipartite. edge_index=2 selects the normalized sorted edge list, not the raw input order. Give (edge_index,edge,left_part,right_part,crosses?,whole_partition_proper?).","answer":{"kind":"expression","value":"(2, [2, 3], 0, 0, No, No)","explanation":"Read the selected normalized edge and report its two part labels, then retain the independent all-edge properness flag. One crossing edge cannot certify all remaining edges. The exact result, retaining the requested order, is (2, [2, 3], 0, 0, No, No)."},"skills":["One Edge Crossing Is Separate from the Whole Partition Audit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r02-bipartite-graph-0029","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2674113c2d4f57aaf59279a3581742f700218705e7363b711a45b5a6528ff911","family_id":"qf-mod31-bipartite-edge-crossing","seed":202610312029,"semantic_template":"qf-mod31-bipartite-edge-crossing/{\"edge_index\":3,\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"n\":6,\"parts\":[1,1,0,0,0,0]}"},"fields":{"title":"One Edge Crossing Is Separate from the Whole Partition Audit","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 4]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. The total part assignment=[1, 1, 0, 0, 0, 0] uses labels 0 or 1 for every vertex. Its two inverse-image parts are disjoint and cover the whole carrier. Every edge must cross parts; empty parts are allowed in an edgeless graph. Failure of this assignment alone does not prove the graph nonbipartite. edge_index=3 selects the normalized sorted edge list, not the raw input order. Give (edge_index,edge,left_part,right_part,crosses?,whole_partition_proper?).","answer":{"kind":"expression","value":"(3, [3, 4], 0, 0, No, No)","explanation":"Read the selected normalized edge and report its two part labels, then retain the independent all-edge properness flag. One crossing edge cannot certify all remaining edges. The exact result, retaining the requested order, is (3, [3, 4], 0, 0, No, No)."},"skills":["One Edge Crossing Is Separate from the Whole Partition Audit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r02-bipartite-graph-0030","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"96f1133c6c3bb05a98b1bd25a0bbd56d5c17afba90b745163e50b0373ad4353e","family_id":"qf-mod31-bipartite-edge-crossing","seed":202610312030,"semantic_template":"qf-mod31-bipartite-edge-crossing/{\"edge_index\":0,\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"n\":6,\"parts\":[0,0,1,0,0,0]}"},"fields":{"title":"One Edge Crossing Is Separate from the Whole Partition Audit","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 4]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. The total part assignment=[0, 0, 1, 0, 0, 0] uses labels 0 or 1 for every vertex. Its two inverse-image parts are disjoint and cover the whole carrier. Every edge must cross parts; empty parts are allowed in an edgeless graph. Failure of this assignment alone does not prove the graph nonbipartite. edge_index=0 selects the normalized sorted edge list, not the raw input order. Give (edge_index,edge,left_part,right_part,crosses?,whole_partition_proper?).","answer":{"kind":"expression","value":"(0, [0, 1], 0, 0, No, No)","explanation":"Read the selected normalized edge and report its two part labels, then retain the independent all-edge properness flag. One crossing edge cannot certify all remaining edges. The exact result, retaining the requested order, is (0, [0, 1], 0, 0, No, No)."},"skills":["One Edge Crossing Is Separate from the Whole Partition Audit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r02-bipartite-graph-0031","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9fa5c89078a41c29d64198d2460c1e6ef940e19fd6b7b2c7da758f16d2e9fe84","family_id":"qf-mod31-bipartite-edge-crossing","seed":202610312031,"semantic_template":"qf-mod31-bipartite-edge-crossing/{\"edge_index\":1,\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"n\":6,\"parts\":[1,0,1,0,0,0]}"},"fields":{"title":"One Edge Crossing Is Separate from the Whole Partition Audit","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 4]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. The total part assignment=[1, 0, 1, 0, 0, 0] uses labels 0 or 1 for every vertex. Its two inverse-image parts are disjoint and cover the whole carrier. Every edge must cross parts; empty parts are allowed in an edgeless graph. Failure of this assignment alone does not prove the graph nonbipartite. edge_index=1 selects the normalized sorted edge list, not the raw input order. Give (edge_index,edge,left_part,right_part,crosses?,whole_partition_proper?).","answer":{"kind":"expression","value":"(1, [1, 2], 0, 1, Yes, No)","explanation":"Read the selected normalized edge and report its two part labels, then retain the independent all-edge properness flag. One crossing edge cannot certify all remaining edges. The exact result, retaining the requested order, is (1, [1, 2], 0, 1, Yes, No)."},"skills":["One Edge Crossing Is Separate from the Whole Partition Audit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r02-bipartite-graph-0032","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"777d6cf510c33b5e2c0bd6ff1bf6ceef89c4ddfa10b1cc96a96e6109956979e4","family_id":"qf-mod31-bipartite-edge-crossing","seed":202610312032,"semantic_template":"qf-mod31-bipartite-edge-crossing/{\"edge_index\":2,\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"n\":6,\"parts\":[0,1,1,0,0,0]}"},"fields":{"title":"One Edge Crossing Is Separate from the Whole Partition Audit","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 4]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. The total part assignment=[0, 1, 1, 0, 0, 0] uses labels 0 or 1 for every vertex. Its two inverse-image parts are disjoint and cover the whole carrier. Every edge must cross parts; empty parts are allowed in an edgeless graph. Failure of this assignment alone does not prove the graph nonbipartite. edge_index=2 selects the normalized sorted edge list, not the raw input order. Give (edge_index,edge,left_part,right_part,crosses?,whole_partition_proper?).","answer":{"kind":"expression","value":"(2, [2, 3], 1, 0, Yes, No)","explanation":"Read the selected normalized edge and report its two part labels, then retain the independent all-edge properness flag. One crossing edge cannot certify all remaining edges. The exact result, retaining the requested order, is (2, [2, 3], 1, 0, Yes, No)."},"skills":["One Edge Crossing Is Separate from the Whole Partition Audit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r02-bipartite-graph-0033","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cd790582fe6891ff6eba6ccf8a3439cfb2c38071c210ecdf6eedd2a2996f2263","family_id":"qf-mod31-bipartite-edge-crossing","seed":202610312033,"semantic_template":"qf-mod31-bipartite-edge-crossing/{\"edge_index\":3,\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"n\":6,\"parts\":[1,1,1,0,0,0]}"},"fields":{"title":"One Edge Crossing Is Separate from the Whole Partition Audit","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 4]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. The total part assignment=[1, 1, 1, 0, 0, 0] uses labels 0 or 1 for every vertex. Its two inverse-image parts are disjoint and cover the whole carrier. Every edge must cross parts; empty parts are allowed in an edgeless graph. Failure of this assignment alone does not prove the graph nonbipartite. edge_index=3 selects the normalized sorted edge list, not the raw input order. Give (edge_index,edge,left_part,right_part,crosses?,whole_partition_proper?).","answer":{"kind":"expression","value":"(3, [3, 4], 0, 0, No, No)","explanation":"Read the selected normalized edge and report its two part labels, then retain the independent all-edge properness flag. One crossing edge cannot certify all remaining edges. The exact result, retaining the requested order, is (3, [3, 4], 0, 0, No, No)."},"skills":["One Edge Crossing Is Separate from the Whole Partition Audit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r02-bipartite-graph-0034","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6c749e46a14955ed8611e21c0f381402ac3f775d7fcc913fb2715fcec8366c05","family_id":"qf-mod31-bipartite-edge-crossing","seed":202610312034,"semantic_template":"qf-mod31-bipartite-edge-crossing/{\"edge_index\":0,\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"n\":6,\"parts\":[0,0,0,1,0,0]}"},"fields":{"title":"One Edge Crossing Is Separate from the Whole Partition Audit","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 4]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. The total part assignment=[0, 0, 0, 1, 0, 0] uses labels 0 or 1 for every vertex. Its two inverse-image parts are disjoint and cover the whole carrier. Every edge must cross parts; empty parts are allowed in an edgeless graph. Failure of this assignment alone does not prove the graph nonbipartite. edge_index=0 selects the normalized sorted edge list, not the raw input order. Give (edge_index,edge,left_part,right_part,crosses?,whole_partition_proper?).","answer":{"kind":"expression","value":"(0, [0, 1], 0, 0, No, No)","explanation":"Read the selected normalized edge and report its two part labels, then retain the independent all-edge properness flag. One crossing edge cannot certify all remaining edges. The exact result, retaining the requested order, is (0, [0, 1], 0, 0, No, No)."},"skills":["One Edge Crossing Is Separate from the Whole Partition Audit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r02-bipartite-graph-0035","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c676daa159e613fde70d41ebe47f86d227addc1c6ab835ddd1df10d1681543cd","family_id":"qf-mod31-bipartite-edge-crossing","seed":202610312035,"semantic_template":"qf-mod31-bipartite-edge-crossing/{\"edge_index\":1,\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"n\":6,\"parts\":[1,0,0,1,0,0]}"},"fields":{"title":"One Edge Crossing Is Separate from the Whole Partition Audit","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 4]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. The total part assignment=[1, 0, 0, 1, 0, 0] uses labels 0 or 1 for every vertex. Its two inverse-image parts are disjoint and cover the whole carrier. Every edge must cross parts; empty parts are allowed in an edgeless graph. Failure of this assignment alone does not prove the graph nonbipartite. edge_index=1 selects the normalized sorted edge list, not the raw input order. Give (edge_index,edge,left_part,right_part,crosses?,whole_partition_proper?).","answer":{"kind":"expression","value":"(1, [1, 2], 0, 0, No, No)","explanation":"Read the selected normalized edge and report its two part labels, then retain the independent all-edge properness flag. One crossing edge cannot certify all remaining edges. The exact result, retaining the requested order, is (1, [1, 2], 0, 0, No, No)."},"skills":["One Edge Crossing Is Separate from the Whole Partition Audit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r02-bipartite-graph-0036","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"00a0ae225be07c021b43950b9730de13a4fc34c09d9b09474a753e8bbdf22304","family_id":"qf-mod31-bipartite-edge-crossing","seed":202610312036,"semantic_template":"qf-mod31-bipartite-edge-crossing/{\"edge_index\":2,\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"n\":6,\"parts\":[0,1,0,1,0,0]}"},"fields":{"title":"One Edge Crossing Is Separate from the Whole Partition Audit","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 4]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. The total part assignment=[0, 1, 0, 1, 0, 0] uses labels 0 or 1 for every vertex. Its two inverse-image parts are disjoint and cover the whole carrier. Every edge must cross parts; empty parts are allowed in an edgeless graph. Failure of this assignment alone does not prove the graph nonbipartite. edge_index=2 selects the normalized sorted edge list, not the raw input order. Give (edge_index,edge,left_part,right_part,crosses?,whole_partition_proper?).","answer":{"kind":"expression","value":"(2, [2, 3], 0, 1, Yes, Yes)","explanation":"Read the selected normalized edge and report its two part labels, then retain the independent all-edge properness flag. One crossing edge cannot certify all remaining edges. The exact result, retaining the requested order, is (2, [2, 3], 0, 1, Yes, Yes)."},"skills":["One Edge Crossing Is Separate from the Whole Partition Audit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r02-bipartite-graph-0037","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8f88a6c2d1ca76d194398825e2c38c53e5b59645cc0a31ff557af6b421d99a4f","family_id":"qf-mod31-bipartite-edge-crossing","seed":202610312037,"semantic_template":"qf-mod31-bipartite-edge-crossing/{\"edge_index\":3,\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"n\":6,\"parts\":[1,1,0,1,0,0]}"},"fields":{"title":"One Edge Crossing Is Separate from the Whole Partition Audit","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 4]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. The total part assignment=[1, 1, 0, 1, 0, 0] uses labels 0 or 1 for every vertex. Its two inverse-image parts are disjoint and cover the whole carrier. Every edge must cross parts; empty parts are allowed in an edgeless graph. Failure of this assignment alone does not prove the graph nonbipartite. edge_index=3 selects the normalized sorted edge list, not the raw input order. Give (edge_index,edge,left_part,right_part,crosses?,whole_partition_proper?).","answer":{"kind":"expression","value":"(3, [3, 4], 1, 0, Yes, No)","explanation":"Read the selected normalized edge and report its two part labels, then retain the independent all-edge properness flag. One crossing edge cannot certify all remaining edges. The exact result, retaining the requested order, is (3, [3, 4], 1, 0, Yes, No)."},"skills":["One Edge Crossing Is Separate from the Whole Partition Audit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r02-bipartite-graph-0038","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9a3267e657af3026d41c92ae58c50f7412e368a6c5a6bc74c37d656e33f7155b","family_id":"qf-mod31-bipartite-edge-crossing","seed":202610312038,"semantic_template":"qf-mod31-bipartite-edge-crossing/{\"edge_index\":0,\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"n\":6,\"parts\":[0,0,1,1,0,0]}"},"fields":{"title":"One Edge Crossing Is Separate from the Whole Partition Audit","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 4]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. The total part assignment=[0, 0, 1, 1, 0, 0] uses labels 0 or 1 for every vertex. Its two inverse-image parts are disjoint and cover the whole carrier. Every edge must cross parts; empty parts are allowed in an edgeless graph. Failure of this assignment alone does not prove the graph nonbipartite. edge_index=0 selects the normalized sorted edge list, not the raw input order. Give (edge_index,edge,left_part,right_part,crosses?,whole_partition_proper?).","answer":{"kind":"expression","value":"(0, [0, 1], 0, 0, No, No)","explanation":"Read the selected normalized edge and report its two part labels, then retain the independent all-edge properness flag. One crossing edge cannot certify all remaining edges. The exact result, retaining the requested order, is (0, [0, 1], 0, 0, No, No)."},"skills":["One Edge Crossing Is Separate from the Whole Partition Audit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r02-bipartite-graph-0039","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5d66790494dd7ecaa238476c850a33a23a57944fa8398f610a5f47b04b9b3cac","family_id":"qf-mod31-bipartite-edge-crossing","seed":202610312039,"semantic_template":"qf-mod31-bipartite-edge-crossing/{\"edge_index\":1,\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"n\":6,\"parts\":[1,0,1,1,0,0]}"},"fields":{"title":"One Edge Crossing Is Separate from the Whole Partition Audit","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 4]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. The total part assignment=[1, 0, 1, 1, 0, 0] uses labels 0 or 1 for every vertex. Its two inverse-image parts are disjoint and cover the whole carrier. Every edge must cross parts; empty parts are allowed in an edgeless graph. Failure of this assignment alone does not prove the graph nonbipartite. edge_index=1 selects the normalized sorted edge list, not the raw input order. Give (edge_index,edge,left_part,right_part,crosses?,whole_partition_proper?).","answer":{"kind":"expression","value":"(1, [1, 2], 0, 1, Yes, No)","explanation":"Read the selected normalized edge and report its two part labels, then retain the independent all-edge properness flag. One crossing edge cannot certify all remaining edges. The exact result, retaining the requested order, is (1, [1, 2], 0, 1, Yes, No)."},"skills":["One Edge Crossing Is Separate from the Whole Partition Audit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r02-bipartite-graph-0040","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"35db0b0e95a5bec1664ea3b5a1cc5c71362ed7b632e791292b589ecdcf4acd59","family_id":"qf-mod31-bipartite-edge-crossing","seed":202610312040,"semantic_template":"qf-mod31-bipartite-edge-crossing/{\"edge_index\":2,\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"n\":6,\"parts\":[0,1,1,1,0,0]}"},"fields":{"title":"One Edge Crossing Is Separate from the Whole Partition Audit","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 4]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. The total part assignment=[0, 1, 1, 1, 0, 0] uses labels 0 or 1 for every vertex. Its two inverse-image parts are disjoint and cover the whole carrier. Every edge must cross parts; empty parts are allowed in an edgeless graph. Failure of this assignment alone does not prove the graph nonbipartite. edge_index=2 selects the normalized sorted edge list, not the raw input order. Give (edge_index,edge,left_part,right_part,crosses?,whole_partition_proper?).","answer":{"kind":"expression","value":"(2, [2, 3], 1, 1, No, No)","explanation":"Read the selected normalized edge and report its two part labels, then retain the independent all-edge properness flag. One crossing edge cannot certify all remaining edges. The exact result, retaining the requested order, is (2, [2, 3], 1, 1, No, No)."},"skills":["One Edge Crossing Is Separate from the Whole Partition Audit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r02-bipartite-graph-0041","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0c9a5e712fbd6fa6dab3864218d4d3c2cd2d896b2e3f3e76feb27f80631fdf9b","family_id":"qf-mod31-bipartite-edge-crossing","seed":202610312041,"semantic_template":"qf-mod31-bipartite-edge-crossing/{\"edge_index\":3,\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"n\":6,\"parts\":[1,1,1,1,0,0]}"},"fields":{"title":"One Edge Crossing Is Separate from the Whole Partition Audit","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 4]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. The total part assignment=[1, 1, 1, 1, 0, 0] uses labels 0 or 1 for every vertex. Its two inverse-image parts are disjoint and cover the whole carrier. Every edge must cross parts; empty parts are allowed in an edgeless graph. Failure of this assignment alone does not prove the graph nonbipartite. edge_index=3 selects the normalized sorted edge list, not the raw input order. Give (edge_index,edge,left_part,right_part,crosses?,whole_partition_proper?).","answer":{"kind":"expression","value":"(3, [3, 4], 1, 0, Yes, No)","explanation":"Read the selected normalized edge and report its two part labels, then retain the independent all-edge properness flag. One crossing edge cannot certify all remaining edges. The exact result, retaining the requested order, is (3, [3, 4], 1, 0, Yes, No)."},"skills":["One Edge Crossing Is Separate from the Whole Partition Audit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r02-bipartite-graph-0042","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"737cbe1636e1c49f3e87aa8f14075258d52febbf3fba2115f44a37a998a2a676","family_id":"qf-mod31-bipartite-edge-crossing","seed":202610312042,"semantic_template":"qf-mod31-bipartite-edge-crossing/{\"edge_index\":0,\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"n\":6,\"parts\":[0,0,0,0,1,0]}"},"fields":{"title":"One Edge Crossing Is Separate from the Whole Partition Audit","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 4]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. The total part assignment=[0, 0, 0, 0, 1, 0] uses labels 0 or 1 for every vertex. Its two inverse-image parts are disjoint and cover the whole carrier. Every edge must cross parts; empty parts are allowed in an edgeless graph. Failure of this assignment alone does not prove the graph nonbipartite. edge_index=0 selects the normalized sorted edge list, not the raw input order. Give (edge_index,edge,left_part,right_part,crosses?,whole_partition_proper?).","answer":{"kind":"expression","value":"(0, [0, 1], 0, 0, No, No)","explanation":"Read the selected normalized edge and report its two part labels, then retain the independent all-edge properness flag. One crossing edge cannot certify all remaining edges. The exact result, retaining the requested order, is (0, [0, 1], 0, 0, No, No)."},"skills":["One Edge Crossing Is Separate from the Whole Partition Audit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r02-bipartite-graph-0043","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"543a44d20b51136ad0adda8d389a8f481a58e9d38cba35c11e51e87d8b0255e7","family_id":"qf-mod31-bipartite-edge-crossing","seed":202610312043,"semantic_template":"qf-mod31-bipartite-edge-crossing/{\"edge_index\":1,\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"n\":6,\"parts\":[1,0,0,0,1,0]}"},"fields":{"title":"One Edge Crossing Is Separate from the Whole Partition Audit","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 4]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. The total part assignment=[1, 0, 0, 0, 1, 0] uses labels 0 or 1 for every vertex. Its two inverse-image parts are disjoint and cover the whole carrier. Every edge must cross parts; empty parts are allowed in an edgeless graph. Failure of this assignment alone does not prove the graph nonbipartite. edge_index=1 selects the normalized sorted edge list, not the raw input order. Give (edge_index,edge,left_part,right_part,crosses?,whole_partition_proper?).","answer":{"kind":"expression","value":"(1, [1, 2], 0, 0, No, No)","explanation":"Read the selected normalized edge and report its two part labels, then retain the independent all-edge properness flag. One crossing edge cannot certify all remaining edges. The exact result, retaining the requested order, is (1, [1, 2], 0, 0, No, No)."},"skills":["One Edge Crossing Is Separate from the Whole Partition Audit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r02-bipartite-graph-0044","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a01ca7a72a56fd2631196830a07b1295231ef2778bf7410a59e22173a60bb978","family_id":"qf-mod31-bipartite-edge-crossing","seed":202610312044,"semantic_template":"qf-mod31-bipartite-edge-crossing/{\"edge_index\":2,\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"n\":6,\"parts\":[0,1,0,0,1,0]}"},"fields":{"title":"One Edge Crossing Is Separate from the Whole Partition Audit","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 4]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. The total part assignment=[0, 1, 0, 0, 1, 0] uses labels 0 or 1 for every vertex. Its two inverse-image parts are disjoint and cover the whole carrier. Every edge must cross parts; empty parts are allowed in an edgeless graph. Failure of this assignment alone does not prove the graph nonbipartite. edge_index=2 selects the normalized sorted edge list, not the raw input order. Give (edge_index,edge,left_part,right_part,crosses?,whole_partition_proper?).","answer":{"kind":"expression","value":"(2, [2, 3], 0, 0, No, No)","explanation":"Read the selected normalized edge and report its two part labels, then retain the independent all-edge properness flag. One crossing edge cannot certify all remaining edges. The exact result, retaining the requested order, is (2, [2, 3], 0, 0, No, No)."},"skills":["One Edge Crossing Is Separate from the Whole Partition Audit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r02-bipartite-graph-0045","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"873d3b33db6a67a17603694d6c5e87d5c2afdd766dd49f6dd072fd525a6b4e37","family_id":"qf-mod31-bipartite-edge-crossing","seed":202610312045,"semantic_template":"qf-mod31-bipartite-edge-crossing/{\"edge_index\":3,\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"n\":6,\"parts\":[1,1,0,0,1,0]}"},"fields":{"title":"One Edge Crossing Is Separate from the Whole Partition Audit","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 4]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. The total part assignment=[1, 1, 0, 0, 1, 0] uses labels 0 or 1 for every vertex. Its two inverse-image parts are disjoint and cover the whole carrier. Every edge must cross parts; empty parts are allowed in an edgeless graph. Failure of this assignment alone does not prove the graph nonbipartite. edge_index=3 selects the normalized sorted edge list, not the raw input order. Give (edge_index,edge,left_part,right_part,crosses?,whole_partition_proper?).","answer":{"kind":"expression","value":"(3, [3, 4], 0, 1, Yes, No)","explanation":"Read the selected normalized edge and report its two part labels, then retain the independent all-edge properness flag. One crossing edge cannot certify all remaining edges. The exact result, retaining the requested order, is (3, [3, 4], 0, 1, Yes, No)."},"skills":["One Edge Crossing Is Separate from the Whole Partition Audit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r02-bipartite-graph-0046","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dfd7ca9f06e15d604b07239961d06771b453af8daeab5b0c777a39c290856056","family_id":"qf-mod31-bipartite-edge-crossing","seed":202610312046,"semantic_template":"qf-mod31-bipartite-edge-crossing/{\"edge_index\":0,\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"n\":6,\"parts\":[0,0,1,0,1,0]}"},"fields":{"title":"One Edge Crossing Is Separate from the Whole Partition Audit","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 4]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. The total part assignment=[0, 0, 1, 0, 1, 0] uses labels 0 or 1 for every vertex. Its two inverse-image parts are disjoint and cover the whole carrier. Every edge must cross parts; empty parts are allowed in an edgeless graph. Failure of this assignment alone does not prove the graph nonbipartite. edge_index=0 selects the normalized sorted edge list, not the raw input order. Give (edge_index,edge,left_part,right_part,crosses?,whole_partition_proper?).","answer":{"kind":"expression","value":"(0, [0, 1], 0, 0, No, No)","explanation":"Read the selected normalized edge and report its two part labels, then retain the independent all-edge properness flag. One crossing edge cannot certify all remaining edges. The exact result, retaining the requested order, is (0, [0, 1], 0, 0, No, No)."},"skills":["One Edge Crossing Is Separate from the Whole Partition Audit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r02-bipartite-graph-0047","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"00b0ea61ee11f88a9a9ab33a5293a53c80165dec61a90f186693430504986dda","family_id":"qf-mod31-bipartite-edge-crossing","seed":202610312047,"semantic_template":"qf-mod31-bipartite-edge-crossing/{\"edge_index\":1,\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"n\":6,\"parts\":[1,0,1,0,1,0]}"},"fields":{"title":"One Edge Crossing Is Separate from the Whole Partition Audit","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 4]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. The total part assignment=[1, 0, 1, 0, 1, 0] uses labels 0 or 1 for every vertex. Its two inverse-image parts are disjoint and cover the whole carrier. Every edge must cross parts; empty parts are allowed in an edgeless graph. Failure of this assignment alone does not prove the graph nonbipartite. edge_index=1 selects the normalized sorted edge list, not the raw input order. Give (edge_index,edge,left_part,right_part,crosses?,whole_partition_proper?).","answer":{"kind":"expression","value":"(1, [1, 2], 0, 1, Yes, Yes)","explanation":"Read the selected normalized edge and report its two part labels, then retain the independent all-edge properness flag. One crossing edge cannot certify all remaining edges. The exact result, retaining the requested order, is (1, [1, 2], 0, 1, Yes, Yes)."},"skills":["One Edge Crossing Is Separate from the Whole Partition Audit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r02-bipartite-graph-0048","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f11b705cc74f50da77ef5470eb6a3fa966bcdc6975cfffd8246b08af744afda2","family_id":"qf-mod31-bipartite-edge-crossing","seed":202610312048,"semantic_template":"qf-mod31-bipartite-edge-crossing/{\"edge_index\":2,\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"n\":6,\"parts\":[0,1,1,0,1,0]}"},"fields":{"title":"One Edge Crossing Is Separate from the Whole Partition Audit","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 4]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. The total part assignment=[0, 1, 1, 0, 1, 0] uses labels 0 or 1 for every vertex. Its two inverse-image parts are disjoint and cover the whole carrier. Every edge must cross parts; empty parts are allowed in an edgeless graph. Failure of this assignment alone does not prove the graph nonbipartite. edge_index=2 selects the normalized sorted edge list, not the raw input order. Give (edge_index,edge,left_part,right_part,crosses?,whole_partition_proper?).","answer":{"kind":"expression","value":"(2, [2, 3], 1, 0, Yes, No)","explanation":"Read the selected normalized edge and report its two part labels, then retain the independent all-edge properness flag. One crossing edge cannot certify all remaining edges. The exact result, retaining the requested order, is (2, [2, 3], 1, 0, Yes, No)."},"skills":["One Edge Crossing Is Separate from the Whole Partition Audit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r02-bipartite-graph-0049","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"327e4ab0fc7ba811ca38cd79bf024373a0b49243b191652d2533004fed5e02d8","family_id":"qf-mod31-bipartite-edge-crossing","seed":202610312049,"semantic_template":"qf-mod31-bipartite-edge-crossing/{\"edge_index\":3,\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"n\":6,\"parts\":[1,1,1,0,1,0]}"},"fields":{"title":"One Edge Crossing Is Separate from the Whole Partition Audit","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 4]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. The total part assignment=[1, 1, 1, 0, 1, 0] uses labels 0 or 1 for every vertex. Its two inverse-image parts are disjoint and cover the whole carrier. Every edge must cross parts; empty parts are allowed in an edgeless graph. Failure of this assignment alone does not prove the graph nonbipartite. edge_index=3 selects the normalized sorted edge list, not the raw input order. Give (edge_index,edge,left_part,right_part,crosses?,whole_partition_proper?).","answer":{"kind":"expression","value":"(3, [3, 4], 0, 1, Yes, No)","explanation":"Read the selected normalized edge and report its two part labels, then retain the independent all-edge properness flag. One crossing edge cannot certify all remaining edges. The exact result, retaining the requested order, is (3, [3, 4], 0, 1, Yes, No)."},"skills":["One Edge Crossing Is Separate from the Whole Partition Audit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r02-bipartite-graph-0050","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f177840bda6c52097b22df43a2055b940cfab5ccf9dc58123571ad3a5e72ae96","family_id":"qf-mod31-bipartite-edge-crossing","seed":202610312050,"semantic_template":"qf-mod31-bipartite-edge-crossing/{\"edge_index\":0,\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"n\":6,\"parts\":[0,0,0,1,1,0]}"},"fields":{"title":"One Edge Crossing Is Separate from the Whole Partition Audit","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 4]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. The total part assignment=[0, 0, 0, 1, 1, 0] uses labels 0 or 1 for every vertex. Its two inverse-image parts are disjoint and cover the whole carrier. Every edge must cross parts; empty parts are allowed in an edgeless graph. Failure of this assignment alone does not prove the graph nonbipartite. edge_index=0 selects the normalized sorted edge list, not the raw input order. Give (edge_index,edge,left_part,right_part,crosses?,whole_partition_proper?).","answer":{"kind":"expression","value":"(0, [0, 1], 0, 0, No, No)","explanation":"Read the selected normalized edge and report its two part labels, then retain the independent all-edge properness flag. One crossing edge cannot certify all remaining edges. The exact result, retaining the requested order, is (0, [0, 1], 0, 0, No, No)."},"skills":["One Edge Crossing Is Separate from the Whole Partition Audit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r02-bipartite-graph-0051","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3644059cf7bcaad98abdd16cda273b2f0ca7c1f88128aab7ff5110626888e6c2","family_id":"qf-mod31-bipartite-odd-cycle","seed":202610312051,"semantic_template":"qf-mod31-bipartite-odd-cycle/{\"cycle\":[0,1,2],\"edges\":[[0,1],[0,2],[1,2]],\"n\":6}"},"fields":{"title":"An Actual Simple Odd Cycle Certifies Nonbipartiteness","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [0, 2]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. Ordered cycle vertices=[0, 1, 2], with the closing edge from the last back to the first implicit. Vertices must be distinct, the cycle length odd and at least 3, and every cycle edge present. Give (cycle,cycle_edges,length,certificate?,odd_cycle_proves_nonbipartite). Normalize each edge's endpoints but retain traversal order in cycle_edges.","answer":{"kind":"expression","value":"([0, 1, 2], [[0, 1], [1, 2], [0, 2]], 3, Yes, odd_cycle_proves_nonbipartite)","explanation":"Alternating between two parts around an odd cycle cannot return consistently to the initial part. The certificate requires the full actual cycle, not merely a failed candidate partition or an incomplete walk. The exact result, retaining the requested order, is ([0, 1, 2], [[0, 1], [1, 2], [0, 2]], 3, Yes, odd_cycle_proves_nonbipartite)."},"skills":["An Actual Simple Odd Cycle Certifies Nonbipartiteness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r02-bipartite-graph-0052","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2ee5a97a65965727810e241f8a82dfcd598914190ab62154fb66c2f1e313a86d","family_id":"qf-mod31-bipartite-odd-cycle","seed":202610312052,"semantic_template":"qf-mod31-bipartite-odd-cycle/{\"cycle\":[0,1,2],\"edges\":[[0,1],[0,2],[1,2],[3,4]],\"n\":6}"},"fields":{"title":"An Actual Simple Odd Cycle Certifies Nonbipartiteness","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [0, 2], [3, 4]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. Ordered cycle vertices=[0, 1, 2], with the closing edge from the last back to the first implicit. Vertices must be distinct, the cycle length odd and at least 3, and every cycle edge present. Give (cycle,cycle_edges,length,certificate?,odd_cycle_proves_nonbipartite). Normalize each edge's endpoints but retain traversal order in cycle_edges.","answer":{"kind":"expression","value":"([0, 1, 2], [[0, 1], [1, 2], [0, 2]], 3, Yes, odd_cycle_proves_nonbipartite)","explanation":"Alternating between two parts around an odd cycle cannot return consistently to the initial part. The certificate requires the full actual cycle, not merely a failed candidate partition or an incomplete walk. The exact result, retaining the requested order, is ([0, 1, 2], [[0, 1], [1, 2], [0, 2]], 3, Yes, odd_cycle_proves_nonbipartite)."},"skills":["An Actual Simple Odd Cycle Certifies Nonbipartiteness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r02-bipartite-graph-0053","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"aa293a9f8393f27690ebac51970f1e3604e9bf99c7d0fb838cff6cdfb34a5aab","family_id":"qf-mod31-bipartite-odd-cycle","seed":202610312053,"semantic_template":"qf-mod31-bipartite-odd-cycle/{\"cycle\":[0,1,2],\"edges\":[[0,1],[0,2],[1,2],[4,5]],\"n\":6}"},"fields":{"title":"An Actual Simple Odd Cycle Certifies Nonbipartiteness","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [0, 2], [4, 5]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. Ordered cycle vertices=[0, 1, 2], with the closing edge from the last back to the first implicit. Vertices must be distinct, the cycle length odd and at least 3, and every cycle edge present. Give (cycle,cycle_edges,length,certificate?,odd_cycle_proves_nonbipartite). Normalize each edge's endpoints but retain traversal order in cycle_edges.","answer":{"kind":"expression","value":"([0, 1, 2], [[0, 1], [1, 2], [0, 2]], 3, Yes, odd_cycle_proves_nonbipartite)","explanation":"Alternating between two parts around an odd cycle cannot return consistently to the initial part. The certificate requires the full actual cycle, not merely a failed candidate partition or an incomplete walk. The exact result, retaining the requested order, is ([0, 1, 2], [[0, 1], [1, 2], [0, 2]], 3, Yes, odd_cycle_proves_nonbipartite)."},"skills":["An Actual Simple Odd Cycle Certifies Nonbipartiteness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r02-bipartite-graph-0054","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"864ac442d01f8e767354ae968518e94735fe0d4ecc43c6e0cd405c25db978236","family_id":"qf-mod31-bipartite-odd-cycle","seed":202610312054,"semantic_template":"qf-mod31-bipartite-odd-cycle/{\"cycle\":[0,1,2],\"edges\":[[0,1],[0,2],[1,2],[3,4],[4,5]],\"n\":6}"},"fields":{"title":"An Actual Simple Odd Cycle Certifies Nonbipartiteness","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [0, 2], [3, 4], [4, 5]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. Ordered cycle vertices=[0, 1, 2], with the closing edge from the last back to the first implicit. Vertices must be distinct, the cycle length odd and at least 3, and every cycle edge present. Give (cycle,cycle_edges,length,certificate?,odd_cycle_proves_nonbipartite). Normalize each edge's endpoints but retain traversal order in cycle_edges.","answer":{"kind":"expression","value":"([0, 1, 2], [[0, 1], [1, 2], [0, 2]], 3, Yes, odd_cycle_proves_nonbipartite)","explanation":"Alternating between two parts around an odd cycle cannot return consistently to the initial part. The certificate requires the full actual cycle, not merely a failed candidate partition or an incomplete walk. The exact result, retaining the requested order, is ([0, 1, 2], [[0, 1], [1, 2], [0, 2]], 3, Yes, odd_cycle_proves_nonbipartite)."},"skills":["An Actual Simple Odd Cycle Certifies Nonbipartiteness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r02-bipartite-graph-0055","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"70b41af940df98ff1e6cf40fcf54f57d10ed613a1d0c6e9212f1722f8be7f35e","family_id":"qf-mod31-bipartite-odd-cycle","seed":202610312055,"semantic_template":"qf-mod31-bipartite-odd-cycle/{\"cycle\":[0,1,2],\"edges\":[[0,1],[0,2],[1,2],[3,5]],\"n\":6}"},"fields":{"title":"An Actual Simple Odd Cycle Certifies Nonbipartiteness","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [0, 2], [3, 5]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. Ordered cycle vertices=[0, 1, 2], with the closing edge from the last back to the first implicit. Vertices must be distinct, the cycle length odd and at least 3, and every cycle edge present. Give (cycle,cycle_edges,length,certificate?,odd_cycle_proves_nonbipartite). Normalize each edge's endpoints but retain traversal order in cycle_edges.","answer":{"kind":"expression","value":"([0, 1, 2], [[0, 1], [1, 2], [0, 2]], 3, Yes, odd_cycle_proves_nonbipartite)","explanation":"Alternating between two parts around an odd cycle cannot return consistently to the initial part. The certificate requires the full actual cycle, not merely a failed candidate partition or an incomplete walk. The exact result, retaining the requested order, is ([0, 1, 2], [[0, 1], [1, 2], [0, 2]], 3, Yes, odd_cycle_proves_nonbipartite)."},"skills":["An Actual Simple Odd Cycle Certifies Nonbipartiteness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r02-bipartite-graph-0056","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"48cffe4e7e66c11f118efa5632835aa215f6ad2634ada0402ff6573bd0517fb3","family_id":"qf-mod31-bipartite-odd-cycle","seed":202610312056,"semantic_template":"qf-mod31-bipartite-odd-cycle/{\"cycle\":[0,1,2],\"edges\":[[0,1],[0,2],[1,2],[3,4],[3,5]],\"n\":6}"},"fields":{"title":"An Actual Simple Odd Cycle Certifies Nonbipartiteness","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [0, 2], [3, 4], [3, 5]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. Ordered cycle vertices=[0, 1, 2], with the closing edge from the last back to the first implicit. Vertices must be distinct, the cycle length odd and at least 3, and every cycle edge present. Give (cycle,cycle_edges,length,certificate?,odd_cycle_proves_nonbipartite). Normalize each edge's endpoints but retain traversal order in cycle_edges.","answer":{"kind":"expression","value":"([0, 1, 2], [[0, 1], [1, 2], [0, 2]], 3, Yes, odd_cycle_proves_nonbipartite)","explanation":"Alternating between two parts around an odd cycle cannot return consistently to the initial part. The certificate requires the full actual cycle, not merely a failed candidate partition or an incomplete walk. The exact result, retaining the requested order, is ([0, 1, 2], [[0, 1], [1, 2], [0, 2]], 3, Yes, odd_cycle_proves_nonbipartite)."},"skills":["An Actual Simple Odd Cycle Certifies Nonbipartiteness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r02-bipartite-graph-0057","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bc873238e80599327c45fd4b20a38441f4965da4713f64e147b5591e61bb4c89","family_id":"qf-mod31-bipartite-odd-cycle","seed":202610312057,"semantic_template":"qf-mod31-bipartite-odd-cycle/{\"cycle\":[0,1,2],\"edges\":[[0,1],[0,2],[1,2],[3,5],[4,5]],\"n\":6}"},"fields":{"title":"An Actual Simple Odd Cycle Certifies Nonbipartiteness","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [0, 2], [4, 5], [3, 5]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. Ordered cycle vertices=[0, 1, 2], with the closing edge from the last back to the first implicit. Vertices must be distinct, the cycle length odd and at least 3, and every cycle edge present. Give (cycle,cycle_edges,length,certificate?,odd_cycle_proves_nonbipartite). Normalize each edge's endpoints but retain traversal order in cycle_edges.","answer":{"kind":"expression","value":"([0, 1, 2], [[0, 1], [1, 2], [0, 2]], 3, Yes, odd_cycle_proves_nonbipartite)","explanation":"Alternating between two parts around an odd cycle cannot return consistently to the initial part. The certificate requires the full actual cycle, not merely a failed candidate partition or an incomplete walk. The exact result, retaining the requested order, is ([0, 1, 2], [[0, 1], [1, 2], [0, 2]], 3, Yes, odd_cycle_proves_nonbipartite)."},"skills":["An Actual Simple Odd Cycle Certifies Nonbipartiteness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r02-bipartite-graph-0058","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f34eb527a633e80646e5c08a53e93153af9db799b362c98b331d2655e27c868b","family_id":"qf-mod31-bipartite-odd-cycle","seed":202610312058,"semantic_template":"qf-mod31-bipartite-odd-cycle/{\"cycle\":[0,1,2],\"edges\":[[0,1],[0,2],[1,2],[3,4],[3,5],[4,5]],\"n\":6}"},"fields":{"title":"An Actual Simple Odd Cycle Certifies Nonbipartiteness","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [0, 2], [3, 4], [4, 5], [3, 5]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. Ordered cycle vertices=[0, 1, 2], with the closing edge from the last back to the first implicit. Vertices must be distinct, the cycle length odd and at least 3, and every cycle edge present. Give (cycle,cycle_edges,length,certificate?,odd_cycle_proves_nonbipartite). Normalize each edge's endpoints but retain traversal order in cycle_edges.","answer":{"kind":"expression","value":"([0, 1, 2], [[0, 1], [1, 2], [0, 2]], 3, Yes, odd_cycle_proves_nonbipartite)","explanation":"Alternating between two parts around an odd cycle cannot return consistently to the initial part. The certificate requires the full actual cycle, not merely a failed candidate partition or an incomplete walk. The exact result, retaining the requested order, is ([0, 1, 2], [[0, 1], [1, 2], [0, 2]], 3, Yes, odd_cycle_proves_nonbipartite)."},"skills":["An Actual Simple Odd Cycle Certifies Nonbipartiteness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r02-bipartite-graph-0059","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"de2741d1ebb675b09ee5e9bba2405ba52a392927db23e2ddb8ee48565ee98f67","family_id":"qf-mod31-bipartite-odd-cycle","seed":202610312059,"semantic_template":"qf-mod31-bipartite-odd-cycle/{\"cycle\":[0,1,2],\"edges\":[[0,1],[0,2],[0,3],[1,2]],\"n\":6}"},"fields":{"title":"An Actual Simple Odd Cycle Certifies Nonbipartiteness","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [0, 2], [0, 3]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. Ordered cycle vertices=[0, 1, 2], with the closing edge from the last back to the first implicit. Vertices must be distinct, the cycle length odd and at least 3, and every cycle edge present. Give (cycle,cycle_edges,length,certificate?,odd_cycle_proves_nonbipartite). Normalize each edge's endpoints but retain traversal order in cycle_edges.","answer":{"kind":"expression","value":"([0, 1, 2], [[0, 1], [1, 2], [0, 2]], 3, Yes, odd_cycle_proves_nonbipartite)","explanation":"Alternating between two parts around an odd cycle cannot return consistently to the initial part. The certificate requires the full actual cycle, not merely a failed candidate partition or an incomplete walk. The exact result, retaining the requested order, is ([0, 1, 2], [[0, 1], [1, 2], [0, 2]], 3, Yes, odd_cycle_proves_nonbipartite)."},"skills":["An Actual Simple Odd Cycle Certifies Nonbipartiteness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r02-bipartite-graph-0060","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"595518a05e289685addc2a1d92839374757052e891c29c964c0e534e90a21c8c","family_id":"qf-mod31-bipartite-odd-cycle","seed":202610312060,"semantic_template":"qf-mod31-bipartite-odd-cycle/{\"cycle\":[0,1,2],\"edges\":[[0,1],[0,2],[0,3],[1,2],[3,4]],\"n\":6}"},"fields":{"title":"An Actual Simple Odd Cycle Certifies Nonbipartiteness","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [0, 2], [3, 4], [0, 3]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. Ordered cycle vertices=[0, 1, 2], with the closing edge from the last back to the first implicit. Vertices must be distinct, the cycle length odd and at least 3, and every cycle edge present. Give (cycle,cycle_edges,length,certificate?,odd_cycle_proves_nonbipartite). Normalize each edge's endpoints but retain traversal order in cycle_edges.","answer":{"kind":"expression","value":"([0, 1, 2], [[0, 1], [1, 2], [0, 2]], 3, Yes, odd_cycle_proves_nonbipartite)","explanation":"Alternating between two parts around an odd cycle cannot return consistently to the initial part. The certificate requires the full actual cycle, not merely a failed candidate partition or an incomplete walk. The exact result, retaining the requested order, is ([0, 1, 2], [[0, 1], [1, 2], [0, 2]], 3, Yes, odd_cycle_proves_nonbipartite)."},"skills":["An Actual Simple Odd Cycle Certifies Nonbipartiteness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r02-bipartite-graph-0061","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"393eb90ee6193f4a71b3e0af165967345a7e67323f9ea8eb9615fbad77346b71","family_id":"qf-mod31-bipartite-odd-cycle","seed":202610312061,"semantic_template":"qf-mod31-bipartite-odd-cycle/{\"cycle\":[0,1,2],\"edges\":[[0,1],[0,2],[0,3],[1,2],[4,5]],\"n\":6}"},"fields":{"title":"An Actual Simple Odd Cycle Certifies Nonbipartiteness","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [0, 2], [4, 5], [0, 3]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. Ordered cycle vertices=[0, 1, 2], with the closing edge from the last back to the first implicit. Vertices must be distinct, the cycle length odd and at least 3, and every cycle edge present. Give (cycle,cycle_edges,length,certificate?,odd_cycle_proves_nonbipartite). Normalize each edge's endpoints but retain traversal order in cycle_edges.","answer":{"kind":"expression","value":"([0, 1, 2], [[0, 1], [1, 2], [0, 2]], 3, Yes, odd_cycle_proves_nonbipartite)","explanation":"Alternating between two parts around an odd cycle cannot return consistently to the initial part. The certificate requires the full actual cycle, not merely a failed candidate partition or an incomplete walk. The exact result, retaining the requested order, is ([0, 1, 2], [[0, 1], [1, 2], [0, 2]], 3, Yes, odd_cycle_proves_nonbipartite)."},"skills":["An Actual Simple Odd Cycle Certifies Nonbipartiteness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r02-bipartite-graph-0062","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"16df33611e704adb2835b0e16fece98eb72619a84779639a15a65e5b2e6183bf","family_id":"qf-mod31-bipartite-odd-cycle","seed":202610312062,"semantic_template":"qf-mod31-bipartite-odd-cycle/{\"cycle\":[0,1,2],\"edges\":[[0,1],[0,2],[0,3],[1,2],[3,4],[4,5]],\"n\":6}"},"fields":{"title":"An Actual Simple Odd Cycle Certifies Nonbipartiteness","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [0, 2], [3, 4], [4, 5], [0, 3]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. Ordered cycle vertices=[0, 1, 2], with the closing edge from the last back to the first implicit. Vertices must be distinct, the cycle length odd and at least 3, and every cycle edge present. Give (cycle,cycle_edges,length,certificate?,odd_cycle_proves_nonbipartite). Normalize each edge's endpoints but retain traversal order in cycle_edges.","answer":{"kind":"expression","value":"([0, 1, 2], [[0, 1], [1, 2], [0, 2]], 3, Yes, odd_cycle_proves_nonbipartite)","explanation":"Alternating between two parts around an odd cycle cannot return consistently to the initial part. The certificate requires the full actual cycle, not merely a failed candidate partition or an incomplete walk. The exact result, retaining the requested order, is ([0, 1, 2], [[0, 1], [1, 2], [0, 2]], 3, Yes, odd_cycle_proves_nonbipartite)."},"skills":["An Actual Simple Odd Cycle Certifies Nonbipartiteness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r02-bipartite-graph-0063","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cb43d737e65130d3ecc9e36f2d8ae2a55b157805c9453517d1ba12fbfc4c1d8d","family_id":"qf-mod31-bipartite-odd-cycle","seed":202610312063,"semantic_template":"qf-mod31-bipartite-odd-cycle/{\"cycle\":[0,1,2],\"edges\":[[0,1],[0,2],[0,3],[1,2],[3,5]],\"n\":6}"},"fields":{"title":"An Actual Simple Odd Cycle Certifies Nonbipartiteness","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [0, 2], [3, 5], [0, 3]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. Ordered cycle vertices=[0, 1, 2], with the closing edge from the last back to the first implicit. Vertices must be distinct, the cycle length odd and at least 3, and every cycle edge present. Give (cycle,cycle_edges,length,certificate?,odd_cycle_proves_nonbipartite). Normalize each edge's endpoints but retain traversal order in cycle_edges.","answer":{"kind":"expression","value":"([0, 1, 2], [[0, 1], [1, 2], [0, 2]], 3, Yes, odd_cycle_proves_nonbipartite)","explanation":"Alternating between two parts around an odd cycle cannot return consistently to the initial part. The certificate requires the full actual cycle, not merely a failed candidate partition or an incomplete walk. The exact result, retaining the requested order, is ([0, 1, 2], [[0, 1], [1, 2], [0, 2]], 3, Yes, odd_cycle_proves_nonbipartite)."},"skills":["An Actual Simple Odd Cycle Certifies Nonbipartiteness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r02-bipartite-graph-0064","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3a1eb7222b8ff63d41e502f1e1b45c57b31ea3b8ee298f7168aee44aa76d7b60","family_id":"qf-mod31-bipartite-odd-cycle","seed":202610312064,"semantic_template":"qf-mod31-bipartite-odd-cycle/{\"cycle\":[0,1,2],\"edges\":[[0,1],[0,2],[0,3],[1,2],[3,4],[3,5]],\"n\":6}"},"fields":{"title":"An Actual Simple Odd Cycle Certifies Nonbipartiteness","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [0, 2], [3, 4], [3, 5], [0, 3]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. Ordered cycle vertices=[0, 1, 2], with the closing edge from the last back to the first implicit. Vertices must be distinct, the cycle length odd and at least 3, and every cycle edge present. Give (cycle,cycle_edges,length,certificate?,odd_cycle_proves_nonbipartite). Normalize each edge's endpoints but retain traversal order in cycle_edges.","answer":{"kind":"expression","value":"([0, 1, 2], [[0, 1], [1, 2], [0, 2]], 3, Yes, odd_cycle_proves_nonbipartite)","explanation":"Alternating between two parts around an odd cycle cannot return consistently to the initial part. The certificate requires the full actual cycle, not merely a failed candidate partition or an incomplete walk. The exact result, retaining the requested order, is ([0, 1, 2], [[0, 1], [1, 2], [0, 2]], 3, Yes, odd_cycle_proves_nonbipartite)."},"skills":["An Actual Simple Odd Cycle Certifies Nonbipartiteness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r02-bipartite-graph-0065","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"23696e2bfd950553b51840cca31a2e03ff23c5a6c071fd4d30cd7783fbe42e5c","family_id":"qf-mod31-bipartite-odd-cycle","seed":202610312065,"semantic_template":"qf-mod31-bipartite-odd-cycle/{\"cycle\":[0,1,2],\"edges\":[[0,1],[0,2],[0,3],[1,2],[3,5],[4,5]],\"n\":6}"},"fields":{"title":"An Actual Simple Odd Cycle Certifies Nonbipartiteness","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [0, 2], [4, 5], [3, 5], [0, 3]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. Ordered cycle vertices=[0, 1, 2], with the closing edge from the last back to the first implicit. Vertices must be distinct, the cycle length odd and at least 3, and every cycle edge present. Give (cycle,cycle_edges,length,certificate?,odd_cycle_proves_nonbipartite). Normalize each edge's endpoints but retain traversal order in cycle_edges.","answer":{"kind":"expression","value":"([0, 1, 2], [[0, 1], [1, 2], [0, 2]], 3, Yes, odd_cycle_proves_nonbipartite)","explanation":"Alternating between two parts around an odd cycle cannot return consistently to the initial part. The certificate requires the full actual cycle, not merely a failed candidate partition or an incomplete walk. The exact result, retaining the requested order, is ([0, 1, 2], [[0, 1], [1, 2], [0, 2]], 3, Yes, odd_cycle_proves_nonbipartite)."},"skills":["An Actual Simple Odd Cycle Certifies Nonbipartiteness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r02-bipartite-graph-0066","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2bfc63f45a486d7af455249a92c128f0d5f2209a8ae02751259650fc6b692a8a","family_id":"qf-mod31-bipartite-odd-cycle","seed":202610312066,"semantic_template":"qf-mod31-bipartite-odd-cycle/{\"cycle\":[0,1,2],\"edges\":[[0,1],[0,2],[0,3],[1,2],[3,4],[3,5],[4,5]],\"n\":6}"},"fields":{"title":"An Actual Simple Odd Cycle Certifies Nonbipartiteness","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [0, 2], [3, 4], [4, 5], [3, 5], [0, 3]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. Ordered cycle vertices=[0, 1, 2], with the closing edge from the last back to the first implicit. Vertices must be distinct, the cycle length odd and at least 3, and every cycle edge present. Give (cycle,cycle_edges,length,certificate?,odd_cycle_proves_nonbipartite). Normalize each edge's endpoints but retain traversal order in cycle_edges.","answer":{"kind":"expression","value":"([0, 1, 2], [[0, 1], [1, 2], [0, 2]], 3, Yes, odd_cycle_proves_nonbipartite)","explanation":"Alternating between two parts around an odd cycle cannot return consistently to the initial part. The certificate requires the full actual cycle, not merely a failed candidate partition or an incomplete walk. The exact result, retaining the requested order, is ([0, 1, 2], [[0, 1], [1, 2], [0, 2]], 3, Yes, odd_cycle_proves_nonbipartite)."},"skills":["An Actual Simple Odd Cycle Certifies Nonbipartiteness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r02-bipartite-graph-0067","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e617b264b24f8b515048bc83e8b4d9cf97d0c05f73d4f21530c16af57771a2a3","family_id":"qf-mod31-bipartite-odd-cycle","seed":202610312067,"semantic_template":"qf-mod31-bipartite-odd-cycle/{\"cycle\":[0,1,2],\"edges\":[[0,1],[0,2],[1,2],[1,4]],\"n\":6}"},"fields":{"title":"An Actual Simple Odd Cycle Certifies Nonbipartiteness","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [0, 2], [1, 4]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. Ordered cycle vertices=[0, 1, 2], with the closing edge from the last back to the first implicit. Vertices must be distinct, the cycle length odd and at least 3, and every cycle edge present. Give (cycle,cycle_edges,length,certificate?,odd_cycle_proves_nonbipartite). Normalize each edge's endpoints but retain traversal order in cycle_edges.","answer":{"kind":"expression","value":"([0, 1, 2], [[0, 1], [1, 2], [0, 2]], 3, Yes, odd_cycle_proves_nonbipartite)","explanation":"Alternating between two parts around an odd cycle cannot return consistently to the initial part. The certificate requires the full actual cycle, not merely a failed candidate partition or an incomplete walk. The exact result, retaining the requested order, is ([0, 1, 2], [[0, 1], [1, 2], [0, 2]], 3, Yes, odd_cycle_proves_nonbipartite)."},"skills":["An Actual Simple Odd Cycle Certifies Nonbipartiteness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r02-bipartite-graph-0068","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8133d4eda2081c9724e39be9b2410ddd110b425881ec378a7f11ea09a3a034ea","family_id":"qf-mod31-bipartite-odd-cycle","seed":202610312068,"semantic_template":"qf-mod31-bipartite-odd-cycle/{\"cycle\":[0,1,2],\"edges\":[[0,1],[0,2],[1,2],[1,4],[3,4]],\"n\":6}"},"fields":{"title":"An Actual Simple Odd Cycle Certifies Nonbipartiteness","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [0, 2], [3, 4], [1, 4]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. Ordered cycle vertices=[0, 1, 2], with the closing edge from the last back to the first implicit. Vertices must be distinct, the cycle length odd and at least 3, and every cycle edge present. Give (cycle,cycle_edges,length,certificate?,odd_cycle_proves_nonbipartite). Normalize each edge's endpoints but retain traversal order in cycle_edges.","answer":{"kind":"expression","value":"([0, 1, 2], [[0, 1], [1, 2], [0, 2]], 3, Yes, odd_cycle_proves_nonbipartite)","explanation":"Alternating between two parts around an odd cycle cannot return consistently to the initial part. The certificate requires the full actual cycle, not merely a failed candidate partition or an incomplete walk. The exact result, retaining the requested order, is ([0, 1, 2], [[0, 1], [1, 2], [0, 2]], 3, Yes, odd_cycle_proves_nonbipartite)."},"skills":["An Actual Simple Odd Cycle Certifies Nonbipartiteness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r02-bipartite-graph-0069","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ba092e57305257ff3662d9bed1e9d647fe5e21c0142812506144133832e75c4f","family_id":"qf-mod31-bipartite-odd-cycle","seed":202610312069,"semantic_template":"qf-mod31-bipartite-odd-cycle/{\"cycle\":[0,1,2],\"edges\":[[0,1],[0,2],[1,2],[1,4],[4,5]],\"n\":6}"},"fields":{"title":"An Actual Simple Odd Cycle Certifies Nonbipartiteness","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [0, 2], [4, 5], [1, 4]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. Ordered cycle vertices=[0, 1, 2], with the closing edge from the last back to the first implicit. Vertices must be distinct, the cycle length odd and at least 3, and every cycle edge present. Give (cycle,cycle_edges,length,certificate?,odd_cycle_proves_nonbipartite). Normalize each edge's endpoints but retain traversal order in cycle_edges.","answer":{"kind":"expression","value":"([0, 1, 2], [[0, 1], [1, 2], [0, 2]], 3, Yes, odd_cycle_proves_nonbipartite)","explanation":"Alternating between two parts around an odd cycle cannot return consistently to the initial part. The certificate requires the full actual cycle, not merely a failed candidate partition or an incomplete walk. The exact result, retaining the requested order, is ([0, 1, 2], [[0, 1], [1, 2], [0, 2]], 3, Yes, odd_cycle_proves_nonbipartite)."},"skills":["An Actual Simple Odd Cycle Certifies Nonbipartiteness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r02-bipartite-graph-0070","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e0894c44b77ec5f8acfadea62d0b026c8f5fe38b2009af44c7b588a63f441e9e","family_id":"qf-mod31-bipartite-odd-cycle","seed":202610312070,"semantic_template":"qf-mod31-bipartite-odd-cycle/{\"cycle\":[0,1,2],\"edges\":[[0,1],[0,2],[1,2],[1,4],[3,4],[4,5]],\"n\":6}"},"fields":{"title":"An Actual Simple Odd Cycle Certifies Nonbipartiteness","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [0, 2], [3, 4], [4, 5], [1, 4]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. Ordered cycle vertices=[0, 1, 2], with the closing edge from the last back to the first implicit. Vertices must be distinct, the cycle length odd and at least 3, and every cycle edge present. Give (cycle,cycle_edges,length,certificate?,odd_cycle_proves_nonbipartite). Normalize each edge's endpoints but retain traversal order in cycle_edges.","answer":{"kind":"expression","value":"([0, 1, 2], [[0, 1], [1, 2], [0, 2]], 3, Yes, odd_cycle_proves_nonbipartite)","explanation":"Alternating between two parts around an odd cycle cannot return consistently to the initial part. The certificate requires the full actual cycle, not merely a failed candidate partition or an incomplete walk. The exact result, retaining the requested order, is ([0, 1, 2], [[0, 1], [1, 2], [0, 2]], 3, Yes, odd_cycle_proves_nonbipartite)."},"skills":["An Actual Simple Odd Cycle Certifies Nonbipartiteness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r02-bipartite-graph-0071","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0ce864f10e3c26f8401dea18d6aafd6ea4e47e79957ba235a79a704d6669d714","family_id":"qf-mod31-bipartite-odd-cycle","seed":202610312071,"semantic_template":"qf-mod31-bipartite-odd-cycle/{\"cycle\":[0,1,2],\"edges\":[[0,1],[0,2],[1,2],[1,4],[3,5]],\"n\":6}"},"fields":{"title":"An Actual Simple Odd Cycle Certifies Nonbipartiteness","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [0, 2], [3, 5], [1, 4]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. Ordered cycle vertices=[0, 1, 2], with the closing edge from the last back to the first implicit. Vertices must be distinct, the cycle length odd and at least 3, and every cycle edge present. Give (cycle,cycle_edges,length,certificate?,odd_cycle_proves_nonbipartite). Normalize each edge's endpoints but retain traversal order in cycle_edges.","answer":{"kind":"expression","value":"([0, 1, 2], [[0, 1], [1, 2], [0, 2]], 3, Yes, odd_cycle_proves_nonbipartite)","explanation":"Alternating between two parts around an odd cycle cannot return consistently to the initial part. The certificate requires the full actual cycle, not merely a failed candidate partition or an incomplete walk. The exact result, retaining the requested order, is ([0, 1, 2], [[0, 1], [1, 2], [0, 2]], 3, Yes, odd_cycle_proves_nonbipartite)."},"skills":["An Actual Simple Odd Cycle Certifies Nonbipartiteness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r02-bipartite-graph-0072","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3cbf6b99c186be96420d8771de7d292a2794f4e34afd2fe605f0df14b69d7406","family_id":"qf-mod31-bipartite-odd-cycle","seed":202610312072,"semantic_template":"qf-mod31-bipartite-odd-cycle/{\"cycle\":[0,1,2],\"edges\":[[0,1],[0,2],[1,2],[1,4],[3,4],[3,5]],\"n\":6}"},"fields":{"title":"An Actual Simple Odd Cycle Certifies Nonbipartiteness","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [0, 2], [3, 4], [3, 5], [1, 4]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. Ordered cycle vertices=[0, 1, 2], with the closing edge from the last back to the first implicit. Vertices must be distinct, the cycle length odd and at least 3, and every cycle edge present. Give (cycle,cycle_edges,length,certificate?,odd_cycle_proves_nonbipartite). Normalize each edge's endpoints but retain traversal order in cycle_edges.","answer":{"kind":"expression","value":"([0, 1, 2], [[0, 1], [1, 2], [0, 2]], 3, Yes, odd_cycle_proves_nonbipartite)","explanation":"Alternating between two parts around an odd cycle cannot return consistently to the initial part. The certificate requires the full actual cycle, not merely a failed candidate partition or an incomplete walk. The exact result, retaining the requested order, is ([0, 1, 2], [[0, 1], [1, 2], [0, 2]], 3, Yes, odd_cycle_proves_nonbipartite)."},"skills":["An Actual Simple Odd Cycle Certifies Nonbipartiteness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r02-bipartite-graph-0073","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"746537b270f0e566e06212048f97ea04e94b72590bf4870369680bd83ddcf656","family_id":"qf-mod31-bipartite-odd-cycle","seed":202610312073,"semantic_template":"qf-mod31-bipartite-odd-cycle/{\"cycle\":[0,1,2],\"edges\":[[0,1],[0,2],[1,2],[1,4],[3,5],[4,5]],\"n\":6}"},"fields":{"title":"An Actual Simple Odd Cycle Certifies Nonbipartiteness","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [0, 2], [4, 5], [3, 5], [1, 4]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. Ordered cycle vertices=[0, 1, 2], with the closing edge from the last back to the first implicit. Vertices must be distinct, the cycle length odd and at least 3, and every cycle edge present. Give (cycle,cycle_edges,length,certificate?,odd_cycle_proves_nonbipartite). Normalize each edge's endpoints but retain traversal order in cycle_edges.","answer":{"kind":"expression","value":"([0, 1, 2], [[0, 1], [1, 2], [0, 2]], 3, Yes, odd_cycle_proves_nonbipartite)","explanation":"Alternating between two parts around an odd cycle cannot return consistently to the initial part. The certificate requires the full actual cycle, not merely a failed candidate partition or an incomplete walk. The exact result, retaining the requested order, is ([0, 1, 2], [[0, 1], [1, 2], [0, 2]], 3, Yes, odd_cycle_proves_nonbipartite)."},"skills":["An Actual Simple Odd Cycle Certifies Nonbipartiteness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r02-bipartite-graph-0074","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b47be3b878a534426b7cd855e95d5d1607c82f8f3feb0b9887dad73d4fe22153","family_id":"qf-mod31-bipartite-odd-cycle","seed":202610312074,"semantic_template":"qf-mod31-bipartite-odd-cycle/{\"cycle\":[0,1,2],\"edges\":[[0,1],[0,2],[1,2],[1,4],[3,4],[3,5],[4,5]],\"n\":6}"},"fields":{"title":"An Actual Simple Odd Cycle Certifies Nonbipartiteness","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [0, 2], [3, 4], [4, 5], [3, 5], [1, 4]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. Ordered cycle vertices=[0, 1, 2], with the closing edge from the last back to the first implicit. Vertices must be distinct, the cycle length odd and at least 3, and every cycle edge present. Give (cycle,cycle_edges,length,certificate?,odd_cycle_proves_nonbipartite). Normalize each edge's endpoints but retain traversal order in cycle_edges.","answer":{"kind":"expression","value":"([0, 1, 2], [[0, 1], [1, 2], [0, 2]], 3, Yes, odd_cycle_proves_nonbipartite)","explanation":"Alternating between two parts around an odd cycle cannot return consistently to the initial part. The certificate requires the full actual cycle, not merely a failed candidate partition or an incomplete walk. The exact result, retaining the requested order, is ([0, 1, 2], [[0, 1], [1, 2], [0, 2]], 3, Yes, odd_cycle_proves_nonbipartite)."},"skills":["An Actual Simple Odd Cycle Certifies Nonbipartiteness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r02-bipartite-graph-0075","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8e1de618f233f23fbe47c0775870f18e2f533a8001c56bc8a1c3fd08a59d2c47","family_id":"qf-mod31-bipartite-odd-cycle","seed":202610312075,"semantic_template":"qf-mod31-bipartite-odd-cycle/{\"cycle\":[0,1,2],\"edges\":[[0,1],[0,2],[0,3],[1,2],[1,4]],\"n\":6}"},"fields":{"title":"An Actual Simple Odd Cycle Certifies Nonbipartiteness","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [0, 2], [0, 3], [1, 4]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. Ordered cycle vertices=[0, 1, 2], with the closing edge from the last back to the first implicit. Vertices must be distinct, the cycle length odd and at least 3, and every cycle edge present. Give (cycle,cycle_edges,length,certificate?,odd_cycle_proves_nonbipartite). Normalize each edge's endpoints but retain traversal order in cycle_edges.","answer":{"kind":"expression","value":"([0, 1, 2], [[0, 1], [1, 2], [0, 2]], 3, Yes, odd_cycle_proves_nonbipartite)","explanation":"Alternating between two parts around an odd cycle cannot return consistently to the initial part. The certificate requires the full actual cycle, not merely a failed candidate partition or an incomplete walk. The exact result, retaining the requested order, is ([0, 1, 2], [[0, 1], [1, 2], [0, 2]], 3, Yes, odd_cycle_proves_nonbipartite)."},"skills":["An Actual Simple Odd Cycle Certifies Nonbipartiteness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r02-bipartite-graph-0076","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2d9eac664b532085dfee5c4e9fbf5a1073541e1b95cf6f8fd33f134d1b0a1aa0","family_id":"qf-mod31-bipartite-isolated-boundary","seed":202610312076,"semantic_template":"qf-mod31-bipartite-isolated-boundary/{\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"n\":6,\"parts\":[0,0,0,0,0,0],\"vertex\":5}"},"fields":{"title":"Isolated Vertices Still Belong to Exactly One Part","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 4]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. The total part assignment=[0, 0, 0, 0, 0, 0] uses labels 0 or 1 for every vertex. Its two inverse-image parts are disjoint and cover the whole carrier. Every edge must cross parts; empty parts are allowed in an edgeless graph. Failure of this assignment alone does not prove the graph nonbipartite. vertex=5 must be isolated. Give (vertex,incident_edges,assigned_part,whole_partition_proper?,isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph).","answer":{"kind":"expression","value":"(5, [], 0, No, isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph)","explanation":"Confirm there are no incident edges before reading the part of this vertex. It can be assigned either part without affecting edge constraints, but it must not be omitted from the total assignment. The exact result, retaining the requested order, is (5, [], 0, No, isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph)."},"skills":["Isolated Vertices Still Belong to Exactly One Part"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r02-bipartite-graph-0077","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3485391483b2fd291fcb722de9969b934c6b9c7a446d27244f71bdff1d221d52","family_id":"qf-mod31-bipartite-isolated-boundary","seed":202610312077,"semantic_template":"qf-mod31-bipartite-isolated-boundary/{\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"n\":6,\"parts\":[1,0,0,0,0,0],\"vertex\":5}"},"fields":{"title":"Isolated Vertices Still Belong to Exactly One Part","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 4]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. The total part assignment=[1, 0, 0, 0, 0, 0] uses labels 0 or 1 for every vertex. Its two inverse-image parts are disjoint and cover the whole carrier. Every edge must cross parts; empty parts are allowed in an edgeless graph. Failure of this assignment alone does not prove the graph nonbipartite. vertex=5 must be isolated. Give (vertex,incident_edges,assigned_part,whole_partition_proper?,isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph).","answer":{"kind":"expression","value":"(5, [], 0, No, isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph)","explanation":"Confirm there are no incident edges before reading the part of this vertex. It can be assigned either part without affecting edge constraints, but it must not be omitted from the total assignment. The exact result, retaining the requested order, is (5, [], 0, No, isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph)."},"skills":["Isolated Vertices Still Belong to Exactly One Part"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r02-bipartite-graph-0078","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f9da3af7169f7a10a860889ebd6cc80808023d9bd98be048afd5d0a9137e0639","family_id":"qf-mod31-bipartite-isolated-boundary","seed":202610312078,"semantic_template":"qf-mod31-bipartite-isolated-boundary/{\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"n\":6,\"parts\":[0,1,0,0,0,0],\"vertex\":5}"},"fields":{"title":"Isolated Vertices Still Belong to Exactly One Part","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 4]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. The total part assignment=[0, 1, 0, 0, 0, 0] uses labels 0 or 1 for every vertex. Its two inverse-image parts are disjoint and cover the whole carrier. Every edge must cross parts; empty parts are allowed in an edgeless graph. Failure of this assignment alone does not prove the graph nonbipartite. vertex=5 must be isolated. Give (vertex,incident_edges,assigned_part,whole_partition_proper?,isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph).","answer":{"kind":"expression","value":"(5, [], 0, No, isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph)","explanation":"Confirm there are no incident edges before reading the part of this vertex. It can be assigned either part without affecting edge constraints, but it must not be omitted from the total assignment. The exact result, retaining the requested order, is (5, [], 0, No, isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph)."},"skills":["Isolated Vertices Still Belong to Exactly One Part"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r02-bipartite-graph-0079","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"75f49fe7428cf32bb1e0e7e53d772f0912ec04c1382b0ce09bbd98c19d851b51","family_id":"qf-mod31-bipartite-isolated-boundary","seed":202610312079,"semantic_template":"qf-mod31-bipartite-isolated-boundary/{\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"n\":6,\"parts\":[1,1,0,0,0,0],\"vertex\":5}"},"fields":{"title":"Isolated Vertices Still Belong to Exactly One Part","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 4]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. The total part assignment=[1, 1, 0, 0, 0, 0] uses labels 0 or 1 for every vertex. Its two inverse-image parts are disjoint and cover the whole carrier. Every edge must cross parts; empty parts are allowed in an edgeless graph. Failure of this assignment alone does not prove the graph nonbipartite. vertex=5 must be isolated. Give (vertex,incident_edges,assigned_part,whole_partition_proper?,isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph).","answer":{"kind":"expression","value":"(5, [], 0, No, isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph)","explanation":"Confirm there are no incident edges before reading the part of this vertex. It can be assigned either part without affecting edge constraints, but it must not be omitted from the total assignment. The exact result, retaining the requested order, is (5, [], 0, No, isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph)."},"skills":["Isolated Vertices Still Belong to Exactly One Part"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r02-bipartite-graph-0080","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2c26b94a0b8df91f9af8a9ba74229fea38c0f3de10ee5b830f5b716fb23f3214","family_id":"qf-mod31-bipartite-isolated-boundary","seed":202610312080,"semantic_template":"qf-mod31-bipartite-isolated-boundary/{\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"n\":6,\"parts\":[0,0,1,0,0,0],\"vertex\":5}"},"fields":{"title":"Isolated Vertices Still Belong to Exactly One Part","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 4]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. The total part assignment=[0, 0, 1, 0, 0, 0] uses labels 0 or 1 for every vertex. Its two inverse-image parts are disjoint and cover the whole carrier. Every edge must cross parts; empty parts are allowed in an edgeless graph. Failure of this assignment alone does not prove the graph nonbipartite. vertex=5 must be isolated. Give (vertex,incident_edges,assigned_part,whole_partition_proper?,isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph).","answer":{"kind":"expression","value":"(5, [], 0, No, isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph)","explanation":"Confirm there are no incident edges before reading the part of this vertex. It can be assigned either part without affecting edge constraints, but it must not be omitted from the total assignment. The exact result, retaining the requested order, is (5, [], 0, No, isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph)."},"skills":["Isolated Vertices Still Belong to Exactly One Part"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r02-bipartite-graph-0081","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ead999747055903049f3a7af5b34f4f95a3f2a5dd61a91d3ac49822d65d5cade","family_id":"qf-mod31-bipartite-isolated-boundary","seed":202610312081,"semantic_template":"qf-mod31-bipartite-isolated-boundary/{\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"n\":6,\"parts\":[1,0,1,0,0,0],\"vertex\":5}"},"fields":{"title":"Isolated Vertices Still Belong to Exactly One Part","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 4]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. The total part assignment=[1, 0, 1, 0, 0, 0] uses labels 0 or 1 for every vertex. Its two inverse-image parts are disjoint and cover the whole carrier. Every edge must cross parts; empty parts are allowed in an edgeless graph. Failure of this assignment alone does not prove the graph nonbipartite. vertex=5 must be isolated. Give (vertex,incident_edges,assigned_part,whole_partition_proper?,isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph).","answer":{"kind":"expression","value":"(5, [], 0, No, isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph)","explanation":"Confirm there are no incident edges before reading the part of this vertex. It can be assigned either part without affecting edge constraints, but it must not be omitted from the total assignment. The exact result, retaining the requested order, is (5, [], 0, No, isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph)."},"skills":["Isolated Vertices Still Belong to Exactly One Part"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r02-bipartite-graph-0082","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f2457046d45aee064d7c88a978229ad23a4c2e326f1da6100db0aa9699745b16","family_id":"qf-mod31-bipartite-isolated-boundary","seed":202610312082,"semantic_template":"qf-mod31-bipartite-isolated-boundary/{\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"n\":6,\"parts\":[0,1,1,0,0,0],\"vertex\":5}"},"fields":{"title":"Isolated Vertices Still Belong to Exactly One Part","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 4]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. The total part assignment=[0, 1, 1, 0, 0, 0] uses labels 0 or 1 for every vertex. Its two inverse-image parts are disjoint and cover the whole carrier. Every edge must cross parts; empty parts are allowed in an edgeless graph. Failure of this assignment alone does not prove the graph nonbipartite. vertex=5 must be isolated. Give (vertex,incident_edges,assigned_part,whole_partition_proper?,isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph).","answer":{"kind":"expression","value":"(5, [], 0, No, isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph)","explanation":"Confirm there are no incident edges before reading the part of this vertex. It can be assigned either part without affecting edge constraints, but it must not be omitted from the total assignment. The exact result, retaining the requested order, is (5, [], 0, No, isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph)."},"skills":["Isolated Vertices Still Belong to Exactly One Part"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r02-bipartite-graph-0083","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"de099a556cf65797f3279da301b436ab582248d2e21a4d5f03b5031c25b964af","family_id":"qf-mod31-bipartite-isolated-boundary","seed":202610312083,"semantic_template":"qf-mod31-bipartite-isolated-boundary/{\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"n\":6,\"parts\":[1,1,1,0,0,0],\"vertex\":5}"},"fields":{"title":"Isolated Vertices Still Belong to Exactly One Part","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 4]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. The total part assignment=[1, 1, 1, 0, 0, 0] uses labels 0 or 1 for every vertex. Its two inverse-image parts are disjoint and cover the whole carrier. Every edge must cross parts; empty parts are allowed in an edgeless graph. Failure of this assignment alone does not prove the graph nonbipartite. vertex=5 must be isolated. Give (vertex,incident_edges,assigned_part,whole_partition_proper?,isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph).","answer":{"kind":"expression","value":"(5, [], 0, No, isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph)","explanation":"Confirm there are no incident edges before reading the part of this vertex. It can be assigned either part without affecting edge constraints, but it must not be omitted from the total assignment. The exact result, retaining the requested order, is (5, [], 0, No, isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph)."},"skills":["Isolated Vertices Still Belong to Exactly One Part"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r02-bipartite-graph-0084","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ac076e7842bc7eb12b0fdf5dc7fbdc500f65058f57188cf37ff873a5c59288a4","family_id":"qf-mod31-bipartite-isolated-boundary","seed":202610312084,"semantic_template":"qf-mod31-bipartite-isolated-boundary/{\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"n\":6,\"parts\":[0,0,0,1,0,0],\"vertex\":5}"},"fields":{"title":"Isolated Vertices Still Belong to Exactly One Part","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 4]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. The total part assignment=[0, 0, 0, 1, 0, 0] uses labels 0 or 1 for every vertex. Its two inverse-image parts are disjoint and cover the whole carrier. Every edge must cross parts; empty parts are allowed in an edgeless graph. Failure of this assignment alone does not prove the graph nonbipartite. vertex=5 must be isolated. Give (vertex,incident_edges,assigned_part,whole_partition_proper?,isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph).","answer":{"kind":"expression","value":"(5, [], 0, No, isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph)","explanation":"Confirm there are no incident edges before reading the part of this vertex. It can be assigned either part without affecting edge constraints, but it must not be omitted from the total assignment. The exact result, retaining the requested order, is (5, [], 0, No, isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph)."},"skills":["Isolated Vertices Still Belong to Exactly One Part"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r02-bipartite-graph-0085","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9391721d459d62ec938d6d2a5a098f816764570d15e73daf83916b81a9ccfbca","family_id":"qf-mod31-bipartite-isolated-boundary","seed":202610312085,"semantic_template":"qf-mod31-bipartite-isolated-boundary/{\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"n\":6,\"parts\":[1,0,0,1,0,0],\"vertex\":5}"},"fields":{"title":"Isolated Vertices Still Belong to Exactly One Part","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 4]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. The total part assignment=[1, 0, 0, 1, 0, 0] uses labels 0 or 1 for every vertex. Its two inverse-image parts are disjoint and cover the whole carrier. Every edge must cross parts; empty parts are allowed in an edgeless graph. Failure of this assignment alone does not prove the graph nonbipartite. vertex=5 must be isolated. Give (vertex,incident_edges,assigned_part,whole_partition_proper?,isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph).","answer":{"kind":"expression","value":"(5, [], 0, No, isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph)","explanation":"Confirm there are no incident edges before reading the part of this vertex. It can be assigned either part without affecting edge constraints, but it must not be omitted from the total assignment. The exact result, retaining the requested order, is (5, [], 0, No, isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph)."},"skills":["Isolated Vertices Still Belong to Exactly One Part"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r02-bipartite-graph-0086","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8cadebe53fd31f9a26b42327f6f8b95318f913c7a4110dd7a9df4a7f3142fbd0","family_id":"qf-mod31-bipartite-isolated-boundary","seed":202610312086,"semantic_template":"qf-mod31-bipartite-isolated-boundary/{\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"n\":6,\"parts\":[0,1,0,1,0,0],\"vertex\":5}"},"fields":{"title":"Isolated Vertices Still Belong to Exactly One Part","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 4]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. The total part assignment=[0, 1, 0, 1, 0, 0] uses labels 0 or 1 for every vertex. Its two inverse-image parts are disjoint and cover the whole carrier. Every edge must cross parts; empty parts are allowed in an edgeless graph. Failure of this assignment alone does not prove the graph nonbipartite. vertex=5 must be isolated. Give (vertex,incident_edges,assigned_part,whole_partition_proper?,isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph).","answer":{"kind":"expression","value":"(5, [], 0, Yes, isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph)","explanation":"Confirm there are no incident edges before reading the part of this vertex. It can be assigned either part without affecting edge constraints, but it must not be omitted from the total assignment. The exact result, retaining the requested order, is (5, [], 0, Yes, isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph)."},"skills":["Isolated Vertices Still Belong to Exactly One Part"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r02-bipartite-graph-0087","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"944a4a252d866b3dd414a1f6e20c91f809bcee38a64d8d96b31d2d9031e6222f","family_id":"qf-mod31-bipartite-isolated-boundary","seed":202610312087,"semantic_template":"qf-mod31-bipartite-isolated-boundary/{\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"n\":6,\"parts\":[1,1,0,1,0,0],\"vertex\":5}"},"fields":{"title":"Isolated Vertices Still Belong to Exactly One Part","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 4]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. The total part assignment=[1, 1, 0, 1, 0, 0] uses labels 0 or 1 for every vertex. Its two inverse-image parts are disjoint and cover the whole carrier. Every edge must cross parts; empty parts are allowed in an edgeless graph. Failure of this assignment alone does not prove the graph nonbipartite. vertex=5 must be isolated. Give (vertex,incident_edges,assigned_part,whole_partition_proper?,isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph).","answer":{"kind":"expression","value":"(5, [], 0, No, isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph)","explanation":"Confirm there are no incident edges before reading the part of this vertex. It can be assigned either part without affecting edge constraints, but it must not be omitted from the total assignment. The exact result, retaining the requested order, is (5, [], 0, No, isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph)."},"skills":["Isolated Vertices Still Belong to Exactly One Part"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r02-bipartite-graph-0088","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ca1b6ded034690899c6f221cbbe0414f27ecc2c26c5f158250aac2ab10cce4a9","family_id":"qf-mod31-bipartite-isolated-boundary","seed":202610312088,"semantic_template":"qf-mod31-bipartite-isolated-boundary/{\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"n\":6,\"parts\":[0,0,1,1,0,0],\"vertex\":5}"},"fields":{"title":"Isolated Vertices Still Belong to Exactly One Part","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 4]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. The total part assignment=[0, 0, 1, 1, 0, 0] uses labels 0 or 1 for every vertex. Its two inverse-image parts are disjoint and cover the whole carrier. Every edge must cross parts; empty parts are allowed in an edgeless graph. Failure of this assignment alone does not prove the graph nonbipartite. vertex=5 must be isolated. Give (vertex,incident_edges,assigned_part,whole_partition_proper?,isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph).","answer":{"kind":"expression","value":"(5, [], 0, No, isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph)","explanation":"Confirm there are no incident edges before reading the part of this vertex. It can be assigned either part without affecting edge constraints, but it must not be omitted from the total assignment. The exact result, retaining the requested order, is (5, [], 0, No, isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph)."},"skills":["Isolated Vertices Still Belong to Exactly One Part"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r02-bipartite-graph-0089","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b63905e2f95e5ca292763a871513225c46ba601722e9e2e569754b63a37c8828","family_id":"qf-mod31-bipartite-isolated-boundary","seed":202610312089,"semantic_template":"qf-mod31-bipartite-isolated-boundary/{\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"n\":6,\"parts\":[1,0,1,1,0,0],\"vertex\":5}"},"fields":{"title":"Isolated Vertices Still Belong to Exactly One Part","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 4]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. The total part assignment=[1, 0, 1, 1, 0, 0] uses labels 0 or 1 for every vertex. Its two inverse-image parts are disjoint and cover the whole carrier. Every edge must cross parts; empty parts are allowed in an edgeless graph. Failure of this assignment alone does not prove the graph nonbipartite. vertex=5 must be isolated. Give (vertex,incident_edges,assigned_part,whole_partition_proper?,isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph).","answer":{"kind":"expression","value":"(5, [], 0, No, isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph)","explanation":"Confirm there are no incident edges before reading the part of this vertex. It can be assigned either part without affecting edge constraints, but it must not be omitted from the total assignment. The exact result, retaining the requested order, is (5, [], 0, No, isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph)."},"skills":["Isolated Vertices Still Belong to Exactly One Part"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r02-bipartite-graph-0090","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ecea78241a106d1160304972d6ce64769fc79444a9b763ce39ee0ba22bee2952","family_id":"qf-mod31-bipartite-isolated-boundary","seed":202610312090,"semantic_template":"qf-mod31-bipartite-isolated-boundary/{\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"n\":6,\"parts\":[0,1,1,1,0,0],\"vertex\":5}"},"fields":{"title":"Isolated Vertices Still Belong to Exactly One Part","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 4]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. The total part assignment=[0, 1, 1, 1, 0, 0] uses labels 0 or 1 for every vertex. Its two inverse-image parts are disjoint and cover the whole carrier. Every edge must cross parts; empty parts are allowed in an edgeless graph. Failure of this assignment alone does not prove the graph nonbipartite. vertex=5 must be isolated. Give (vertex,incident_edges,assigned_part,whole_partition_proper?,isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph).","answer":{"kind":"expression","value":"(5, [], 0, No, isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph)","explanation":"Confirm there are no incident edges before reading the part of this vertex. It can be assigned either part without affecting edge constraints, but it must not be omitted from the total assignment. The exact result, retaining the requested order, is (5, [], 0, No, isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph)."},"skills":["Isolated Vertices Still Belong to Exactly One Part"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r02-bipartite-graph-0091","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"be3059f7d8fa9be01333ceb372beef51598afc9ac7f2ea1debe5b8b322fba862","family_id":"qf-mod31-bipartite-isolated-boundary","seed":202610312091,"semantic_template":"qf-mod31-bipartite-isolated-boundary/{\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"n\":6,\"parts\":[1,1,1,1,0,0],\"vertex\":5}"},"fields":{"title":"Isolated Vertices Still Belong to Exactly One Part","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 4]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. The total part assignment=[1, 1, 1, 1, 0, 0] uses labels 0 or 1 for every vertex. Its two inverse-image parts are disjoint and cover the whole carrier. Every edge must cross parts; empty parts are allowed in an edgeless graph. Failure of this assignment alone does not prove the graph nonbipartite. vertex=5 must be isolated. Give (vertex,incident_edges,assigned_part,whole_partition_proper?,isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph).","answer":{"kind":"expression","value":"(5, [], 0, No, isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph)","explanation":"Confirm there are no incident edges before reading the part of this vertex. It can be assigned either part without affecting edge constraints, but it must not be omitted from the total assignment. The exact result, retaining the requested order, is (5, [], 0, No, isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph)."},"skills":["Isolated Vertices Still Belong to Exactly One Part"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r02-bipartite-graph-0092","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b252fb065d8f4cd103b38459cc9e09e538de99530dafc7574f6a2c03b159edf4","family_id":"qf-mod31-bipartite-isolated-boundary","seed":202610312092,"semantic_template":"qf-mod31-bipartite-isolated-boundary/{\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"n\":6,\"parts\":[0,0,0,0,1,0],\"vertex\":5}"},"fields":{"title":"Isolated Vertices Still Belong to Exactly One Part","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 4]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. The total part assignment=[0, 0, 0, 0, 1, 0] uses labels 0 or 1 for every vertex. Its two inverse-image parts are disjoint and cover the whole carrier. Every edge must cross parts; empty parts are allowed in an edgeless graph. Failure of this assignment alone does not prove the graph nonbipartite. vertex=5 must be isolated. Give (vertex,incident_edges,assigned_part,whole_partition_proper?,isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph).","answer":{"kind":"expression","value":"(5, [], 0, No, isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph)","explanation":"Confirm there are no incident edges before reading the part of this vertex. It can be assigned either part without affecting edge constraints, but it must not be omitted from the total assignment. The exact result, retaining the requested order, is (5, [], 0, No, isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph)."},"skills":["Isolated Vertices Still Belong to Exactly One Part"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r02-bipartite-graph-0093","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d16df836110a11faa24141a8bf1e7dc91d33754d4da63c4262593105ab3b8de3","family_id":"qf-mod31-bipartite-isolated-boundary","seed":202610312093,"semantic_template":"qf-mod31-bipartite-isolated-boundary/{\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"n\":6,\"parts\":[1,0,0,0,1,0],\"vertex\":5}"},"fields":{"title":"Isolated Vertices Still Belong to Exactly One Part","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 4]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. The total part assignment=[1, 0, 0, 0, 1, 0] uses labels 0 or 1 for every vertex. Its two inverse-image parts are disjoint and cover the whole carrier. Every edge must cross parts; empty parts are allowed in an edgeless graph. Failure of this assignment alone does not prove the graph nonbipartite. vertex=5 must be isolated. Give (vertex,incident_edges,assigned_part,whole_partition_proper?,isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph).","answer":{"kind":"expression","value":"(5, [], 0, No, isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph)","explanation":"Confirm there are no incident edges before reading the part of this vertex. It can be assigned either part without affecting edge constraints, but it must not be omitted from the total assignment. The exact result, retaining the requested order, is (5, [], 0, No, isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph)."},"skills":["Isolated Vertices Still Belong to Exactly One Part"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r02-bipartite-graph-0094","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c8c350717fbcf32d1d1521bbdd65a555214163d194159c56b1347cf084d9ecea","family_id":"qf-mod31-bipartite-isolated-boundary","seed":202610312094,"semantic_template":"qf-mod31-bipartite-isolated-boundary/{\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"n\":6,\"parts\":[0,1,0,0,1,0],\"vertex\":5}"},"fields":{"title":"Isolated Vertices Still Belong to Exactly One Part","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 4]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. The total part assignment=[0, 1, 0, 0, 1, 0] uses labels 0 or 1 for every vertex. Its two inverse-image parts are disjoint and cover the whole carrier. Every edge must cross parts; empty parts are allowed in an edgeless graph. Failure of this assignment alone does not prove the graph nonbipartite. vertex=5 must be isolated. Give (vertex,incident_edges,assigned_part,whole_partition_proper?,isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph).","answer":{"kind":"expression","value":"(5, [], 0, No, isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph)","explanation":"Confirm there are no incident edges before reading the part of this vertex. It can be assigned either part without affecting edge constraints, but it must not be omitted from the total assignment. The exact result, retaining the requested order, is (5, [], 0, No, isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph)."},"skills":["Isolated Vertices Still Belong to Exactly One Part"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r02-bipartite-graph-0095","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"06a2c29316e8a1e996d3857072f4f39f81fe3466827d917bedf8a424d6e004cf","family_id":"qf-mod31-bipartite-isolated-boundary","seed":202610312095,"semantic_template":"qf-mod31-bipartite-isolated-boundary/{\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"n\":6,\"parts\":[1,1,0,0,1,0],\"vertex\":5}"},"fields":{"title":"Isolated Vertices Still Belong to Exactly One Part","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 4]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. The total part assignment=[1, 1, 0, 0, 1, 0] uses labels 0 or 1 for every vertex. Its two inverse-image parts are disjoint and cover the whole carrier. Every edge must cross parts; empty parts are allowed in an edgeless graph. Failure of this assignment alone does not prove the graph nonbipartite. vertex=5 must be isolated. Give (vertex,incident_edges,assigned_part,whole_partition_proper?,isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph).","answer":{"kind":"expression","value":"(5, [], 0, No, isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph)","explanation":"Confirm there are no incident edges before reading the part of this vertex. It can be assigned either part without affecting edge constraints, but it must not be omitted from the total assignment. The exact result, retaining the requested order, is (5, [], 0, No, isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph)."},"skills":["Isolated Vertices Still Belong to Exactly One Part"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r02-bipartite-graph-0096","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"297125200211ccdf88e9048013975985a5304e8ec07a9e90664d417943c91eaa","family_id":"qf-mod31-bipartite-isolated-boundary","seed":202610312096,"semantic_template":"qf-mod31-bipartite-isolated-boundary/{\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"n\":6,\"parts\":[0,0,1,0,1,0],\"vertex\":5}"},"fields":{"title":"Isolated Vertices Still Belong to Exactly One Part","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 4]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. The total part assignment=[0, 0, 1, 0, 1, 0] uses labels 0 or 1 for every vertex. Its two inverse-image parts are disjoint and cover the whole carrier. Every edge must cross parts; empty parts are allowed in an edgeless graph. Failure of this assignment alone does not prove the graph nonbipartite. vertex=5 must be isolated. Give (vertex,incident_edges,assigned_part,whole_partition_proper?,isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph).","answer":{"kind":"expression","value":"(5, [], 0, No, isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph)","explanation":"Confirm there are no incident edges before reading the part of this vertex. It can be assigned either part without affecting edge constraints, but it must not be omitted from the total assignment. The exact result, retaining the requested order, is (5, [], 0, No, isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph)."},"skills":["Isolated Vertices Still Belong to Exactly One Part"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r02-bipartite-graph-0097","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ac0b8fe7de9992640994b36464ddd19c45100976d2b9e55fae9c6494457b15d4","family_id":"qf-mod31-bipartite-isolated-boundary","seed":202610312097,"semantic_template":"qf-mod31-bipartite-isolated-boundary/{\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"n\":6,\"parts\":[1,0,1,0,1,0],\"vertex\":5}"},"fields":{"title":"Isolated Vertices Still Belong to Exactly One Part","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 4]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. The total part assignment=[1, 0, 1, 0, 1, 0] uses labels 0 or 1 for every vertex. Its two inverse-image parts are disjoint and cover the whole carrier. Every edge must cross parts; empty parts are allowed in an edgeless graph. Failure of this assignment alone does not prove the graph nonbipartite. vertex=5 must be isolated. Give (vertex,incident_edges,assigned_part,whole_partition_proper?,isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph).","answer":{"kind":"expression","value":"(5, [], 0, Yes, isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph)","explanation":"Confirm there are no incident edges before reading the part of this vertex. It can be assigned either part without affecting edge constraints, but it must not be omitted from the total assignment. The exact result, retaining the requested order, is (5, [], 0, Yes, isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph)."},"skills":["Isolated Vertices Still Belong to Exactly One Part"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r02-bipartite-graph-0098","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"592a771d6d9fa2c25c2950afaa385125bbad8ba11d8f72de570537983bf91258","family_id":"qf-mod31-bipartite-isolated-boundary","seed":202610312098,"semantic_template":"qf-mod31-bipartite-isolated-boundary/{\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"n\":6,\"parts\":[0,1,1,0,1,0],\"vertex\":5}"},"fields":{"title":"Isolated Vertices Still Belong to Exactly One Part","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 4]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. The total part assignment=[0, 1, 1, 0, 1, 0] uses labels 0 or 1 for every vertex. Its two inverse-image parts are disjoint and cover the whole carrier. Every edge must cross parts; empty parts are allowed in an edgeless graph. Failure of this assignment alone does not prove the graph nonbipartite. vertex=5 must be isolated. Give (vertex,incident_edges,assigned_part,whole_partition_proper?,isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph).","answer":{"kind":"expression","value":"(5, [], 0, No, isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph)","explanation":"Confirm there are no incident edges before reading the part of this vertex. It can be assigned either part without affecting edge constraints, but it must not be omitted from the total assignment. The exact result, retaining the requested order, is (5, [], 0, No, isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph)."},"skills":["Isolated Vertices Still Belong to Exactly One Part"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r02-bipartite-graph-0099","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5fc1757c0d2e5f69d155f778ebfc0d02d37bac934806da58f41da3742a340cf3","family_id":"qf-mod31-bipartite-isolated-boundary","seed":202610312099,"semantic_template":"qf-mod31-bipartite-isolated-boundary/{\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"n\":6,\"parts\":[1,1,1,0,1,0],\"vertex\":5}"},"fields":{"title":"Isolated Vertices Still Belong to Exactly One Part","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 4]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. The total part assignment=[1, 1, 1, 0, 1, 0] uses labels 0 or 1 for every vertex. Its two inverse-image parts are disjoint and cover the whole carrier. Every edge must cross parts; empty parts are allowed in an edgeless graph. Failure of this assignment alone does not prove the graph nonbipartite. vertex=5 must be isolated. Give (vertex,incident_edges,assigned_part,whole_partition_proper?,isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph).","answer":{"kind":"expression","value":"(5, [], 0, No, isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph)","explanation":"Confirm there are no incident edges before reading the part of this vertex. It can be assigned either part without affecting edge constraints, but it must not be omitted from the total assignment. The exact result, retaining the requested order, is (5, [], 0, No, isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph)."},"skills":["Isolated Vertices Still Belong to Exactly One Part"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r02-bipartite-graph-0100","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cf27c5db09da32bec4d629a978e4c347f056544f8efbaea06cf4d8fc264a4295","family_id":"qf-mod31-bipartite-isolated-boundary","seed":202610312100,"semantic_template":"qf-mod31-bipartite-isolated-boundary/{\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"n\":6,\"parts\":[0,0,0,1,1,0],\"vertex\":5}"},"fields":{"title":"Isolated Vertices Still Belong to Exactly One Part","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 4]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. The total part assignment=[0, 0, 0, 1, 1, 0] uses labels 0 or 1 for every vertex. Its two inverse-image parts are disjoint and cover the whole carrier. Every edge must cross parts; empty parts are allowed in an edgeless graph. Failure of this assignment alone does not prove the graph nonbipartite. vertex=5 must be isolated. Give (vertex,incident_edges,assigned_part,whole_partition_proper?,isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph).","answer":{"kind":"expression","value":"(5, [], 0, No, isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph)","explanation":"Confirm there are no incident edges before reading the part of this vertex. It can be assigned either part without affecting edge constraints, but it must not be omitted from the total assignment. The exact result, retaining the requested order, is (5, [], 0, No, isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph)."},"skills":["Isolated Vertices Still Belong to Exactly One Part"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}