{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r05-graph-coloring-0001","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8ab90bd0ca6aec8d454705745404d5893389d8c8395421836d6781774e67e7bf","family_id":"qf-mod31-coloring-proper-audit","seed":202610315001,"semantic_template":"qf-mod31-coloring-proper-audit/{\"colors\":[0,1,0,1,0,1],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"k\":2,\"n\":6}"},"fields":{"title":"Proper k-Coloring Means Edge Properness and At Most k Used Labels","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], [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. Total vertex colors=[0, 1, 0, 1, 0, 1] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. k=2. Give (increasing_used_colors,normalized_conflicting_edges,edge_proper?,at_most_k?,proper_k_coloring?,proper_k_coloring_uses_at_most_k_colors).","answer":{"kind":"expression","value":"([0, 1], [], Yes, Yes, Yes, proper_k_coloring_uses_at_most_k_colors)","explanation":"Audit every normalized edge and count distinct used labels separately. Both conditions are required; one passing coloring is not a proof of minimum chromatic number. The exact result, retaining the requested order, is ([0, 1], [], Yes, Yes, Yes, proper_k_coloring_uses_at_most_k_colors)."},"skills":["Proper k-Coloring Means Edge Properness and At Most k Used Labels"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r05-graph-coloring-0002","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"90e1aa68f9782acb620974c4866c45169047dd0ddf86e76baaf9bd775c4d6efd","family_id":"qf-mod31-coloring-proper-audit","seed":202610315002,"semantic_template":"qf-mod31-coloring-proper-audit/{\"colors\":[1,2,1,2,1,2],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"k\":3,\"n\":6}"},"fields":{"title":"Proper k-Coloring Means Edge Properness and At Most k Used Labels","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], [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. Total vertex colors=[1, 2, 1, 2, 1, 2] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. k=3. Give (increasing_used_colors,normalized_conflicting_edges,edge_proper?,at_most_k?,proper_k_coloring?,proper_k_coloring_uses_at_most_k_colors).","answer":{"kind":"expression","value":"([1, 2], [], Yes, Yes, Yes, proper_k_coloring_uses_at_most_k_colors)","explanation":"Audit every normalized edge and count distinct used labels separately. Both conditions are required; one passing coloring is not a proof of minimum chromatic number. The exact result, retaining the requested order, is ([1, 2], [], Yes, Yes, Yes, proper_k_coloring_uses_at_most_k_colors)."},"skills":["Proper k-Coloring Means Edge Properness and At Most k Used Labels"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r05-graph-coloring-0003","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7a9f53ea7a3360896b64bd612417e075e0121badb402abd6e1f6efc36ca92dc3","family_id":"qf-mod31-coloring-proper-audit","seed":202610315003,"semantic_template":"qf-mod31-coloring-proper-audit/{\"colors\":[2,3,2,3,2,3],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"k\":4,\"n\":6}"},"fields":{"title":"Proper k-Coloring Means Edge Properness and At Most k Used Labels","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], [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. Total vertex colors=[2, 3, 2, 3, 2, 3] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. k=4. Give (increasing_used_colors,normalized_conflicting_edges,edge_proper?,at_most_k?,proper_k_coloring?,proper_k_coloring_uses_at_most_k_colors).","answer":{"kind":"expression","value":"([2, 3], [], Yes, Yes, Yes, proper_k_coloring_uses_at_most_k_colors)","explanation":"Audit every normalized edge and count distinct used labels separately. Both conditions are required; one passing coloring is not a proof of minimum chromatic number. The exact result, retaining the requested order, is ([2, 3], [], Yes, Yes, Yes, proper_k_coloring_uses_at_most_k_colors)."},"skills":["Proper k-Coloring Means Edge Properness and At Most k Used Labels"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r05-graph-coloring-0004","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0dad8c6bb7746d922a7286eb7355b5241271225860bdfe538f6f2a17cef1453d","family_id":"qf-mod31-coloring-proper-audit","seed":202610315004,"semantic_template":"qf-mod31-coloring-proper-audit/{\"colors\":[3,4,3,4,3,4],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"k\":2,\"n\":6}"},"fields":{"title":"Proper k-Coloring Means Edge Properness and At Most k Used Labels","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], [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. Total vertex colors=[3, 4, 3, 4, 3, 4] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. k=2. Give (increasing_used_colors,normalized_conflicting_edges,edge_proper?,at_most_k?,proper_k_coloring?,proper_k_coloring_uses_at_most_k_colors).","answer":{"kind":"expression","value":"([3, 4], [], Yes, Yes, Yes, proper_k_coloring_uses_at_most_k_colors)","explanation":"Audit every normalized edge and count distinct used labels separately. Both conditions are required; one passing coloring is not a proof of minimum chromatic number. The exact result, retaining the requested order, is ([3, 4], [], Yes, Yes, Yes, proper_k_coloring_uses_at_most_k_colors)."},"skills":["Proper k-Coloring Means Edge Properness and At Most k Used Labels"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r05-graph-coloring-0005","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"03899a19a8132a0dcea08fb79d6ed4c2fbd8ff0dabbf967fe8926217bcdddd1a","family_id":"qf-mod31-coloring-proper-audit","seed":202610315005,"semantic_template":"qf-mod31-coloring-proper-audit/{\"colors\":[4,5,4,5,4,5],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"k\":3,\"n\":6}"},"fields":{"title":"Proper k-Coloring Means Edge Properness and At Most k Used Labels","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], [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. Total vertex colors=[4, 5, 4, 5, 4, 5] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. k=3. Give (increasing_used_colors,normalized_conflicting_edges,edge_proper?,at_most_k?,proper_k_coloring?,proper_k_coloring_uses_at_most_k_colors).","answer":{"kind":"expression","value":"([4, 5], [], Yes, Yes, Yes, proper_k_coloring_uses_at_most_k_colors)","explanation":"Audit every normalized edge and count distinct used labels separately. Both conditions are required; one passing coloring is not a proof of minimum chromatic number. The exact result, retaining the requested order, is ([4, 5], [], Yes, Yes, Yes, proper_k_coloring_uses_at_most_k_colors)."},"skills":["Proper k-Coloring Means Edge Properness and At Most k Used Labels"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r05-graph-coloring-0006","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f665efafd59fa53f767de5db889e793b2235ffc8d86356c080265b8b4034e4f0","family_id":"qf-mod31-coloring-proper-audit","seed":202610315006,"semantic_template":"qf-mod31-coloring-proper-audit/{\"colors\":[5,6,5,6,5,6],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"k\":4,\"n\":6}"},"fields":{"title":"Proper k-Coloring Means Edge Properness and At Most k Used Labels","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], [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. Total vertex colors=[5, 6, 5, 6, 5, 6] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. k=4. Give (increasing_used_colors,normalized_conflicting_edges,edge_proper?,at_most_k?,proper_k_coloring?,proper_k_coloring_uses_at_most_k_colors).","answer":{"kind":"expression","value":"([5, 6], [], Yes, Yes, Yes, proper_k_coloring_uses_at_most_k_colors)","explanation":"Audit every normalized edge and count distinct used labels separately. Both conditions are required; one passing coloring is not a proof of minimum chromatic number. The exact result, retaining the requested order, is ([5, 6], [], Yes, Yes, Yes, proper_k_coloring_uses_at_most_k_colors)."},"skills":["Proper k-Coloring Means Edge Properness and At Most k Used Labels"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r05-graph-coloring-0007","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3b0aeee7db73442a88ed1ec92c846f9b520fbb9608843dd8e8577acf0213b73a","family_id":"qf-mod31-coloring-proper-audit","seed":202610315007,"semantic_template":"qf-mod31-coloring-proper-audit/{\"colors\":[6,7,6,7,6,7],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"k\":2,\"n\":6}"},"fields":{"title":"Proper k-Coloring Means Edge Properness and At Most k Used Labels","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], [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. Total vertex colors=[6, 7, 6, 7, 6, 7] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. k=2. Give (increasing_used_colors,normalized_conflicting_edges,edge_proper?,at_most_k?,proper_k_coloring?,proper_k_coloring_uses_at_most_k_colors).","answer":{"kind":"expression","value":"([6, 7], [], Yes, Yes, Yes, proper_k_coloring_uses_at_most_k_colors)","explanation":"Audit every normalized edge and count distinct used labels separately. Both conditions are required; one passing coloring is not a proof of minimum chromatic number. The exact result, retaining the requested order, is ([6, 7], [], Yes, Yes, Yes, proper_k_coloring_uses_at_most_k_colors)."},"skills":["Proper k-Coloring Means Edge Properness and At Most k Used Labels"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r05-graph-coloring-0008","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b6ee6cd66c8ae75a2288e5c431d2822f6b63372bcc3b613d29847a3549a28a12","family_id":"qf-mod31-coloring-proper-audit","seed":202610315008,"semantic_template":"qf-mod31-coloring-proper-audit/{\"colors\":[7,8,7,8,7,8],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"k\":3,\"n\":6}"},"fields":{"title":"Proper k-Coloring Means Edge Properness and At Most k Used Labels","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], [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. Total vertex colors=[7, 8, 7, 8, 7, 8] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. k=3. Give (increasing_used_colors,normalized_conflicting_edges,edge_proper?,at_most_k?,proper_k_coloring?,proper_k_coloring_uses_at_most_k_colors).","answer":{"kind":"expression","value":"([7, 8], [], Yes, Yes, Yes, proper_k_coloring_uses_at_most_k_colors)","explanation":"Audit every normalized edge and count distinct used labels separately. Both conditions are required; one passing coloring is not a proof of minimum chromatic number. The exact result, retaining the requested order, is ([7, 8], [], Yes, Yes, Yes, proper_k_coloring_uses_at_most_k_colors)."},"skills":["Proper k-Coloring Means Edge Properness and At Most k Used Labels"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r05-graph-coloring-0009","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f5582c9ce94516f4ab43334f229e110c42f8cfff469ef9772d8eb81a30435a30","family_id":"qf-mod31-coloring-proper-audit","seed":202610315009,"semantic_template":"qf-mod31-coloring-proper-audit/{\"colors\":[8,9,8,9,8,9],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"k\":4,\"n\":6}"},"fields":{"title":"Proper k-Coloring Means Edge Properness and At Most k Used Labels","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], [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. Total vertex colors=[8, 9, 8, 9, 8, 9] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. k=4. Give (increasing_used_colors,normalized_conflicting_edges,edge_proper?,at_most_k?,proper_k_coloring?,proper_k_coloring_uses_at_most_k_colors).","answer":{"kind":"expression","value":"([8, 9], [], Yes, Yes, Yes, proper_k_coloring_uses_at_most_k_colors)","explanation":"Audit every normalized edge and count distinct used labels separately. Both conditions are required; one passing coloring is not a proof of minimum chromatic number. The exact result, retaining the requested order, is ([8, 9], [], Yes, Yes, Yes, proper_k_coloring_uses_at_most_k_colors)."},"skills":["Proper k-Coloring Means Edge Properness and At Most k Used Labels"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r05-graph-coloring-0010","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2b13ed514dabe87e52d359c4cb9c3e1c157a9991301ec492785687c9d9cdf996","family_id":"qf-mod31-coloring-proper-audit","seed":202610315010,"semantic_template":"qf-mod31-coloring-proper-audit/{\"colors\":[9,10,9,10,9,10],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"k\":2,\"n\":6}"},"fields":{"title":"Proper k-Coloring Means Edge Properness and At Most k Used Labels","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], [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. Total vertex colors=[9, 10, 9, 10, 9, 10] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. k=2. Give (increasing_used_colors,normalized_conflicting_edges,edge_proper?,at_most_k?,proper_k_coloring?,proper_k_coloring_uses_at_most_k_colors).","answer":{"kind":"expression","value":"([9, 10], [], Yes, Yes, Yes, proper_k_coloring_uses_at_most_k_colors)","explanation":"Audit every normalized edge and count distinct used labels separately. Both conditions are required; one passing coloring is not a proof of minimum chromatic number. The exact result, retaining the requested order, is ([9, 10], [], Yes, Yes, Yes, proper_k_coloring_uses_at_most_k_colors)."},"skills":["Proper k-Coloring Means Edge Properness and At Most k Used Labels"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r05-graph-coloring-0011","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"de5679cb660eebdf23af43b0cbee3e5d16ba10df1dfebf4ebef6a278f503c145","family_id":"qf-mod31-coloring-proper-audit","seed":202610315011,"semantic_template":"qf-mod31-coloring-proper-audit/{\"colors\":[10,11,10,11,10,11],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"k\":3,\"n\":6}"},"fields":{"title":"Proper k-Coloring Means Edge Properness and At Most k Used Labels","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], [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. Total vertex colors=[10, 11, 10, 11, 10, 11] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. k=3. Give (increasing_used_colors,normalized_conflicting_edges,edge_proper?,at_most_k?,proper_k_coloring?,proper_k_coloring_uses_at_most_k_colors).","answer":{"kind":"expression","value":"([10, 11], [], Yes, Yes, Yes, proper_k_coloring_uses_at_most_k_colors)","explanation":"Audit every normalized edge and count distinct used labels separately. Both conditions are required; one passing coloring is not a proof of minimum chromatic number. The exact result, retaining the requested order, is ([10, 11], [], Yes, Yes, Yes, proper_k_coloring_uses_at_most_k_colors)."},"skills":["Proper k-Coloring Means Edge Properness and At Most k Used Labels"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r05-graph-coloring-0012","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1edb0163de449c05d7866eb1962063f37dcfc7900c3e4a239dce2d84d4974cf6","family_id":"qf-mod31-coloring-proper-audit","seed":202610315012,"semantic_template":"qf-mod31-coloring-proper-audit/{\"colors\":[11,12,11,12,11,12],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"k\":4,\"n\":6}"},"fields":{"title":"Proper k-Coloring Means Edge Properness and At Most k Used Labels","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], [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. Total vertex colors=[11, 12, 11, 12, 11, 12] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. k=4. Give (increasing_used_colors,normalized_conflicting_edges,edge_proper?,at_most_k?,proper_k_coloring?,proper_k_coloring_uses_at_most_k_colors).","answer":{"kind":"expression","value":"([11, 12], [], Yes, Yes, Yes, proper_k_coloring_uses_at_most_k_colors)","explanation":"Audit every normalized edge and count distinct used labels separately. Both conditions are required; one passing coloring is not a proof of minimum chromatic number. The exact result, retaining the requested order, is ([11, 12], [], Yes, Yes, Yes, proper_k_coloring_uses_at_most_k_colors)."},"skills":["Proper k-Coloring Means Edge Properness and At Most k Used Labels"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r05-graph-coloring-0013","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6a9bf3e9f65b41026cadaa43447e0b9fdbec23d45c7a6da3d60aed7ddff8e5f7","family_id":"qf-mod31-coloring-proper-audit","seed":202610315013,"semantic_template":"qf-mod31-coloring-proper-audit/{\"colors\":[12,13,12,13,12,13],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"k\":2,\"n\":6}"},"fields":{"title":"Proper k-Coloring Means Edge Properness and At Most k Used Labels","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], [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. Total vertex colors=[12, 13, 12, 13, 12, 13] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. k=2. Give (increasing_used_colors,normalized_conflicting_edges,edge_proper?,at_most_k?,proper_k_coloring?,proper_k_coloring_uses_at_most_k_colors).","answer":{"kind":"expression","value":"([12, 13], [], Yes, Yes, Yes, proper_k_coloring_uses_at_most_k_colors)","explanation":"Audit every normalized edge and count distinct used labels separately. Both conditions are required; one passing coloring is not a proof of minimum chromatic number. The exact result, retaining the requested order, is ([12, 13], [], Yes, Yes, Yes, proper_k_coloring_uses_at_most_k_colors)."},"skills":["Proper k-Coloring Means Edge Properness and At Most k Used Labels"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r05-graph-coloring-0014","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"10f798b3c6c1f17580f7fa7109bd46c90202baaaa7ef291c380be280b5635470","family_id":"qf-mod31-coloring-proper-audit","seed":202610315014,"semantic_template":"qf-mod31-coloring-proper-audit/{\"colors\":[13,14,13,14,13,14],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"k\":3,\"n\":6}"},"fields":{"title":"Proper k-Coloring Means Edge Properness and At Most k Used Labels","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], [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. Total vertex colors=[13, 14, 13, 14, 13, 14] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. k=3. Give (increasing_used_colors,normalized_conflicting_edges,edge_proper?,at_most_k?,proper_k_coloring?,proper_k_coloring_uses_at_most_k_colors).","answer":{"kind":"expression","value":"([13, 14], [], Yes, Yes, Yes, proper_k_coloring_uses_at_most_k_colors)","explanation":"Audit every normalized edge and count distinct used labels separately. Both conditions are required; one passing coloring is not a proof of minimum chromatic number. The exact result, retaining the requested order, is ([13, 14], [], Yes, Yes, Yes, proper_k_coloring_uses_at_most_k_colors)."},"skills":["Proper k-Coloring Means Edge Properness and At Most k Used Labels"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r05-graph-coloring-0015","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"afd726c6b80c3b5294ca270f9eb4fe79b8d55ed0ff9b44c7209cfd2280c86b6e","family_id":"qf-mod31-coloring-proper-audit","seed":202610315015,"semantic_template":"qf-mod31-coloring-proper-audit/{\"colors\":[14,15,14,15,14,15],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"k\":4,\"n\":6}"},"fields":{"title":"Proper k-Coloring Means Edge Properness and At Most k Used Labels","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], [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. Total vertex colors=[14, 15, 14, 15, 14, 15] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. k=4. Give (increasing_used_colors,normalized_conflicting_edges,edge_proper?,at_most_k?,proper_k_coloring?,proper_k_coloring_uses_at_most_k_colors).","answer":{"kind":"expression","value":"([14, 15], [], Yes, Yes, Yes, proper_k_coloring_uses_at_most_k_colors)","explanation":"Audit every normalized edge and count distinct used labels separately. Both conditions are required; one passing coloring is not a proof of minimum chromatic number. The exact result, retaining the requested order, is ([14, 15], [], Yes, Yes, Yes, proper_k_coloring_uses_at_most_k_colors)."},"skills":["Proper k-Coloring Means Edge Properness and At Most k Used Labels"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r05-graph-coloring-0016","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"430089f90e73123844fa053c026b6fd238fdb5bd77cf52a50f1cc01fb13908b0","family_id":"qf-mod31-coloring-proper-audit","seed":202610315016,"semantic_template":"qf-mod31-coloring-proper-audit/{\"colors\":[15,16,15,16,15,16],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"k\":2,\"n\":6}"},"fields":{"title":"Proper k-Coloring Means Edge Properness and At Most k Used Labels","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], [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. Total vertex colors=[15, 16, 15, 16, 15, 16] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. k=2. Give (increasing_used_colors,normalized_conflicting_edges,edge_proper?,at_most_k?,proper_k_coloring?,proper_k_coloring_uses_at_most_k_colors).","answer":{"kind":"expression","value":"([15, 16], [], Yes, Yes, Yes, proper_k_coloring_uses_at_most_k_colors)","explanation":"Audit every normalized edge and count distinct used labels separately. Both conditions are required; one passing coloring is not a proof of minimum chromatic number. The exact result, retaining the requested order, is ([15, 16], [], Yes, Yes, Yes, proper_k_coloring_uses_at_most_k_colors)."},"skills":["Proper k-Coloring Means Edge Properness and At Most k Used Labels"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r05-graph-coloring-0017","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7a732a1967ce356ec7002ca09f1c41613f6cc8ba56073f7666c9f0f2946bfe4c","family_id":"qf-mod31-coloring-proper-audit","seed":202610315017,"semantic_template":"qf-mod31-coloring-proper-audit/{\"colors\":[16,17,16,17,16,17],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"k\":3,\"n\":6}"},"fields":{"title":"Proper k-Coloring Means Edge Properness and At Most k Used Labels","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], [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. Total vertex colors=[16, 17, 16, 17, 16, 17] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. k=3. Give (increasing_used_colors,normalized_conflicting_edges,edge_proper?,at_most_k?,proper_k_coloring?,proper_k_coloring_uses_at_most_k_colors).","answer":{"kind":"expression","value":"([16, 17], [], Yes, Yes, Yes, proper_k_coloring_uses_at_most_k_colors)","explanation":"Audit every normalized edge and count distinct used labels separately. Both conditions are required; one passing coloring is not a proof of minimum chromatic number. The exact result, retaining the requested order, is ([16, 17], [], Yes, Yes, Yes, proper_k_coloring_uses_at_most_k_colors)."},"skills":["Proper k-Coloring Means Edge Properness and At Most k Used Labels"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r05-graph-coloring-0018","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4dffad9ba2454043b3d77417478ecd355aa8d8a2e5cdd2ee0aad5b13be24fee6","family_id":"qf-mod31-coloring-proper-audit","seed":202610315018,"semantic_template":"qf-mod31-coloring-proper-audit/{\"colors\":[17,18,17,18,17,18],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"k\":4,\"n\":6}"},"fields":{"title":"Proper k-Coloring Means Edge Properness and At Most k Used Labels","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], [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. Total vertex colors=[17, 18, 17, 18, 17, 18] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. k=4. Give (increasing_used_colors,normalized_conflicting_edges,edge_proper?,at_most_k?,proper_k_coloring?,proper_k_coloring_uses_at_most_k_colors).","answer":{"kind":"expression","value":"([17, 18], [], Yes, Yes, Yes, proper_k_coloring_uses_at_most_k_colors)","explanation":"Audit every normalized edge and count distinct used labels separately. Both conditions are required; one passing coloring is not a proof of minimum chromatic number. The exact result, retaining the requested order, is ([17, 18], [], Yes, Yes, Yes, proper_k_coloring_uses_at_most_k_colors)."},"skills":["Proper k-Coloring Means Edge Properness and At Most k Used Labels"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r05-graph-coloring-0019","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d5585d0711752e65c77b9e8452382b8d0a602aa250a9f816d0ef9f2b83f71a13","family_id":"qf-mod31-coloring-proper-audit","seed":202610315019,"semantic_template":"qf-mod31-coloring-proper-audit/{\"colors\":[18,19,18,19,18,19],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"k\":2,\"n\":6}"},"fields":{"title":"Proper k-Coloring Means Edge Properness and At Most k Used Labels","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], [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. Total vertex colors=[18, 19, 18, 19, 18, 19] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. k=2. Give (increasing_used_colors,normalized_conflicting_edges,edge_proper?,at_most_k?,proper_k_coloring?,proper_k_coloring_uses_at_most_k_colors).","answer":{"kind":"expression","value":"([18, 19], [], Yes, Yes, Yes, proper_k_coloring_uses_at_most_k_colors)","explanation":"Audit every normalized edge and count distinct used labels separately. Both conditions are required; one passing coloring is not a proof of minimum chromatic number. The exact result, retaining the requested order, is ([18, 19], [], Yes, Yes, Yes, proper_k_coloring_uses_at_most_k_colors)."},"skills":["Proper k-Coloring Means Edge Properness and At Most k Used Labels"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r05-graph-coloring-0020","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e1aaf20b1882baf5b768723a758961e781ce56feceae8eb137d94ec5a012a69c","family_id":"qf-mod31-coloring-proper-audit","seed":202610315020,"semantic_template":"qf-mod31-coloring-proper-audit/{\"colors\":[19,20,19,20,19,20],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"k\":3,\"n\":6}"},"fields":{"title":"Proper k-Coloring Means Edge Properness and At Most k Used Labels","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], [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. Total vertex colors=[19, 20, 19, 20, 19, 20] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. k=3. Give (increasing_used_colors,normalized_conflicting_edges,edge_proper?,at_most_k?,proper_k_coloring?,proper_k_coloring_uses_at_most_k_colors).","answer":{"kind":"expression","value":"([19, 20], [], Yes, Yes, Yes, proper_k_coloring_uses_at_most_k_colors)","explanation":"Audit every normalized edge and count distinct used labels separately. Both conditions are required; one passing coloring is not a proof of minimum chromatic number. The exact result, retaining the requested order, is ([19, 20], [], Yes, Yes, Yes, proper_k_coloring_uses_at_most_k_colors)."},"skills":["Proper k-Coloring Means Edge Properness and At Most k Used Labels"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r05-graph-coloring-0021","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ca1d7ca9bb49b5c3b7592d20b1dbd189a7403c75d0e0bbddc3dd6b447c1ddf99","family_id":"qf-mod31-coloring-proper-audit","seed":202610315021,"semantic_template":"qf-mod31-coloring-proper-audit/{\"colors\":[20,21,20,21,20,21],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"k\":4,\"n\":6}"},"fields":{"title":"Proper k-Coloring Means Edge Properness and At Most k Used Labels","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], [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. Total vertex colors=[20, 21, 20, 21, 20, 21] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. k=4. Give (increasing_used_colors,normalized_conflicting_edges,edge_proper?,at_most_k?,proper_k_coloring?,proper_k_coloring_uses_at_most_k_colors).","answer":{"kind":"expression","value":"([20, 21], [], Yes, Yes, Yes, proper_k_coloring_uses_at_most_k_colors)","explanation":"Audit every normalized edge and count distinct used labels separately. Both conditions are required; one passing coloring is not a proof of minimum chromatic number. The exact result, retaining the requested order, is ([20, 21], [], Yes, Yes, Yes, proper_k_coloring_uses_at_most_k_colors)."},"skills":["Proper k-Coloring Means Edge Properness and At Most k Used Labels"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r05-graph-coloring-0022","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1ca8f6e13c28dcaf7592f475d59dfc7a857f99d62fa097e0316ced3bbed35fa5","family_id":"qf-mod31-coloring-proper-audit","seed":202610315022,"semantic_template":"qf-mod31-coloring-proper-audit/{\"colors\":[21,22,21,22,21,22],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"k\":2,\"n\":6}"},"fields":{"title":"Proper k-Coloring Means Edge Properness and At Most k Used Labels","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], [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. Total vertex colors=[21, 22, 21, 22, 21, 22] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. k=2. Give (increasing_used_colors,normalized_conflicting_edges,edge_proper?,at_most_k?,proper_k_coloring?,proper_k_coloring_uses_at_most_k_colors).","answer":{"kind":"expression","value":"([21, 22], [], Yes, Yes, Yes, proper_k_coloring_uses_at_most_k_colors)","explanation":"Audit every normalized edge and count distinct used labels separately. Both conditions are required; one passing coloring is not a proof of minimum chromatic number. The exact result, retaining the requested order, is ([21, 22], [], Yes, Yes, Yes, proper_k_coloring_uses_at_most_k_colors)."},"skills":["Proper k-Coloring Means Edge Properness and At Most k Used Labels"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r05-graph-coloring-0023","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"70218456f63af01e957a1996e6b7d25d60edb822775b70f7077e56522bf87128","family_id":"qf-mod31-coloring-proper-audit","seed":202610315023,"semantic_template":"qf-mod31-coloring-proper-audit/{\"colors\":[22,23,22,23,22,23],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"k\":3,\"n\":6}"},"fields":{"title":"Proper k-Coloring Means Edge Properness and At Most k Used Labels","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], [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. Total vertex colors=[22, 23, 22, 23, 22, 23] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. k=3. Give (increasing_used_colors,normalized_conflicting_edges,edge_proper?,at_most_k?,proper_k_coloring?,proper_k_coloring_uses_at_most_k_colors).","answer":{"kind":"expression","value":"([22, 23], [], Yes, Yes, Yes, proper_k_coloring_uses_at_most_k_colors)","explanation":"Audit every normalized edge and count distinct used labels separately. Both conditions are required; one passing coloring is not a proof of minimum chromatic number. The exact result, retaining the requested order, is ([22, 23], [], Yes, Yes, Yes, proper_k_coloring_uses_at_most_k_colors)."},"skills":["Proper k-Coloring Means Edge Properness and At Most k Used Labels"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r05-graph-coloring-0024","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f8ee6d20a68127ea39b3ef3b8caab0dea708821b80dd318932dd5abc06b43119","family_id":"qf-mod31-coloring-proper-audit","seed":202610315024,"semantic_template":"qf-mod31-coloring-proper-audit/{\"colors\":[23,24,23,24,23,24],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"k\":4,\"n\":6}"},"fields":{"title":"Proper k-Coloring Means Edge Properness and At Most k Used Labels","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], [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. Total vertex colors=[23, 24, 23, 24, 23, 24] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. k=4. Give (increasing_used_colors,normalized_conflicting_edges,edge_proper?,at_most_k?,proper_k_coloring?,proper_k_coloring_uses_at_most_k_colors).","answer":{"kind":"expression","value":"([23, 24], [], Yes, Yes, Yes, proper_k_coloring_uses_at_most_k_colors)","explanation":"Audit every normalized edge and count distinct used labels separately. Both conditions are required; one passing coloring is not a proof of minimum chromatic number. The exact result, retaining the requested order, is ([23, 24], [], Yes, Yes, Yes, proper_k_coloring_uses_at_most_k_colors)."},"skills":["Proper k-Coloring Means Edge Properness and At Most k Used Labels"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r05-graph-coloring-0025","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"39dff0199ebf89ba4a9ce5aa0f3fd6a5ff6993fad462042ec19d3f9d2de08026","family_id":"qf-mod31-coloring-proper-audit","seed":202610315025,"semantic_template":"qf-mod31-coloring-proper-audit/{\"colors\":[24,25,24,25,24,25],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"k\":2,\"n\":6}"},"fields":{"title":"Proper k-Coloring Means Edge Properness and At Most k Used Labels","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], [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. Total vertex colors=[24, 25, 24, 25, 24, 25] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. k=2. Give (increasing_used_colors,normalized_conflicting_edges,edge_proper?,at_most_k?,proper_k_coloring?,proper_k_coloring_uses_at_most_k_colors).","answer":{"kind":"expression","value":"([24, 25], [], Yes, Yes, Yes, proper_k_coloring_uses_at_most_k_colors)","explanation":"Audit every normalized edge and count distinct used labels separately. Both conditions are required; one passing coloring is not a proof of minimum chromatic number. The exact result, retaining the requested order, is ([24, 25], [], Yes, Yes, Yes, proper_k_coloring_uses_at_most_k_colors)."},"skills":["Proper k-Coloring Means Edge Properness and At Most k Used Labels"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r05-graph-coloring-0026","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b0e94f1037acacf81bdad0ea1dd271b00c537cd0a810cd21c1115542bcc1d1ff","family_id":"qf-mod31-coloring-conflict","seed":202610315026,"semantic_template":"qf-mod31-coloring-conflict/{\"colors\":[0,0,1,0,1,0],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"n\":6}"},"fields":{"title":"List Every Monochromatic Edge, Not Every Repeated Color","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], [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. Total vertex colors=[0, 0, 1, 0, 1, 0] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. Give (conflicting_edges,count,proper?), with edges in normalized sorted order.","answer":{"kind":"expression","value":"([[0, 1]], 1, No)","explanation":"Only same-colored endpoints joined by an edge are conflicts. Reusing a color on nonadjacent vertices is allowed. The exact result, retaining the requested order, is ([[0, 1]], 1, No)."},"skills":["List Every Monochromatic Edge, Not Every Repeated Color"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions 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-r05-graph-coloring-0027","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c031a5143c6ed2f6e2327f5454d66978e5e0521534a83e23fc4d173220a4a4e2","family_id":"qf-mod31-coloring-conflict","seed":202610315027,"semantic_template":"qf-mod31-coloring-conflict/{\"colors\":[1,1,2,1,2,1],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"n\":6}"},"fields":{"title":"List Every Monochromatic Edge, Not Every Repeated Color","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], [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. Total vertex colors=[1, 1, 2, 1, 2, 1] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. Give (conflicting_edges,count,proper?), with edges in normalized sorted order.","answer":{"kind":"expression","value":"([[0, 1]], 1, No)","explanation":"Only same-colored endpoints joined by an edge are conflicts. Reusing a color on nonadjacent vertices is allowed. The exact result, retaining the requested order, is ([[0, 1]], 1, No)."},"skills":["List Every Monochromatic Edge, Not Every Repeated Color"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions 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-r05-graph-coloring-0028","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e4d43f761ad3f650bd360014cdfbaa39c2438584d3a31c578d97f9053f8733a6","family_id":"qf-mod31-coloring-conflict","seed":202610315028,"semantic_template":"qf-mod31-coloring-conflict/{\"colors\":[2,2,3,2,3,2],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"n\":6}"},"fields":{"title":"List Every Monochromatic Edge, Not Every Repeated Color","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], [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. Total vertex colors=[2, 2, 3, 2, 3, 2] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. Give (conflicting_edges,count,proper?), with edges in normalized sorted order.","answer":{"kind":"expression","value":"([[0, 1]], 1, No)","explanation":"Only same-colored endpoints joined by an edge are conflicts. Reusing a color on nonadjacent vertices is allowed. The exact result, retaining the requested order, is ([[0, 1]], 1, No)."},"skills":["List Every Monochromatic Edge, Not Every Repeated Color"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions 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-r05-graph-coloring-0029","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fcf75327abd0660a3e37e0522a681a330c11f8f278d4989b30fccd321630d4d9","family_id":"qf-mod31-coloring-conflict","seed":202610315029,"semantic_template":"qf-mod31-coloring-conflict/{\"colors\":[3,3,4,3,4,3],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"n\":6}"},"fields":{"title":"List Every Monochromatic Edge, Not Every Repeated Color","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], [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. Total vertex colors=[3, 3, 4, 3, 4, 3] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. Give (conflicting_edges,count,proper?), with edges in normalized sorted order.","answer":{"kind":"expression","value":"([[0, 1]], 1, No)","explanation":"Only same-colored endpoints joined by an edge are conflicts. Reusing a color on nonadjacent vertices is allowed. The exact result, retaining the requested order, is ([[0, 1]], 1, No)."},"skills":["List Every Monochromatic Edge, Not Every Repeated Color"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions 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-r05-graph-coloring-0030","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8d0b2caab4d2ce3b5b3450a848ed2aa642de8ae1051800ae14df0b66ed6a762e","family_id":"qf-mod31-coloring-conflict","seed":202610315030,"semantic_template":"qf-mod31-coloring-conflict/{\"colors\":[4,4,5,4,5,4],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"n\":6}"},"fields":{"title":"List Every Monochromatic Edge, Not Every Repeated Color","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], [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. Total vertex colors=[4, 4, 5, 4, 5, 4] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. Give (conflicting_edges,count,proper?), with edges in normalized sorted order.","answer":{"kind":"expression","value":"([[0, 1]], 1, No)","explanation":"Only same-colored endpoints joined by an edge are conflicts. Reusing a color on nonadjacent vertices is allowed. The exact result, retaining the requested order, is ([[0, 1]], 1, No)."},"skills":["List Every Monochromatic Edge, Not Every Repeated Color"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions 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-r05-graph-coloring-0031","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"59f2f30c978c84e37feeea8d56d27ef23242df7a0ad1fb3a17a61c7a4f31fe26","family_id":"qf-mod31-coloring-conflict","seed":202610315031,"semantic_template":"qf-mod31-coloring-conflict/{\"colors\":[5,5,6,5,6,5],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"n\":6}"},"fields":{"title":"List Every Monochromatic Edge, Not Every Repeated Color","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], [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. Total vertex colors=[5, 5, 6, 5, 6, 5] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. Give (conflicting_edges,count,proper?), with edges in normalized sorted order.","answer":{"kind":"expression","value":"([[0, 1]], 1, No)","explanation":"Only same-colored endpoints joined by an edge are conflicts. Reusing a color on nonadjacent vertices is allowed. The exact result, retaining the requested order, is ([[0, 1]], 1, No)."},"skills":["List Every Monochromatic Edge, Not Every Repeated Color"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions 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-r05-graph-coloring-0032","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"095f7319a36aa673dbcc81c6c9307ec33277e9929b83c62ad69c82e5396f8990","family_id":"qf-mod31-coloring-conflict","seed":202610315032,"semantic_template":"qf-mod31-coloring-conflict/{\"colors\":[6,6,7,6,7,6],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"n\":6}"},"fields":{"title":"List Every Monochromatic Edge, Not Every Repeated Color","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], [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. Total vertex colors=[6, 6, 7, 6, 7, 6] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. Give (conflicting_edges,count,proper?), with edges in normalized sorted order.","answer":{"kind":"expression","value":"([[0, 1]], 1, No)","explanation":"Only same-colored endpoints joined by an edge are conflicts. Reusing a color on nonadjacent vertices is allowed. The exact result, retaining the requested order, is ([[0, 1]], 1, No)."},"skills":["List Every Monochromatic Edge, Not Every Repeated Color"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions 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-r05-graph-coloring-0033","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2385b29944f953e8cd5ade873935745365a5d6dc2c4eee4c4463ae361fad2287","family_id":"qf-mod31-coloring-conflict","seed":202610315033,"semantic_template":"qf-mod31-coloring-conflict/{\"colors\":[7,7,8,7,8,7],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"n\":6}"},"fields":{"title":"List Every Monochromatic Edge, Not Every Repeated Color","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], [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. Total vertex colors=[7, 7, 8, 7, 8, 7] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. Give (conflicting_edges,count,proper?), with edges in normalized sorted order.","answer":{"kind":"expression","value":"([[0, 1]], 1, No)","explanation":"Only same-colored endpoints joined by an edge are conflicts. Reusing a color on nonadjacent vertices is allowed. The exact result, retaining the requested order, is ([[0, 1]], 1, No)."},"skills":["List Every Monochromatic Edge, Not Every Repeated Color"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions 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-r05-graph-coloring-0034","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"402309d32bc625364b8ac9b784716bd7f9aacffc48ecf1063a61ac654add95e0","family_id":"qf-mod31-coloring-conflict","seed":202610315034,"semantic_template":"qf-mod31-coloring-conflict/{\"colors\":[8,8,9,8,9,8],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"n\":6}"},"fields":{"title":"List Every Monochromatic Edge, Not Every Repeated Color","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], [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. Total vertex colors=[8, 8, 9, 8, 9, 8] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. Give (conflicting_edges,count,proper?), with edges in normalized sorted order.","answer":{"kind":"expression","value":"([[0, 1]], 1, No)","explanation":"Only same-colored endpoints joined by an edge are conflicts. Reusing a color on nonadjacent vertices is allowed. The exact result, retaining the requested order, is ([[0, 1]], 1, No)."},"skills":["List Every Monochromatic Edge, Not Every Repeated Color"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions 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-r05-graph-coloring-0035","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1366d1738cae4d8e2d0bce34f4126ff3164b8789f50f853f32c615c0eb16d020","family_id":"qf-mod31-coloring-conflict","seed":202610315035,"semantic_template":"qf-mod31-coloring-conflict/{\"colors\":[9,9,10,9,10,9],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"n\":6}"},"fields":{"title":"List Every Monochromatic Edge, Not Every Repeated Color","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], [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. Total vertex colors=[9, 9, 10, 9, 10, 9] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. Give (conflicting_edges,count,proper?), with edges in normalized sorted order.","answer":{"kind":"expression","value":"([[0, 1]], 1, No)","explanation":"Only same-colored endpoints joined by an edge are conflicts. Reusing a color on nonadjacent vertices is allowed. The exact result, retaining the requested order, is ([[0, 1]], 1, No)."},"skills":["List Every Monochromatic Edge, Not Every Repeated Color"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions 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-r05-graph-coloring-0036","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f3b8a983c7d6c8b2a12f73b90ca08790b485449fce077ba0a52bd275ab042fef","family_id":"qf-mod31-coloring-conflict","seed":202610315036,"semantic_template":"qf-mod31-coloring-conflict/{\"colors\":[10,10,11,10,11,10],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"n\":6}"},"fields":{"title":"List Every Monochromatic Edge, Not Every Repeated Color","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], [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. Total vertex colors=[10, 10, 11, 10, 11, 10] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. Give (conflicting_edges,count,proper?), with edges in normalized sorted order.","answer":{"kind":"expression","value":"([[0, 1]], 1, No)","explanation":"Only same-colored endpoints joined by an edge are conflicts. Reusing a color on nonadjacent vertices is allowed. The exact result, retaining the requested order, is ([[0, 1]], 1, No)."},"skills":["List Every Monochromatic Edge, Not Every Repeated Color"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions 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-r05-graph-coloring-0037","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a325324ceb7a263fea821e3bf6556a3845e4333f65d27f01dbc07b92c8486890","family_id":"qf-mod31-coloring-conflict","seed":202610315037,"semantic_template":"qf-mod31-coloring-conflict/{\"colors\":[11,11,12,11,12,11],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"n\":6}"},"fields":{"title":"List Every Monochromatic Edge, Not Every Repeated Color","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], [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. Total vertex colors=[11, 11, 12, 11, 12, 11] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. Give (conflicting_edges,count,proper?), with edges in normalized sorted order.","answer":{"kind":"expression","value":"([[0, 1]], 1, No)","explanation":"Only same-colored endpoints joined by an edge are conflicts. Reusing a color on nonadjacent vertices is allowed. The exact result, retaining the requested order, is ([[0, 1]], 1, No)."},"skills":["List Every Monochromatic Edge, Not Every Repeated Color"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions 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-r05-graph-coloring-0038","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dc9864fed2cabd439338bf3644abcf5ac73a7872a2b677c6a854ce67783223a1","family_id":"qf-mod31-coloring-conflict","seed":202610315038,"semantic_template":"qf-mod31-coloring-conflict/{\"colors\":[12,12,13,12,13,12],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"n\":6}"},"fields":{"title":"List Every Monochromatic Edge, Not Every Repeated Color","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], [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. Total vertex colors=[12, 12, 13, 12, 13, 12] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. Give (conflicting_edges,count,proper?), with edges in normalized sorted order.","answer":{"kind":"expression","value":"([[0, 1]], 1, No)","explanation":"Only same-colored endpoints joined by an edge are conflicts. Reusing a color on nonadjacent vertices is allowed. The exact result, retaining the requested order, is ([[0, 1]], 1, No)."},"skills":["List Every Monochromatic Edge, Not Every Repeated Color"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions 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-r05-graph-coloring-0039","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bef6ec0eeec8bcd7bdfd69f5bb7dbb900ca8e2fc327d89b5c98d501cce587fc5","family_id":"qf-mod31-coloring-conflict","seed":202610315039,"semantic_template":"qf-mod31-coloring-conflict/{\"colors\":[13,13,14,13,14,13],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"n\":6}"},"fields":{"title":"List Every Monochromatic Edge, Not Every Repeated Color","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], [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. Total vertex colors=[13, 13, 14, 13, 14, 13] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. Give (conflicting_edges,count,proper?), with edges in normalized sorted order.","answer":{"kind":"expression","value":"([[0, 1]], 1, No)","explanation":"Only same-colored endpoints joined by an edge are conflicts. Reusing a color on nonadjacent vertices is allowed. The exact result, retaining the requested order, is ([[0, 1]], 1, No)."},"skills":["List Every Monochromatic Edge, Not Every Repeated Color"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions 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-r05-graph-coloring-0040","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ddbd114571029c4e4cdf428945cc43774a87fdeb5436aac1c95981a2a8ff6151","family_id":"qf-mod31-coloring-conflict","seed":202610315040,"semantic_template":"qf-mod31-coloring-conflict/{\"colors\":[14,14,15,14,15,14],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"n\":6}"},"fields":{"title":"List Every Monochromatic Edge, Not Every Repeated Color","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], [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. Total vertex colors=[14, 14, 15, 14, 15, 14] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. Give (conflicting_edges,count,proper?), with edges in normalized sorted order.","answer":{"kind":"expression","value":"([[0, 1]], 1, No)","explanation":"Only same-colored endpoints joined by an edge are conflicts. Reusing a color on nonadjacent vertices is allowed. The exact result, retaining the requested order, is ([[0, 1]], 1, No)."},"skills":["List Every Monochromatic Edge, Not Every Repeated Color"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions 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-r05-graph-coloring-0041","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9a4b2638635a4b14feb817fb00f10ab6dbc384f545d5644bd9250e270f57ca61","family_id":"qf-mod31-coloring-conflict","seed":202610315041,"semantic_template":"qf-mod31-coloring-conflict/{\"colors\":[15,15,16,15,16,15],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"n\":6}"},"fields":{"title":"List Every Monochromatic Edge, Not Every Repeated Color","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], [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. Total vertex colors=[15, 15, 16, 15, 16, 15] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. Give (conflicting_edges,count,proper?), with edges in normalized sorted order.","answer":{"kind":"expression","value":"([[0, 1]], 1, No)","explanation":"Only same-colored endpoints joined by an edge are conflicts. Reusing a color on nonadjacent vertices is allowed. The exact result, retaining the requested order, is ([[0, 1]], 1, No)."},"skills":["List Every Monochromatic Edge, Not Every Repeated Color"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions 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-r05-graph-coloring-0042","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1ae3baa7be3dae3bf467619a489935e44c05942902038556e8764ea9cab8f1a3","family_id":"qf-mod31-coloring-conflict","seed":202610315042,"semantic_template":"qf-mod31-coloring-conflict/{\"colors\":[16,16,17,16,17,16],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"n\":6}"},"fields":{"title":"List Every Monochromatic Edge, Not Every Repeated Color","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], [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. Total vertex colors=[16, 16, 17, 16, 17, 16] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. Give (conflicting_edges,count,proper?), with edges in normalized sorted order.","answer":{"kind":"expression","value":"([[0, 1]], 1, No)","explanation":"Only same-colored endpoints joined by an edge are conflicts. Reusing a color on nonadjacent vertices is allowed. The exact result, retaining the requested order, is ([[0, 1]], 1, No)."},"skills":["List Every Monochromatic Edge, Not Every Repeated Color"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions 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-r05-graph-coloring-0043","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b8022faa1b7cf733e84dff8643a0514a853b9743d2ea92d6e4b0eaf2ce232195","family_id":"qf-mod31-coloring-conflict","seed":202610315043,"semantic_template":"qf-mod31-coloring-conflict/{\"colors\":[17,17,18,17,18,17],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"n\":6}"},"fields":{"title":"List Every Monochromatic Edge, Not Every Repeated Color","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], [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. Total vertex colors=[17, 17, 18, 17, 18, 17] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. Give (conflicting_edges,count,proper?), with edges in normalized sorted order.","answer":{"kind":"expression","value":"([[0, 1]], 1, No)","explanation":"Only same-colored endpoints joined by an edge are conflicts. Reusing a color on nonadjacent vertices is allowed. The exact result, retaining the requested order, is ([[0, 1]], 1, No)."},"skills":["List Every Monochromatic Edge, Not Every Repeated Color"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions 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-r05-graph-coloring-0044","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"58d1f15b8f07ad24b87e796668b177fd618c8720f60a611c4027a30713fdc0d7","family_id":"qf-mod31-coloring-conflict","seed":202610315044,"semantic_template":"qf-mod31-coloring-conflict/{\"colors\":[18,18,19,18,19,18],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"n\":6}"},"fields":{"title":"List Every Monochromatic Edge, Not Every Repeated Color","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], [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. Total vertex colors=[18, 18, 19, 18, 19, 18] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. Give (conflicting_edges,count,proper?), with edges in normalized sorted order.","answer":{"kind":"expression","value":"([[0, 1]], 1, No)","explanation":"Only same-colored endpoints joined by an edge are conflicts. Reusing a color on nonadjacent vertices is allowed. The exact result, retaining the requested order, is ([[0, 1]], 1, No)."},"skills":["List Every Monochromatic Edge, Not Every Repeated Color"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions 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-r05-graph-coloring-0045","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d3c04f17d18063795c20ea41485e4f6c32ae1dd44b462f4c2f57da185cd098bc","family_id":"qf-mod31-coloring-conflict","seed":202610315045,"semantic_template":"qf-mod31-coloring-conflict/{\"colors\":[19,19,20,19,20,19],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"n\":6}"},"fields":{"title":"List Every Monochromatic Edge, Not Every Repeated Color","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], [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. Total vertex colors=[19, 19, 20, 19, 20, 19] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. Give (conflicting_edges,count,proper?), with edges in normalized sorted order.","answer":{"kind":"expression","value":"([[0, 1]], 1, No)","explanation":"Only same-colored endpoints joined by an edge are conflicts. Reusing a color on nonadjacent vertices is allowed. The exact result, retaining the requested order, is ([[0, 1]], 1, No)."},"skills":["List Every Monochromatic Edge, Not Every Repeated Color"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions 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-r05-graph-coloring-0046","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2fc5a0f7de3f6f846e54f20f5314ef698c30e72e62b5c8c117ca26ffcdf81710","family_id":"qf-mod31-coloring-conflict","seed":202610315046,"semantic_template":"qf-mod31-coloring-conflict/{\"colors\":[20,20,21,20,21,20],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"n\":6}"},"fields":{"title":"List Every Monochromatic Edge, Not Every Repeated Color","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], [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. Total vertex colors=[20, 20, 21, 20, 21, 20] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. Give (conflicting_edges,count,proper?), with edges in normalized sorted order.","answer":{"kind":"expression","value":"([[0, 1]], 1, No)","explanation":"Only same-colored endpoints joined by an edge are conflicts. Reusing a color on nonadjacent vertices is allowed. The exact result, retaining the requested order, is ([[0, 1]], 1, No)."},"skills":["List Every Monochromatic Edge, Not Every Repeated Color"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions 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-r05-graph-coloring-0047","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"428ba9665a4a4a3d786d1e741f4814d67f115a30870685beb0815ecd9bd89db5","family_id":"qf-mod31-coloring-conflict","seed":202610315047,"semantic_template":"qf-mod31-coloring-conflict/{\"colors\":[21,21,22,21,22,21],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"n\":6}"},"fields":{"title":"List Every Monochromatic Edge, Not Every Repeated Color","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], [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. Total vertex colors=[21, 21, 22, 21, 22, 21] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. Give (conflicting_edges,count,proper?), with edges in normalized sorted order.","answer":{"kind":"expression","value":"([[0, 1]], 1, No)","explanation":"Only same-colored endpoints joined by an edge are conflicts. Reusing a color on nonadjacent vertices is allowed. The exact result, retaining the requested order, is ([[0, 1]], 1, No)."},"skills":["List Every Monochromatic Edge, Not Every Repeated Color"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions 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-r05-graph-coloring-0048","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b68828df11f068ac0fd5711ed722e36f5f76a8a206c8a155cf27f9ab6befbaed","family_id":"qf-mod31-coloring-conflict","seed":202610315048,"semantic_template":"qf-mod31-coloring-conflict/{\"colors\":[22,22,23,22,23,22],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"n\":6}"},"fields":{"title":"List Every Monochromatic Edge, Not Every Repeated Color","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], [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. Total vertex colors=[22, 22, 23, 22, 23, 22] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. Give (conflicting_edges,count,proper?), with edges in normalized sorted order.","answer":{"kind":"expression","value":"([[0, 1]], 1, No)","explanation":"Only same-colored endpoints joined by an edge are conflicts. Reusing a color on nonadjacent vertices is allowed. The exact result, retaining the requested order, is ([[0, 1]], 1, No)."},"skills":["List Every Monochromatic Edge, Not Every Repeated Color"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions 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-r05-graph-coloring-0049","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"70d40613f094725b2d77daba96262060f428602661f5b43db71046bd7629bed4","family_id":"qf-mod31-coloring-conflict","seed":202610315049,"semantic_template":"qf-mod31-coloring-conflict/{\"colors\":[23,23,24,23,24,23],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"n\":6}"},"fields":{"title":"List Every Monochromatic Edge, Not Every Repeated Color","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], [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. Total vertex colors=[23, 23, 24, 23, 24, 23] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. Give (conflicting_edges,count,proper?), with edges in normalized sorted order.","answer":{"kind":"expression","value":"([[0, 1]], 1, No)","explanation":"Only same-colored endpoints joined by an edge are conflicts. Reusing a color on nonadjacent vertices is allowed. The exact result, retaining the requested order, is ([[0, 1]], 1, No)."},"skills":["List Every Monochromatic Edge, Not Every Repeated Color"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions 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-r05-graph-coloring-0050","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8856f89d634f6e9589dd32f27e4f70b7c1c5de30e2107c6732f34ec33661c83e","family_id":"qf-mod31-coloring-conflict","seed":202610315050,"semantic_template":"qf-mod31-coloring-conflict/{\"colors\":[24,24,25,24,25,24],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"n\":6}"},"fields":{"title":"List Every Monochromatic Edge, Not Every Repeated Color","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], [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. Total vertex colors=[24, 24, 25, 24, 25, 24] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. Give (conflicting_edges,count,proper?), with edges in normalized sorted order.","answer":{"kind":"expression","value":"([[0, 1]], 1, No)","explanation":"Only same-colored endpoints joined by an edge are conflicts. Reusing a color on nonadjacent vertices is allowed. The exact result, retaining the requested order, is ([[0, 1]], 1, No)."},"skills":["List Every Monochromatic Edge, Not Every Repeated Color"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions 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-r05-graph-coloring-0051","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0cba700a2a19def007bb5bec01d996e2349b730abac7f29a54221687f5be5fcf","family_id":"qf-mod31-coloring-upper-bound","seed":202610315051,"semantic_template":"qf-mod31-coloring-upper-bound/{\"colors\":[0,1,0,1,0,1],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"k\":2,\"n\":6}"},"fields":{"title":"A Nonoptimal Valid Coloring Still Supplies an Upper Bound","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], [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. Total vertex colors=[0, 1, 0, 1, 0, 1] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. Proposed k=2. Give (used_count,k,proper?,certificate?,conclusion,upper_bound_not_exact_optimum). A valid certificate yields chromatic_number_at_most_k with the supplied numeric k; otherwise no_upper_bound_from_this_coloring.","answer":{"kind":"expression","value":"(2, 2, Yes, Yes, chromatic_number_at_most_2, upper_bound_not_exact_optimum)","explanation":"Conjoin all-edge properness with used_count≤k. A loose k may be sufficient without being optimal, and failure of this coloring gives no general lower-bound certificate. The exact result, retaining the requested order, is (2, 2, Yes, Yes, chromatic_number_at_most_2, upper_bound_not_exact_optimum)."},"skills":["A Nonoptimal Valid Coloring Still Supplies an Upper Bound"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions 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-r05-graph-coloring-0052","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d8e715e5bfdf5e6b2d2beff5c348a983c1b73d11e5c87a152b0469d6e69543bb","family_id":"qf-mod31-coloring-upper-bound","seed":202610315052,"semantic_template":"qf-mod31-coloring-upper-bound/{\"colors\":[1,2,1,2,1,2],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"k\":3,\"n\":6}"},"fields":{"title":"A Nonoptimal Valid Coloring Still Supplies an Upper Bound","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], [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. Total vertex colors=[1, 2, 1, 2, 1, 2] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. Proposed k=3. Give (used_count,k,proper?,certificate?,conclusion,upper_bound_not_exact_optimum). A valid certificate yields chromatic_number_at_most_k with the supplied numeric k; otherwise no_upper_bound_from_this_coloring.","answer":{"kind":"expression","value":"(2, 3, Yes, Yes, chromatic_number_at_most_3, upper_bound_not_exact_optimum)","explanation":"Conjoin all-edge properness with used_count≤k. A loose k may be sufficient without being optimal, and failure of this coloring gives no general lower-bound certificate. The exact result, retaining the requested order, is (2, 3, Yes, Yes, chromatic_number_at_most_3, upper_bound_not_exact_optimum)."},"skills":["A Nonoptimal Valid Coloring Still Supplies an Upper Bound"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions 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-r05-graph-coloring-0053","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2087d236aa77dce429386e51a8a653127607026c00481f693b59620a836db381","family_id":"qf-mod31-coloring-upper-bound","seed":202610315053,"semantic_template":"qf-mod31-coloring-upper-bound/{\"colors\":[2,3,2,3,2,3],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"k\":4,\"n\":6}"},"fields":{"title":"A Nonoptimal Valid Coloring Still Supplies an Upper Bound","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], [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. Total vertex colors=[2, 3, 2, 3, 2, 3] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. Proposed k=4. Give (used_count,k,proper?,certificate?,conclusion,upper_bound_not_exact_optimum). A valid certificate yields chromatic_number_at_most_k with the supplied numeric k; otherwise no_upper_bound_from_this_coloring.","answer":{"kind":"expression","value":"(2, 4, Yes, Yes, chromatic_number_at_most_4, upper_bound_not_exact_optimum)","explanation":"Conjoin all-edge properness with used_count≤k. A loose k may be sufficient without being optimal, and failure of this coloring gives no general lower-bound certificate. The exact result, retaining the requested order, is (2, 4, Yes, Yes, chromatic_number_at_most_4, upper_bound_not_exact_optimum)."},"skills":["A Nonoptimal Valid Coloring Still Supplies an Upper Bound"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions 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-r05-graph-coloring-0054","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5714a47c2d9a46c75806aee70ddf5e1975dc82d223886cc2ed03ff99cf2b42f1","family_id":"qf-mod31-coloring-upper-bound","seed":202610315054,"semantic_template":"qf-mod31-coloring-upper-bound/{\"colors\":[3,4,3,4,3,4],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"k\":2,\"n\":6}"},"fields":{"title":"A Nonoptimal Valid Coloring Still Supplies an Upper Bound","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], [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. Total vertex colors=[3, 4, 3, 4, 3, 4] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. Proposed k=2. Give (used_count,k,proper?,certificate?,conclusion,upper_bound_not_exact_optimum). A valid certificate yields chromatic_number_at_most_k with the supplied numeric k; otherwise no_upper_bound_from_this_coloring.","answer":{"kind":"expression","value":"(2, 2, Yes, Yes, chromatic_number_at_most_2, upper_bound_not_exact_optimum)","explanation":"Conjoin all-edge properness with used_count≤k. A loose k may be sufficient without being optimal, and failure of this coloring gives no general lower-bound certificate. The exact result, retaining the requested order, is (2, 2, Yes, Yes, chromatic_number_at_most_2, upper_bound_not_exact_optimum)."},"skills":["A Nonoptimal Valid Coloring Still Supplies an Upper Bound"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions 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-r05-graph-coloring-0055","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cd0c4b247c72b4aa006e2923d01aaa3576e00a0be478aa08a7a7213bb2309a97","family_id":"qf-mod31-coloring-upper-bound","seed":202610315055,"semantic_template":"qf-mod31-coloring-upper-bound/{\"colors\":[4,5,4,5,4,5],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"k\":3,\"n\":6}"},"fields":{"title":"A Nonoptimal Valid Coloring Still Supplies an Upper Bound","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], [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. Total vertex colors=[4, 5, 4, 5, 4, 5] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. Proposed k=3. Give (used_count,k,proper?,certificate?,conclusion,upper_bound_not_exact_optimum). A valid certificate yields chromatic_number_at_most_k with the supplied numeric k; otherwise no_upper_bound_from_this_coloring.","answer":{"kind":"expression","value":"(2, 3, Yes, Yes, chromatic_number_at_most_3, upper_bound_not_exact_optimum)","explanation":"Conjoin all-edge properness with used_count≤k. A loose k may be sufficient without being optimal, and failure of this coloring gives no general lower-bound certificate. The exact result, retaining the requested order, is (2, 3, Yes, Yes, chromatic_number_at_most_3, upper_bound_not_exact_optimum)."},"skills":["A Nonoptimal Valid Coloring Still Supplies an Upper Bound"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions 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-r05-graph-coloring-0056","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"24c51c427c9f93a0ba2fb54af3563b33c84338a2fba183abfb5d399aeafb3b11","family_id":"qf-mod31-coloring-upper-bound","seed":202610315056,"semantic_template":"qf-mod31-coloring-upper-bound/{\"colors\":[5,6,5,6,5,6],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"k\":4,\"n\":6}"},"fields":{"title":"A Nonoptimal Valid Coloring Still Supplies an Upper Bound","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], [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. Total vertex colors=[5, 6, 5, 6, 5, 6] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. Proposed k=4. Give (used_count,k,proper?,certificate?,conclusion,upper_bound_not_exact_optimum). A valid certificate yields chromatic_number_at_most_k with the supplied numeric k; otherwise no_upper_bound_from_this_coloring.","answer":{"kind":"expression","value":"(2, 4, Yes, Yes, chromatic_number_at_most_4, upper_bound_not_exact_optimum)","explanation":"Conjoin all-edge properness with used_count≤k. A loose k may be sufficient without being optimal, and failure of this coloring gives no general lower-bound certificate. The exact result, retaining the requested order, is (2, 4, Yes, Yes, chromatic_number_at_most_4, upper_bound_not_exact_optimum)."},"skills":["A Nonoptimal Valid Coloring Still Supplies an Upper Bound"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions 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-r05-graph-coloring-0057","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5b8cc03d9a4c0deba9353e21f5dba3f186f39a2daf11bb1a87c221cdd266a23a","family_id":"qf-mod31-coloring-upper-bound","seed":202610315057,"semantic_template":"qf-mod31-coloring-upper-bound/{\"colors\":[6,7,6,7,6,7],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"k\":2,\"n\":6}"},"fields":{"title":"A Nonoptimal Valid Coloring Still Supplies an Upper Bound","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], [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. Total vertex colors=[6, 7, 6, 7, 6, 7] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. Proposed k=2. Give (used_count,k,proper?,certificate?,conclusion,upper_bound_not_exact_optimum). A valid certificate yields chromatic_number_at_most_k with the supplied numeric k; otherwise no_upper_bound_from_this_coloring.","answer":{"kind":"expression","value":"(2, 2, Yes, Yes, chromatic_number_at_most_2, upper_bound_not_exact_optimum)","explanation":"Conjoin all-edge properness with used_count≤k. A loose k may be sufficient without being optimal, and failure of this coloring gives no general lower-bound certificate. The exact result, retaining the requested order, is (2, 2, Yes, Yes, chromatic_number_at_most_2, upper_bound_not_exact_optimum)."},"skills":["A Nonoptimal Valid Coloring Still Supplies an Upper Bound"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions 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-r05-graph-coloring-0058","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"99eb9ae7bfee986c567db6251659ec6c8a41ed2c157412920748457898843793","family_id":"qf-mod31-coloring-upper-bound","seed":202610315058,"semantic_template":"qf-mod31-coloring-upper-bound/{\"colors\":[7,8,7,8,7,8],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"k\":3,\"n\":6}"},"fields":{"title":"A Nonoptimal Valid Coloring Still Supplies an Upper Bound","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], [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. Total vertex colors=[7, 8, 7, 8, 7, 8] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. Proposed k=3. Give (used_count,k,proper?,certificate?,conclusion,upper_bound_not_exact_optimum). A valid certificate yields chromatic_number_at_most_k with the supplied numeric k; otherwise no_upper_bound_from_this_coloring.","answer":{"kind":"expression","value":"(2, 3, Yes, Yes, chromatic_number_at_most_3, upper_bound_not_exact_optimum)","explanation":"Conjoin all-edge properness with used_count≤k. A loose k may be sufficient without being optimal, and failure of this coloring gives no general lower-bound certificate. The exact result, retaining the requested order, is (2, 3, Yes, Yes, chromatic_number_at_most_3, upper_bound_not_exact_optimum)."},"skills":["A Nonoptimal Valid Coloring Still Supplies an Upper Bound"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions 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-r05-graph-coloring-0059","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d795d38a2a6e1dc7c15a8dfd114b27288c4da95e3a65237db6ec4ab0df366a5a","family_id":"qf-mod31-coloring-upper-bound","seed":202610315059,"semantic_template":"qf-mod31-coloring-upper-bound/{\"colors\":[8,9,8,9,8,9],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"k\":4,\"n\":6}"},"fields":{"title":"A Nonoptimal Valid Coloring Still Supplies an Upper Bound","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], [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. Total vertex colors=[8, 9, 8, 9, 8, 9] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. Proposed k=4. Give (used_count,k,proper?,certificate?,conclusion,upper_bound_not_exact_optimum). A valid certificate yields chromatic_number_at_most_k with the supplied numeric k; otherwise no_upper_bound_from_this_coloring.","answer":{"kind":"expression","value":"(2, 4, Yes, Yes, chromatic_number_at_most_4, upper_bound_not_exact_optimum)","explanation":"Conjoin all-edge properness with used_count≤k. A loose k may be sufficient without being optimal, and failure of this coloring gives no general lower-bound certificate. The exact result, retaining the requested order, is (2, 4, Yes, Yes, chromatic_number_at_most_4, upper_bound_not_exact_optimum)."},"skills":["A Nonoptimal Valid Coloring Still Supplies an Upper Bound"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions 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-r05-graph-coloring-0060","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a85bdb2f21c7868baee3a3b66833329248c0c0bcd65e05a693f25d1cbe79b412","family_id":"qf-mod31-coloring-upper-bound","seed":202610315060,"semantic_template":"qf-mod31-coloring-upper-bound/{\"colors\":[9,10,9,10,9,10],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"k\":2,\"n\":6}"},"fields":{"title":"A Nonoptimal Valid Coloring Still Supplies an Upper Bound","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], [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. Total vertex colors=[9, 10, 9, 10, 9, 10] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. Proposed k=2. Give (used_count,k,proper?,certificate?,conclusion,upper_bound_not_exact_optimum). A valid certificate yields chromatic_number_at_most_k with the supplied numeric k; otherwise no_upper_bound_from_this_coloring.","answer":{"kind":"expression","value":"(2, 2, Yes, Yes, chromatic_number_at_most_2, upper_bound_not_exact_optimum)","explanation":"Conjoin all-edge properness with used_count≤k. A loose k may be sufficient without being optimal, and failure of this coloring gives no general lower-bound certificate. The exact result, retaining the requested order, is (2, 2, Yes, Yes, chromatic_number_at_most_2, upper_bound_not_exact_optimum)."},"skills":["A Nonoptimal Valid Coloring Still Supplies an Upper Bound"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions 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-r05-graph-coloring-0061","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8ac5c56bc1d4be71508277aefe6692a48316f715e0601c456f7ee2cac7cfed3e","family_id":"qf-mod31-coloring-upper-bound","seed":202610315061,"semantic_template":"qf-mod31-coloring-upper-bound/{\"colors\":[10,11,10,11,10,11],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"k\":3,\"n\":6}"},"fields":{"title":"A Nonoptimal Valid Coloring Still Supplies an Upper Bound","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], [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. Total vertex colors=[10, 11, 10, 11, 10, 11] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. Proposed k=3. Give (used_count,k,proper?,certificate?,conclusion,upper_bound_not_exact_optimum). A valid certificate yields chromatic_number_at_most_k with the supplied numeric k; otherwise no_upper_bound_from_this_coloring.","answer":{"kind":"expression","value":"(2, 3, Yes, Yes, chromatic_number_at_most_3, upper_bound_not_exact_optimum)","explanation":"Conjoin all-edge properness with used_count≤k. A loose k may be sufficient without being optimal, and failure of this coloring gives no general lower-bound certificate. The exact result, retaining the requested order, is (2, 3, Yes, Yes, chromatic_number_at_most_3, upper_bound_not_exact_optimum)."},"skills":["A Nonoptimal Valid Coloring Still Supplies an Upper Bound"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions 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-r05-graph-coloring-0062","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"159c4dec178968f44867923a74da41aa5d15e999cc4c67c34fa7a2de814727e3","family_id":"qf-mod31-coloring-upper-bound","seed":202610315062,"semantic_template":"qf-mod31-coloring-upper-bound/{\"colors\":[11,12,11,12,11,12],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"k\":4,\"n\":6}"},"fields":{"title":"A Nonoptimal Valid Coloring Still Supplies an Upper Bound","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], [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. Total vertex colors=[11, 12, 11, 12, 11, 12] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. Proposed k=4. Give (used_count,k,proper?,certificate?,conclusion,upper_bound_not_exact_optimum). A valid certificate yields chromatic_number_at_most_k with the supplied numeric k; otherwise no_upper_bound_from_this_coloring.","answer":{"kind":"expression","value":"(2, 4, Yes, Yes, chromatic_number_at_most_4, upper_bound_not_exact_optimum)","explanation":"Conjoin all-edge properness with used_count≤k. A loose k may be sufficient without being optimal, and failure of this coloring gives no general lower-bound certificate. The exact result, retaining the requested order, is (2, 4, Yes, Yes, chromatic_number_at_most_4, upper_bound_not_exact_optimum)."},"skills":["A Nonoptimal Valid Coloring Still Supplies an Upper Bound"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions 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-r05-graph-coloring-0063","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3aa5ea43c58267fbac50627c9f8c3aa44cc7a38e8731132d74575bcf3dfb8629","family_id":"qf-mod31-coloring-upper-bound","seed":202610315063,"semantic_template":"qf-mod31-coloring-upper-bound/{\"colors\":[12,13,12,13,12,13],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"k\":2,\"n\":6}"},"fields":{"title":"A Nonoptimal Valid Coloring Still Supplies an Upper Bound","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], [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. Total vertex colors=[12, 13, 12, 13, 12, 13] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. Proposed k=2. Give (used_count,k,proper?,certificate?,conclusion,upper_bound_not_exact_optimum). A valid certificate yields chromatic_number_at_most_k with the supplied numeric k; otherwise no_upper_bound_from_this_coloring.","answer":{"kind":"expression","value":"(2, 2, Yes, Yes, chromatic_number_at_most_2, upper_bound_not_exact_optimum)","explanation":"Conjoin all-edge properness with used_count≤k. A loose k may be sufficient without being optimal, and failure of this coloring gives no general lower-bound certificate. The exact result, retaining the requested order, is (2, 2, Yes, Yes, chromatic_number_at_most_2, upper_bound_not_exact_optimum)."},"skills":["A Nonoptimal Valid Coloring Still Supplies an Upper Bound"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions 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-r05-graph-coloring-0064","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"45f4d961c4eec4db3f52fbc99529b83ed5ed3b0f5aa7168df9429e1136ff12d3","family_id":"qf-mod31-coloring-upper-bound","seed":202610315064,"semantic_template":"qf-mod31-coloring-upper-bound/{\"colors\":[13,14,13,14,13,14],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"k\":3,\"n\":6}"},"fields":{"title":"A Nonoptimal Valid Coloring Still Supplies an Upper Bound","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], [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. Total vertex colors=[13, 14, 13, 14, 13, 14] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. Proposed k=3. Give (used_count,k,proper?,certificate?,conclusion,upper_bound_not_exact_optimum). A valid certificate yields chromatic_number_at_most_k with the supplied numeric k; otherwise no_upper_bound_from_this_coloring.","answer":{"kind":"expression","value":"(2, 3, Yes, Yes, chromatic_number_at_most_3, upper_bound_not_exact_optimum)","explanation":"Conjoin all-edge properness with used_count≤k. A loose k may be sufficient without being optimal, and failure of this coloring gives no general lower-bound certificate. The exact result, retaining the requested order, is (2, 3, Yes, Yes, chromatic_number_at_most_3, upper_bound_not_exact_optimum)."},"skills":["A Nonoptimal Valid Coloring Still Supplies an Upper Bound"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions 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-r05-graph-coloring-0065","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b165cadb4152e296ec6e1608a6b4f9934fc8a57f6e24aed2eb6297469c698e02","family_id":"qf-mod31-coloring-upper-bound","seed":202610315065,"semantic_template":"qf-mod31-coloring-upper-bound/{\"colors\":[14,15,14,15,14,15],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"k\":4,\"n\":6}"},"fields":{"title":"A Nonoptimal Valid Coloring Still Supplies an Upper Bound","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], [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. Total vertex colors=[14, 15, 14, 15, 14, 15] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. Proposed k=4. Give (used_count,k,proper?,certificate?,conclusion,upper_bound_not_exact_optimum). A valid certificate yields chromatic_number_at_most_k with the supplied numeric k; otherwise no_upper_bound_from_this_coloring.","answer":{"kind":"expression","value":"(2, 4, Yes, Yes, chromatic_number_at_most_4, upper_bound_not_exact_optimum)","explanation":"Conjoin all-edge properness with used_count≤k. A loose k may be sufficient without being optimal, and failure of this coloring gives no general lower-bound certificate. The exact result, retaining the requested order, is (2, 4, Yes, Yes, chromatic_number_at_most_4, upper_bound_not_exact_optimum)."},"skills":["A Nonoptimal Valid Coloring Still Supplies an Upper Bound"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions 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-r05-graph-coloring-0066","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"32d6bd65c359a461c0c8180c7200e056227d00724a9ac0c12c77ee4a9e0a8e05","family_id":"qf-mod31-coloring-upper-bound","seed":202610315066,"semantic_template":"qf-mod31-coloring-upper-bound/{\"colors\":[15,16,15,16,15,16],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"k\":2,\"n\":6}"},"fields":{"title":"A Nonoptimal Valid Coloring Still Supplies an Upper Bound","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], [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. Total vertex colors=[15, 16, 15, 16, 15, 16] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. Proposed k=2. Give (used_count,k,proper?,certificate?,conclusion,upper_bound_not_exact_optimum). A valid certificate yields chromatic_number_at_most_k with the supplied numeric k; otherwise no_upper_bound_from_this_coloring.","answer":{"kind":"expression","value":"(2, 2, Yes, Yes, chromatic_number_at_most_2, upper_bound_not_exact_optimum)","explanation":"Conjoin all-edge properness with used_count≤k. A loose k may be sufficient without being optimal, and failure of this coloring gives no general lower-bound certificate. The exact result, retaining the requested order, is (2, 2, Yes, Yes, chromatic_number_at_most_2, upper_bound_not_exact_optimum)."},"skills":["A Nonoptimal Valid Coloring Still Supplies an Upper Bound"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions 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-r05-graph-coloring-0067","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8d52b2cb5fff81e2f561dcc1ba382c5c531648126ce4957e7611357ff3e63f77","family_id":"qf-mod31-coloring-upper-bound","seed":202610315067,"semantic_template":"qf-mod31-coloring-upper-bound/{\"colors\":[16,17,16,17,16,17],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"k\":3,\"n\":6}"},"fields":{"title":"A Nonoptimal Valid Coloring Still Supplies an Upper Bound","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], [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. Total vertex colors=[16, 17, 16, 17, 16, 17] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. Proposed k=3. Give (used_count,k,proper?,certificate?,conclusion,upper_bound_not_exact_optimum). A valid certificate yields chromatic_number_at_most_k with the supplied numeric k; otherwise no_upper_bound_from_this_coloring.","answer":{"kind":"expression","value":"(2, 3, Yes, Yes, chromatic_number_at_most_3, upper_bound_not_exact_optimum)","explanation":"Conjoin all-edge properness with used_count≤k. A loose k may be sufficient without being optimal, and failure of this coloring gives no general lower-bound certificate. The exact result, retaining the requested order, is (2, 3, Yes, Yes, chromatic_number_at_most_3, upper_bound_not_exact_optimum)."},"skills":["A Nonoptimal Valid Coloring Still Supplies an Upper Bound"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions 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-r05-graph-coloring-0068","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0b79a7a155df907bef08e02b6aa9c43385c142debffc64fd8ad655640475f063","family_id":"qf-mod31-coloring-upper-bound","seed":202610315068,"semantic_template":"qf-mod31-coloring-upper-bound/{\"colors\":[17,18,17,18,17,18],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"k\":4,\"n\":6}"},"fields":{"title":"A Nonoptimal Valid Coloring Still Supplies an Upper Bound","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], [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. Total vertex colors=[17, 18, 17, 18, 17, 18] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. Proposed k=4. Give (used_count,k,proper?,certificate?,conclusion,upper_bound_not_exact_optimum). A valid certificate yields chromatic_number_at_most_k with the supplied numeric k; otherwise no_upper_bound_from_this_coloring.","answer":{"kind":"expression","value":"(2, 4, Yes, Yes, chromatic_number_at_most_4, upper_bound_not_exact_optimum)","explanation":"Conjoin all-edge properness with used_count≤k. A loose k may be sufficient without being optimal, and failure of this coloring gives no general lower-bound certificate. The exact result, retaining the requested order, is (2, 4, Yes, Yes, chromatic_number_at_most_4, upper_bound_not_exact_optimum)."},"skills":["A Nonoptimal Valid Coloring Still Supplies an Upper Bound"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions 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-r05-graph-coloring-0069","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ca0bf7e6c91a9b4f19fba3fbeb700238f7c3b450cc85180df60ce69a7f799e3f","family_id":"qf-mod31-coloring-upper-bound","seed":202610315069,"semantic_template":"qf-mod31-coloring-upper-bound/{\"colors\":[18,19,18,19,18,19],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"k\":2,\"n\":6}"},"fields":{"title":"A Nonoptimal Valid Coloring Still Supplies an Upper Bound","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], [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. Total vertex colors=[18, 19, 18, 19, 18, 19] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. Proposed k=2. Give (used_count,k,proper?,certificate?,conclusion,upper_bound_not_exact_optimum). A valid certificate yields chromatic_number_at_most_k with the supplied numeric k; otherwise no_upper_bound_from_this_coloring.","answer":{"kind":"expression","value":"(2, 2, Yes, Yes, chromatic_number_at_most_2, upper_bound_not_exact_optimum)","explanation":"Conjoin all-edge properness with used_count≤k. A loose k may be sufficient without being optimal, and failure of this coloring gives no general lower-bound certificate. The exact result, retaining the requested order, is (2, 2, Yes, Yes, chromatic_number_at_most_2, upper_bound_not_exact_optimum)."},"skills":["A Nonoptimal Valid Coloring Still Supplies an Upper Bound"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions 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-r05-graph-coloring-0070","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5805b16e6f5bbdcbeab7ea0fbed138e0745f57474a5af78f0151d16a10f157cf","family_id":"qf-mod31-coloring-upper-bound","seed":202610315070,"semantic_template":"qf-mod31-coloring-upper-bound/{\"colors\":[19,20,19,20,19,20],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"k\":3,\"n\":6}"},"fields":{"title":"A Nonoptimal Valid Coloring Still Supplies an Upper Bound","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], [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. Total vertex colors=[19, 20, 19, 20, 19, 20] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. Proposed k=3. Give (used_count,k,proper?,certificate?,conclusion,upper_bound_not_exact_optimum). A valid certificate yields chromatic_number_at_most_k with the supplied numeric k; otherwise no_upper_bound_from_this_coloring.","answer":{"kind":"expression","value":"(2, 3, Yes, Yes, chromatic_number_at_most_3, upper_bound_not_exact_optimum)","explanation":"Conjoin all-edge properness with used_count≤k. A loose k may be sufficient without being optimal, and failure of this coloring gives no general lower-bound certificate. The exact result, retaining the requested order, is (2, 3, Yes, Yes, chromatic_number_at_most_3, upper_bound_not_exact_optimum)."},"skills":["A Nonoptimal Valid Coloring Still Supplies an Upper Bound"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions 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-r05-graph-coloring-0071","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1dd13a2f003c960ff4a44723747fef573ff02318e6b2a702465f653988ed07df","family_id":"qf-mod31-coloring-upper-bound","seed":202610315071,"semantic_template":"qf-mod31-coloring-upper-bound/{\"colors\":[20,21,20,21,20,21],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"k\":4,\"n\":6}"},"fields":{"title":"A Nonoptimal Valid Coloring Still Supplies an Upper Bound","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], [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. Total vertex colors=[20, 21, 20, 21, 20, 21] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. Proposed k=4. Give (used_count,k,proper?,certificate?,conclusion,upper_bound_not_exact_optimum). A valid certificate yields chromatic_number_at_most_k with the supplied numeric k; otherwise no_upper_bound_from_this_coloring.","answer":{"kind":"expression","value":"(2, 4, Yes, Yes, chromatic_number_at_most_4, upper_bound_not_exact_optimum)","explanation":"Conjoin all-edge properness with used_count≤k. A loose k may be sufficient without being optimal, and failure of this coloring gives no general lower-bound certificate. The exact result, retaining the requested order, is (2, 4, Yes, Yes, chromatic_number_at_most_4, upper_bound_not_exact_optimum)."},"skills":["A Nonoptimal Valid Coloring Still Supplies an Upper Bound"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions 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-r05-graph-coloring-0072","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8c5525a39214c588bbf9543798f5c8622e9d1821d5087a180ca257e33b383e73","family_id":"qf-mod31-coloring-upper-bound","seed":202610315072,"semantic_template":"qf-mod31-coloring-upper-bound/{\"colors\":[21,22,21,22,21,22],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"k\":2,\"n\":6}"},"fields":{"title":"A Nonoptimal Valid Coloring Still Supplies an Upper Bound","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], [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. Total vertex colors=[21, 22, 21, 22, 21, 22] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. Proposed k=2. Give (used_count,k,proper?,certificate?,conclusion,upper_bound_not_exact_optimum). A valid certificate yields chromatic_number_at_most_k with the supplied numeric k; otherwise no_upper_bound_from_this_coloring.","answer":{"kind":"expression","value":"(2, 2, Yes, Yes, chromatic_number_at_most_2, upper_bound_not_exact_optimum)","explanation":"Conjoin all-edge properness with used_count≤k. A loose k may be sufficient without being optimal, and failure of this coloring gives no general lower-bound certificate. The exact result, retaining the requested order, is (2, 2, Yes, Yes, chromatic_number_at_most_2, upper_bound_not_exact_optimum)."},"skills":["A Nonoptimal Valid Coloring Still Supplies an Upper Bound"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions 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-r05-graph-coloring-0073","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"739e5e83c7613f4719dfbc3fdce43a0fc5d856c7f1a7aac73bec0f6ed62dea84","family_id":"qf-mod31-coloring-upper-bound","seed":202610315073,"semantic_template":"qf-mod31-coloring-upper-bound/{\"colors\":[22,23,22,23,22,23],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"k\":3,\"n\":6}"},"fields":{"title":"A Nonoptimal Valid Coloring Still Supplies an Upper Bound","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], [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. Total vertex colors=[22, 23, 22, 23, 22, 23] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. Proposed k=3. Give (used_count,k,proper?,certificate?,conclusion,upper_bound_not_exact_optimum). A valid certificate yields chromatic_number_at_most_k with the supplied numeric k; otherwise no_upper_bound_from_this_coloring.","answer":{"kind":"expression","value":"(2, 3, Yes, Yes, chromatic_number_at_most_3, upper_bound_not_exact_optimum)","explanation":"Conjoin all-edge properness with used_count≤k. A loose k may be sufficient without being optimal, and failure of this coloring gives no general lower-bound certificate. The exact result, retaining the requested order, is (2, 3, Yes, Yes, chromatic_number_at_most_3, upper_bound_not_exact_optimum)."},"skills":["A Nonoptimal Valid Coloring Still Supplies an Upper Bound"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions 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-r05-graph-coloring-0074","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4bf01fc3a04a420f09584ac09598f7065a9f234a9ff91377d0e69814bf7bf41d","family_id":"qf-mod31-coloring-upper-bound","seed":202610315074,"semantic_template":"qf-mod31-coloring-upper-bound/{\"colors\":[23,24,23,24,23,24],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"k\":4,\"n\":6}"},"fields":{"title":"A Nonoptimal Valid Coloring Still Supplies an Upper Bound","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], [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. Total vertex colors=[23, 24, 23, 24, 23, 24] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. Proposed k=4. Give (used_count,k,proper?,certificate?,conclusion,upper_bound_not_exact_optimum). A valid certificate yields chromatic_number_at_most_k with the supplied numeric k; otherwise no_upper_bound_from_this_coloring.","answer":{"kind":"expression","value":"(2, 4, Yes, Yes, chromatic_number_at_most_4, upper_bound_not_exact_optimum)","explanation":"Conjoin all-edge properness with used_count≤k. A loose k may be sufficient without being optimal, and failure of this coloring gives no general lower-bound certificate. The exact result, retaining the requested order, is (2, 4, Yes, Yes, chromatic_number_at_most_4, upper_bound_not_exact_optimum)."},"skills":["A Nonoptimal Valid Coloring Still Supplies an Upper Bound"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions 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-r05-graph-coloring-0075","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a3c96cef2ba8ac10502531a5516e639e57edf27854a85b673f165d6b8db5c8d2","family_id":"qf-mod31-coloring-upper-bound","seed":202610315075,"semantic_template":"qf-mod31-coloring-upper-bound/{\"colors\":[24,25,24,25,24,25],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"k\":2,\"n\":6}"},"fields":{"title":"A Nonoptimal Valid Coloring Still Supplies an Upper Bound","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], [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. Total vertex colors=[24, 25, 24, 25, 24, 25] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. Proposed k=2. Give (used_count,k,proper?,certificate?,conclusion,upper_bound_not_exact_optimum). A valid certificate yields chromatic_number_at_most_k with the supplied numeric k; otherwise no_upper_bound_from_this_coloring.","answer":{"kind":"expression","value":"(2, 2, Yes, Yes, chromatic_number_at_most_2, upper_bound_not_exact_optimum)","explanation":"Conjoin all-edge properness with used_count≤k. A loose k may be sufficient without being optimal, and failure of this coloring gives no general lower-bound certificate. The exact result, retaining the requested order, is (2, 2, Yes, Yes, chromatic_number_at_most_2, upper_bound_not_exact_optimum)."},"skills":["A Nonoptimal Valid Coloring Still Supplies an Upper Bound"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions 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-r05-graph-coloring-0076","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"27b8919c07e06002005130c6a02d55ad273b36a69b7f02d4cb714f6ad74838a7","family_id":"qf-mod31-coloring-relabel-invariance","seed":202610315076,"semantic_template":"qf-mod31-coloring-relabel-invariance/{\"colors\":[0,1,0,1,0,1],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"n\":6,\"relabel\":[10,40]}"},"fields":{"title":"Injective Renaming of Used Colors Preserves the Vertex Partition","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], [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. Total vertex colors=[0, 1, 0, 1, 0, 1] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. Color relabel table=[10, 40] is indexed by the original color labels and must be injective on the labels actually used. Give (colors,recolored,proper_before?,proper_after?,same_conflicts?,color_names_do_not_change_vertex_partition).","answer":{"kind":"expression","value":"([0, 1, 0, 1, 0, 1], [10, 40, 10, 40, 10, 40], Yes, Yes, Yes, color_names_do_not_change_vertex_partition)","explanation":"Apply the table to every vertex in original order. Distinct used colors must remain distinct, so exactly the same edges remain monochromatic. Even the conflicts of a failed coloring are preserved, not repaired. The exact result, retaining the requested order, is ([0, 1, 0, 1, 0, 1], [10, 40, 10, 40, 10, 40], Yes, Yes, Yes, color_names_do_not_change_vertex_partition)."},"skills":["Injective Renaming of Used Colors Preserves the Vertex Partition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions 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-r05-graph-coloring-0077","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"057c593a0bf0b00236ef41b04b3976ca60f7e770b6dc95fe1b2f09905f883fb6","family_id":"qf-mod31-coloring-relabel-invariance","seed":202610315077,"semantic_template":"qf-mod31-coloring-relabel-invariance/{\"colors\":[0,1,0,1,0,1],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"n\":6,\"relabel\":[11,41]}"},"fields":{"title":"Injective Renaming of Used Colors Preserves the Vertex Partition","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], [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. Total vertex colors=[0, 1, 0, 1, 0, 1] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. Color relabel table=[11, 41] is indexed by the original color labels and must be injective on the labels actually used. Give (colors,recolored,proper_before?,proper_after?,same_conflicts?,color_names_do_not_change_vertex_partition).","answer":{"kind":"expression","value":"([0, 1, 0, 1, 0, 1], [11, 41, 11, 41, 11, 41], Yes, Yes, Yes, color_names_do_not_change_vertex_partition)","explanation":"Apply the table to every vertex in original order. Distinct used colors must remain distinct, so exactly the same edges remain monochromatic. Even the conflicts of a failed coloring are preserved, not repaired. The exact result, retaining the requested order, is ([0, 1, 0, 1, 0, 1], [11, 41, 11, 41, 11, 41], Yes, Yes, Yes, color_names_do_not_change_vertex_partition)."},"skills":["Injective Renaming of Used Colors Preserves the Vertex Partition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions 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-r05-graph-coloring-0078","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"44df60fca0ca2b8abdaeb0b0ef43dcbd1dea206cc55ab3ed57608d296933ced6","family_id":"qf-mod31-coloring-relabel-invariance","seed":202610315078,"semantic_template":"qf-mod31-coloring-relabel-invariance/{\"colors\":[0,1,0,1,0,1],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"n\":6,\"relabel\":[12,42]}"},"fields":{"title":"Injective Renaming of Used Colors Preserves the Vertex Partition","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], [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. Total vertex colors=[0, 1, 0, 1, 0, 1] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. Color relabel table=[12, 42] is indexed by the original color labels and must be injective on the labels actually used. Give (colors,recolored,proper_before?,proper_after?,same_conflicts?,color_names_do_not_change_vertex_partition).","answer":{"kind":"expression","value":"([0, 1, 0, 1, 0, 1], [12, 42, 12, 42, 12, 42], Yes, Yes, Yes, color_names_do_not_change_vertex_partition)","explanation":"Apply the table to every vertex in original order. Distinct used colors must remain distinct, so exactly the same edges remain monochromatic. Even the conflicts of a failed coloring are preserved, not repaired. The exact result, retaining the requested order, is ([0, 1, 0, 1, 0, 1], [12, 42, 12, 42, 12, 42], Yes, Yes, Yes, color_names_do_not_change_vertex_partition)."},"skills":["Injective Renaming of Used Colors Preserves the Vertex Partition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions 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-r05-graph-coloring-0079","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9b35cdb7a1fe20aeb5bc71b234c5e7db218cd571a5dab76ca4d1e33d85eb14df","family_id":"qf-mod31-coloring-relabel-invariance","seed":202610315079,"semantic_template":"qf-mod31-coloring-relabel-invariance/{\"colors\":[0,1,0,1,0,1],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"n\":6,\"relabel\":[13,43]}"},"fields":{"title":"Injective Renaming of Used Colors Preserves the Vertex Partition","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], [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. Total vertex colors=[0, 1, 0, 1, 0, 1] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. Color relabel table=[13, 43] is indexed by the original color labels and must be injective on the labels actually used. Give (colors,recolored,proper_before?,proper_after?,same_conflicts?,color_names_do_not_change_vertex_partition).","answer":{"kind":"expression","value":"([0, 1, 0, 1, 0, 1], [13, 43, 13, 43, 13, 43], Yes, Yes, Yes, color_names_do_not_change_vertex_partition)","explanation":"Apply the table to every vertex in original order. Distinct used colors must remain distinct, so exactly the same edges remain monochromatic. Even the conflicts of a failed coloring are preserved, not repaired. The exact result, retaining the requested order, is ([0, 1, 0, 1, 0, 1], [13, 43, 13, 43, 13, 43], Yes, Yes, Yes, color_names_do_not_change_vertex_partition)."},"skills":["Injective Renaming of Used Colors Preserves the Vertex Partition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions 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-r05-graph-coloring-0080","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"95429d2da2cd81d264f599d9cb1b1accd34b93e21e100d67667ebfdbcf5fe472","family_id":"qf-mod31-coloring-relabel-invariance","seed":202610315080,"semantic_template":"qf-mod31-coloring-relabel-invariance/{\"colors\":[0,1,0,1,0,1],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"n\":6,\"relabel\":[14,44]}"},"fields":{"title":"Injective Renaming of Used Colors Preserves the Vertex Partition","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], [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. Total vertex colors=[0, 1, 0, 1, 0, 1] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. Color relabel table=[14, 44] is indexed by the original color labels and must be injective on the labels actually used. Give (colors,recolored,proper_before?,proper_after?,same_conflicts?,color_names_do_not_change_vertex_partition).","answer":{"kind":"expression","value":"([0, 1, 0, 1, 0, 1], [14, 44, 14, 44, 14, 44], Yes, Yes, Yes, color_names_do_not_change_vertex_partition)","explanation":"Apply the table to every vertex in original order. Distinct used colors must remain distinct, so exactly the same edges remain monochromatic. Even the conflicts of a failed coloring are preserved, not repaired. The exact result, retaining the requested order, is ([0, 1, 0, 1, 0, 1], [14, 44, 14, 44, 14, 44], Yes, Yes, Yes, color_names_do_not_change_vertex_partition)."},"skills":["Injective Renaming of Used Colors Preserves the Vertex Partition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions 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-r05-graph-coloring-0081","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"19c09204700d86ec000e1ee8ffafed8cfd20232ce84477102a1c668586ab75f0","family_id":"qf-mod31-coloring-relabel-invariance","seed":202610315081,"semantic_template":"qf-mod31-coloring-relabel-invariance/{\"colors\":[0,1,0,1,0,1],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"n\":6,\"relabel\":[15,45]}"},"fields":{"title":"Injective Renaming of Used Colors Preserves the Vertex Partition","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], [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. Total vertex colors=[0, 1, 0, 1, 0, 1] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. Color relabel table=[15, 45] is indexed by the original color labels and must be injective on the labels actually used. Give (colors,recolored,proper_before?,proper_after?,same_conflicts?,color_names_do_not_change_vertex_partition).","answer":{"kind":"expression","value":"([0, 1, 0, 1, 0, 1], [15, 45, 15, 45, 15, 45], Yes, Yes, Yes, color_names_do_not_change_vertex_partition)","explanation":"Apply the table to every vertex in original order. Distinct used colors must remain distinct, so exactly the same edges remain monochromatic. Even the conflicts of a failed coloring are preserved, not repaired. The exact result, retaining the requested order, is ([0, 1, 0, 1, 0, 1], [15, 45, 15, 45, 15, 45], Yes, Yes, Yes, color_names_do_not_change_vertex_partition)."},"skills":["Injective Renaming of Used Colors Preserves the Vertex Partition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions 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-r05-graph-coloring-0082","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ba51643cd2b3f20d32ccdd644cadb7dd183aace871b35351cce0f23384763a78","family_id":"qf-mod31-coloring-relabel-invariance","seed":202610315082,"semantic_template":"qf-mod31-coloring-relabel-invariance/{\"colors\":[0,1,0,1,0,1],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"n\":6,\"relabel\":[16,46]}"},"fields":{"title":"Injective Renaming of Used Colors Preserves the Vertex Partition","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], [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. Total vertex colors=[0, 1, 0, 1, 0, 1] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. Color relabel table=[16, 46] is indexed by the original color labels and must be injective on the labels actually used. Give (colors,recolored,proper_before?,proper_after?,same_conflicts?,color_names_do_not_change_vertex_partition).","answer":{"kind":"expression","value":"([0, 1, 0, 1, 0, 1], [16, 46, 16, 46, 16, 46], Yes, Yes, Yes, color_names_do_not_change_vertex_partition)","explanation":"Apply the table to every vertex in original order. Distinct used colors must remain distinct, so exactly the same edges remain monochromatic. Even the conflicts of a failed coloring are preserved, not repaired. The exact result, retaining the requested order, is ([0, 1, 0, 1, 0, 1], [16, 46, 16, 46, 16, 46], Yes, Yes, Yes, color_names_do_not_change_vertex_partition)."},"skills":["Injective Renaming of Used Colors Preserves the Vertex Partition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions 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-r05-graph-coloring-0083","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"63b35a229cfc0bfcb6e2bfe0ca02e99107225586ebb6c2f0afdc9befd27b27cb","family_id":"qf-mod31-coloring-relabel-invariance","seed":202610315083,"semantic_template":"qf-mod31-coloring-relabel-invariance/{\"colors\":[0,1,0,1,0,1],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"n\":6,\"relabel\":[17,47]}"},"fields":{"title":"Injective Renaming of Used Colors Preserves the Vertex Partition","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], [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. Total vertex colors=[0, 1, 0, 1, 0, 1] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. Color relabel table=[17, 47] is indexed by the original color labels and must be injective on the labels actually used. Give (colors,recolored,proper_before?,proper_after?,same_conflicts?,color_names_do_not_change_vertex_partition).","answer":{"kind":"expression","value":"([0, 1, 0, 1, 0, 1], [17, 47, 17, 47, 17, 47], Yes, Yes, Yes, color_names_do_not_change_vertex_partition)","explanation":"Apply the table to every vertex in original order. Distinct used colors must remain distinct, so exactly the same edges remain monochromatic. Even the conflicts of a failed coloring are preserved, not repaired. The exact result, retaining the requested order, is ([0, 1, 0, 1, 0, 1], [17, 47, 17, 47, 17, 47], Yes, Yes, Yes, color_names_do_not_change_vertex_partition)."},"skills":["Injective Renaming of Used Colors Preserves the Vertex Partition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions 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-r05-graph-coloring-0084","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d3c325ce083d6d8fb6fbf0e3232cba9cb912b5bbad34486f702634761a4c8472","family_id":"qf-mod31-coloring-relabel-invariance","seed":202610315084,"semantic_template":"qf-mod31-coloring-relabel-invariance/{\"colors\":[0,1,0,1,0,1],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"n\":6,\"relabel\":[18,48]}"},"fields":{"title":"Injective Renaming of Used Colors Preserves the Vertex Partition","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], [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. Total vertex colors=[0, 1, 0, 1, 0, 1] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. Color relabel table=[18, 48] is indexed by the original color labels and must be injective on the labels actually used. Give (colors,recolored,proper_before?,proper_after?,same_conflicts?,color_names_do_not_change_vertex_partition).","answer":{"kind":"expression","value":"([0, 1, 0, 1, 0, 1], [18, 48, 18, 48, 18, 48], Yes, Yes, Yes, color_names_do_not_change_vertex_partition)","explanation":"Apply the table to every vertex in original order. Distinct used colors must remain distinct, so exactly the same edges remain monochromatic. Even the conflicts of a failed coloring are preserved, not repaired. The exact result, retaining the requested order, is ([0, 1, 0, 1, 0, 1], [18, 48, 18, 48, 18, 48], Yes, Yes, Yes, color_names_do_not_change_vertex_partition)."},"skills":["Injective Renaming of Used Colors Preserves the Vertex Partition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions 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-r05-graph-coloring-0085","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c1dd6c93fbe1a94a8a17c8e33c93fd888c1f0e15365be01dff7b755de66b513a","family_id":"qf-mod31-coloring-relabel-invariance","seed":202610315085,"semantic_template":"qf-mod31-coloring-relabel-invariance/{\"colors\":[0,1,0,1,0,1],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"n\":6,\"relabel\":[19,49]}"},"fields":{"title":"Injective Renaming of Used Colors Preserves the Vertex Partition","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], [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. Total vertex colors=[0, 1, 0, 1, 0, 1] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. Color relabel table=[19, 49] is indexed by the original color labels and must be injective on the labels actually used. Give (colors,recolored,proper_before?,proper_after?,same_conflicts?,color_names_do_not_change_vertex_partition).","answer":{"kind":"expression","value":"([0, 1, 0, 1, 0, 1], [19, 49, 19, 49, 19, 49], Yes, Yes, Yes, color_names_do_not_change_vertex_partition)","explanation":"Apply the table to every vertex in original order. Distinct used colors must remain distinct, so exactly the same edges remain monochromatic. Even the conflicts of a failed coloring are preserved, not repaired. The exact result, retaining the requested order, is ([0, 1, 0, 1, 0, 1], [19, 49, 19, 49, 19, 49], Yes, Yes, Yes, color_names_do_not_change_vertex_partition)."},"skills":["Injective Renaming of Used Colors Preserves the Vertex Partition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions 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-r05-graph-coloring-0086","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"47463d050641666fbd2636f9f3f50f0b3b25dbc0373e1defef877be7f3ec358b","family_id":"qf-mod31-coloring-relabel-invariance","seed":202610315086,"semantic_template":"qf-mod31-coloring-relabel-invariance/{\"colors\":[0,1,0,1,0,1],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"n\":6,\"relabel\":[20,50]}"},"fields":{"title":"Injective Renaming of Used Colors Preserves the Vertex Partition","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], [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. Total vertex colors=[0, 1, 0, 1, 0, 1] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. Color relabel table=[20, 50] is indexed by the original color labels and must be injective on the labels actually used. Give (colors,recolored,proper_before?,proper_after?,same_conflicts?,color_names_do_not_change_vertex_partition).","answer":{"kind":"expression","value":"([0, 1, 0, 1, 0, 1], [20, 50, 20, 50, 20, 50], Yes, Yes, Yes, color_names_do_not_change_vertex_partition)","explanation":"Apply the table to every vertex in original order. Distinct used colors must remain distinct, so exactly the same edges remain monochromatic. Even the conflicts of a failed coloring are preserved, not repaired. The exact result, retaining the requested order, is ([0, 1, 0, 1, 0, 1], [20, 50, 20, 50, 20, 50], Yes, Yes, Yes, color_names_do_not_change_vertex_partition)."},"skills":["Injective Renaming of Used Colors Preserves the Vertex Partition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions 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-r05-graph-coloring-0087","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"72a8060dbed7ff1f8add4133bf1ed81b3902fca092f594768e0d95a619e03972","family_id":"qf-mod31-coloring-relabel-invariance","seed":202610315087,"semantic_template":"qf-mod31-coloring-relabel-invariance/{\"colors\":[0,1,0,1,0,1],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"n\":6,\"relabel\":[21,51]}"},"fields":{"title":"Injective Renaming of Used Colors Preserves the Vertex Partition","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], [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. Total vertex colors=[0, 1, 0, 1, 0, 1] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. Color relabel table=[21, 51] is indexed by the original color labels and must be injective on the labels actually used. Give (colors,recolored,proper_before?,proper_after?,same_conflicts?,color_names_do_not_change_vertex_partition).","answer":{"kind":"expression","value":"([0, 1, 0, 1, 0, 1], [21, 51, 21, 51, 21, 51], Yes, Yes, Yes, color_names_do_not_change_vertex_partition)","explanation":"Apply the table to every vertex in original order. Distinct used colors must remain distinct, so exactly the same edges remain monochromatic. Even the conflicts of a failed coloring are preserved, not repaired. The exact result, retaining the requested order, is ([0, 1, 0, 1, 0, 1], [21, 51, 21, 51, 21, 51], Yes, Yes, Yes, color_names_do_not_change_vertex_partition)."},"skills":["Injective Renaming of Used Colors Preserves the Vertex Partition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions 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-r05-graph-coloring-0088","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3da29f09097f50dd7a7f454d522dafa30a27a3aa0dc7eddcef00a616b85da0bb","family_id":"qf-mod31-coloring-relabel-invariance","seed":202610315088,"semantic_template":"qf-mod31-coloring-relabel-invariance/{\"colors\":[0,1,0,1,0,1],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"n\":6,\"relabel\":[22,52]}"},"fields":{"title":"Injective Renaming of Used Colors Preserves the Vertex Partition","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], [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. Total vertex colors=[0, 1, 0, 1, 0, 1] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. Color relabel table=[22, 52] is indexed by the original color labels and must be injective on the labels actually used. Give (colors,recolored,proper_before?,proper_after?,same_conflicts?,color_names_do_not_change_vertex_partition).","answer":{"kind":"expression","value":"([0, 1, 0, 1, 0, 1], [22, 52, 22, 52, 22, 52], Yes, Yes, Yes, color_names_do_not_change_vertex_partition)","explanation":"Apply the table to every vertex in original order. Distinct used colors must remain distinct, so exactly the same edges remain monochromatic. Even the conflicts of a failed coloring are preserved, not repaired. The exact result, retaining the requested order, is ([0, 1, 0, 1, 0, 1], [22, 52, 22, 52, 22, 52], Yes, Yes, Yes, color_names_do_not_change_vertex_partition)."},"skills":["Injective Renaming of Used Colors Preserves the Vertex Partition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions 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-r05-graph-coloring-0089","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"10e425f2af4ae37060254107bc2f5bd9eb3eaeb0ffef9cf353d1f5b03b5ee098","family_id":"qf-mod31-coloring-relabel-invariance","seed":202610315089,"semantic_template":"qf-mod31-coloring-relabel-invariance/{\"colors\":[0,1,0,1,0,1],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"n\":6,\"relabel\":[23,53]}"},"fields":{"title":"Injective Renaming of Used Colors Preserves the Vertex Partition","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], [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. Total vertex colors=[0, 1, 0, 1, 0, 1] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. Color relabel table=[23, 53] is indexed by the original color labels and must be injective on the labels actually used. Give (colors,recolored,proper_before?,proper_after?,same_conflicts?,color_names_do_not_change_vertex_partition).","answer":{"kind":"expression","value":"([0, 1, 0, 1, 0, 1], [23, 53, 23, 53, 23, 53], Yes, Yes, Yes, color_names_do_not_change_vertex_partition)","explanation":"Apply the table to every vertex in original order. Distinct used colors must remain distinct, so exactly the same edges remain monochromatic. Even the conflicts of a failed coloring are preserved, not repaired. The exact result, retaining the requested order, is ([0, 1, 0, 1, 0, 1], [23, 53, 23, 53, 23, 53], Yes, Yes, Yes, color_names_do_not_change_vertex_partition)."},"skills":["Injective Renaming of Used Colors Preserves the Vertex Partition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions 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-r05-graph-coloring-0090","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7095f275ad92788f294978a4c6ee3bc117bb22cec82c81c71501e561ebd3fd01","family_id":"qf-mod31-coloring-relabel-invariance","seed":202610315090,"semantic_template":"qf-mod31-coloring-relabel-invariance/{\"colors\":[0,1,0,1,0,1],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"n\":6,\"relabel\":[24,54]}"},"fields":{"title":"Injective Renaming of Used Colors Preserves the Vertex Partition","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], [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. Total vertex colors=[0, 1, 0, 1, 0, 1] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. Color relabel table=[24, 54] is indexed by the original color labels and must be injective on the labels actually used. Give (colors,recolored,proper_before?,proper_after?,same_conflicts?,color_names_do_not_change_vertex_partition).","answer":{"kind":"expression","value":"([0, 1, 0, 1, 0, 1], [24, 54, 24, 54, 24, 54], Yes, Yes, Yes, color_names_do_not_change_vertex_partition)","explanation":"Apply the table to every vertex in original order. Distinct used colors must remain distinct, so exactly the same edges remain monochromatic. Even the conflicts of a failed coloring are preserved, not repaired. The exact result, retaining the requested order, is ([0, 1, 0, 1, 0, 1], [24, 54, 24, 54, 24, 54], Yes, Yes, Yes, color_names_do_not_change_vertex_partition)."},"skills":["Injective Renaming of Used Colors Preserves the Vertex Partition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions 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-r05-graph-coloring-0091","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f59f536a1e3c7f6c09310d68d82c14a930425bfa1444379d2fe0024bcc9d6653","family_id":"qf-mod31-coloring-relabel-invariance","seed":202610315091,"semantic_template":"qf-mod31-coloring-relabel-invariance/{\"colors\":[0,1,0,1,0,1],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"n\":6,\"relabel\":[25,55]}"},"fields":{"title":"Injective Renaming of Used Colors Preserves the Vertex Partition","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], [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. Total vertex colors=[0, 1, 0, 1, 0, 1] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. Color relabel table=[25, 55] is indexed by the original color labels and must be injective on the labels actually used. Give (colors,recolored,proper_before?,proper_after?,same_conflicts?,color_names_do_not_change_vertex_partition).","answer":{"kind":"expression","value":"([0, 1, 0, 1, 0, 1], [25, 55, 25, 55, 25, 55], Yes, Yes, Yes, color_names_do_not_change_vertex_partition)","explanation":"Apply the table to every vertex in original order. Distinct used colors must remain distinct, so exactly the same edges remain monochromatic. Even the conflicts of a failed coloring are preserved, not repaired. The exact result, retaining the requested order, is ([0, 1, 0, 1, 0, 1], [25, 55, 25, 55, 25, 55], Yes, Yes, Yes, color_names_do_not_change_vertex_partition)."},"skills":["Injective Renaming of Used Colors Preserves the Vertex Partition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions 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-r05-graph-coloring-0092","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"243026f45226fa13d06fce8b64b8e8d8c5b4f845c55e506a449b51800fda8b9b","family_id":"qf-mod31-coloring-relabel-invariance","seed":202610315092,"semantic_template":"qf-mod31-coloring-relabel-invariance/{\"colors\":[0,1,0,1,0,1],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"n\":6,\"relabel\":[26,56]}"},"fields":{"title":"Injective Renaming of Used Colors Preserves the Vertex Partition","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], [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. Total vertex colors=[0, 1, 0, 1, 0, 1] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. Color relabel table=[26, 56] is indexed by the original color labels and must be injective on the labels actually used. Give (colors,recolored,proper_before?,proper_after?,same_conflicts?,color_names_do_not_change_vertex_partition).","answer":{"kind":"expression","value":"([0, 1, 0, 1, 0, 1], [26, 56, 26, 56, 26, 56], Yes, Yes, Yes, color_names_do_not_change_vertex_partition)","explanation":"Apply the table to every vertex in original order. Distinct used colors must remain distinct, so exactly the same edges remain monochromatic. Even the conflicts of a failed coloring are preserved, not repaired. The exact result, retaining the requested order, is ([0, 1, 0, 1, 0, 1], [26, 56, 26, 56, 26, 56], Yes, Yes, Yes, color_names_do_not_change_vertex_partition)."},"skills":["Injective Renaming of Used Colors Preserves the Vertex Partition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions 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-r05-graph-coloring-0093","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"be8a1c813f3b69d4463a0163730c96000c8fdd8b675c50912b17c5976ac72f41","family_id":"qf-mod31-coloring-relabel-invariance","seed":202610315093,"semantic_template":"qf-mod31-coloring-relabel-invariance/{\"colors\":[0,1,0,1,0,1],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"n\":6,\"relabel\":[27,57]}"},"fields":{"title":"Injective Renaming of Used Colors Preserves the Vertex Partition","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], [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. Total vertex colors=[0, 1, 0, 1, 0, 1] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. Color relabel table=[27, 57] is indexed by the original color labels and must be injective on the labels actually used. Give (colors,recolored,proper_before?,proper_after?,same_conflicts?,color_names_do_not_change_vertex_partition).","answer":{"kind":"expression","value":"([0, 1, 0, 1, 0, 1], [27, 57, 27, 57, 27, 57], Yes, Yes, Yes, color_names_do_not_change_vertex_partition)","explanation":"Apply the table to every vertex in original order. Distinct used colors must remain distinct, so exactly the same edges remain monochromatic. Even the conflicts of a failed coloring are preserved, not repaired. The exact result, retaining the requested order, is ([0, 1, 0, 1, 0, 1], [27, 57, 27, 57, 27, 57], Yes, Yes, Yes, color_names_do_not_change_vertex_partition)."},"skills":["Injective Renaming of Used Colors Preserves the Vertex Partition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions 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-r05-graph-coloring-0094","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"66abce12606612567e022b956a1cacffa2244d22f5bc1069c52e5e6c95e64efa","family_id":"qf-mod31-coloring-relabel-invariance","seed":202610315094,"semantic_template":"qf-mod31-coloring-relabel-invariance/{\"colors\":[0,1,0,1,0,1],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"n\":6,\"relabel\":[28,58]}"},"fields":{"title":"Injective Renaming of Used Colors Preserves the Vertex Partition","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], [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. Total vertex colors=[0, 1, 0, 1, 0, 1] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. Color relabel table=[28, 58] is indexed by the original color labels and must be injective on the labels actually used. Give (colors,recolored,proper_before?,proper_after?,same_conflicts?,color_names_do_not_change_vertex_partition).","answer":{"kind":"expression","value":"([0, 1, 0, 1, 0, 1], [28, 58, 28, 58, 28, 58], Yes, Yes, Yes, color_names_do_not_change_vertex_partition)","explanation":"Apply the table to every vertex in original order. Distinct used colors must remain distinct, so exactly the same edges remain monochromatic. Even the conflicts of a failed coloring are preserved, not repaired. The exact result, retaining the requested order, is ([0, 1, 0, 1, 0, 1], [28, 58, 28, 58, 28, 58], Yes, Yes, Yes, color_names_do_not_change_vertex_partition)."},"skills":["Injective Renaming of Used Colors Preserves the Vertex Partition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions 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-r05-graph-coloring-0095","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0941124d718948f553fdad24a8e1a00ba98c4ab5c7327986c529d0512996670d","family_id":"qf-mod31-coloring-relabel-invariance","seed":202610315095,"semantic_template":"qf-mod31-coloring-relabel-invariance/{\"colors\":[0,1,0,1,0,1],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"n\":6,\"relabel\":[29,59]}"},"fields":{"title":"Injective Renaming of Used Colors Preserves the Vertex Partition","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], [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. Total vertex colors=[0, 1, 0, 1, 0, 1] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. Color relabel table=[29, 59] is indexed by the original color labels and must be injective on the labels actually used. Give (colors,recolored,proper_before?,proper_after?,same_conflicts?,color_names_do_not_change_vertex_partition).","answer":{"kind":"expression","value":"([0, 1, 0, 1, 0, 1], [29, 59, 29, 59, 29, 59], Yes, Yes, Yes, color_names_do_not_change_vertex_partition)","explanation":"Apply the table to every vertex in original order. Distinct used colors must remain distinct, so exactly the same edges remain monochromatic. Even the conflicts of a failed coloring are preserved, not repaired. The exact result, retaining the requested order, is ([0, 1, 0, 1, 0, 1], [29, 59, 29, 59, 29, 59], Yes, Yes, Yes, color_names_do_not_change_vertex_partition)."},"skills":["Injective Renaming of Used Colors Preserves the Vertex Partition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions 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-r05-graph-coloring-0096","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dfe41f08f1ab5412dfb13256408847fa691496cc82e663578bc6a60f002c66a5","family_id":"qf-mod31-coloring-relabel-invariance","seed":202610315096,"semantic_template":"qf-mod31-coloring-relabel-invariance/{\"colors\":[0,1,0,1,0,1],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"n\":6,\"relabel\":[30,60]}"},"fields":{"title":"Injective Renaming of Used Colors Preserves the Vertex Partition","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], [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. Total vertex colors=[0, 1, 0, 1, 0, 1] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. Color relabel table=[30, 60] is indexed by the original color labels and must be injective on the labels actually used. Give (colors,recolored,proper_before?,proper_after?,same_conflicts?,color_names_do_not_change_vertex_partition).","answer":{"kind":"expression","value":"([0, 1, 0, 1, 0, 1], [30, 60, 30, 60, 30, 60], Yes, Yes, Yes, color_names_do_not_change_vertex_partition)","explanation":"Apply the table to every vertex in original order. Distinct used colors must remain distinct, so exactly the same edges remain monochromatic. Even the conflicts of a failed coloring are preserved, not repaired. The exact result, retaining the requested order, is ([0, 1, 0, 1, 0, 1], [30, 60, 30, 60, 30, 60], Yes, Yes, Yes, color_names_do_not_change_vertex_partition)."},"skills":["Injective Renaming of Used Colors Preserves the Vertex Partition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions 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-r05-graph-coloring-0097","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1a090c3637cf7f3c5c68ba4b7989e051fabb1ce1b409ad9f131d78285eab518d","family_id":"qf-mod31-coloring-relabel-invariance","seed":202610315097,"semantic_template":"qf-mod31-coloring-relabel-invariance/{\"colors\":[0,1,0,1,0,1],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"n\":6,\"relabel\":[31,61]}"},"fields":{"title":"Injective Renaming of Used Colors Preserves the Vertex Partition","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], [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. Total vertex colors=[0, 1, 0, 1, 0, 1] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. Color relabel table=[31, 61] is indexed by the original color labels and must be injective on the labels actually used. Give (colors,recolored,proper_before?,proper_after?,same_conflicts?,color_names_do_not_change_vertex_partition).","answer":{"kind":"expression","value":"([0, 1, 0, 1, 0, 1], [31, 61, 31, 61, 31, 61], Yes, Yes, Yes, color_names_do_not_change_vertex_partition)","explanation":"Apply the table to every vertex in original order. Distinct used colors must remain distinct, so exactly the same edges remain monochromatic. Even the conflicts of a failed coloring are preserved, not repaired. The exact result, retaining the requested order, is ([0, 1, 0, 1, 0, 1], [31, 61, 31, 61, 31, 61], Yes, Yes, Yes, color_names_do_not_change_vertex_partition)."},"skills":["Injective Renaming of Used Colors Preserves the Vertex Partition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions 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-r05-graph-coloring-0098","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a5bd6c5975cb0161f036499049b8610eef97d4a2dac3286920d1f177d6cce42a","family_id":"qf-mod31-coloring-relabel-invariance","seed":202610315098,"semantic_template":"qf-mod31-coloring-relabel-invariance/{\"colors\":[0,1,0,1,0,1],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"n\":6,\"relabel\":[32,62]}"},"fields":{"title":"Injective Renaming of Used Colors Preserves the Vertex Partition","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], [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. Total vertex colors=[0, 1, 0, 1, 0, 1] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. Color relabel table=[32, 62] is indexed by the original color labels and must be injective on the labels actually used. Give (colors,recolored,proper_before?,proper_after?,same_conflicts?,color_names_do_not_change_vertex_partition).","answer":{"kind":"expression","value":"([0, 1, 0, 1, 0, 1], [32, 62, 32, 62, 32, 62], Yes, Yes, Yes, color_names_do_not_change_vertex_partition)","explanation":"Apply the table to every vertex in original order. Distinct used colors must remain distinct, so exactly the same edges remain monochromatic. Even the conflicts of a failed coloring are preserved, not repaired. The exact result, retaining the requested order, is ([0, 1, 0, 1, 0, 1], [32, 62, 32, 62, 32, 62], Yes, Yes, Yes, color_names_do_not_change_vertex_partition)."},"skills":["Injective Renaming of Used Colors Preserves the Vertex Partition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions 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-r05-graph-coloring-0099","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"74f38df3a4d7f3af4069a19bd3d7b2710def54f9ae5a2fd5becc0b61eb4450de","family_id":"qf-mod31-coloring-relabel-invariance","seed":202610315099,"semantic_template":"qf-mod31-coloring-relabel-invariance/{\"colors\":[0,1,0,1,0,1],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"n\":6,\"relabel\":[33,63]}"},"fields":{"title":"Injective Renaming of Used Colors Preserves the Vertex Partition","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], [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. Total vertex colors=[0, 1, 0, 1, 0, 1] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. Color relabel table=[33, 63] is indexed by the original color labels and must be injective on the labels actually used. Give (colors,recolored,proper_before?,proper_after?,same_conflicts?,color_names_do_not_change_vertex_partition).","answer":{"kind":"expression","value":"([0, 1, 0, 1, 0, 1], [33, 63, 33, 63, 33, 63], Yes, Yes, Yes, color_names_do_not_change_vertex_partition)","explanation":"Apply the table to every vertex in original order. Distinct used colors must remain distinct, so exactly the same edges remain monochromatic. Even the conflicts of a failed coloring are preserved, not repaired. The exact result, retaining the requested order, is ([0, 1, 0, 1, 0, 1], [33, 63, 33, 63, 33, 63], Yes, Yes, Yes, color_names_do_not_change_vertex_partition)."},"skills":["Injective Renaming of Used Colors Preserves the Vertex Partition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions 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-r05-graph-coloring-0100","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fc4c93ee723755e3e7f90a398c076dd1877449082faf59999a2b32ce0a3d7bd6","family_id":"qf-mod31-coloring-relabel-invariance","seed":202610315100,"semantic_template":"qf-mod31-coloring-relabel-invariance/{\"colors\":[0,1,0,1,0,1],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"n\":6,\"relabel\":[34,64]}"},"fields":{"title":"Injective Renaming of Used Colors Preserves the Vertex Partition","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], [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. Total vertex colors=[0, 1, 0, 1, 0, 1] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. Color relabel table=[34, 64] is indexed by the original color labels and must be injective on the labels actually used. Give (colors,recolored,proper_before?,proper_after?,same_conflicts?,color_names_do_not_change_vertex_partition).","answer":{"kind":"expression","value":"([0, 1, 0, 1, 0, 1], [34, 64, 34, 64, 34, 64], Yes, Yes, Yes, color_names_do_not_change_vertex_partition)","explanation":"Apply the table to every vertex in original order. Distinct used colors must remain distinct, so exactly the same edges remain monochromatic. Even the conflicts of a failed coloring are preserved, not repaired. The exact result, retaining the requested order, is ([0, 1, 0, 1, 0, 1], [34, 64, 34, 64, 34, 64], Yes, Yes, Yes, color_names_do_not_change_vertex_partition)."},"skills":["Injective Renaming of Used Colors Preserves the Vertex Partition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}