{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r07-0001","batch_id":"batch-20261015-advance-20261119-r07-optimization-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0599d5089c076eab426e7cf41651049c6e19a73c4d2ba59d227d17b42c219a19","family_id":"qf-mod50-optdeep-local-global","seed":202611197001,"semantic_template":"optdeep-local-global|{\"candidate\":\"A\",\"edges\":[[\"A\",\"B\"],[\"A\",\"C\"],[\"B\",\"D\"],[\"C\",\"D\"]],\"points\":[\"A\",\"B\",\"C\",\"D\"],\"values\":[[1,1],[2,1],[3,1],[4,1]]}"},"fields":{"title":"Declared-Neighborhood Locality and Whole-Domain Globality Are Separate","stem":"Use the complete finite feasible carrier=[A, B, C, D] with aligned exact objective values=[1, 2, 3, 4], candidate=A. Complete undirected neighborhood edges=[[A, B], [A, C], [B, D], [C, D]]. Give (complete_objective_table, undirected_neighborhood_edges, candidate, candidate_value, complete_neighbor_values, neighborhood_local_minimum, global_minimum, all_global_minimizers, comparison_scope).","answer":{"kind":"expression","value":"([[A, 1], [B, 2], [C, 3], [D, 4]], [[A, B], [A, C], [B, D], [C, D]], A, 1, [[B, 2], [C, 3]], Yes, Yes, [A], local_uses_complete_declared_neighborhood_global_uses_complete_feasible_domain)","explanation":"Compare the candidate with every declared neighbor for graph-local minimality and with every feasible point for global minimality. Retain every global minimizer, including ties. Graph-locality is not a local optimum in an unspecified continuous topology, and a partial list is not a whole-domain certificate. The exact result, retaining the requested order, is ([[A, 1], [B, 2], [C, 3], [D, 4]], [[A, B], [A, C], [B, D], [C, D]], A, 1, [[B, 2], [C, 3]], Yes, Yes, [A], local_uses_complete_declared_neighborhood_global_uses_complete_feasible_domain)."},"skills":["Declared-Neighborhood Locality and Whole-Domain Globality Are Separate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r07-0002","batch_id":"batch-20261015-advance-20261119-r07-optimization-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dc52a302b5361b5c416cd5cac12bc2f8dfed9441688c61dcf953c7b0455c2a1a","family_id":"qf-mod50-optdeep-local-global","seed":202611197002,"semantic_template":"optdeep-local-global|{\"candidate\":\"A\",\"edges\":[[\"A\",\"B\"],[\"A\",\"C\"],[\"B\",\"D\"],[\"C\",\"D\"]],\"points\":[\"A\",\"B\",\"C\",\"D\"],\"values\":[[2,1],[4,1],[6,1],[8,1]]}"},"fields":{"title":"Declared-Neighborhood Locality and Whole-Domain Globality Are Separate","stem":"Use the complete finite feasible carrier=[A, B, C, D] with aligned exact objective values=[2, 4, 6, 8], candidate=A. Complete undirected neighborhood edges=[[A, B], [A, C], [B, D], [C, D]]. Give (complete_objective_table, undirected_neighborhood_edges, candidate, candidate_value, complete_neighbor_values, neighborhood_local_minimum, global_minimum, all_global_minimizers, comparison_scope).","answer":{"kind":"expression","value":"([[A, 2], [B, 4], [C, 6], [D, 8]], [[A, B], [A, C], [B, D], [C, D]], A, 2, [[B, 4], [C, 6]], Yes, Yes, [A], local_uses_complete_declared_neighborhood_global_uses_complete_feasible_domain)","explanation":"Compare the candidate with every declared neighbor for graph-local minimality and with every feasible point for global minimality. Retain every global minimizer, including ties. Graph-locality is not a local optimum in an unspecified continuous topology, and a partial list is not a whole-domain certificate. The exact result, retaining the requested order, is ([[A, 2], [B, 4], [C, 6], [D, 8]], [[A, B], [A, C], [B, D], [C, D]], A, 2, [[B, 4], [C, 6]], Yes, Yes, [A], local_uses_complete_declared_neighborhood_global_uses_complete_feasible_domain)."},"skills":["Declared-Neighborhood Locality and Whole-Domain Globality Are Separate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r07-0003","batch_id":"batch-20261015-advance-20261119-r07-optimization-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bfb1675433f6e9cd33752cfcb56ecc0eab663137bbe85ff47b9b7d6e49fa6243","family_id":"qf-mod50-optdeep-local-global","seed":202611197003,"semantic_template":"optdeep-local-global|{\"candidate\":\"A\",\"edges\":[[\"A\",\"B\"],[\"A\",\"C\"],[\"B\",\"D\"],[\"C\",\"D\"]],\"points\":[\"A\",\"B\",\"C\",\"D\"],\"values\":[[3,1],[6,1],[9,1],[12,1]]}"},"fields":{"title":"Declared-Neighborhood Locality and Whole-Domain Globality Are Separate","stem":"Use the complete finite feasible carrier=[A, B, C, D] with aligned exact objective values=[3, 6, 9, 12], candidate=A. Complete undirected neighborhood edges=[[A, B], [A, C], [B, D], [C, D]]. Give (complete_objective_table, undirected_neighborhood_edges, candidate, candidate_value, complete_neighbor_values, neighborhood_local_minimum, global_minimum, all_global_minimizers, comparison_scope).","answer":{"kind":"expression","value":"([[A, 3], [B, 6], [C, 9], [D, 12]], [[A, B], [A, C], [B, D], [C, D]], A, 3, [[B, 6], [C, 9]], Yes, Yes, [A], local_uses_complete_declared_neighborhood_global_uses_complete_feasible_domain)","explanation":"Compare the candidate with every declared neighbor for graph-local minimality and with every feasible point for global minimality. Retain every global minimizer, including ties. Graph-locality is not a local optimum in an unspecified continuous topology, and a partial list is not a whole-domain certificate. The exact result, retaining the requested order, is ([[A, 3], [B, 6], [C, 9], [D, 12]], [[A, B], [A, C], [B, D], [C, D]], A, 3, [[B, 6], [C, 9]], Yes, Yes, [A], local_uses_complete_declared_neighborhood_global_uses_complete_feasible_domain)."},"skills":["Declared-Neighborhood Locality and Whole-Domain Globality Are Separate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r07-0004","batch_id":"batch-20261015-advance-20261119-r07-optimization-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4abe9e9d2b6fce6a861da45dc95b383c449d0dbc930dc803b73d4930bd092db4","family_id":"qf-mod50-optdeep-local-global","seed":202611197004,"semantic_template":"optdeep-local-global|{\"candidate\":\"A\",\"edges\":[[\"A\",\"B\"],[\"A\",\"C\"],[\"B\",\"D\"],[\"C\",\"D\"]],\"points\":[\"A\",\"B\",\"C\",\"D\"],\"values\":[[4,1],[8,1],[12,1],[16,1]]}"},"fields":{"title":"Declared-Neighborhood Locality and Whole-Domain Globality Are Separate","stem":"Use the complete finite feasible carrier=[A, B, C, D] with aligned exact objective values=[4, 8, 12, 16], candidate=A. Complete undirected neighborhood edges=[[A, B], [A, C], [B, D], [C, D]]. Give (complete_objective_table, undirected_neighborhood_edges, candidate, candidate_value, complete_neighbor_values, neighborhood_local_minimum, global_minimum, all_global_minimizers, comparison_scope).","answer":{"kind":"expression","value":"([[A, 4], [B, 8], [C, 12], [D, 16]], [[A, B], [A, C], [B, D], [C, D]], A, 4, [[B, 8], [C, 12]], Yes, Yes, [A], local_uses_complete_declared_neighborhood_global_uses_complete_feasible_domain)","explanation":"Compare the candidate with every declared neighbor for graph-local minimality and with every feasible point for global minimality. Retain every global minimizer, including ties. Graph-locality is not a local optimum in an unspecified continuous topology, and a partial list is not a whole-domain certificate. The exact result, retaining the requested order, is ([[A, 4], [B, 8], [C, 12], [D, 16]], [[A, B], [A, C], [B, D], [C, D]], A, 4, [[B, 8], [C, 12]], Yes, Yes, [A], local_uses_complete_declared_neighborhood_global_uses_complete_feasible_domain)."},"skills":["Declared-Neighborhood Locality and Whole-Domain Globality Are Separate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r07-0005","batch_id":"batch-20261015-advance-20261119-r07-optimization-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7808cce4d7291a5a784f1e2d0e232ddc077e496d7ebf6f35f3c44f152808ac91","family_id":"qf-mod50-optdeep-local-global","seed":202611197005,"semantic_template":"optdeep-local-global|{\"candidate\":\"A\",\"edges\":[[\"A\",\"B\"],[\"A\",\"C\"],[\"B\",\"D\"],[\"C\",\"D\"]],\"points\":[\"A\",\"B\",\"C\",\"D\"],\"values\":[[5,1],[10,1],[15,1],[20,1]]}"},"fields":{"title":"Declared-Neighborhood Locality and Whole-Domain Globality Are Separate","stem":"Use the complete finite feasible carrier=[A, B, C, D] with aligned exact objective values=[5, 10, 15, 20], candidate=A. Complete undirected neighborhood edges=[[A, B], [A, C], [B, D], [C, D]]. Give (complete_objective_table, undirected_neighborhood_edges, candidate, candidate_value, complete_neighbor_values, neighborhood_local_minimum, global_minimum, all_global_minimizers, comparison_scope).","answer":{"kind":"expression","value":"([[A, 5], [B, 10], [C, 15], [D, 20]], [[A, B], [A, C], [B, D], [C, D]], A, 5, [[B, 10], [C, 15]], Yes, Yes, [A], local_uses_complete_declared_neighborhood_global_uses_complete_feasible_domain)","explanation":"Compare the candidate with every declared neighbor for graph-local minimality and with every feasible point for global minimality. Retain every global minimizer, including ties. Graph-locality is not a local optimum in an unspecified continuous topology, and a partial list is not a whole-domain certificate. The exact result, retaining the requested order, is ([[A, 5], [B, 10], [C, 15], [D, 20]], [[A, B], [A, C], [B, D], [C, D]], A, 5, [[B, 10], [C, 15]], Yes, Yes, [A], local_uses_complete_declared_neighborhood_global_uses_complete_feasible_domain)."},"skills":["Declared-Neighborhood Locality and Whole-Domain Globality Are Separate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r07-0006","batch_id":"batch-20261015-advance-20261119-r07-optimization-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"995e6016ce139a1b73cc7e79f17e197864dedd63e5186c76dea2a265d0a107ff","family_id":"qf-mod50-optdeep-local-global","seed":202611197006,"semantic_template":"optdeep-local-global|{\"candidate\":\"A\",\"edges\":[[\"A\",\"B\"],[\"A\",\"C\"],[\"B\",\"D\"],[\"C\",\"D\"]],\"points\":[\"A\",\"B\",\"C\",\"D\"],\"values\":[[6,1],[12,1],[18,1],[24,1]]}"},"fields":{"title":"Declared-Neighborhood Locality and Whole-Domain Globality Are Separate","stem":"Use the complete finite feasible carrier=[A, B, C, D] with aligned exact objective values=[6, 12, 18, 24], candidate=A. Complete undirected neighborhood edges=[[A, B], [A, C], [B, D], [C, D]]. Give (complete_objective_table, undirected_neighborhood_edges, candidate, candidate_value, complete_neighbor_values, neighborhood_local_minimum, global_minimum, all_global_minimizers, comparison_scope).","answer":{"kind":"expression","value":"([[A, 6], [B, 12], [C, 18], [D, 24]], [[A, B], [A, C], [B, D], [C, D]], A, 6, [[B, 12], [C, 18]], Yes, Yes, [A], local_uses_complete_declared_neighborhood_global_uses_complete_feasible_domain)","explanation":"Compare the candidate with every declared neighbor for graph-local minimality and with every feasible point for global minimality. Retain every global minimizer, including ties. Graph-locality is not a local optimum in an unspecified continuous topology, and a partial list is not a whole-domain certificate. The exact result, retaining the requested order, is ([[A, 6], [B, 12], [C, 18], [D, 24]], [[A, B], [A, C], [B, D], [C, D]], A, 6, [[B, 12], [C, 18]], Yes, Yes, [A], local_uses_complete_declared_neighborhood_global_uses_complete_feasible_domain)."},"skills":["Declared-Neighborhood Locality and Whole-Domain Globality Are Separate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r07-0007","batch_id":"batch-20261015-advance-20261119-r07-optimization-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"280d7009f5e6290f7de9a2c965ecf4cd269b994dba7c0afc6a8fdaab69a33160","family_id":"qf-mod50-optdeep-local-global","seed":202611197007,"semantic_template":"optdeep-local-global|{\"candidate\":\"A\",\"edges\":[[\"A\",\"B\"],[\"A\",\"C\"],[\"B\",\"D\"],[\"C\",\"D\"]],\"points\":[\"A\",\"B\",\"C\",\"D\"],\"values\":[[7,1],[14,1],[21,1],[28,1]]}"},"fields":{"title":"Declared-Neighborhood Locality and Whole-Domain Globality Are Separate","stem":"Use the complete finite feasible carrier=[A, B, C, D] with aligned exact objective values=[7, 14, 21, 28], candidate=A. Complete undirected neighborhood edges=[[A, B], [A, C], [B, D], [C, D]]. Give (complete_objective_table, undirected_neighborhood_edges, candidate, candidate_value, complete_neighbor_values, neighborhood_local_minimum, global_minimum, all_global_minimizers, comparison_scope).","answer":{"kind":"expression","value":"([[A, 7], [B, 14], [C, 21], [D, 28]], [[A, B], [A, C], [B, D], [C, D]], A, 7, [[B, 14], [C, 21]], Yes, Yes, [A], local_uses_complete_declared_neighborhood_global_uses_complete_feasible_domain)","explanation":"Compare the candidate with every declared neighbor for graph-local minimality and with every feasible point for global minimality. Retain every global minimizer, including ties. Graph-locality is not a local optimum in an unspecified continuous topology, and a partial list is not a whole-domain certificate. The exact result, retaining the requested order, is ([[A, 7], [B, 14], [C, 21], [D, 28]], [[A, B], [A, C], [B, D], [C, D]], A, 7, [[B, 14], [C, 21]], Yes, Yes, [A], local_uses_complete_declared_neighborhood_global_uses_complete_feasible_domain)."},"skills":["Declared-Neighborhood Locality and Whole-Domain Globality Are Separate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r07-0008","batch_id":"batch-20261015-advance-20261119-r07-optimization-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dae79c8e581f7010e4c0e04ccb3fd895125f27eb6b12761ed37fc965f40c79d2","family_id":"qf-mod50-optdeep-local-global","seed":202611197008,"semantic_template":"optdeep-local-global|{\"candidate\":\"A\",\"edges\":[[\"A\",\"B\"],[\"A\",\"C\"],[\"B\",\"D\"],[\"C\",\"D\"]],\"points\":[\"A\",\"B\",\"C\",\"D\"],\"values\":[[8,1],[16,1],[24,1],[32,1]]}"},"fields":{"title":"Declared-Neighborhood Locality and Whole-Domain Globality Are Separate","stem":"Use the complete finite feasible carrier=[A, B, C, D] with aligned exact objective values=[8, 16, 24, 32], candidate=A. Complete undirected neighborhood edges=[[A, B], [A, C], [B, D], [C, D]]. Give (complete_objective_table, undirected_neighborhood_edges, candidate, candidate_value, complete_neighbor_values, neighborhood_local_minimum, global_minimum, all_global_minimizers, comparison_scope).","answer":{"kind":"expression","value":"([[A, 8], [B, 16], [C, 24], [D, 32]], [[A, B], [A, C], [B, D], [C, D]], A, 8, [[B, 16], [C, 24]], Yes, Yes, [A], local_uses_complete_declared_neighborhood_global_uses_complete_feasible_domain)","explanation":"Compare the candidate with every declared neighbor for graph-local minimality and with every feasible point for global minimality. Retain every global minimizer, including ties. Graph-locality is not a local optimum in an unspecified continuous topology, and a partial list is not a whole-domain certificate. The exact result, retaining the requested order, is ([[A, 8], [B, 16], [C, 24], [D, 32]], [[A, B], [A, C], [B, D], [C, D]], A, 8, [[B, 16], [C, 24]], Yes, Yes, [A], local_uses_complete_declared_neighborhood_global_uses_complete_feasible_domain)."},"skills":["Declared-Neighborhood Locality and Whole-Domain Globality Are Separate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r07-0009","batch_id":"batch-20261015-advance-20261119-r07-optimization-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"12ff856a9b87fcaf7790734d69bc01dc2a8344aa763e7930829563754a0910ec","family_id":"qf-mod50-optdeep-local-global","seed":202611197009,"semantic_template":"optdeep-local-global|{\"candidate\":\"A\",\"edges\":[[\"A\",\"B\"],[\"A\",\"C\"],[\"B\",\"D\"],[\"C\",\"D\"]],\"points\":[\"A\",\"B\",\"C\",\"D\"],\"values\":[[9,1],[18,1],[27,1],[36,1]]}"},"fields":{"title":"Declared-Neighborhood Locality and Whole-Domain Globality Are Separate","stem":"Use the complete finite feasible carrier=[A, B, C, D] with aligned exact objective values=[9, 18, 27, 36], candidate=A. Complete undirected neighborhood edges=[[A, B], [A, C], [B, D], [C, D]]. Give (complete_objective_table, undirected_neighborhood_edges, candidate, candidate_value, complete_neighbor_values, neighborhood_local_minimum, global_minimum, all_global_minimizers, comparison_scope).","answer":{"kind":"expression","value":"([[A, 9], [B, 18], [C, 27], [D, 36]], [[A, B], [A, C], [B, D], [C, D]], A, 9, [[B, 18], [C, 27]], Yes, Yes, [A], local_uses_complete_declared_neighborhood_global_uses_complete_feasible_domain)","explanation":"Compare the candidate with every declared neighbor for graph-local minimality and with every feasible point for global minimality. Retain every global minimizer, including ties. Graph-locality is not a local optimum in an unspecified continuous topology, and a partial list is not a whole-domain certificate. The exact result, retaining the requested order, is ([[A, 9], [B, 18], [C, 27], [D, 36]], [[A, B], [A, C], [B, D], [C, D]], A, 9, [[B, 18], [C, 27]], Yes, Yes, [A], local_uses_complete_declared_neighborhood_global_uses_complete_feasible_domain)."},"skills":["Declared-Neighborhood Locality and Whole-Domain Globality Are Separate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r07-0010","batch_id":"batch-20261015-advance-20261119-r07-optimization-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3c8ca13bbf6e032e8dae6d87a2696c2a2cfbdc5dc3232d606fad184572651c97","family_id":"qf-mod50-optdeep-local-global","seed":202611197010,"semantic_template":"optdeep-local-global|{\"candidate\":\"A\",\"edges\":[[\"A\",\"B\"],[\"A\",\"C\"],[\"B\",\"D\"],[\"C\",\"D\"]],\"points\":[\"A\",\"B\",\"C\",\"D\"],\"values\":[[10,1],[20,1],[30,1],[40,1]]}"},"fields":{"title":"Declared-Neighborhood Locality and Whole-Domain Globality Are Separate","stem":"Use the complete finite feasible carrier=[A, B, C, D] with aligned exact objective values=[10, 20, 30, 40], candidate=A. Complete undirected neighborhood edges=[[A, B], [A, C], [B, D], [C, D]]. Give (complete_objective_table, undirected_neighborhood_edges, candidate, candidate_value, complete_neighbor_values, neighborhood_local_minimum, global_minimum, all_global_minimizers, comparison_scope).","answer":{"kind":"expression","value":"([[A, 10], [B, 20], [C, 30], [D, 40]], [[A, B], [A, C], [B, D], [C, D]], A, 10, [[B, 20], [C, 30]], Yes, Yes, [A], local_uses_complete_declared_neighborhood_global_uses_complete_feasible_domain)","explanation":"Compare the candidate with every declared neighbor for graph-local minimality and with every feasible point for global minimality. Retain every global minimizer, including ties. Graph-locality is not a local optimum in an unspecified continuous topology, and a partial list is not a whole-domain certificate. The exact result, retaining the requested order, is ([[A, 10], [B, 20], [C, 30], [D, 40]], [[A, B], [A, C], [B, D], [C, D]], A, 10, [[B, 20], [C, 30]], Yes, Yes, [A], local_uses_complete_declared_neighborhood_global_uses_complete_feasible_domain)."},"skills":["Declared-Neighborhood Locality and Whole-Domain Globality Are Separate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r07-0011","batch_id":"batch-20261015-advance-20261119-r07-optimization-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1a80758ebe5b6f5497db946a0b33a28a09d2a3907f48b2adf20215623c961729","family_id":"qf-mod50-optdeep-local-global","seed":202611197011,"semantic_template":"optdeep-local-global|{\"candidate\":\"A\",\"edges\":[[\"A\",\"B\"],[\"A\",\"C\"],[\"B\",\"D\"],[\"C\",\"D\"]],\"points\":[\"A\",\"B\",\"C\",\"D\"],\"values\":[[11,1],[22,1],[33,1],[44,1]]}"},"fields":{"title":"Declared-Neighborhood Locality and Whole-Domain Globality Are Separate","stem":"Use the complete finite feasible carrier=[A, B, C, D] with aligned exact objective values=[11, 22, 33, 44], candidate=A. Complete undirected neighborhood edges=[[A, B], [A, C], [B, D], [C, D]]. Give (complete_objective_table, undirected_neighborhood_edges, candidate, candidate_value, complete_neighbor_values, neighborhood_local_minimum, global_minimum, all_global_minimizers, comparison_scope).","answer":{"kind":"expression","value":"([[A, 11], [B, 22], [C, 33], [D, 44]], [[A, B], [A, C], [B, D], [C, D]], A, 11, [[B, 22], [C, 33]], Yes, Yes, [A], local_uses_complete_declared_neighborhood_global_uses_complete_feasible_domain)","explanation":"Compare the candidate with every declared neighbor for graph-local minimality and with every feasible point for global minimality. Retain every global minimizer, including ties. Graph-locality is not a local optimum in an unspecified continuous topology, and a partial list is not a whole-domain certificate. The exact result, retaining the requested order, is ([[A, 11], [B, 22], [C, 33], [D, 44]], [[A, B], [A, C], [B, D], [C, D]], A, 11, [[B, 22], [C, 33]], Yes, Yes, [A], local_uses_complete_declared_neighborhood_global_uses_complete_feasible_domain)."},"skills":["Declared-Neighborhood Locality and Whole-Domain Globality Are Separate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r07-0012","batch_id":"batch-20261015-advance-20261119-r07-optimization-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c549817d6042c3adac4c24ad60ae267fe42447f128cf68f31828ac3c1ccc8930","family_id":"qf-mod50-optdeep-local-global","seed":202611197012,"semantic_template":"optdeep-local-global|{\"candidate\":\"A\",\"edges\":[[\"A\",\"B\"],[\"A\",\"C\"],[\"B\",\"D\"],[\"C\",\"D\"]],\"points\":[\"A\",\"B\",\"C\",\"D\"],\"values\":[[12,1],[24,1],[36,1],[48,1]]}"},"fields":{"title":"Declared-Neighborhood Locality and Whole-Domain Globality Are Separate","stem":"Use the complete finite feasible carrier=[A, B, C, D] with aligned exact objective values=[12, 24, 36, 48], candidate=A. Complete undirected neighborhood edges=[[A, B], [A, C], [B, D], [C, D]]. Give (complete_objective_table, undirected_neighborhood_edges, candidate, candidate_value, complete_neighbor_values, neighborhood_local_minimum, global_minimum, all_global_minimizers, comparison_scope).","answer":{"kind":"expression","value":"([[A, 12], [B, 24], [C, 36], [D, 48]], [[A, B], [A, C], [B, D], [C, D]], A, 12, [[B, 24], [C, 36]], Yes, Yes, [A], local_uses_complete_declared_neighborhood_global_uses_complete_feasible_domain)","explanation":"Compare the candidate with every declared neighbor for graph-local minimality and with every feasible point for global minimality. Retain every global minimizer, including ties. Graph-locality is not a local optimum in an unspecified continuous topology, and a partial list is not a whole-domain certificate. The exact result, retaining the requested order, is ([[A, 12], [B, 24], [C, 36], [D, 48]], [[A, B], [A, C], [B, D], [C, D]], A, 12, [[B, 24], [C, 36]], Yes, Yes, [A], local_uses_complete_declared_neighborhood_global_uses_complete_feasible_domain)."},"skills":["Declared-Neighborhood Locality and Whole-Domain Globality Are Separate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r07-0013","batch_id":"batch-20261015-advance-20261119-r07-optimization-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6e3b1f501031c0baea1a821211e18b7ae07cab3b4a7a1be87bc7f4a3960355e1","family_id":"qf-mod50-optdeep-local-global","seed":202611197013,"semantic_template":"optdeep-local-global|{\"candidate\":\"A\",\"edges\":[[\"A\",\"B\"],[\"A\",\"C\"],[\"B\",\"D\"],[\"C\",\"D\"]],\"points\":[\"A\",\"B\",\"C\",\"D\"],\"values\":[[13,1],[26,1],[39,1],[52,1]]}"},"fields":{"title":"Declared-Neighborhood Locality and Whole-Domain Globality Are Separate","stem":"Use the complete finite feasible carrier=[A, B, C, D] with aligned exact objective values=[13, 26, 39, 52], candidate=A. Complete undirected neighborhood edges=[[A, B], [A, C], [B, D], [C, D]]. Give (complete_objective_table, undirected_neighborhood_edges, candidate, candidate_value, complete_neighbor_values, neighborhood_local_minimum, global_minimum, all_global_minimizers, comparison_scope).","answer":{"kind":"expression","value":"([[A, 13], [B, 26], [C, 39], [D, 52]], [[A, B], [A, C], [B, D], [C, D]], A, 13, [[B, 26], [C, 39]], Yes, Yes, [A], local_uses_complete_declared_neighborhood_global_uses_complete_feasible_domain)","explanation":"Compare the candidate with every declared neighbor for graph-local minimality and with every feasible point for global minimality. Retain every global minimizer, including ties. Graph-locality is not a local optimum in an unspecified continuous topology, and a partial list is not a whole-domain certificate. The exact result, retaining the requested order, is ([[A, 13], [B, 26], [C, 39], [D, 52]], [[A, B], [A, C], [B, D], [C, D]], A, 13, [[B, 26], [C, 39]], Yes, Yes, [A], local_uses_complete_declared_neighborhood_global_uses_complete_feasible_domain)."},"skills":["Declared-Neighborhood Locality and Whole-Domain Globality Are Separate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r07-0014","batch_id":"batch-20261015-advance-20261119-r07-optimization-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"77820a6d23779e5572d8a802e308df570bcd2641cfdf1f46d09618711301557d","family_id":"qf-mod50-optdeep-local-global","seed":202611197014,"semantic_template":"optdeep-local-global|{\"candidate\":\"A\",\"edges\":[[\"A\",\"B\"],[\"A\",\"C\"],[\"B\",\"D\"],[\"C\",\"D\"]],\"points\":[\"A\",\"B\",\"C\",\"D\"],\"values\":[[14,1],[28,1],[42,1],[56,1]]}"},"fields":{"title":"Declared-Neighborhood Locality and Whole-Domain Globality Are Separate","stem":"Use the complete finite feasible carrier=[A, B, C, D] with aligned exact objective values=[14, 28, 42, 56], candidate=A. Complete undirected neighborhood edges=[[A, B], [A, C], [B, D], [C, D]]. Give (complete_objective_table, undirected_neighborhood_edges, candidate, candidate_value, complete_neighbor_values, neighborhood_local_minimum, global_minimum, all_global_minimizers, comparison_scope).","answer":{"kind":"expression","value":"([[A, 14], [B, 28], [C, 42], [D, 56]], [[A, B], [A, C], [B, D], [C, D]], A, 14, [[B, 28], [C, 42]], Yes, Yes, [A], local_uses_complete_declared_neighborhood_global_uses_complete_feasible_domain)","explanation":"Compare the candidate with every declared neighbor for graph-local minimality and with every feasible point for global minimality. Retain every global minimizer, including ties. Graph-locality is not a local optimum in an unspecified continuous topology, and a partial list is not a whole-domain certificate. The exact result, retaining the requested order, is ([[A, 14], [B, 28], [C, 42], [D, 56]], [[A, B], [A, C], [B, D], [C, D]], A, 14, [[B, 28], [C, 42]], Yes, Yes, [A], local_uses_complete_declared_neighborhood_global_uses_complete_feasible_domain)."},"skills":["Declared-Neighborhood Locality and Whole-Domain Globality Are Separate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r07-0015","batch_id":"batch-20261015-advance-20261119-r07-optimization-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"09b16f680c4389f03442ea80853b562c0d25dcc3f60cb8b901b3bd389ebd8d2a","family_id":"qf-mod50-optdeep-local-global","seed":202611197015,"semantic_template":"optdeep-local-global|{\"candidate\":\"A\",\"edges\":[[\"A\",\"B\"],[\"A\",\"C\"],[\"B\",\"D\"],[\"C\",\"D\"]],\"points\":[\"A\",\"B\",\"C\",\"D\"],\"values\":[[15,1],[30,1],[45,1],[60,1]]}"},"fields":{"title":"Declared-Neighborhood Locality and Whole-Domain Globality Are Separate","stem":"Use the complete finite feasible carrier=[A, B, C, D] with aligned exact objective values=[15, 30, 45, 60], candidate=A. Complete undirected neighborhood edges=[[A, B], [A, C], [B, D], [C, D]]. Give (complete_objective_table, undirected_neighborhood_edges, candidate, candidate_value, complete_neighbor_values, neighborhood_local_minimum, global_minimum, all_global_minimizers, comparison_scope).","answer":{"kind":"expression","value":"([[A, 15], [B, 30], [C, 45], [D, 60]], [[A, B], [A, C], [B, D], [C, D]], A, 15, [[B, 30], [C, 45]], Yes, Yes, [A], local_uses_complete_declared_neighborhood_global_uses_complete_feasible_domain)","explanation":"Compare the candidate with every declared neighbor for graph-local minimality and with every feasible point for global minimality. Retain every global minimizer, including ties. Graph-locality is not a local optimum in an unspecified continuous topology, and a partial list is not a whole-domain certificate. The exact result, retaining the requested order, is ([[A, 15], [B, 30], [C, 45], [D, 60]], [[A, B], [A, C], [B, D], [C, D]], A, 15, [[B, 30], [C, 45]], Yes, Yes, [A], local_uses_complete_declared_neighborhood_global_uses_complete_feasible_domain)."},"skills":["Declared-Neighborhood Locality and Whole-Domain Globality Are Separate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r07-0016","batch_id":"batch-20261015-advance-20261119-r07-optimization-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0ae99d34147e7cbd41e434d5fec4db564e9b35514b99ce39d3f943ead1d16fd1","family_id":"qf-mod50-optdeep-local-global","seed":202611197016,"semantic_template":"optdeep-local-global|{\"candidate\":\"A\",\"edges\":[[\"A\",\"B\"],[\"A\",\"C\"],[\"B\",\"D\"],[\"C\",\"D\"]],\"points\":[\"A\",\"B\",\"C\",\"D\"],\"values\":[[16,1],[32,1],[48,1],[64,1]]}"},"fields":{"title":"Declared-Neighborhood Locality and Whole-Domain Globality Are Separate","stem":"Use the complete finite feasible carrier=[A, B, C, D] with aligned exact objective values=[16, 32, 48, 64], candidate=A. Complete undirected neighborhood edges=[[A, B], [A, C], [B, D], [C, D]]. Give (complete_objective_table, undirected_neighborhood_edges, candidate, candidate_value, complete_neighbor_values, neighborhood_local_minimum, global_minimum, all_global_minimizers, comparison_scope).","answer":{"kind":"expression","value":"([[A, 16], [B, 32], [C, 48], [D, 64]], [[A, B], [A, C], [B, D], [C, D]], A, 16, [[B, 32], [C, 48]], Yes, Yes, [A], local_uses_complete_declared_neighborhood_global_uses_complete_feasible_domain)","explanation":"Compare the candidate with every declared neighbor for graph-local minimality and with every feasible point for global minimality. Retain every global minimizer, including ties. Graph-locality is not a local optimum in an unspecified continuous topology, and a partial list is not a whole-domain certificate. The exact result, retaining the requested order, is ([[A, 16], [B, 32], [C, 48], [D, 64]], [[A, B], [A, C], [B, D], [C, D]], A, 16, [[B, 32], [C, 48]], Yes, Yes, [A], local_uses_complete_declared_neighborhood_global_uses_complete_feasible_domain)."},"skills":["Declared-Neighborhood Locality and Whole-Domain Globality Are Separate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r07-0017","batch_id":"batch-20261015-advance-20261119-r07-optimization-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ce4a8200758a73a734ebf242ec7a234fcf7e977b278c5154e6a96f277fb9ddaf","family_id":"qf-mod50-optdeep-local-global","seed":202611197017,"semantic_template":"optdeep-local-global|{\"candidate\":\"A\",\"edges\":[[\"A\",\"B\"],[\"A\",\"C\"],[\"B\",\"D\"],[\"C\",\"D\"]],\"points\":[\"A\",\"B\",\"C\",\"D\"],\"values\":[[17,1],[34,1],[51,1],[68,1]]}"},"fields":{"title":"Declared-Neighborhood Locality and Whole-Domain Globality Are Separate","stem":"Use the complete finite feasible carrier=[A, B, C, D] with aligned exact objective values=[17, 34, 51, 68], candidate=A. Complete undirected neighborhood edges=[[A, B], [A, C], [B, D], [C, D]]. Give (complete_objective_table, undirected_neighborhood_edges, candidate, candidate_value, complete_neighbor_values, neighborhood_local_minimum, global_minimum, all_global_minimizers, comparison_scope).","answer":{"kind":"expression","value":"([[A, 17], [B, 34], [C, 51], [D, 68]], [[A, B], [A, C], [B, D], [C, D]], A, 17, [[B, 34], [C, 51]], Yes, Yes, [A], local_uses_complete_declared_neighborhood_global_uses_complete_feasible_domain)","explanation":"Compare the candidate with every declared neighbor for graph-local minimality and with every feasible point for global minimality. Retain every global minimizer, including ties. Graph-locality is not a local optimum in an unspecified continuous topology, and a partial list is not a whole-domain certificate. The exact result, retaining the requested order, is ([[A, 17], [B, 34], [C, 51], [D, 68]], [[A, B], [A, C], [B, D], [C, D]], A, 17, [[B, 34], [C, 51]], Yes, Yes, [A], local_uses_complete_declared_neighborhood_global_uses_complete_feasible_domain)."},"skills":["Declared-Neighborhood Locality and Whole-Domain Globality Are Separate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r07-0018","batch_id":"batch-20261015-advance-20261119-r07-optimization-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"27bbec5019c141105c9ce4fb8f59b05aa3ebdfec99d14dc76e0fa20263a5e319","family_id":"qf-mod50-optdeep-local-global","seed":202611197018,"semantic_template":"optdeep-local-global|{\"candidate\":\"A\",\"edges\":[[\"A\",\"B\"],[\"A\",\"C\"],[\"B\",\"D\"],[\"C\",\"D\"]],\"points\":[\"A\",\"B\",\"C\",\"D\"],\"values\":[[18,1],[36,1],[54,1],[72,1]]}"},"fields":{"title":"Declared-Neighborhood Locality and Whole-Domain Globality Are Separate","stem":"Use the complete finite feasible carrier=[A, B, C, D] with aligned exact objective values=[18, 36, 54, 72], candidate=A. Complete undirected neighborhood edges=[[A, B], [A, C], [B, D], [C, D]]. Give (complete_objective_table, undirected_neighborhood_edges, candidate, candidate_value, complete_neighbor_values, neighborhood_local_minimum, global_minimum, all_global_minimizers, comparison_scope).","answer":{"kind":"expression","value":"([[A, 18], [B, 36], [C, 54], [D, 72]], [[A, B], [A, C], [B, D], [C, D]], A, 18, [[B, 36], [C, 54]], Yes, Yes, [A], local_uses_complete_declared_neighborhood_global_uses_complete_feasible_domain)","explanation":"Compare the candidate with every declared neighbor for graph-local minimality and with every feasible point for global minimality. Retain every global minimizer, including ties. Graph-locality is not a local optimum in an unspecified continuous topology, and a partial list is not a whole-domain certificate. The exact result, retaining the requested order, is ([[A, 18], [B, 36], [C, 54], [D, 72]], [[A, B], [A, C], [B, D], [C, D]], A, 18, [[B, 36], [C, 54]], Yes, Yes, [A], local_uses_complete_declared_neighborhood_global_uses_complete_feasible_domain)."},"skills":["Declared-Neighborhood Locality and Whole-Domain Globality Are Separate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r07-0019","batch_id":"batch-20261015-advance-20261119-r07-optimization-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"27e5a7cd047518a28987cfa60774c36cd192c56ff8300accf6cc1b9c4516c4f9","family_id":"qf-mod50-optdeep-local-global","seed":202611197019,"semantic_template":"optdeep-local-global|{\"candidate\":\"A\",\"edges\":[[\"A\",\"B\"],[\"A\",\"C\"],[\"B\",\"D\"],[\"C\",\"D\"]],\"points\":[\"A\",\"B\",\"C\",\"D\"],\"values\":[[19,1],[38,1],[57,1],[76,1]]}"},"fields":{"title":"Declared-Neighborhood Locality and Whole-Domain Globality Are Separate","stem":"Use the complete finite feasible carrier=[A, B, C, D] with aligned exact objective values=[19, 38, 57, 76], candidate=A. Complete undirected neighborhood edges=[[A, B], [A, C], [B, D], [C, D]]. Give (complete_objective_table, undirected_neighborhood_edges, candidate, candidate_value, complete_neighbor_values, neighborhood_local_minimum, global_minimum, all_global_minimizers, comparison_scope).","answer":{"kind":"expression","value":"([[A, 19], [B, 38], [C, 57], [D, 76]], [[A, B], [A, C], [B, D], [C, D]], A, 19, [[B, 38], [C, 57]], Yes, Yes, [A], local_uses_complete_declared_neighborhood_global_uses_complete_feasible_domain)","explanation":"Compare the candidate with every declared neighbor for graph-local minimality and with every feasible point for global minimality. Retain every global minimizer, including ties. Graph-locality is not a local optimum in an unspecified continuous topology, and a partial list is not a whole-domain certificate. The exact result, retaining the requested order, is ([[A, 19], [B, 38], [C, 57], [D, 76]], [[A, B], [A, C], [B, D], [C, D]], A, 19, [[B, 38], [C, 57]], Yes, Yes, [A], local_uses_complete_declared_neighborhood_global_uses_complete_feasible_domain)."},"skills":["Declared-Neighborhood Locality and Whole-Domain Globality Are Separate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r07-0020","batch_id":"batch-20261015-advance-20261119-r07-optimization-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7bea3bb187f50b6b8389ecbf29fe5d4324692618e3c1d3a355ddc82ca1affe3c","family_id":"qf-mod50-optdeep-local-global","seed":202611197020,"semantic_template":"optdeep-local-global|{\"candidate\":\"A\",\"edges\":[[\"A\",\"B\"],[\"A\",\"C\"],[\"B\",\"D\"],[\"C\",\"D\"]],\"points\":[\"A\",\"B\",\"C\",\"D\"],\"values\":[[20,1],[40,1],[60,1],[80,1]]}"},"fields":{"title":"Declared-Neighborhood Locality and Whole-Domain Globality Are Separate","stem":"Use the complete finite feasible carrier=[A, B, C, D] with aligned exact objective values=[20, 40, 60, 80], candidate=A. Complete undirected neighborhood edges=[[A, B], [A, C], [B, D], [C, D]]. Give (complete_objective_table, undirected_neighborhood_edges, candidate, candidate_value, complete_neighbor_values, neighborhood_local_minimum, global_minimum, all_global_minimizers, comparison_scope).","answer":{"kind":"expression","value":"([[A, 20], [B, 40], [C, 60], [D, 80]], [[A, B], [A, C], [B, D], [C, D]], A, 20, [[B, 40], [C, 60]], Yes, Yes, [A], local_uses_complete_declared_neighborhood_global_uses_complete_feasible_domain)","explanation":"Compare the candidate with every declared neighbor for graph-local minimality and with every feasible point for global minimality. Retain every global minimizer, including ties. Graph-locality is not a local optimum in an unspecified continuous topology, and a partial list is not a whole-domain certificate. The exact result, retaining the requested order, is ([[A, 20], [B, 40], [C, 60], [D, 80]], [[A, B], [A, C], [B, D], [C, D]], A, 20, [[B, 40], [C, 60]], Yes, Yes, [A], local_uses_complete_declared_neighborhood_global_uses_complete_feasible_domain)."},"skills":["Declared-Neighborhood Locality and Whole-Domain Globality Are Separate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r07-0021","batch_id":"batch-20261015-advance-20261119-r07-optimization-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0022cba58de3b8b6724a6b2d1ec7a872d19c5336afb81672a59ec7a06650dd4f","family_id":"qf-mod50-optdeep-local-global","seed":202611197021,"semantic_template":"optdeep-local-global|{\"candidate\":\"A\",\"edges\":[[\"A\",\"B\"],[\"A\",\"C\"],[\"B\",\"D\"],[\"C\",\"D\"]],\"points\":[\"A\",\"B\",\"C\",\"D\"],\"values\":[[21,1],[42,1],[63,1],[84,1]]}"},"fields":{"title":"Declared-Neighborhood Locality and Whole-Domain Globality Are Separate","stem":"Use the complete finite feasible carrier=[A, B, C, D] with aligned exact objective values=[21, 42, 63, 84], candidate=A. Complete undirected neighborhood edges=[[A, B], [A, C], [B, D], [C, D]]. Give (complete_objective_table, undirected_neighborhood_edges, candidate, candidate_value, complete_neighbor_values, neighborhood_local_minimum, global_minimum, all_global_minimizers, comparison_scope).","answer":{"kind":"expression","value":"([[A, 21], [B, 42], [C, 63], [D, 84]], [[A, B], [A, C], [B, D], [C, D]], A, 21, [[B, 42], [C, 63]], Yes, Yes, [A], local_uses_complete_declared_neighborhood_global_uses_complete_feasible_domain)","explanation":"Compare the candidate with every declared neighbor for graph-local minimality and with every feasible point for global minimality. Retain every global minimizer, including ties. Graph-locality is not a local optimum in an unspecified continuous topology, and a partial list is not a whole-domain certificate. The exact result, retaining the requested order, is ([[A, 21], [B, 42], [C, 63], [D, 84]], [[A, B], [A, C], [B, D], [C, D]], A, 21, [[B, 42], [C, 63]], Yes, Yes, [A], local_uses_complete_declared_neighborhood_global_uses_complete_feasible_domain)."},"skills":["Declared-Neighborhood Locality and Whole-Domain Globality Are Separate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r07-0022","batch_id":"batch-20261015-advance-20261119-r07-optimization-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a79a401d4fad87dac0bf3665e01b563841d3dc0a5301a38e494e0be8a0ba22ba","family_id":"qf-mod50-optdeep-local-global","seed":202611197022,"semantic_template":"optdeep-local-global|{\"candidate\":\"A\",\"edges\":[[\"A\",\"B\"],[\"A\",\"C\"],[\"B\",\"D\"],[\"C\",\"D\"]],\"points\":[\"A\",\"B\",\"C\",\"D\"],\"values\":[[22,1],[44,1],[66,1],[88,1]]}"},"fields":{"title":"Declared-Neighborhood Locality and Whole-Domain Globality Are Separate","stem":"Use the complete finite feasible carrier=[A, B, C, D] with aligned exact objective values=[22, 44, 66, 88], candidate=A. Complete undirected neighborhood edges=[[A, B], [A, C], [B, D], [C, D]]. Give (complete_objective_table, undirected_neighborhood_edges, candidate, candidate_value, complete_neighbor_values, neighborhood_local_minimum, global_minimum, all_global_minimizers, comparison_scope).","answer":{"kind":"expression","value":"([[A, 22], [B, 44], [C, 66], [D, 88]], [[A, B], [A, C], [B, D], [C, D]], A, 22, [[B, 44], [C, 66]], Yes, Yes, [A], local_uses_complete_declared_neighborhood_global_uses_complete_feasible_domain)","explanation":"Compare the candidate with every declared neighbor for graph-local minimality and with every feasible point for global minimality. Retain every global minimizer, including ties. Graph-locality is not a local optimum in an unspecified continuous topology, and a partial list is not a whole-domain certificate. The exact result, retaining the requested order, is ([[A, 22], [B, 44], [C, 66], [D, 88]], [[A, B], [A, C], [B, D], [C, D]], A, 22, [[B, 44], [C, 66]], Yes, Yes, [A], local_uses_complete_declared_neighborhood_global_uses_complete_feasible_domain)."},"skills":["Declared-Neighborhood Locality and Whole-Domain Globality Are Separate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r07-0023","batch_id":"batch-20261015-advance-20261119-r07-optimization-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"13b82cf81c6502961387830d3bc7184dc76c41aa81c54c18812f19e5b6ad7d3b","family_id":"qf-mod50-optdeep-local-global","seed":202611197023,"semantic_template":"optdeep-local-global|{\"candidate\":\"A\",\"edges\":[[\"A\",\"B\"],[\"A\",\"C\"],[\"B\",\"D\"],[\"C\",\"D\"]],\"points\":[\"A\",\"B\",\"C\",\"D\"],\"values\":[[23,1],[46,1],[69,1],[92,1]]}"},"fields":{"title":"Declared-Neighborhood Locality and Whole-Domain Globality Are Separate","stem":"Use the complete finite feasible carrier=[A, B, C, D] with aligned exact objective values=[23, 46, 69, 92], candidate=A. Complete undirected neighborhood edges=[[A, B], [A, C], [B, D], [C, D]]. Give (complete_objective_table, undirected_neighborhood_edges, candidate, candidate_value, complete_neighbor_values, neighborhood_local_minimum, global_minimum, all_global_minimizers, comparison_scope).","answer":{"kind":"expression","value":"([[A, 23], [B, 46], [C, 69], [D, 92]], [[A, B], [A, C], [B, D], [C, D]], A, 23, [[B, 46], [C, 69]], Yes, Yes, [A], local_uses_complete_declared_neighborhood_global_uses_complete_feasible_domain)","explanation":"Compare the candidate with every declared neighbor for graph-local minimality and with every feasible point for global minimality. Retain every global minimizer, including ties. Graph-locality is not a local optimum in an unspecified continuous topology, and a partial list is not a whole-domain certificate. The exact result, retaining the requested order, is ([[A, 23], [B, 46], [C, 69], [D, 92]], [[A, B], [A, C], [B, D], [C, D]], A, 23, [[B, 46], [C, 69]], Yes, Yes, [A], local_uses_complete_declared_neighborhood_global_uses_complete_feasible_domain)."},"skills":["Declared-Neighborhood Locality and Whole-Domain Globality Are Separate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r07-0024","batch_id":"batch-20261015-advance-20261119-r07-optimization-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"12b5c8e932f230c1fbe8e099ef8d4a19bd508c247ea009523955447aa27aa381","family_id":"qf-mod50-optdeep-local-global","seed":202611197024,"semantic_template":"optdeep-local-global|{\"candidate\":\"A\",\"edges\":[[\"A\",\"B\"],[\"A\",\"C\"],[\"B\",\"D\"],[\"C\",\"D\"]],\"points\":[\"A\",\"B\",\"C\",\"D\"],\"values\":[[24,1],[48,1],[72,1],[96,1]]}"},"fields":{"title":"Declared-Neighborhood Locality and Whole-Domain Globality Are Separate","stem":"Use the complete finite feasible carrier=[A, B, C, D] with aligned exact objective values=[24, 48, 72, 96], candidate=A. Complete undirected neighborhood edges=[[A, B], [A, C], [B, D], [C, D]]. Give (complete_objective_table, undirected_neighborhood_edges, candidate, candidate_value, complete_neighbor_values, neighborhood_local_minimum, global_minimum, all_global_minimizers, comparison_scope).","answer":{"kind":"expression","value":"([[A, 24], [B, 48], [C, 72], [D, 96]], [[A, B], [A, C], [B, D], [C, D]], A, 24, [[B, 48], [C, 72]], Yes, Yes, [A], local_uses_complete_declared_neighborhood_global_uses_complete_feasible_domain)","explanation":"Compare the candidate with every declared neighbor for graph-local minimality and with every feasible point for global minimality. Retain every global minimizer, including ties. Graph-locality is not a local optimum in an unspecified continuous topology, and a partial list is not a whole-domain certificate. The exact result, retaining the requested order, is ([[A, 24], [B, 48], [C, 72], [D, 96]], [[A, B], [A, C], [B, D], [C, D]], A, 24, [[B, 48], [C, 72]], Yes, Yes, [A], local_uses_complete_declared_neighborhood_global_uses_complete_feasible_domain)."},"skills":["Declared-Neighborhood Locality and Whole-Domain Globality Are Separate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r07-0025","batch_id":"batch-20261015-advance-20261119-r07-optimization-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4a185594969e23fc4c039d75e0b6d2bb4c0706b5fdc7154d30765cbde66331e2","family_id":"qf-mod50-optdeep-local-global","seed":202611197025,"semantic_template":"optdeep-local-global|{\"candidate\":\"A\",\"edges\":[[\"A\",\"B\"],[\"A\",\"C\"],[\"B\",\"D\"],[\"C\",\"D\"]],\"points\":[\"A\",\"B\",\"C\",\"D\"],\"values\":[[25,1],[50,1],[75,1],[100,1]]}"},"fields":{"title":"Declared-Neighborhood Locality and Whole-Domain Globality Are Separate","stem":"Use the complete finite feasible carrier=[A, B, C, D] with aligned exact objective values=[25, 50, 75, 100], candidate=A. Complete undirected neighborhood edges=[[A, B], [A, C], [B, D], [C, D]]. Give (complete_objective_table, undirected_neighborhood_edges, candidate, candidate_value, complete_neighbor_values, neighborhood_local_minimum, global_minimum, all_global_minimizers, comparison_scope).","answer":{"kind":"expression","value":"([[A, 25], [B, 50], [C, 75], [D, 100]], [[A, B], [A, C], [B, D], [C, D]], A, 25, [[B, 50], [C, 75]], Yes, Yes, [A], local_uses_complete_declared_neighborhood_global_uses_complete_feasible_domain)","explanation":"Compare the candidate with every declared neighbor for graph-local minimality and with every feasible point for global minimality. Retain every global minimizer, including ties. Graph-locality is not a local optimum in an unspecified continuous topology, and a partial list is not a whole-domain certificate. The exact result, retaining the requested order, is ([[A, 25], [B, 50], [C, 75], [D, 100]], [[A, B], [A, C], [B, D], [C, D]], A, 25, [[B, 50], [C, 75]], Yes, Yes, [A], local_uses_complete_declared_neighborhood_global_uses_complete_feasible_domain)."},"skills":["Declared-Neighborhood Locality and Whole-Domain Globality Are Separate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r07-0026","batch_id":"batch-20261015-advance-20261119-r07-optimization-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b5941bb0453fb1c684ef4d1e0a9489e744c04d47f748c113ab7159d97dba61b8","family_id":"qf-mod50-optdeep-improvement-witness","seed":202611197026,"semantic_template":"optdeep-improvement-witness|{\"candidate\":\"A\",\"challenger\":\"B\",\"points\":[\"A\",\"B\",\"C\",\"D\"],\"values\":[[3,1],[1,1],[4,1],[5,1]]}"},"fields":{"title":"One Strictly Better Feasible Point Refutes Candidate Global Optimality","stem":"Use the complete finite feasible carrier=[A, B, C, D] with aligned exact objective values=[3, 1, 4, 5], candidate=A. The declared feasible challenger is=B. Give (complete_objective_table, candidate, candidate_value, challenger, challenger_value, strict_feasible_improvement, candidate_global_optimality_claim, refutation_scope).","answer":{"kind":"expression","value":"([[A, 3], [B, 1], [C, 4], [D, 5]], A, 3, B, 1, Yes, No, one_strictly_better_feasible_point_refutes_candidate_global_optimality)","explanation":"Verify the challenger is distinct and has a strictly lower exact objective value. This is a refutation of the candidate global claim, not a proof that the challenger is globally optimal or that the candidate fails every neighborhood-local condition. The exact result, retaining the requested order, is ([[A, 3], [B, 1], [C, 4], [D, 5]], A, 3, B, 1, Yes, No, one_strictly_better_feasible_point_refutes_candidate_global_optimality)."},"skills":["One Strictly Better Feasible Point Refutes Candidate Global Optimality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r07-0027","batch_id":"batch-20261015-advance-20261119-r07-optimization-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c82f4eed1da46b88ab2c8d466c17c9a5e39a648c9d0ac87c5f9e5e2e620fcf2d","family_id":"qf-mod50-optdeep-improvement-witness","seed":202611197027,"semantic_template":"optdeep-improvement-witness|{\"candidate\":\"A\",\"challenger\":\"B\",\"points\":[\"A\",\"B\",\"C\",\"D\"],\"values\":[[6,1],[2,1],[8,1],[10,1]]}"},"fields":{"title":"One Strictly Better Feasible Point Refutes Candidate Global Optimality","stem":"Use the complete finite feasible carrier=[A, B, C, D] with aligned exact objective values=[6, 2, 8, 10], candidate=A. The declared feasible challenger is=B. Give (complete_objective_table, candidate, candidate_value, challenger, challenger_value, strict_feasible_improvement, candidate_global_optimality_claim, refutation_scope).","answer":{"kind":"expression","value":"([[A, 6], [B, 2], [C, 8], [D, 10]], A, 6, B, 2, Yes, No, one_strictly_better_feasible_point_refutes_candidate_global_optimality)","explanation":"Verify the challenger is distinct and has a strictly lower exact objective value. This is a refutation of the candidate global claim, not a proof that the challenger is globally optimal or that the candidate fails every neighborhood-local condition. The exact result, retaining the requested order, is ([[A, 6], [B, 2], [C, 8], [D, 10]], A, 6, B, 2, Yes, No, one_strictly_better_feasible_point_refutes_candidate_global_optimality)."},"skills":["One Strictly Better Feasible Point Refutes Candidate Global Optimality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r07-0028","batch_id":"batch-20261015-advance-20261119-r07-optimization-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"33cd4416b40fd650ad019505bce1e00838dbcbdc0ad475c28cbba142ed8fec4e","family_id":"qf-mod50-optdeep-improvement-witness","seed":202611197028,"semantic_template":"optdeep-improvement-witness|{\"candidate\":\"A\",\"challenger\":\"B\",\"points\":[\"A\",\"B\",\"C\",\"D\"],\"values\":[[9,1],[3,1],[12,1],[15,1]]}"},"fields":{"title":"One Strictly Better Feasible Point Refutes Candidate Global Optimality","stem":"Use the complete finite feasible carrier=[A, B, C, D] with aligned exact objective values=[9, 3, 12, 15], candidate=A. The declared feasible challenger is=B. Give (complete_objective_table, candidate, candidate_value, challenger, challenger_value, strict_feasible_improvement, candidate_global_optimality_claim, refutation_scope).","answer":{"kind":"expression","value":"([[A, 9], [B, 3], [C, 12], [D, 15]], A, 9, B, 3, Yes, No, one_strictly_better_feasible_point_refutes_candidate_global_optimality)","explanation":"Verify the challenger is distinct and has a strictly lower exact objective value. This is a refutation of the candidate global claim, not a proof that the challenger is globally optimal or that the candidate fails every neighborhood-local condition. The exact result, retaining the requested order, is ([[A, 9], [B, 3], [C, 12], [D, 15]], A, 9, B, 3, Yes, No, one_strictly_better_feasible_point_refutes_candidate_global_optimality)."},"skills":["One Strictly Better Feasible Point Refutes Candidate Global Optimality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r07-0029","batch_id":"batch-20261015-advance-20261119-r07-optimization-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"71f3b34689f8d62ae816ba2f637f04bb0f6bc5c512700b64fbf7df589a02cf09","family_id":"qf-mod50-optdeep-improvement-witness","seed":202611197029,"semantic_template":"optdeep-improvement-witness|{\"candidate\":\"A\",\"challenger\":\"B\",\"points\":[\"A\",\"B\",\"C\",\"D\"],\"values\":[[12,1],[4,1],[16,1],[20,1]]}"},"fields":{"title":"One Strictly Better Feasible Point Refutes Candidate Global Optimality","stem":"Use the complete finite feasible carrier=[A, B, C, D] with aligned exact objective values=[12, 4, 16, 20], candidate=A. The declared feasible challenger is=B. Give (complete_objective_table, candidate, candidate_value, challenger, challenger_value, strict_feasible_improvement, candidate_global_optimality_claim, refutation_scope).","answer":{"kind":"expression","value":"([[A, 12], [B, 4], [C, 16], [D, 20]], A, 12, B, 4, Yes, No, one_strictly_better_feasible_point_refutes_candidate_global_optimality)","explanation":"Verify the challenger is distinct and has a strictly lower exact objective value. This is a refutation of the candidate global claim, not a proof that the challenger is globally optimal or that the candidate fails every neighborhood-local condition. The exact result, retaining the requested order, is ([[A, 12], [B, 4], [C, 16], [D, 20]], A, 12, B, 4, Yes, No, one_strictly_better_feasible_point_refutes_candidate_global_optimality)."},"skills":["One Strictly Better Feasible Point Refutes Candidate Global Optimality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r07-0030","batch_id":"batch-20261015-advance-20261119-r07-optimization-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"73c45371d5a37b6b32706d5374d9ff279d2be3a238f80fa3c04666abf36f125e","family_id":"qf-mod50-optdeep-improvement-witness","seed":202611197030,"semantic_template":"optdeep-improvement-witness|{\"candidate\":\"A\",\"challenger\":\"B\",\"points\":[\"A\",\"B\",\"C\",\"D\"],\"values\":[[15,1],[5,1],[20,1],[25,1]]}"},"fields":{"title":"One Strictly Better Feasible Point Refutes Candidate Global Optimality","stem":"Use the complete finite feasible carrier=[A, B, C, D] with aligned exact objective values=[15, 5, 20, 25], candidate=A. The declared feasible challenger is=B. Give (complete_objective_table, candidate, candidate_value, challenger, challenger_value, strict_feasible_improvement, candidate_global_optimality_claim, refutation_scope).","answer":{"kind":"expression","value":"([[A, 15], [B, 5], [C, 20], [D, 25]], A, 15, B, 5, Yes, No, one_strictly_better_feasible_point_refutes_candidate_global_optimality)","explanation":"Verify the challenger is distinct and has a strictly lower exact objective value. This is a refutation of the candidate global claim, not a proof that the challenger is globally optimal or that the candidate fails every neighborhood-local condition. The exact result, retaining the requested order, is ([[A, 15], [B, 5], [C, 20], [D, 25]], A, 15, B, 5, Yes, No, one_strictly_better_feasible_point_refutes_candidate_global_optimality)."},"skills":["One Strictly Better Feasible Point Refutes Candidate Global Optimality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r07-0031","batch_id":"batch-20261015-advance-20261119-r07-optimization-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"56ff5cf1f674a9748a8a9eed9d1d1263a71cddc373dd36482d1f26ddf765c4b3","family_id":"qf-mod50-optdeep-improvement-witness","seed":202611197031,"semantic_template":"optdeep-improvement-witness|{\"candidate\":\"A\",\"challenger\":\"B\",\"points\":[\"A\",\"B\",\"C\",\"D\"],\"values\":[[18,1],[6,1],[24,1],[30,1]]}"},"fields":{"title":"One Strictly Better Feasible Point Refutes Candidate Global Optimality","stem":"Use the complete finite feasible carrier=[A, B, C, D] with aligned exact objective values=[18, 6, 24, 30], candidate=A. The declared feasible challenger is=B. Give (complete_objective_table, candidate, candidate_value, challenger, challenger_value, strict_feasible_improvement, candidate_global_optimality_claim, refutation_scope).","answer":{"kind":"expression","value":"([[A, 18], [B, 6], [C, 24], [D, 30]], A, 18, B, 6, Yes, No, one_strictly_better_feasible_point_refutes_candidate_global_optimality)","explanation":"Verify the challenger is distinct and has a strictly lower exact objective value. This is a refutation of the candidate global claim, not a proof that the challenger is globally optimal or that the candidate fails every neighborhood-local condition. The exact result, retaining the requested order, is ([[A, 18], [B, 6], [C, 24], [D, 30]], A, 18, B, 6, Yes, No, one_strictly_better_feasible_point_refutes_candidate_global_optimality)."},"skills":["One Strictly Better Feasible Point Refutes Candidate Global Optimality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r07-0032","batch_id":"batch-20261015-advance-20261119-r07-optimization-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8cb3ed4d712b901df77605d86673634a9eb0d918442e6d7d50b2975dad7c9699","family_id":"qf-mod50-optdeep-improvement-witness","seed":202611197032,"semantic_template":"optdeep-improvement-witness|{\"candidate\":\"A\",\"challenger\":\"B\",\"points\":[\"A\",\"B\",\"C\",\"D\"],\"values\":[[21,1],[7,1],[28,1],[35,1]]}"},"fields":{"title":"One Strictly Better Feasible Point Refutes Candidate Global Optimality","stem":"Use the complete finite feasible carrier=[A, B, C, D] with aligned exact objective values=[21, 7, 28, 35], candidate=A. The declared feasible challenger is=B. Give (complete_objective_table, candidate, candidate_value, challenger, challenger_value, strict_feasible_improvement, candidate_global_optimality_claim, refutation_scope).","answer":{"kind":"expression","value":"([[A, 21], [B, 7], [C, 28], [D, 35]], A, 21, B, 7, Yes, No, one_strictly_better_feasible_point_refutes_candidate_global_optimality)","explanation":"Verify the challenger is distinct and has a strictly lower exact objective value. This is a refutation of the candidate global claim, not a proof that the challenger is globally optimal or that the candidate fails every neighborhood-local condition. The exact result, retaining the requested order, is ([[A, 21], [B, 7], [C, 28], [D, 35]], A, 21, B, 7, Yes, No, one_strictly_better_feasible_point_refutes_candidate_global_optimality)."},"skills":["One Strictly Better Feasible Point Refutes Candidate Global Optimality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r07-0033","batch_id":"batch-20261015-advance-20261119-r07-optimization-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d6ef3e2e8cd52aeb4bd8d731aaa4b7e74f1e64484736068971789ccefba2f5bc","family_id":"qf-mod50-optdeep-improvement-witness","seed":202611197033,"semantic_template":"optdeep-improvement-witness|{\"candidate\":\"A\",\"challenger\":\"B\",\"points\":[\"A\",\"B\",\"C\",\"D\"],\"values\":[[24,1],[8,1],[32,1],[40,1]]}"},"fields":{"title":"One Strictly Better Feasible Point Refutes Candidate Global Optimality","stem":"Use the complete finite feasible carrier=[A, B, C, D] with aligned exact objective values=[24, 8, 32, 40], candidate=A. The declared feasible challenger is=B. Give (complete_objective_table, candidate, candidate_value, challenger, challenger_value, strict_feasible_improvement, candidate_global_optimality_claim, refutation_scope).","answer":{"kind":"expression","value":"([[A, 24], [B, 8], [C, 32], [D, 40]], A, 24, B, 8, Yes, No, one_strictly_better_feasible_point_refutes_candidate_global_optimality)","explanation":"Verify the challenger is distinct and has a strictly lower exact objective value. This is a refutation of the candidate global claim, not a proof that the challenger is globally optimal or that the candidate fails every neighborhood-local condition. The exact result, retaining the requested order, is ([[A, 24], [B, 8], [C, 32], [D, 40]], A, 24, B, 8, Yes, No, one_strictly_better_feasible_point_refutes_candidate_global_optimality)."},"skills":["One Strictly Better Feasible Point Refutes Candidate Global Optimality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r07-0034","batch_id":"batch-20261015-advance-20261119-r07-optimization-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e019140f5c50aaf65f3d1a6dc19b2e3ed841aeaac67309e15502f6fdb4dc0d17","family_id":"qf-mod50-optdeep-improvement-witness","seed":202611197034,"semantic_template":"optdeep-improvement-witness|{\"candidate\":\"A\",\"challenger\":\"B\",\"points\":[\"A\",\"B\",\"C\",\"D\"],\"values\":[[27,1],[9,1],[36,1],[45,1]]}"},"fields":{"title":"One Strictly Better Feasible Point Refutes Candidate Global Optimality","stem":"Use the complete finite feasible carrier=[A, B, C, D] with aligned exact objective values=[27, 9, 36, 45], candidate=A. The declared feasible challenger is=B. Give (complete_objective_table, candidate, candidate_value, challenger, challenger_value, strict_feasible_improvement, candidate_global_optimality_claim, refutation_scope).","answer":{"kind":"expression","value":"([[A, 27], [B, 9], [C, 36], [D, 45]], A, 27, B, 9, Yes, No, one_strictly_better_feasible_point_refutes_candidate_global_optimality)","explanation":"Verify the challenger is distinct and has a strictly lower exact objective value. This is a refutation of the candidate global claim, not a proof that the challenger is globally optimal or that the candidate fails every neighborhood-local condition. The exact result, retaining the requested order, is ([[A, 27], [B, 9], [C, 36], [D, 45]], A, 27, B, 9, Yes, No, one_strictly_better_feasible_point_refutes_candidate_global_optimality)."},"skills":["One Strictly Better Feasible Point Refutes Candidate Global Optimality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r07-0035","batch_id":"batch-20261015-advance-20261119-r07-optimization-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3c992b6a95c45c750c85386481bc38b21cdd56d68d25f1fe6462b44674e7e08f","family_id":"qf-mod50-optdeep-improvement-witness","seed":202611197035,"semantic_template":"optdeep-improvement-witness|{\"candidate\":\"A\",\"challenger\":\"B\",\"points\":[\"A\",\"B\",\"C\",\"D\"],\"values\":[[30,1],[10,1],[40,1],[50,1]]}"},"fields":{"title":"One Strictly Better Feasible Point Refutes Candidate Global Optimality","stem":"Use the complete finite feasible carrier=[A, B, C, D] with aligned exact objective values=[30, 10, 40, 50], candidate=A. The declared feasible challenger is=B. Give (complete_objective_table, candidate, candidate_value, challenger, challenger_value, strict_feasible_improvement, candidate_global_optimality_claim, refutation_scope).","answer":{"kind":"expression","value":"([[A, 30], [B, 10], [C, 40], [D, 50]], A, 30, B, 10, Yes, No, one_strictly_better_feasible_point_refutes_candidate_global_optimality)","explanation":"Verify the challenger is distinct and has a strictly lower exact objective value. This is a refutation of the candidate global claim, not a proof that the challenger is globally optimal or that the candidate fails every neighborhood-local condition. The exact result, retaining the requested order, is ([[A, 30], [B, 10], [C, 40], [D, 50]], A, 30, B, 10, Yes, No, one_strictly_better_feasible_point_refutes_candidate_global_optimality)."},"skills":["One Strictly Better Feasible Point Refutes Candidate Global Optimality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r07-0036","batch_id":"batch-20261015-advance-20261119-r07-optimization-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1451db270d336acac0c64d8aaf43ef4b3016a28e888e5b72f4981178db621d14","family_id":"qf-mod50-optdeep-improvement-witness","seed":202611197036,"semantic_template":"optdeep-improvement-witness|{\"candidate\":\"A\",\"challenger\":\"B\",\"points\":[\"A\",\"B\",\"C\",\"D\"],\"values\":[[33,1],[11,1],[44,1],[55,1]]}"},"fields":{"title":"One Strictly Better Feasible Point Refutes Candidate Global Optimality","stem":"Use the complete finite feasible carrier=[A, B, C, D] with aligned exact objective values=[33, 11, 44, 55], candidate=A. The declared feasible challenger is=B. Give (complete_objective_table, candidate, candidate_value, challenger, challenger_value, strict_feasible_improvement, candidate_global_optimality_claim, refutation_scope).","answer":{"kind":"expression","value":"([[A, 33], [B, 11], [C, 44], [D, 55]], A, 33, B, 11, Yes, No, one_strictly_better_feasible_point_refutes_candidate_global_optimality)","explanation":"Verify the challenger is distinct and has a strictly lower exact objective value. This is a refutation of the candidate global claim, not a proof that the challenger is globally optimal or that the candidate fails every neighborhood-local condition. The exact result, retaining the requested order, is ([[A, 33], [B, 11], [C, 44], [D, 55]], A, 33, B, 11, Yes, No, one_strictly_better_feasible_point_refutes_candidate_global_optimality)."},"skills":["One Strictly Better Feasible Point Refutes Candidate Global Optimality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r07-0037","batch_id":"batch-20261015-advance-20261119-r07-optimization-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6d43ae05c7a3b4e5a1c42bdff1bde3fba07d015914b6292384b0e10c92dab026","family_id":"qf-mod50-optdeep-improvement-witness","seed":202611197037,"semantic_template":"optdeep-improvement-witness|{\"candidate\":\"A\",\"challenger\":\"B\",\"points\":[\"A\",\"B\",\"C\",\"D\"],\"values\":[[36,1],[12,1],[48,1],[60,1]]}"},"fields":{"title":"One Strictly Better Feasible Point Refutes Candidate Global Optimality","stem":"Use the complete finite feasible carrier=[A, B, C, D] with aligned exact objective values=[36, 12, 48, 60], candidate=A. The declared feasible challenger is=B. Give (complete_objective_table, candidate, candidate_value, challenger, challenger_value, strict_feasible_improvement, candidate_global_optimality_claim, refutation_scope).","answer":{"kind":"expression","value":"([[A, 36], [B, 12], [C, 48], [D, 60]], A, 36, B, 12, Yes, No, one_strictly_better_feasible_point_refutes_candidate_global_optimality)","explanation":"Verify the challenger is distinct and has a strictly lower exact objective value. This is a refutation of the candidate global claim, not a proof that the challenger is globally optimal or that the candidate fails every neighborhood-local condition. The exact result, retaining the requested order, is ([[A, 36], [B, 12], [C, 48], [D, 60]], A, 36, B, 12, Yes, No, one_strictly_better_feasible_point_refutes_candidate_global_optimality)."},"skills":["One Strictly Better Feasible Point Refutes Candidate Global Optimality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r07-0038","batch_id":"batch-20261015-advance-20261119-r07-optimization-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3bacbde8fc309f28ad425c28524e5c8ffa994ff37327e062e468d4cb75c45e59","family_id":"qf-mod50-optdeep-improvement-witness","seed":202611197038,"semantic_template":"optdeep-improvement-witness|{\"candidate\":\"A\",\"challenger\":\"B\",\"points\":[\"A\",\"B\",\"C\",\"D\"],\"values\":[[39,1],[13,1],[52,1],[65,1]]}"},"fields":{"title":"One Strictly Better Feasible Point Refutes Candidate Global Optimality","stem":"Use the complete finite feasible carrier=[A, B, C, D] with aligned exact objective values=[39, 13, 52, 65], candidate=A. The declared feasible challenger is=B. Give (complete_objective_table, candidate, candidate_value, challenger, challenger_value, strict_feasible_improvement, candidate_global_optimality_claim, refutation_scope).","answer":{"kind":"expression","value":"([[A, 39], [B, 13], [C, 52], [D, 65]], A, 39, B, 13, Yes, No, one_strictly_better_feasible_point_refutes_candidate_global_optimality)","explanation":"Verify the challenger is distinct and has a strictly lower exact objective value. This is a refutation of the candidate global claim, not a proof that the challenger is globally optimal or that the candidate fails every neighborhood-local condition. The exact result, retaining the requested order, is ([[A, 39], [B, 13], [C, 52], [D, 65]], A, 39, B, 13, Yes, No, one_strictly_better_feasible_point_refutes_candidate_global_optimality)."},"skills":["One Strictly Better Feasible Point Refutes Candidate Global Optimality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r07-0039","batch_id":"batch-20261015-advance-20261119-r07-optimization-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"17524a098ab9bd51d106ac1ba2e4762c72f98f6cf97c8edf9177810e7b4bad6a","family_id":"qf-mod50-optdeep-improvement-witness","seed":202611197039,"semantic_template":"optdeep-improvement-witness|{\"candidate\":\"A\",\"challenger\":\"B\",\"points\":[\"A\",\"B\",\"C\",\"D\"],\"values\":[[42,1],[14,1],[56,1],[70,1]]}"},"fields":{"title":"One Strictly Better Feasible Point Refutes Candidate Global Optimality","stem":"Use the complete finite feasible carrier=[A, B, C, D] with aligned exact objective values=[42, 14, 56, 70], candidate=A. The declared feasible challenger is=B. Give (complete_objective_table, candidate, candidate_value, challenger, challenger_value, strict_feasible_improvement, candidate_global_optimality_claim, refutation_scope).","answer":{"kind":"expression","value":"([[A, 42], [B, 14], [C, 56], [D, 70]], A, 42, B, 14, Yes, No, one_strictly_better_feasible_point_refutes_candidate_global_optimality)","explanation":"Verify the challenger is distinct and has a strictly lower exact objective value. This is a refutation of the candidate global claim, not a proof that the challenger is globally optimal or that the candidate fails every neighborhood-local condition. The exact result, retaining the requested order, is ([[A, 42], [B, 14], [C, 56], [D, 70]], A, 42, B, 14, Yes, No, one_strictly_better_feasible_point_refutes_candidate_global_optimality)."},"skills":["One Strictly Better Feasible Point Refutes Candidate Global Optimality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r07-0040","batch_id":"batch-20261015-advance-20261119-r07-optimization-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1cefdc456821f64a9a1837bda114a71bb652a9bf0be81ee70702443a4ee9fb8a","family_id":"qf-mod50-optdeep-improvement-witness","seed":202611197040,"semantic_template":"optdeep-improvement-witness|{\"candidate\":\"A\",\"challenger\":\"B\",\"points\":[\"A\",\"B\",\"C\",\"D\"],\"values\":[[45,1],[15,1],[60,1],[75,1]]}"},"fields":{"title":"One Strictly Better Feasible Point Refutes Candidate Global Optimality","stem":"Use the complete finite feasible carrier=[A, B, C, D] with aligned exact objective values=[45, 15, 60, 75], candidate=A. The declared feasible challenger is=B. Give (complete_objective_table, candidate, candidate_value, challenger, challenger_value, strict_feasible_improvement, candidate_global_optimality_claim, refutation_scope).","answer":{"kind":"expression","value":"([[A, 45], [B, 15], [C, 60], [D, 75]], A, 45, B, 15, Yes, No, one_strictly_better_feasible_point_refutes_candidate_global_optimality)","explanation":"Verify the challenger is distinct and has a strictly lower exact objective value. This is a refutation of the candidate global claim, not a proof that the challenger is globally optimal or that the candidate fails every neighborhood-local condition. The exact result, retaining the requested order, is ([[A, 45], [B, 15], [C, 60], [D, 75]], A, 45, B, 15, Yes, No, one_strictly_better_feasible_point_refutes_candidate_global_optimality)."},"skills":["One Strictly Better Feasible Point Refutes Candidate Global Optimality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r07-0041","batch_id":"batch-20261015-advance-20261119-r07-optimization-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"37ec45fd43a257edee462384a66eb2691562bce54b5bf2dc14d346804080750d","family_id":"qf-mod50-optdeep-improvement-witness","seed":202611197041,"semantic_template":"optdeep-improvement-witness|{\"candidate\":\"A\",\"challenger\":\"B\",\"points\":[\"A\",\"B\",\"C\",\"D\"],\"values\":[[48,1],[16,1],[64,1],[80,1]]}"},"fields":{"title":"One Strictly Better Feasible Point Refutes Candidate Global Optimality","stem":"Use the complete finite feasible carrier=[A, B, C, D] with aligned exact objective values=[48, 16, 64, 80], candidate=A. The declared feasible challenger is=B. Give (complete_objective_table, candidate, candidate_value, challenger, challenger_value, strict_feasible_improvement, candidate_global_optimality_claim, refutation_scope).","answer":{"kind":"expression","value":"([[A, 48], [B, 16], [C, 64], [D, 80]], A, 48, B, 16, Yes, No, one_strictly_better_feasible_point_refutes_candidate_global_optimality)","explanation":"Verify the challenger is distinct and has a strictly lower exact objective value. This is a refutation of the candidate global claim, not a proof that the challenger is globally optimal or that the candidate fails every neighborhood-local condition. The exact result, retaining the requested order, is ([[A, 48], [B, 16], [C, 64], [D, 80]], A, 48, B, 16, Yes, No, one_strictly_better_feasible_point_refutes_candidate_global_optimality)."},"skills":["One Strictly Better Feasible Point Refutes Candidate Global Optimality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r07-0042","batch_id":"batch-20261015-advance-20261119-r07-optimization-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7d597c4c38ae0a8556ef6a425d851fafb54dd9fdd03921f33f417105655fb0cd","family_id":"qf-mod50-optdeep-improvement-witness","seed":202611197042,"semantic_template":"optdeep-improvement-witness|{\"candidate\":\"A\",\"challenger\":\"B\",\"points\":[\"A\",\"B\",\"C\",\"D\"],\"values\":[[51,1],[17,1],[68,1],[85,1]]}"},"fields":{"title":"One Strictly Better Feasible Point Refutes Candidate Global Optimality","stem":"Use the complete finite feasible carrier=[A, B, C, D] with aligned exact objective values=[51, 17, 68, 85], candidate=A. The declared feasible challenger is=B. Give (complete_objective_table, candidate, candidate_value, challenger, challenger_value, strict_feasible_improvement, candidate_global_optimality_claim, refutation_scope).","answer":{"kind":"expression","value":"([[A, 51], [B, 17], [C, 68], [D, 85]], A, 51, B, 17, Yes, No, one_strictly_better_feasible_point_refutes_candidate_global_optimality)","explanation":"Verify the challenger is distinct and has a strictly lower exact objective value. This is a refutation of the candidate global claim, not a proof that the challenger is globally optimal or that the candidate fails every neighborhood-local condition. The exact result, retaining the requested order, is ([[A, 51], [B, 17], [C, 68], [D, 85]], A, 51, B, 17, Yes, No, one_strictly_better_feasible_point_refutes_candidate_global_optimality)."},"skills":["One Strictly Better Feasible Point Refutes Candidate Global Optimality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r07-0043","batch_id":"batch-20261015-advance-20261119-r07-optimization-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c60200d90e0b109e258a2edec69ad906f588abb05451da2f62c44a2c0b860efd","family_id":"qf-mod50-optdeep-improvement-witness","seed":202611197043,"semantic_template":"optdeep-improvement-witness|{\"candidate\":\"A\",\"challenger\":\"B\",\"points\":[\"A\",\"B\",\"C\",\"D\"],\"values\":[[54,1],[18,1],[72,1],[90,1]]}"},"fields":{"title":"One Strictly Better Feasible Point Refutes Candidate Global Optimality","stem":"Use the complete finite feasible carrier=[A, B, C, D] with aligned exact objective values=[54, 18, 72, 90], candidate=A. The declared feasible challenger is=B. Give (complete_objective_table, candidate, candidate_value, challenger, challenger_value, strict_feasible_improvement, candidate_global_optimality_claim, refutation_scope).","answer":{"kind":"expression","value":"([[A, 54], [B, 18], [C, 72], [D, 90]], A, 54, B, 18, Yes, No, one_strictly_better_feasible_point_refutes_candidate_global_optimality)","explanation":"Verify the challenger is distinct and has a strictly lower exact objective value. This is a refutation of the candidate global claim, not a proof that the challenger is globally optimal or that the candidate fails every neighborhood-local condition. The exact result, retaining the requested order, is ([[A, 54], [B, 18], [C, 72], [D, 90]], A, 54, B, 18, Yes, No, one_strictly_better_feasible_point_refutes_candidate_global_optimality)."},"skills":["One Strictly Better Feasible Point Refutes Candidate Global Optimality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r07-0044","batch_id":"batch-20261015-advance-20261119-r07-optimization-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"50d53fe15c9f2e40238e8e518631aa1e483a5e202994a5db12c062991b2bbaf7","family_id":"qf-mod50-optdeep-improvement-witness","seed":202611197044,"semantic_template":"optdeep-improvement-witness|{\"candidate\":\"A\",\"challenger\":\"B\",\"points\":[\"A\",\"B\",\"C\",\"D\"],\"values\":[[57,1],[19,1],[76,1],[95,1]]}"},"fields":{"title":"One Strictly Better Feasible Point Refutes Candidate Global Optimality","stem":"Use the complete finite feasible carrier=[A, B, C, D] with aligned exact objective values=[57, 19, 76, 95], candidate=A. The declared feasible challenger is=B. Give (complete_objective_table, candidate, candidate_value, challenger, challenger_value, strict_feasible_improvement, candidate_global_optimality_claim, refutation_scope).","answer":{"kind":"expression","value":"([[A, 57], [B, 19], [C, 76], [D, 95]], A, 57, B, 19, Yes, No, one_strictly_better_feasible_point_refutes_candidate_global_optimality)","explanation":"Verify the challenger is distinct and has a strictly lower exact objective value. This is a refutation of the candidate global claim, not a proof that the challenger is globally optimal or that the candidate fails every neighborhood-local condition. The exact result, retaining the requested order, is ([[A, 57], [B, 19], [C, 76], [D, 95]], A, 57, B, 19, Yes, No, one_strictly_better_feasible_point_refutes_candidate_global_optimality)."},"skills":["One Strictly Better Feasible Point Refutes Candidate Global Optimality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r07-0045","batch_id":"batch-20261015-advance-20261119-r07-optimization-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"70568dfd88214c938c2f4330ba31512de7b924054b5e1c32c3d6e7e2be42fb7a","family_id":"qf-mod50-optdeep-improvement-witness","seed":202611197045,"semantic_template":"optdeep-improvement-witness|{\"candidate\":\"A\",\"challenger\":\"B\",\"points\":[\"A\",\"B\",\"C\",\"D\"],\"values\":[[60,1],[20,1],[80,1],[100,1]]}"},"fields":{"title":"One Strictly Better Feasible Point Refutes Candidate Global Optimality","stem":"Use the complete finite feasible carrier=[A, B, C, D] with aligned exact objective values=[60, 20, 80, 100], candidate=A. The declared feasible challenger is=B. Give (complete_objective_table, candidate, candidate_value, challenger, challenger_value, strict_feasible_improvement, candidate_global_optimality_claim, refutation_scope).","answer":{"kind":"expression","value":"([[A, 60], [B, 20], [C, 80], [D, 100]], A, 60, B, 20, Yes, No, one_strictly_better_feasible_point_refutes_candidate_global_optimality)","explanation":"Verify the challenger is distinct and has a strictly lower exact objective value. This is a refutation of the candidate global claim, not a proof that the challenger is globally optimal or that the candidate fails every neighborhood-local condition. The exact result, retaining the requested order, is ([[A, 60], [B, 20], [C, 80], [D, 100]], A, 60, B, 20, Yes, No, one_strictly_better_feasible_point_refutes_candidate_global_optimality)."},"skills":["One Strictly Better Feasible Point Refutes Candidate Global Optimality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r07-0046","batch_id":"batch-20261015-advance-20261119-r07-optimization-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"aa404b7fe78072915c957d98bf64ce2f24d89a47cbeeb6ba9109991f83fb8131","family_id":"qf-mod50-optdeep-improvement-witness","seed":202611197046,"semantic_template":"optdeep-improvement-witness|{\"candidate\":\"A\",\"challenger\":\"B\",\"points\":[\"A\",\"B\",\"C\",\"D\"],\"values\":[[63,1],[21,1],[84,1],[105,1]]}"},"fields":{"title":"One Strictly Better Feasible Point Refutes Candidate Global Optimality","stem":"Use the complete finite feasible carrier=[A, B, C, D] with aligned exact objective values=[63, 21, 84, 105], candidate=A. The declared feasible challenger is=B. Give (complete_objective_table, candidate, candidate_value, challenger, challenger_value, strict_feasible_improvement, candidate_global_optimality_claim, refutation_scope).","answer":{"kind":"expression","value":"([[A, 63], [B, 21], [C, 84], [D, 105]], A, 63, B, 21, Yes, No, one_strictly_better_feasible_point_refutes_candidate_global_optimality)","explanation":"Verify the challenger is distinct and has a strictly lower exact objective value. This is a refutation of the candidate global claim, not a proof that the challenger is globally optimal or that the candidate fails every neighborhood-local condition. The exact result, retaining the requested order, is ([[A, 63], [B, 21], [C, 84], [D, 105]], A, 63, B, 21, Yes, No, one_strictly_better_feasible_point_refutes_candidate_global_optimality)."},"skills":["One Strictly Better Feasible Point Refutes Candidate Global Optimality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r07-0047","batch_id":"batch-20261015-advance-20261119-r07-optimization-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"621b3859ef1ce06ad9c40c1cd269923ca0b5e928ce3f62ea3766750d0516777e","family_id":"qf-mod50-optdeep-improvement-witness","seed":202611197047,"semantic_template":"optdeep-improvement-witness|{\"candidate\":\"A\",\"challenger\":\"B\",\"points\":[\"A\",\"B\",\"C\",\"D\"],\"values\":[[66,1],[22,1],[88,1],[110,1]]}"},"fields":{"title":"One Strictly Better Feasible Point Refutes Candidate Global Optimality","stem":"Use the complete finite feasible carrier=[A, B, C, D] with aligned exact objective values=[66, 22, 88, 110], candidate=A. The declared feasible challenger is=B. Give (complete_objective_table, candidate, candidate_value, challenger, challenger_value, strict_feasible_improvement, candidate_global_optimality_claim, refutation_scope).","answer":{"kind":"expression","value":"([[A, 66], [B, 22], [C, 88], [D, 110]], A, 66, B, 22, Yes, No, one_strictly_better_feasible_point_refutes_candidate_global_optimality)","explanation":"Verify the challenger is distinct and has a strictly lower exact objective value. This is a refutation of the candidate global claim, not a proof that the challenger is globally optimal or that the candidate fails every neighborhood-local condition. The exact result, retaining the requested order, is ([[A, 66], [B, 22], [C, 88], [D, 110]], A, 66, B, 22, Yes, No, one_strictly_better_feasible_point_refutes_candidate_global_optimality)."},"skills":["One Strictly Better Feasible Point Refutes Candidate Global Optimality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r07-0048","batch_id":"batch-20261015-advance-20261119-r07-optimization-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f0994674f93c5ab5819a3a087a4da3b3f88be3f161cb7e5bcaa3c73b2c2c0c9a","family_id":"qf-mod50-optdeep-improvement-witness","seed":202611197048,"semantic_template":"optdeep-improvement-witness|{\"candidate\":\"A\",\"challenger\":\"B\",\"points\":[\"A\",\"B\",\"C\",\"D\"],\"values\":[[69,1],[23,1],[92,1],[115,1]]}"},"fields":{"title":"One Strictly Better Feasible Point Refutes Candidate Global Optimality","stem":"Use the complete finite feasible carrier=[A, B, C, D] with aligned exact objective values=[69, 23, 92, 115], candidate=A. The declared feasible challenger is=B. Give (complete_objective_table, candidate, candidate_value, challenger, challenger_value, strict_feasible_improvement, candidate_global_optimality_claim, refutation_scope).","answer":{"kind":"expression","value":"([[A, 69], [B, 23], [C, 92], [D, 115]], A, 69, B, 23, Yes, No, one_strictly_better_feasible_point_refutes_candidate_global_optimality)","explanation":"Verify the challenger is distinct and has a strictly lower exact objective value. This is a refutation of the candidate global claim, not a proof that the challenger is globally optimal or that the candidate fails every neighborhood-local condition. The exact result, retaining the requested order, is ([[A, 69], [B, 23], [C, 92], [D, 115]], A, 69, B, 23, Yes, No, one_strictly_better_feasible_point_refutes_candidate_global_optimality)."},"skills":["One Strictly Better Feasible Point Refutes Candidate Global Optimality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r07-0049","batch_id":"batch-20261015-advance-20261119-r07-optimization-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"63024b28d6911237c8299ffe0c3633a56c85ad2e7b09be989264b30e831e94d7","family_id":"qf-mod50-optdeep-improvement-witness","seed":202611197049,"semantic_template":"optdeep-improvement-witness|{\"candidate\":\"A\",\"challenger\":\"B\",\"points\":[\"A\",\"B\",\"C\",\"D\"],\"values\":[[72,1],[24,1],[96,1],[120,1]]}"},"fields":{"title":"One Strictly Better Feasible Point Refutes Candidate Global Optimality","stem":"Use the complete finite feasible carrier=[A, B, C, D] with aligned exact objective values=[72, 24, 96, 120], candidate=A. The declared feasible challenger is=B. Give (complete_objective_table, candidate, candidate_value, challenger, challenger_value, strict_feasible_improvement, candidate_global_optimality_claim, refutation_scope).","answer":{"kind":"expression","value":"([[A, 72], [B, 24], [C, 96], [D, 120]], A, 72, B, 24, Yes, No, one_strictly_better_feasible_point_refutes_candidate_global_optimality)","explanation":"Verify the challenger is distinct and has a strictly lower exact objective value. This is a refutation of the candidate global claim, not a proof that the challenger is globally optimal or that the candidate fails every neighborhood-local condition. The exact result, retaining the requested order, is ([[A, 72], [B, 24], [C, 96], [D, 120]], A, 72, B, 24, Yes, No, one_strictly_better_feasible_point_refutes_candidate_global_optimality)."},"skills":["One Strictly Better Feasible Point Refutes Candidate Global Optimality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r07-0050","batch_id":"batch-20261015-advance-20261119-r07-optimization-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9004ce33f101e0ef23d09c7a602b49c2fdb599a06019815123ba6f91ed140e75","family_id":"qf-mod50-optdeep-improvement-witness","seed":202611197050,"semantic_template":"optdeep-improvement-witness|{\"candidate\":\"A\",\"challenger\":\"B\",\"points\":[\"A\",\"B\",\"C\",\"D\"],\"values\":[[75,1],[25,1],[100,1],[125,1]]}"},"fields":{"title":"One Strictly Better Feasible Point Refutes Candidate Global Optimality","stem":"Use the complete finite feasible carrier=[A, B, C, D] with aligned exact objective values=[75, 25, 100, 125], candidate=A. The declared feasible challenger is=B. Give (complete_objective_table, candidate, candidate_value, challenger, challenger_value, strict_feasible_improvement, candidate_global_optimality_claim, refutation_scope).","answer":{"kind":"expression","value":"([[A, 75], [B, 25], [C, 100], [D, 125]], A, 75, B, 25, Yes, No, one_strictly_better_feasible_point_refutes_candidate_global_optimality)","explanation":"Verify the challenger is distinct and has a strictly lower exact objective value. This is a refutation of the candidate global claim, not a proof that the challenger is globally optimal or that the candidate fails every neighborhood-local condition. The exact result, retaining the requested order, is ([[A, 75], [B, 25], [C, 100], [D, 125]], A, 75, B, 25, Yes, No, one_strictly_better_feasible_point_refutes_candidate_global_optimality)."},"skills":["One Strictly Better Feasible Point Refutes Candidate Global Optimality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r07-0051","batch_id":"batch-20261015-advance-20261119-r07-optimization-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3444a07448c4e312c7d1d8badbaa8bb5bd2a02c3db7abcfe6e13ef092e263f3a","family_id":"qf-mod50-optdeep-lower-bound-certificate","seed":202611197051,"semantic_template":"optdeep-lower-bound-certificate|{\"candidate\":\"A\",\"lower_bound\":[1,1],\"points\":[\"A\",\"B\",\"C\",\"D\"],\"values\":[[1,1],[2,1],[3,1],[4,1]]}"},"fields":{"title":"A Complete Attained Lower Bound Certifies Globality but Not Automatic Uniqueness","stem":"Use the complete finite feasible carrier=[A, B, C, D] with aligned exact objective values=[1, 2, 3, 4], candidate=A. The declared global lower bound is=1. Give (complete_objective_table, lower_bound, candidate, candidate_value, all_values_above_bound, bound_attained, global_optimum, unique, all_global_minimizers, certificate_scope).","answer":{"kind":"expression","value":"([[A, 1], [B, 2], [C, 3], [D, 4]], 1, A, 1, Yes, Yes, Yes, Yes, [A], complete_lower_bound_plus_attainment_certifies_global_but_not_automatic_uniqueness)","explanation":"Check the bound against every feasible value and exact attainment at the candidate. Retain all minimizers and set the uniqueness flag only if exactly one attains the bound. The certificate applies to this complete finite feasible domain, not unlisted candidates or continuous problems. The exact result, retaining the requested order, is ([[A, 1], [B, 2], [C, 3], [D, 4]], 1, A, 1, Yes, Yes, Yes, Yes, [A], complete_lower_bound_plus_attainment_certifies_global_but_not_automatic_uniqueness)."},"skills":["A Complete Attained Lower Bound Certifies Globality but Not Automatic Uniqueness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r07-0052","batch_id":"batch-20261015-advance-20261119-r07-optimization-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"70d6dcd447e14060edb9a8b0017189cf2eebee9a90c5c6cca03fa84663eb9e2d","family_id":"qf-mod50-optdeep-lower-bound-certificate","seed":202611197052,"semantic_template":"optdeep-lower-bound-certificate|{\"candidate\":\"A\",\"lower_bound\":[2,1],\"points\":[\"A\",\"B\",\"C\",\"D\"],\"values\":[[2,1],[4,1],[6,1],[8,1]]}"},"fields":{"title":"A Complete Attained Lower Bound Certifies Globality but Not Automatic Uniqueness","stem":"Use the complete finite feasible carrier=[A, B, C, D] with aligned exact objective values=[2, 4, 6, 8], candidate=A. The declared global lower bound is=2. Give (complete_objective_table, lower_bound, candidate, candidate_value, all_values_above_bound, bound_attained, global_optimum, unique, all_global_minimizers, certificate_scope).","answer":{"kind":"expression","value":"([[A, 2], [B, 4], [C, 6], [D, 8]], 2, A, 2, Yes, Yes, Yes, Yes, [A], complete_lower_bound_plus_attainment_certifies_global_but_not_automatic_uniqueness)","explanation":"Check the bound against every feasible value and exact attainment at the candidate. Retain all minimizers and set the uniqueness flag only if exactly one attains the bound. The certificate applies to this complete finite feasible domain, not unlisted candidates or continuous problems. The exact result, retaining the requested order, is ([[A, 2], [B, 4], [C, 6], [D, 8]], 2, A, 2, Yes, Yes, Yes, Yes, [A], complete_lower_bound_plus_attainment_certifies_global_but_not_automatic_uniqueness)."},"skills":["A Complete Attained Lower Bound Certifies Globality but Not Automatic Uniqueness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r07-0053","batch_id":"batch-20261015-advance-20261119-r07-optimization-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"19e2a6673abab219574eab6315d231c0b2717a33f180661fb96b91c3023d5624","family_id":"qf-mod50-optdeep-lower-bound-certificate","seed":202611197053,"semantic_template":"optdeep-lower-bound-certificate|{\"candidate\":\"A\",\"lower_bound\":[3,1],\"points\":[\"A\",\"B\",\"C\",\"D\"],\"values\":[[3,1],[6,1],[9,1],[12,1]]}"},"fields":{"title":"A Complete Attained Lower Bound Certifies Globality but Not Automatic Uniqueness","stem":"Use the complete finite feasible carrier=[A, B, C, D] with aligned exact objective values=[3, 6, 9, 12], candidate=A. The declared global lower bound is=3. Give (complete_objective_table, lower_bound, candidate, candidate_value, all_values_above_bound, bound_attained, global_optimum, unique, all_global_minimizers, certificate_scope).","answer":{"kind":"expression","value":"([[A, 3], [B, 6], [C, 9], [D, 12]], 3, A, 3, Yes, Yes, Yes, Yes, [A], complete_lower_bound_plus_attainment_certifies_global_but_not_automatic_uniqueness)","explanation":"Check the bound against every feasible value and exact attainment at the candidate. Retain all minimizers and set the uniqueness flag only if exactly one attains the bound. The certificate applies to this complete finite feasible domain, not unlisted candidates or continuous problems. The exact result, retaining the requested order, is ([[A, 3], [B, 6], [C, 9], [D, 12]], 3, A, 3, Yes, Yes, Yes, Yes, [A], complete_lower_bound_plus_attainment_certifies_global_but_not_automatic_uniqueness)."},"skills":["A Complete Attained Lower Bound Certifies Globality but Not Automatic Uniqueness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r07-0054","batch_id":"batch-20261015-advance-20261119-r07-optimization-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"799f642cd5057452363d46837af8643702a3ea7953fbf957d15975e7052dbb06","family_id":"qf-mod50-optdeep-lower-bound-certificate","seed":202611197054,"semantic_template":"optdeep-lower-bound-certificate|{\"candidate\":\"A\",\"lower_bound\":[4,1],\"points\":[\"A\",\"B\",\"C\",\"D\"],\"values\":[[4,1],[8,1],[12,1],[16,1]]}"},"fields":{"title":"A Complete Attained Lower Bound Certifies Globality but Not Automatic Uniqueness","stem":"Use the complete finite feasible carrier=[A, B, C, D] with aligned exact objective values=[4, 8, 12, 16], candidate=A. The declared global lower bound is=4. Give (complete_objective_table, lower_bound, candidate, candidate_value, all_values_above_bound, bound_attained, global_optimum, unique, all_global_minimizers, certificate_scope).","answer":{"kind":"expression","value":"([[A, 4], [B, 8], [C, 12], [D, 16]], 4, A, 4, Yes, Yes, Yes, Yes, [A], complete_lower_bound_plus_attainment_certifies_global_but_not_automatic_uniqueness)","explanation":"Check the bound against every feasible value and exact attainment at the candidate. Retain all minimizers and set the uniqueness flag only if exactly one attains the bound. The certificate applies to this complete finite feasible domain, not unlisted candidates or continuous problems. The exact result, retaining the requested order, is ([[A, 4], [B, 8], [C, 12], [D, 16]], 4, A, 4, Yes, Yes, Yes, Yes, [A], complete_lower_bound_plus_attainment_certifies_global_but_not_automatic_uniqueness)."},"skills":["A Complete Attained Lower Bound Certifies Globality but Not Automatic Uniqueness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r07-0055","batch_id":"batch-20261015-advance-20261119-r07-optimization-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"eb29afabe5f016814ee40827213f4dd7c6b156980db7c2ff1eb82e14f659a4db","family_id":"qf-mod50-optdeep-lower-bound-certificate","seed":202611197055,"semantic_template":"optdeep-lower-bound-certificate|{\"candidate\":\"A\",\"lower_bound\":[5,1],\"points\":[\"A\",\"B\",\"C\",\"D\"],\"values\":[[5,1],[10,1],[15,1],[20,1]]}"},"fields":{"title":"A Complete Attained Lower Bound Certifies Globality but Not Automatic Uniqueness","stem":"Use the complete finite feasible carrier=[A, B, C, D] with aligned exact objective values=[5, 10, 15, 20], candidate=A. The declared global lower bound is=5. Give (complete_objective_table, lower_bound, candidate, candidate_value, all_values_above_bound, bound_attained, global_optimum, unique, all_global_minimizers, certificate_scope).","answer":{"kind":"expression","value":"([[A, 5], [B, 10], [C, 15], [D, 20]], 5, A, 5, Yes, Yes, Yes, Yes, [A], complete_lower_bound_plus_attainment_certifies_global_but_not_automatic_uniqueness)","explanation":"Check the bound against every feasible value and exact attainment at the candidate. Retain all minimizers and set the uniqueness flag only if exactly one attains the bound. The certificate applies to this complete finite feasible domain, not unlisted candidates or continuous problems. The exact result, retaining the requested order, is ([[A, 5], [B, 10], [C, 15], [D, 20]], 5, A, 5, Yes, Yes, Yes, Yes, [A], complete_lower_bound_plus_attainment_certifies_global_but_not_automatic_uniqueness)."},"skills":["A Complete Attained Lower Bound Certifies Globality but Not Automatic Uniqueness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r07-0056","batch_id":"batch-20261015-advance-20261119-r07-optimization-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"625655d721230ede0a63a2505d3c0b4f31762483eebce55a65ef65de0c733307","family_id":"qf-mod50-optdeep-lower-bound-certificate","seed":202611197056,"semantic_template":"optdeep-lower-bound-certificate|{\"candidate\":\"A\",\"lower_bound\":[6,1],\"points\":[\"A\",\"B\",\"C\",\"D\"],\"values\":[[6,1],[12,1],[18,1],[24,1]]}"},"fields":{"title":"A Complete Attained Lower Bound Certifies Globality but Not Automatic Uniqueness","stem":"Use the complete finite feasible carrier=[A, B, C, D] with aligned exact objective values=[6, 12, 18, 24], candidate=A. The declared global lower bound is=6. Give (complete_objective_table, lower_bound, candidate, candidate_value, all_values_above_bound, bound_attained, global_optimum, unique, all_global_minimizers, certificate_scope).","answer":{"kind":"expression","value":"([[A, 6], [B, 12], [C, 18], [D, 24]], 6, A, 6, Yes, Yes, Yes, Yes, [A], complete_lower_bound_plus_attainment_certifies_global_but_not_automatic_uniqueness)","explanation":"Check the bound against every feasible value and exact attainment at the candidate. Retain all minimizers and set the uniqueness flag only if exactly one attains the bound. The certificate applies to this complete finite feasible domain, not unlisted candidates or continuous problems. The exact result, retaining the requested order, is ([[A, 6], [B, 12], [C, 18], [D, 24]], 6, A, 6, Yes, Yes, Yes, Yes, [A], complete_lower_bound_plus_attainment_certifies_global_but_not_automatic_uniqueness)."},"skills":["A Complete Attained Lower Bound Certifies Globality but Not Automatic Uniqueness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r07-0057","batch_id":"batch-20261015-advance-20261119-r07-optimization-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8be54c8fe9339356c3fe282ffb4c4cb1b46e408ad019bce5e9a6939cf1cd640f","family_id":"qf-mod50-optdeep-lower-bound-certificate","seed":202611197057,"semantic_template":"optdeep-lower-bound-certificate|{\"candidate\":\"A\",\"lower_bound\":[7,1],\"points\":[\"A\",\"B\",\"C\",\"D\"],\"values\":[[7,1],[14,1],[21,1],[28,1]]}"},"fields":{"title":"A Complete Attained Lower Bound Certifies Globality but Not Automatic Uniqueness","stem":"Use the complete finite feasible carrier=[A, B, C, D] with aligned exact objective values=[7, 14, 21, 28], candidate=A. The declared global lower bound is=7. Give (complete_objective_table, lower_bound, candidate, candidate_value, all_values_above_bound, bound_attained, global_optimum, unique, all_global_minimizers, certificate_scope).","answer":{"kind":"expression","value":"([[A, 7], [B, 14], [C, 21], [D, 28]], 7, A, 7, Yes, Yes, Yes, Yes, [A], complete_lower_bound_plus_attainment_certifies_global_but_not_automatic_uniqueness)","explanation":"Check the bound against every feasible value and exact attainment at the candidate. Retain all minimizers and set the uniqueness flag only if exactly one attains the bound. The certificate applies to this complete finite feasible domain, not unlisted candidates or continuous problems. The exact result, retaining the requested order, is ([[A, 7], [B, 14], [C, 21], [D, 28]], 7, A, 7, Yes, Yes, Yes, Yes, [A], complete_lower_bound_plus_attainment_certifies_global_but_not_automatic_uniqueness)."},"skills":["A Complete Attained Lower Bound Certifies Globality but Not Automatic Uniqueness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r07-0058","batch_id":"batch-20261015-advance-20261119-r07-optimization-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"830f8d00a62a9264340355b96b12f8059c8ca05690bcabf4e8aab5f6c600f197","family_id":"qf-mod50-optdeep-lower-bound-certificate","seed":202611197058,"semantic_template":"optdeep-lower-bound-certificate|{\"candidate\":\"A\",\"lower_bound\":[8,1],\"points\":[\"A\",\"B\",\"C\",\"D\"],\"values\":[[8,1],[16,1],[24,1],[32,1]]}"},"fields":{"title":"A Complete Attained Lower Bound Certifies Globality but Not Automatic Uniqueness","stem":"Use the complete finite feasible carrier=[A, B, C, D] with aligned exact objective values=[8, 16, 24, 32], candidate=A. The declared global lower bound is=8. Give (complete_objective_table, lower_bound, candidate, candidate_value, all_values_above_bound, bound_attained, global_optimum, unique, all_global_minimizers, certificate_scope).","answer":{"kind":"expression","value":"([[A, 8], [B, 16], [C, 24], [D, 32]], 8, A, 8, Yes, Yes, Yes, Yes, [A], complete_lower_bound_plus_attainment_certifies_global_but_not_automatic_uniqueness)","explanation":"Check the bound against every feasible value and exact attainment at the candidate. Retain all minimizers and set the uniqueness flag only if exactly one attains the bound. The certificate applies to this complete finite feasible domain, not unlisted candidates or continuous problems. The exact result, retaining the requested order, is ([[A, 8], [B, 16], [C, 24], [D, 32]], 8, A, 8, Yes, Yes, Yes, Yes, [A], complete_lower_bound_plus_attainment_certifies_global_but_not_automatic_uniqueness)."},"skills":["A Complete Attained Lower Bound Certifies Globality but Not Automatic Uniqueness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r07-0059","batch_id":"batch-20261015-advance-20261119-r07-optimization-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"848bfb89f89c9730180e60dc6ae0fbcdbddeaa7b5a3b68246e8e02fb5b633020","family_id":"qf-mod50-optdeep-lower-bound-certificate","seed":202611197059,"semantic_template":"optdeep-lower-bound-certificate|{\"candidate\":\"A\",\"lower_bound\":[9,1],\"points\":[\"A\",\"B\",\"C\",\"D\"],\"values\":[[9,1],[18,1],[27,1],[36,1]]}"},"fields":{"title":"A Complete Attained Lower Bound Certifies Globality but Not Automatic Uniqueness","stem":"Use the complete finite feasible carrier=[A, B, C, D] with aligned exact objective values=[9, 18, 27, 36], candidate=A. The declared global lower bound is=9. Give (complete_objective_table, lower_bound, candidate, candidate_value, all_values_above_bound, bound_attained, global_optimum, unique, all_global_minimizers, certificate_scope).","answer":{"kind":"expression","value":"([[A, 9], [B, 18], [C, 27], [D, 36]], 9, A, 9, Yes, Yes, Yes, Yes, [A], complete_lower_bound_plus_attainment_certifies_global_but_not_automatic_uniqueness)","explanation":"Check the bound against every feasible value and exact attainment at the candidate. Retain all minimizers and set the uniqueness flag only if exactly one attains the bound. The certificate applies to this complete finite feasible domain, not unlisted candidates or continuous problems. The exact result, retaining the requested order, is ([[A, 9], [B, 18], [C, 27], [D, 36]], 9, A, 9, Yes, Yes, Yes, Yes, [A], complete_lower_bound_plus_attainment_certifies_global_but_not_automatic_uniqueness)."},"skills":["A Complete Attained Lower Bound Certifies Globality but Not Automatic Uniqueness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r07-0060","batch_id":"batch-20261015-advance-20261119-r07-optimization-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"aef0394774645ff30c2adbe92fa5ef6a4ea56922e0deb6de1350453b6e29779e","family_id":"qf-mod50-optdeep-lower-bound-certificate","seed":202611197060,"semantic_template":"optdeep-lower-bound-certificate|{\"candidate\":\"A\",\"lower_bound\":[10,1],\"points\":[\"A\",\"B\",\"C\",\"D\"],\"values\":[[10,1],[20,1],[30,1],[40,1]]}"},"fields":{"title":"A Complete Attained Lower Bound Certifies Globality but Not Automatic Uniqueness","stem":"Use the complete finite feasible carrier=[A, B, C, D] with aligned exact objective values=[10, 20, 30, 40], candidate=A. The declared global lower bound is=10. Give (complete_objective_table, lower_bound, candidate, candidate_value, all_values_above_bound, bound_attained, global_optimum, unique, all_global_minimizers, certificate_scope).","answer":{"kind":"expression","value":"([[A, 10], [B, 20], [C, 30], [D, 40]], 10, A, 10, Yes, Yes, Yes, Yes, [A], complete_lower_bound_plus_attainment_certifies_global_but_not_automatic_uniqueness)","explanation":"Check the bound against every feasible value and exact attainment at the candidate. Retain all minimizers and set the uniqueness flag only if exactly one attains the bound. The certificate applies to this complete finite feasible domain, not unlisted candidates or continuous problems. The exact result, retaining the requested order, is ([[A, 10], [B, 20], [C, 30], [D, 40]], 10, A, 10, Yes, Yes, Yes, Yes, [A], complete_lower_bound_plus_attainment_certifies_global_but_not_automatic_uniqueness)."},"skills":["A Complete Attained Lower Bound Certifies Globality but Not Automatic Uniqueness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r07-0061","batch_id":"batch-20261015-advance-20261119-r07-optimization-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"843c657b79425a3fe894e026d33e93509ce15a16e4d8f026dd20865297159cd8","family_id":"qf-mod50-optdeep-lower-bound-certificate","seed":202611197061,"semantic_template":"optdeep-lower-bound-certificate|{\"candidate\":\"A\",\"lower_bound\":[11,1],\"points\":[\"A\",\"B\",\"C\",\"D\"],\"values\":[[11,1],[22,1],[33,1],[44,1]]}"},"fields":{"title":"A Complete Attained Lower Bound Certifies Globality but Not Automatic Uniqueness","stem":"Use the complete finite feasible carrier=[A, B, C, D] with aligned exact objective values=[11, 22, 33, 44], candidate=A. The declared global lower bound is=11. Give (complete_objective_table, lower_bound, candidate, candidate_value, all_values_above_bound, bound_attained, global_optimum, unique, all_global_minimizers, certificate_scope).","answer":{"kind":"expression","value":"([[A, 11], [B, 22], [C, 33], [D, 44]], 11, A, 11, Yes, Yes, Yes, Yes, [A], complete_lower_bound_plus_attainment_certifies_global_but_not_automatic_uniqueness)","explanation":"Check the bound against every feasible value and exact attainment at the candidate. Retain all minimizers and set the uniqueness flag only if exactly one attains the bound. The certificate applies to this complete finite feasible domain, not unlisted candidates or continuous problems. The exact result, retaining the requested order, is ([[A, 11], [B, 22], [C, 33], [D, 44]], 11, A, 11, Yes, Yes, Yes, Yes, [A], complete_lower_bound_plus_attainment_certifies_global_but_not_automatic_uniqueness)."},"skills":["A Complete Attained Lower Bound Certifies Globality but Not Automatic Uniqueness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r07-0062","batch_id":"batch-20261015-advance-20261119-r07-optimization-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2bb90f16f2dcc26eaa0efe1e645550b5bb99b52d4ab5aa33079256634b0ed33b","family_id":"qf-mod50-optdeep-lower-bound-certificate","seed":202611197062,"semantic_template":"optdeep-lower-bound-certificate|{\"candidate\":\"A\",\"lower_bound\":[12,1],\"points\":[\"A\",\"B\",\"C\",\"D\"],\"values\":[[12,1],[24,1],[36,1],[48,1]]}"},"fields":{"title":"A Complete Attained Lower Bound Certifies Globality but Not Automatic Uniqueness","stem":"Use the complete finite feasible carrier=[A, B, C, D] with aligned exact objective values=[12, 24, 36, 48], candidate=A. The declared global lower bound is=12. Give (complete_objective_table, lower_bound, candidate, candidate_value, all_values_above_bound, bound_attained, global_optimum, unique, all_global_minimizers, certificate_scope).","answer":{"kind":"expression","value":"([[A, 12], [B, 24], [C, 36], [D, 48]], 12, A, 12, Yes, Yes, Yes, Yes, [A], complete_lower_bound_plus_attainment_certifies_global_but_not_automatic_uniqueness)","explanation":"Check the bound against every feasible value and exact attainment at the candidate. Retain all minimizers and set the uniqueness flag only if exactly one attains the bound. The certificate applies to this complete finite feasible domain, not unlisted candidates or continuous problems. The exact result, retaining the requested order, is ([[A, 12], [B, 24], [C, 36], [D, 48]], 12, A, 12, Yes, Yes, Yes, Yes, [A], complete_lower_bound_plus_attainment_certifies_global_but_not_automatic_uniqueness)."},"skills":["A Complete Attained Lower Bound Certifies Globality but Not Automatic Uniqueness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r07-0063","batch_id":"batch-20261015-advance-20261119-r07-optimization-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"98a1b03add2013b312bebe738f5df90220e00eecab16f7f8299270cfdb1e6357","family_id":"qf-mod50-optdeep-lower-bound-certificate","seed":202611197063,"semantic_template":"optdeep-lower-bound-certificate|{\"candidate\":\"A\",\"lower_bound\":[13,1],\"points\":[\"A\",\"B\",\"C\",\"D\"],\"values\":[[13,1],[26,1],[39,1],[52,1]]}"},"fields":{"title":"A Complete Attained Lower Bound Certifies Globality but Not Automatic Uniqueness","stem":"Use the complete finite feasible carrier=[A, B, C, D] with aligned exact objective values=[13, 26, 39, 52], candidate=A. The declared global lower bound is=13. Give (complete_objective_table, lower_bound, candidate, candidate_value, all_values_above_bound, bound_attained, global_optimum, unique, all_global_minimizers, certificate_scope).","answer":{"kind":"expression","value":"([[A, 13], [B, 26], [C, 39], [D, 52]], 13, A, 13, Yes, Yes, Yes, Yes, [A], complete_lower_bound_plus_attainment_certifies_global_but_not_automatic_uniqueness)","explanation":"Check the bound against every feasible value and exact attainment at the candidate. Retain all minimizers and set the uniqueness flag only if exactly one attains the bound. The certificate applies to this complete finite feasible domain, not unlisted candidates or continuous problems. The exact result, retaining the requested order, is ([[A, 13], [B, 26], [C, 39], [D, 52]], 13, A, 13, Yes, Yes, Yes, Yes, [A], complete_lower_bound_plus_attainment_certifies_global_but_not_automatic_uniqueness)."},"skills":["A Complete Attained Lower Bound Certifies Globality but Not Automatic Uniqueness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r07-0064","batch_id":"batch-20261015-advance-20261119-r07-optimization-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"18ebb45bcd537c60ee703cccf114a74c327ef3dfb10d9371235866be923155ef","family_id":"qf-mod50-optdeep-lower-bound-certificate","seed":202611197064,"semantic_template":"optdeep-lower-bound-certificate|{\"candidate\":\"A\",\"lower_bound\":[14,1],\"points\":[\"A\",\"B\",\"C\",\"D\"],\"values\":[[14,1],[28,1],[42,1],[56,1]]}"},"fields":{"title":"A Complete Attained Lower Bound Certifies Globality but Not Automatic Uniqueness","stem":"Use the complete finite feasible carrier=[A, B, C, D] with aligned exact objective values=[14, 28, 42, 56], candidate=A. The declared global lower bound is=14. Give (complete_objective_table, lower_bound, candidate, candidate_value, all_values_above_bound, bound_attained, global_optimum, unique, all_global_minimizers, certificate_scope).","answer":{"kind":"expression","value":"([[A, 14], [B, 28], [C, 42], [D, 56]], 14, A, 14, Yes, Yes, Yes, Yes, [A], complete_lower_bound_plus_attainment_certifies_global_but_not_automatic_uniqueness)","explanation":"Check the bound against every feasible value and exact attainment at the candidate. Retain all minimizers and set the uniqueness flag only if exactly one attains the bound. The certificate applies to this complete finite feasible domain, not unlisted candidates or continuous problems. The exact result, retaining the requested order, is ([[A, 14], [B, 28], [C, 42], [D, 56]], 14, A, 14, Yes, Yes, Yes, Yes, [A], complete_lower_bound_plus_attainment_certifies_global_but_not_automatic_uniqueness)."},"skills":["A Complete Attained Lower Bound Certifies Globality but Not Automatic Uniqueness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r07-0065","batch_id":"batch-20261015-advance-20261119-r07-optimization-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"28e8ce166fab0e35a74052f0d27bbcdef3eba1e0b9dc9fa66f33bcf42910b9f7","family_id":"qf-mod50-optdeep-lower-bound-certificate","seed":202611197065,"semantic_template":"optdeep-lower-bound-certificate|{\"candidate\":\"A\",\"lower_bound\":[15,1],\"points\":[\"A\",\"B\",\"C\",\"D\"],\"values\":[[15,1],[30,1],[45,1],[60,1]]}"},"fields":{"title":"A Complete Attained Lower Bound Certifies Globality but Not Automatic Uniqueness","stem":"Use the complete finite feasible carrier=[A, B, C, D] with aligned exact objective values=[15, 30, 45, 60], candidate=A. The declared global lower bound is=15. Give (complete_objective_table, lower_bound, candidate, candidate_value, all_values_above_bound, bound_attained, global_optimum, unique, all_global_minimizers, certificate_scope).","answer":{"kind":"expression","value":"([[A, 15], [B, 30], [C, 45], [D, 60]], 15, A, 15, Yes, Yes, Yes, Yes, [A], complete_lower_bound_plus_attainment_certifies_global_but_not_automatic_uniqueness)","explanation":"Check the bound against every feasible value and exact attainment at the candidate. Retain all minimizers and set the uniqueness flag only if exactly one attains the bound. The certificate applies to this complete finite feasible domain, not unlisted candidates or continuous problems. The exact result, retaining the requested order, is ([[A, 15], [B, 30], [C, 45], [D, 60]], 15, A, 15, Yes, Yes, Yes, Yes, [A], complete_lower_bound_plus_attainment_certifies_global_but_not_automatic_uniqueness)."},"skills":["A Complete Attained Lower Bound Certifies Globality but Not Automatic Uniqueness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r07-0066","batch_id":"batch-20261015-advance-20261119-r07-optimization-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"00daad7a3af73240b0c666088e86d275086379929a3e95a109e3ad2fa73474bd","family_id":"qf-mod50-optdeep-lower-bound-certificate","seed":202611197066,"semantic_template":"optdeep-lower-bound-certificate|{\"candidate\":\"A\",\"lower_bound\":[16,1],\"points\":[\"A\",\"B\",\"C\",\"D\"],\"values\":[[16,1],[32,1],[48,1],[64,1]]}"},"fields":{"title":"A Complete Attained Lower Bound Certifies Globality but Not Automatic Uniqueness","stem":"Use the complete finite feasible carrier=[A, B, C, D] with aligned exact objective values=[16, 32, 48, 64], candidate=A. The declared global lower bound is=16. Give (complete_objective_table, lower_bound, candidate, candidate_value, all_values_above_bound, bound_attained, global_optimum, unique, all_global_minimizers, certificate_scope).","answer":{"kind":"expression","value":"([[A, 16], [B, 32], [C, 48], [D, 64]], 16, A, 16, Yes, Yes, Yes, Yes, [A], complete_lower_bound_plus_attainment_certifies_global_but_not_automatic_uniqueness)","explanation":"Check the bound against every feasible value and exact attainment at the candidate. Retain all minimizers and set the uniqueness flag only if exactly one attains the bound. The certificate applies to this complete finite feasible domain, not unlisted candidates or continuous problems. The exact result, retaining the requested order, is ([[A, 16], [B, 32], [C, 48], [D, 64]], 16, A, 16, Yes, Yes, Yes, Yes, [A], complete_lower_bound_plus_attainment_certifies_global_but_not_automatic_uniqueness)."},"skills":["A Complete Attained Lower Bound Certifies Globality but Not Automatic Uniqueness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r07-0067","batch_id":"batch-20261015-advance-20261119-r07-optimization-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6ddda75784884735c0c4649be275791d5df5495e8984d79a464886c257b9cbf1","family_id":"qf-mod50-optdeep-lower-bound-certificate","seed":202611197067,"semantic_template":"optdeep-lower-bound-certificate|{\"candidate\":\"A\",\"lower_bound\":[17,1],\"points\":[\"A\",\"B\",\"C\",\"D\"],\"values\":[[17,1],[34,1],[51,1],[68,1]]}"},"fields":{"title":"A Complete Attained Lower Bound Certifies Globality but Not Automatic Uniqueness","stem":"Use the complete finite feasible carrier=[A, B, C, D] with aligned exact objective values=[17, 34, 51, 68], candidate=A. The declared global lower bound is=17. Give (complete_objective_table, lower_bound, candidate, candidate_value, all_values_above_bound, bound_attained, global_optimum, unique, all_global_minimizers, certificate_scope).","answer":{"kind":"expression","value":"([[A, 17], [B, 34], [C, 51], [D, 68]], 17, A, 17, Yes, Yes, Yes, Yes, [A], complete_lower_bound_plus_attainment_certifies_global_but_not_automatic_uniqueness)","explanation":"Check the bound against every feasible value and exact attainment at the candidate. Retain all minimizers and set the uniqueness flag only if exactly one attains the bound. The certificate applies to this complete finite feasible domain, not unlisted candidates or continuous problems. The exact result, retaining the requested order, is ([[A, 17], [B, 34], [C, 51], [D, 68]], 17, A, 17, Yes, Yes, Yes, Yes, [A], complete_lower_bound_plus_attainment_certifies_global_but_not_automatic_uniqueness)."},"skills":["A Complete Attained Lower Bound Certifies Globality but Not Automatic Uniqueness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r07-0068","batch_id":"batch-20261015-advance-20261119-r07-optimization-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3437e3c6199d5594feab8016232a1e57e985b999b527f5e39e16c773e5bf28c1","family_id":"qf-mod50-optdeep-lower-bound-certificate","seed":202611197068,"semantic_template":"optdeep-lower-bound-certificate|{\"candidate\":\"A\",\"lower_bound\":[18,1],\"points\":[\"A\",\"B\",\"C\",\"D\"],\"values\":[[18,1],[36,1],[54,1],[72,1]]}"},"fields":{"title":"A Complete Attained Lower Bound Certifies Globality but Not Automatic Uniqueness","stem":"Use the complete finite feasible carrier=[A, B, C, D] with aligned exact objective values=[18, 36, 54, 72], candidate=A. The declared global lower bound is=18. Give (complete_objective_table, lower_bound, candidate, candidate_value, all_values_above_bound, bound_attained, global_optimum, unique, all_global_minimizers, certificate_scope).","answer":{"kind":"expression","value":"([[A, 18], [B, 36], [C, 54], [D, 72]], 18, A, 18, Yes, Yes, Yes, Yes, [A], complete_lower_bound_plus_attainment_certifies_global_but_not_automatic_uniqueness)","explanation":"Check the bound against every feasible value and exact attainment at the candidate. Retain all minimizers and set the uniqueness flag only if exactly one attains the bound. The certificate applies to this complete finite feasible domain, not unlisted candidates or continuous problems. The exact result, retaining the requested order, is ([[A, 18], [B, 36], [C, 54], [D, 72]], 18, A, 18, Yes, Yes, Yes, Yes, [A], complete_lower_bound_plus_attainment_certifies_global_but_not_automatic_uniqueness)."},"skills":["A Complete Attained Lower Bound Certifies Globality but Not Automatic Uniqueness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r07-0069","batch_id":"batch-20261015-advance-20261119-r07-optimization-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9addfe8336de4e805e0d61f5cc035e021562479e48539fff1576997312ebb956","family_id":"qf-mod50-optdeep-lower-bound-certificate","seed":202611197069,"semantic_template":"optdeep-lower-bound-certificate|{\"candidate\":\"A\",\"lower_bound\":[19,1],\"points\":[\"A\",\"B\",\"C\",\"D\"],\"values\":[[19,1],[38,1],[57,1],[76,1]]}"},"fields":{"title":"A Complete Attained Lower Bound Certifies Globality but Not Automatic Uniqueness","stem":"Use the complete finite feasible carrier=[A, B, C, D] with aligned exact objective values=[19, 38, 57, 76], candidate=A. The declared global lower bound is=19. Give (complete_objective_table, lower_bound, candidate, candidate_value, all_values_above_bound, bound_attained, global_optimum, unique, all_global_minimizers, certificate_scope).","answer":{"kind":"expression","value":"([[A, 19], [B, 38], [C, 57], [D, 76]], 19, A, 19, Yes, Yes, Yes, Yes, [A], complete_lower_bound_plus_attainment_certifies_global_but_not_automatic_uniqueness)","explanation":"Check the bound against every feasible value and exact attainment at the candidate. Retain all minimizers and set the uniqueness flag only if exactly one attains the bound. The certificate applies to this complete finite feasible domain, not unlisted candidates or continuous problems. The exact result, retaining the requested order, is ([[A, 19], [B, 38], [C, 57], [D, 76]], 19, A, 19, Yes, Yes, Yes, Yes, [A], complete_lower_bound_plus_attainment_certifies_global_but_not_automatic_uniqueness)."},"skills":["A Complete Attained Lower Bound Certifies Globality but Not Automatic Uniqueness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r07-0070","batch_id":"batch-20261015-advance-20261119-r07-optimization-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c893e5995b0d0674d378c8cb860b6bf3e24df7a270e15cf162dc8248982c6f39","family_id":"qf-mod50-optdeep-lower-bound-certificate","seed":202611197070,"semantic_template":"optdeep-lower-bound-certificate|{\"candidate\":\"A\",\"lower_bound\":[20,1],\"points\":[\"A\",\"B\",\"C\",\"D\"],\"values\":[[20,1],[40,1],[60,1],[80,1]]}"},"fields":{"title":"A Complete Attained Lower Bound Certifies Globality but Not Automatic Uniqueness","stem":"Use the complete finite feasible carrier=[A, B, C, D] with aligned exact objective values=[20, 40, 60, 80], candidate=A. The declared global lower bound is=20. Give (complete_objective_table, lower_bound, candidate, candidate_value, all_values_above_bound, bound_attained, global_optimum, unique, all_global_minimizers, certificate_scope).","answer":{"kind":"expression","value":"([[A, 20], [B, 40], [C, 60], [D, 80]], 20, A, 20, Yes, Yes, Yes, Yes, [A], complete_lower_bound_plus_attainment_certifies_global_but_not_automatic_uniqueness)","explanation":"Check the bound against every feasible value and exact attainment at the candidate. Retain all minimizers and set the uniqueness flag only if exactly one attains the bound. The certificate applies to this complete finite feasible domain, not unlisted candidates or continuous problems. The exact result, retaining the requested order, is ([[A, 20], [B, 40], [C, 60], [D, 80]], 20, A, 20, Yes, Yes, Yes, Yes, [A], complete_lower_bound_plus_attainment_certifies_global_but_not_automatic_uniqueness)."},"skills":["A Complete Attained Lower Bound Certifies Globality but Not Automatic Uniqueness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r07-0071","batch_id":"batch-20261015-advance-20261119-r07-optimization-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5dc52930cbc3615bbea45df89231571d07c2322204b002d7d4c7a02b80c8d35d","family_id":"qf-mod50-optdeep-lower-bound-certificate","seed":202611197071,"semantic_template":"optdeep-lower-bound-certificate|{\"candidate\":\"A\",\"lower_bound\":[21,1],\"points\":[\"A\",\"B\",\"C\",\"D\"],\"values\":[[21,1],[42,1],[63,1],[84,1]]}"},"fields":{"title":"A Complete Attained Lower Bound Certifies Globality but Not Automatic Uniqueness","stem":"Use the complete finite feasible carrier=[A, B, C, D] with aligned exact objective values=[21, 42, 63, 84], candidate=A. The declared global lower bound is=21. Give (complete_objective_table, lower_bound, candidate, candidate_value, all_values_above_bound, bound_attained, global_optimum, unique, all_global_minimizers, certificate_scope).","answer":{"kind":"expression","value":"([[A, 21], [B, 42], [C, 63], [D, 84]], 21, A, 21, Yes, Yes, Yes, Yes, [A], complete_lower_bound_plus_attainment_certifies_global_but_not_automatic_uniqueness)","explanation":"Check the bound against every feasible value and exact attainment at the candidate. Retain all minimizers and set the uniqueness flag only if exactly one attains the bound. The certificate applies to this complete finite feasible domain, not unlisted candidates or continuous problems. The exact result, retaining the requested order, is ([[A, 21], [B, 42], [C, 63], [D, 84]], 21, A, 21, Yes, Yes, Yes, Yes, [A], complete_lower_bound_plus_attainment_certifies_global_but_not_automatic_uniqueness)."},"skills":["A Complete Attained Lower Bound Certifies Globality but Not Automatic Uniqueness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r07-0072","batch_id":"batch-20261015-advance-20261119-r07-optimization-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b854a3a8a9966a2f0c4f0428be1c64d883336d9b735969d4eb92a896170b08a8","family_id":"qf-mod50-optdeep-lower-bound-certificate","seed":202611197072,"semantic_template":"optdeep-lower-bound-certificate|{\"candidate\":\"A\",\"lower_bound\":[22,1],\"points\":[\"A\",\"B\",\"C\",\"D\"],\"values\":[[22,1],[44,1],[66,1],[88,1]]}"},"fields":{"title":"A Complete Attained Lower Bound Certifies Globality but Not Automatic Uniqueness","stem":"Use the complete finite feasible carrier=[A, B, C, D] with aligned exact objective values=[22, 44, 66, 88], candidate=A. The declared global lower bound is=22. Give (complete_objective_table, lower_bound, candidate, candidate_value, all_values_above_bound, bound_attained, global_optimum, unique, all_global_minimizers, certificate_scope).","answer":{"kind":"expression","value":"([[A, 22], [B, 44], [C, 66], [D, 88]], 22, A, 22, Yes, Yes, Yes, Yes, [A], complete_lower_bound_plus_attainment_certifies_global_but_not_automatic_uniqueness)","explanation":"Check the bound against every feasible value and exact attainment at the candidate. Retain all minimizers and set the uniqueness flag only if exactly one attains the bound. The certificate applies to this complete finite feasible domain, not unlisted candidates or continuous problems. The exact result, retaining the requested order, is ([[A, 22], [B, 44], [C, 66], [D, 88]], 22, A, 22, Yes, Yes, Yes, Yes, [A], complete_lower_bound_plus_attainment_certifies_global_but_not_automatic_uniqueness)."},"skills":["A Complete Attained Lower Bound Certifies Globality but Not Automatic Uniqueness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r07-0073","batch_id":"batch-20261015-advance-20261119-r07-optimization-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"534ca60f948d46b8cdd0d64a62601797498e07cb3c2eea55c38d8ab2eb7ed515","family_id":"qf-mod50-optdeep-lower-bound-certificate","seed":202611197073,"semantic_template":"optdeep-lower-bound-certificate|{\"candidate\":\"A\",\"lower_bound\":[23,1],\"points\":[\"A\",\"B\",\"C\",\"D\"],\"values\":[[23,1],[46,1],[69,1],[92,1]]}"},"fields":{"title":"A Complete Attained Lower Bound Certifies Globality but Not Automatic Uniqueness","stem":"Use the complete finite feasible carrier=[A, B, C, D] with aligned exact objective values=[23, 46, 69, 92], candidate=A. The declared global lower bound is=23. Give (complete_objective_table, lower_bound, candidate, candidate_value, all_values_above_bound, bound_attained, global_optimum, unique, all_global_minimizers, certificate_scope).","answer":{"kind":"expression","value":"([[A, 23], [B, 46], [C, 69], [D, 92]], 23, A, 23, Yes, Yes, Yes, Yes, [A], complete_lower_bound_plus_attainment_certifies_global_but_not_automatic_uniqueness)","explanation":"Check the bound against every feasible value and exact attainment at the candidate. Retain all minimizers and set the uniqueness flag only if exactly one attains the bound. The certificate applies to this complete finite feasible domain, not unlisted candidates or continuous problems. The exact result, retaining the requested order, is ([[A, 23], [B, 46], [C, 69], [D, 92]], 23, A, 23, Yes, Yes, Yes, Yes, [A], complete_lower_bound_plus_attainment_certifies_global_but_not_automatic_uniqueness)."},"skills":["A Complete Attained Lower Bound Certifies Globality but Not Automatic Uniqueness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r07-0074","batch_id":"batch-20261015-advance-20261119-r07-optimization-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a47e30b1ec31f69be76fe9bf06bbe72db64d09cf4df1c66189b5511cdff5d484","family_id":"qf-mod50-optdeep-lower-bound-certificate","seed":202611197074,"semantic_template":"optdeep-lower-bound-certificate|{\"candidate\":\"A\",\"lower_bound\":[24,1],\"points\":[\"A\",\"B\",\"C\",\"D\"],\"values\":[[24,1],[48,1],[72,1],[96,1]]}"},"fields":{"title":"A Complete Attained Lower Bound Certifies Globality but Not Automatic Uniqueness","stem":"Use the complete finite feasible carrier=[A, B, C, D] with aligned exact objective values=[24, 48, 72, 96], candidate=A. The declared global lower bound is=24. Give (complete_objective_table, lower_bound, candidate, candidate_value, all_values_above_bound, bound_attained, global_optimum, unique, all_global_minimizers, certificate_scope).","answer":{"kind":"expression","value":"([[A, 24], [B, 48], [C, 72], [D, 96]], 24, A, 24, Yes, Yes, Yes, Yes, [A], complete_lower_bound_plus_attainment_certifies_global_but_not_automatic_uniqueness)","explanation":"Check the bound against every feasible value and exact attainment at the candidate. Retain all minimizers and set the uniqueness flag only if exactly one attains the bound. The certificate applies to this complete finite feasible domain, not unlisted candidates or continuous problems. The exact result, retaining the requested order, is ([[A, 24], [B, 48], [C, 72], [D, 96]], 24, A, 24, Yes, Yes, Yes, Yes, [A], complete_lower_bound_plus_attainment_certifies_global_but_not_automatic_uniqueness)."},"skills":["A Complete Attained Lower Bound Certifies Globality but Not Automatic Uniqueness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r07-0075","batch_id":"batch-20261015-advance-20261119-r07-optimization-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2c9538c95839ce2b2ceb95cadeb66c2a73a0ef9fd00506de6cdd9e54165101cb","family_id":"qf-mod50-optdeep-lower-bound-certificate","seed":202611197075,"semantic_template":"optdeep-lower-bound-certificate|{\"candidate\":\"A\",\"lower_bound\":[25,1],\"points\":[\"A\",\"B\",\"C\",\"D\"],\"values\":[[25,1],[50,1],[75,1],[100,1]]}"},"fields":{"title":"A Complete Attained Lower Bound Certifies Globality but Not Automatic Uniqueness","stem":"Use the complete finite feasible carrier=[A, B, C, D] with aligned exact objective values=[25, 50, 75, 100], candidate=A. The declared global lower bound is=25. Give (complete_objective_table, lower_bound, candidate, candidate_value, all_values_above_bound, bound_attained, global_optimum, unique, all_global_minimizers, certificate_scope).","answer":{"kind":"expression","value":"([[A, 25], [B, 50], [C, 75], [D, 100]], 25, A, 25, Yes, Yes, Yes, Yes, [A], complete_lower_bound_plus_attainment_certifies_global_but_not_automatic_uniqueness)","explanation":"Check the bound against every feasible value and exact attainment at the candidate. Retain all minimizers and set the uniqueness flag only if exactly one attains the bound. The certificate applies to this complete finite feasible domain, not unlisted candidates or continuous problems. The exact result, retaining the requested order, is ([[A, 25], [B, 50], [C, 75], [D, 100]], 25, A, 25, Yes, Yes, Yes, Yes, [A], complete_lower_bound_plus_attainment_certifies_global_but_not_automatic_uniqueness)."},"skills":["A Complete Attained Lower Bound Certifies Globality but Not Automatic Uniqueness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r07-0076","batch_id":"batch-20261015-advance-20261119-r07-optimization-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"580f8292e474bc568998de90d0b491b3369208ce5f120bcf39145bae72196c78","family_id":"qf-mod50-optdeep-boundary","seed":202611197076,"semantic_template":"optdeep-boundary|{\"mode\":\"empty_feasible\",\"model\":{}}"},"fields":{"title":"Empty Domains, Partial Lists, Ties and Local-Only Witnesses Block Different Claims","stem":"Use mode=empty_feasible, complete model={}. Give (mode, feasible_set, optimizer_exists_claim, boundary).","answer":{"kind":"expression","value":"(empty_feasible, [], No, empty_feasible_set_has_no_optimizer)","explanation":"The empty feasible set has no optimizer. A list not established as complete cannot certify globality. Tied minima establish existence but not uniqueness. A graph-local minimum can fail globally; use the actual complete neighborhood and feasible objective table rather than conflating the two comparison scopes. The exact result, retaining the requested order, is (empty_feasible, [], No, empty_feasible_set_has_no_optimizer)."},"skills":["Empty Domains, Partial Lists, Ties and Local-Only Witnesses Block Different Claims"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r07-0077","batch_id":"batch-20261015-advance-20261119-r07-optimization-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d5ffeecabb9eead4719aedd1d9c1d04618e7666a24e79e6a15828099d383ef35","family_id":"qf-mod50-optdeep-boundary","seed":202611197077,"semantic_template":"optdeep-boundary|{\"mode\":\"incomplete_domain\",\"model\":{\"listed_points\":[\"A\",\"B\"],\"values\":[[2,1],[4,1]]}}"},"fields":{"title":"Empty Domains, Partial Lists, Ties and Local-Only Witnesses Block Different Claims","stem":"Use mode=incomplete_domain, complete model={\"listed_points\":[\"A\",\"B\"],\"values\":[2,4]}. Give (mode, listed_objective_rows, global_certificate_claim, boundary).","answer":{"kind":"expression","value":"(incomplete_domain, [[A, 2], [B, 4]], No, an_unverified_partial_list_cannot_support_a_global_claim)","explanation":"The empty feasible set has no optimizer. A list not established as complete cannot certify globality. Tied minima establish existence but not uniqueness. A graph-local minimum can fail globally; use the actual complete neighborhood and feasible objective table rather than conflating the two comparison scopes. The exact result, retaining the requested order, is (incomplete_domain, [[A, 2], [B, 4]], No, an_unverified_partial_list_cannot_support_a_global_claim)."},"skills":["Empty Domains, Partial Lists, Ties and Local-Only Witnesses Block Different Claims"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r07-0078","batch_id":"batch-20261015-advance-20261119-r07-optimization-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"62f3b6eb3a61eda746c6b5e7c5a9a594e941aa99c9c17fa7e0c22180a905657f","family_id":"qf-mod50-optdeep-boundary","seed":202611197078,"semantic_template":"optdeep-boundary|{\"mode\":\"tie_not_unique\",\"model\":{\"points\":[\"A\",\"B\",\"C\"],\"values\":[[3,1],[3,1],[9,1]]}}"},"fields":{"title":"Empty Domains, Partial Lists, Ties and Local-Only Witnesses Block Different Claims","stem":"Use mode=tie_not_unique, complete model={\"points\":[\"A\",\"B\",\"C\"],\"values\":[3,3,9]}. Give (mode, minimum_value, all_global_minimizers, global_optimum_exists, unique_claim, boundary).","answer":{"kind":"expression","value":"(tie_not_unique, 3, [A, B], Yes, No, global_optimum_need_not_be_unique)","explanation":"The empty feasible set has no optimizer. A list not established as complete cannot certify globality. Tied minima establish existence but not uniqueness. A graph-local minimum can fail globally; use the actual complete neighborhood and feasible objective table rather than conflating the two comparison scopes. The exact result, retaining the requested order, is (tie_not_unique, 3, [A, B], Yes, No, global_optimum_need_not_be_unique)."},"skills":["Empty Domains, Partial Lists, Ties and Local-Only Witnesses Block Different Claims"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r07-0079","batch_id":"batch-20261015-advance-20261119-r07-optimization-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"50fefe47ff49a158ca8816a3162683adf01085619e0cb6103fde6c4697d0dc64","family_id":"qf-mod50-optdeep-boundary","seed":202611197079,"semantic_template":"optdeep-boundary|{\"mode\":\"local_not_global\",\"model\":{\"candidate\":\"A\",\"edges\":[[\"A\",\"B\"]],\"points\":[\"A\",\"B\",\"C\",\"D\"],\"values\":[[8,1],[12,1],[4,1],[16,1]]}}"},"fields":{"title":"Empty Domains, Partial Lists, Ties and Local-Only Witnesses Block Different Claims","stem":"Use mode=local_not_global, complete model={\"points\":[\"A\",\"B\",\"C\",\"D\"],\"values\":[8,12,4,16],\"edges\":[[\"A\",\"B\"]],\"candidate\":\"A\"}. Give (mode, candidate, candidate_value, declared_neighbors, neighborhood_local_minimum, global_minimum, boundary).","answer":{"kind":"expression","value":"(local_not_global, A, 8, [B], Yes, No, local_minimum_does_not_imply_global_minimum)","explanation":"The empty feasible set has no optimizer. A list not established as complete cannot certify globality. Tied minima establish existence but not uniqueness. A graph-local minimum can fail globally; use the actual complete neighborhood and feasible objective table rather than conflating the two comparison scopes. The exact result, retaining the requested order, is (local_not_global, A, 8, [B], Yes, No, local_minimum_does_not_imply_global_minimum)."},"skills":["Empty Domains, Partial Lists, Ties and Local-Only Witnesses Block Different Claims"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r07-0080","batch_id":"batch-20261015-advance-20261119-r07-optimization-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d6a61865a72b4272dca285e271a2b95fe7752ee3765743eeb43bd84f13dad239","family_id":"qf-mod50-optdeep-boundary","seed":202611197080,"semantic_template":"optdeep-boundary|{\"mode\":\"incomplete_domain\",\"model\":{\"listed_points\":[\"A\",\"B\"],\"values\":[[5,1],[10,1]]}}"},"fields":{"title":"Empty Domains, Partial Lists, Ties and Local-Only Witnesses Block Different Claims","stem":"Use mode=incomplete_domain, complete model={\"listed_points\":[\"A\",\"B\"],\"values\":[5,10]}. Give (mode, listed_objective_rows, global_certificate_claim, boundary).","answer":{"kind":"expression","value":"(incomplete_domain, [[A, 5], [B, 10]], No, an_unverified_partial_list_cannot_support_a_global_claim)","explanation":"The empty feasible set has no optimizer. A list not established as complete cannot certify globality. Tied minima establish existence but not uniqueness. A graph-local minimum can fail globally; use the actual complete neighborhood and feasible objective table rather than conflating the two comparison scopes. The exact result, retaining the requested order, is (incomplete_domain, [[A, 5], [B, 10]], No, an_unverified_partial_list_cannot_support_a_global_claim)."},"skills":["Empty Domains, Partial Lists, Ties and Local-Only Witnesses Block Different Claims"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r07-0081","batch_id":"batch-20261015-advance-20261119-r07-optimization-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e650e9cc450c38991a5e1ce205a64cd823d8f7c156e3e18f06d0b54e7da7a6bf","family_id":"qf-mod50-optdeep-boundary","seed":202611197081,"semantic_template":"optdeep-boundary|{\"mode\":\"tie_not_unique\",\"model\":{\"points\":[\"A\",\"B\",\"C\"],\"values\":[[6,1],[6,1],[18,1]]}}"},"fields":{"title":"Empty Domains, Partial Lists, Ties and Local-Only Witnesses Block Different Claims","stem":"Use mode=tie_not_unique, complete model={\"points\":[\"A\",\"B\",\"C\"],\"values\":[6,6,18]}. Give (mode, minimum_value, all_global_minimizers, global_optimum_exists, unique_claim, boundary).","answer":{"kind":"expression","value":"(tie_not_unique, 6, [A, B], Yes, No, global_optimum_need_not_be_unique)","explanation":"The empty feasible set has no optimizer. A list not established as complete cannot certify globality. Tied minima establish existence but not uniqueness. A graph-local minimum can fail globally; use the actual complete neighborhood and feasible objective table rather than conflating the two comparison scopes. The exact result, retaining the requested order, is (tie_not_unique, 6, [A, B], Yes, No, global_optimum_need_not_be_unique)."},"skills":["Empty Domains, Partial Lists, Ties and Local-Only Witnesses Block Different Claims"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r07-0082","batch_id":"batch-20261015-advance-20261119-r07-optimization-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3c1820bddd20f5aecf04c3a61815e52ec97eac2f64a7a1fed2903cd24ac31b52","family_id":"qf-mod50-optdeep-boundary","seed":202611197082,"semantic_template":"optdeep-boundary|{\"mode\":\"local_not_global\",\"model\":{\"candidate\":\"A\",\"edges\":[[\"A\",\"B\"]],\"points\":[\"A\",\"B\",\"C\",\"D\"],\"values\":[[14,1],[21,1],[7,1],[28,1]]}}"},"fields":{"title":"Empty Domains, Partial Lists, Ties and Local-Only Witnesses Block Different Claims","stem":"Use mode=local_not_global, complete model={\"points\":[\"A\",\"B\",\"C\",\"D\"],\"values\":[14,21,7,28],\"edges\":[[\"A\",\"B\"]],\"candidate\":\"A\"}. Give (mode, candidate, candidate_value, declared_neighbors, neighborhood_local_minimum, global_minimum, boundary).","answer":{"kind":"expression","value":"(local_not_global, A, 14, [B], Yes, No, local_minimum_does_not_imply_global_minimum)","explanation":"The empty feasible set has no optimizer. A list not established as complete cannot certify globality. Tied minima establish existence but not uniqueness. A graph-local minimum can fail globally; use the actual complete neighborhood and feasible objective table rather than conflating the two comparison scopes. The exact result, retaining the requested order, is (local_not_global, A, 14, [B], Yes, No, local_minimum_does_not_imply_global_minimum)."},"skills":["Empty Domains, Partial Lists, Ties and Local-Only Witnesses Block Different Claims"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r07-0083","batch_id":"batch-20261015-advance-20261119-r07-optimization-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"76aa2ba4c60cd10b52ba8b541bd0e1e5b605c9e85eac55a7fc8b0f59eaf4acff","family_id":"qf-mod50-optdeep-boundary","seed":202611197083,"semantic_template":"optdeep-boundary|{\"mode\":\"incomplete_domain\",\"model\":{\"listed_points\":[\"A\",\"B\"],\"values\":[[8,1],[16,1]]}}"},"fields":{"title":"Empty Domains, Partial Lists, Ties and Local-Only Witnesses Block Different Claims","stem":"Use mode=incomplete_domain, complete model={\"listed_points\":[\"A\",\"B\"],\"values\":[8,16]}. Give (mode, listed_objective_rows, global_certificate_claim, boundary).","answer":{"kind":"expression","value":"(incomplete_domain, [[A, 8], [B, 16]], No, an_unverified_partial_list_cannot_support_a_global_claim)","explanation":"The empty feasible set has no optimizer. A list not established as complete cannot certify globality. Tied minima establish existence but not uniqueness. A graph-local minimum can fail globally; use the actual complete neighborhood and feasible objective table rather than conflating the two comparison scopes. The exact result, retaining the requested order, is (incomplete_domain, [[A, 8], [B, 16]], No, an_unverified_partial_list_cannot_support_a_global_claim)."},"skills":["Empty Domains, Partial Lists, Ties and Local-Only Witnesses Block Different Claims"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r07-0084","batch_id":"batch-20261015-advance-20261119-r07-optimization-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c36d186bbd4d0dd15491d313d9dacbfd49698373c237f757884a6a5a8edc5972","family_id":"qf-mod50-optdeep-boundary","seed":202611197084,"semantic_template":"optdeep-boundary|{\"mode\":\"tie_not_unique\",\"model\":{\"points\":[\"A\",\"B\",\"C\"],\"values\":[[9,1],[9,1],[27,1]]}}"},"fields":{"title":"Empty Domains, Partial Lists, Ties and Local-Only Witnesses Block Different Claims","stem":"Use mode=tie_not_unique, complete model={\"points\":[\"A\",\"B\",\"C\"],\"values\":[9,9,27]}. Give (mode, minimum_value, all_global_minimizers, global_optimum_exists, unique_claim, boundary).","answer":{"kind":"expression","value":"(tie_not_unique, 9, [A, B], Yes, No, global_optimum_need_not_be_unique)","explanation":"The empty feasible set has no optimizer. A list not established as complete cannot certify globality. Tied minima establish existence but not uniqueness. A graph-local minimum can fail globally; use the actual complete neighborhood and feasible objective table rather than conflating the two comparison scopes. The exact result, retaining the requested order, is (tie_not_unique, 9, [A, B], Yes, No, global_optimum_need_not_be_unique)."},"skills":["Empty Domains, Partial Lists, Ties and Local-Only Witnesses Block Different Claims"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r07-0085","batch_id":"batch-20261015-advance-20261119-r07-optimization-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"319699c3cf8520bfcd4434944a125f8280f922e08c29b230311525e4413a4278","family_id":"qf-mod50-optdeep-boundary","seed":202611197085,"semantic_template":"optdeep-boundary|{\"mode\":\"local_not_global\",\"model\":{\"candidate\":\"A\",\"edges\":[[\"A\",\"B\"]],\"points\":[\"A\",\"B\",\"C\",\"D\"],\"values\":[[20,1],[30,1],[10,1],[40,1]]}}"},"fields":{"title":"Empty Domains, Partial Lists, Ties and Local-Only Witnesses Block Different Claims","stem":"Use mode=local_not_global, complete model={\"points\":[\"A\",\"B\",\"C\",\"D\"],\"values\":[20,30,10,40],\"edges\":[[\"A\",\"B\"]],\"candidate\":\"A\"}. Give (mode, candidate, candidate_value, declared_neighbors, neighborhood_local_minimum, global_minimum, boundary).","answer":{"kind":"expression","value":"(local_not_global, A, 20, [B], Yes, No, local_minimum_does_not_imply_global_minimum)","explanation":"The empty feasible set has no optimizer. A list not established as complete cannot certify globality. Tied minima establish existence but not uniqueness. A graph-local minimum can fail globally; use the actual complete neighborhood and feasible objective table rather than conflating the two comparison scopes. The exact result, retaining the requested order, is (local_not_global, A, 20, [B], Yes, No, local_minimum_does_not_imply_global_minimum)."},"skills":["Empty Domains, Partial Lists, Ties and Local-Only Witnesses Block Different Claims"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r07-0086","batch_id":"batch-20261015-advance-20261119-r07-optimization-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"55e7a4dbc855cfedc74613d478eba4194da215e448df0823d3e52fb6e221decb","family_id":"qf-mod50-optdeep-boundary","seed":202611197086,"semantic_template":"optdeep-boundary|{\"mode\":\"incomplete_domain\",\"model\":{\"listed_points\":[\"A\",\"B\"],\"values\":[[11,1],[22,1]]}}"},"fields":{"title":"Empty Domains, Partial Lists, Ties and Local-Only Witnesses Block Different Claims","stem":"Use mode=incomplete_domain, complete model={\"listed_points\":[\"A\",\"B\"],\"values\":[11,22]}. Give (mode, listed_objective_rows, global_certificate_claim, boundary).","answer":{"kind":"expression","value":"(incomplete_domain, [[A, 11], [B, 22]], No, an_unverified_partial_list_cannot_support_a_global_claim)","explanation":"The empty feasible set has no optimizer. A list not established as complete cannot certify globality. Tied minima establish existence but not uniqueness. A graph-local minimum can fail globally; use the actual complete neighborhood and feasible objective table rather than conflating the two comparison scopes. The exact result, retaining the requested order, is (incomplete_domain, [[A, 11], [B, 22]], No, an_unverified_partial_list_cannot_support_a_global_claim)."},"skills":["Empty Domains, Partial Lists, Ties and Local-Only Witnesses Block Different Claims"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r07-0087","batch_id":"batch-20261015-advance-20261119-r07-optimization-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"da0e5864043905c39101785b0cd6ff36655606c91b0ddccabb3408edf21e627a","family_id":"qf-mod50-optdeep-boundary","seed":202611197087,"semantic_template":"optdeep-boundary|{\"mode\":\"tie_not_unique\",\"model\":{\"points\":[\"A\",\"B\",\"C\"],\"values\":[[12,1],[12,1],[36,1]]}}"},"fields":{"title":"Empty Domains, Partial Lists, Ties and Local-Only Witnesses Block Different Claims","stem":"Use mode=tie_not_unique, complete model={\"points\":[\"A\",\"B\",\"C\"],\"values\":[12,12,36]}. Give (mode, minimum_value, all_global_minimizers, global_optimum_exists, unique_claim, boundary).","answer":{"kind":"expression","value":"(tie_not_unique, 12, [A, B], Yes, No, global_optimum_need_not_be_unique)","explanation":"The empty feasible set has no optimizer. A list not established as complete cannot certify globality. Tied minima establish existence but not uniqueness. A graph-local minimum can fail globally; use the actual complete neighborhood and feasible objective table rather than conflating the two comparison scopes. The exact result, retaining the requested order, is (tie_not_unique, 12, [A, B], Yes, No, global_optimum_need_not_be_unique)."},"skills":["Empty Domains, Partial Lists, Ties and Local-Only Witnesses Block Different Claims"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r07-0088","batch_id":"batch-20261015-advance-20261119-r07-optimization-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6f1b93eb16d1015a024fe7a28150df2064d6329b056d453976aae0a84ece0fdb","family_id":"qf-mod50-optdeep-boundary","seed":202611197088,"semantic_template":"optdeep-boundary|{\"mode\":\"local_not_global\",\"model\":{\"candidate\":\"A\",\"edges\":[[\"A\",\"B\"]],\"points\":[\"A\",\"B\",\"C\",\"D\"],\"values\":[[26,1],[39,1],[13,1],[52,1]]}}"},"fields":{"title":"Empty Domains, Partial Lists, Ties and Local-Only Witnesses Block Different Claims","stem":"Use mode=local_not_global, complete model={\"points\":[\"A\",\"B\",\"C\",\"D\"],\"values\":[26,39,13,52],\"edges\":[[\"A\",\"B\"]],\"candidate\":\"A\"}. Give (mode, candidate, candidate_value, declared_neighbors, neighborhood_local_minimum, global_minimum, boundary).","answer":{"kind":"expression","value":"(local_not_global, A, 26, [B], Yes, No, local_minimum_does_not_imply_global_minimum)","explanation":"The empty feasible set has no optimizer. A list not established as complete cannot certify globality. Tied minima establish existence but not uniqueness. A graph-local minimum can fail globally; use the actual complete neighborhood and feasible objective table rather than conflating the two comparison scopes. The exact result, retaining the requested order, is (local_not_global, A, 26, [B], Yes, No, local_minimum_does_not_imply_global_minimum)."},"skills":["Empty Domains, Partial Lists, Ties and Local-Only Witnesses Block Different Claims"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r07-0089","batch_id":"batch-20261015-advance-20261119-r07-optimization-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d5a8c82bb0c3eb7caac5f0c50f0c4702566f83be5f3041f201246e86653b05c5","family_id":"qf-mod50-optdeep-boundary","seed":202611197089,"semantic_template":"optdeep-boundary|{\"mode\":\"incomplete_domain\",\"model\":{\"listed_points\":[\"A\",\"B\"],\"values\":[[14,1],[28,1]]}}"},"fields":{"title":"Empty Domains, Partial Lists, Ties and Local-Only Witnesses Block Different Claims","stem":"Use mode=incomplete_domain, complete model={\"listed_points\":[\"A\",\"B\"],\"values\":[14,28]}. Give (mode, listed_objective_rows, global_certificate_claim, boundary).","answer":{"kind":"expression","value":"(incomplete_domain, [[A, 14], [B, 28]], No, an_unverified_partial_list_cannot_support_a_global_claim)","explanation":"The empty feasible set has no optimizer. A list not established as complete cannot certify globality. Tied minima establish existence but not uniqueness. A graph-local minimum can fail globally; use the actual complete neighborhood and feasible objective table rather than conflating the two comparison scopes. The exact result, retaining the requested order, is (incomplete_domain, [[A, 14], [B, 28]], No, an_unverified_partial_list_cannot_support_a_global_claim)."},"skills":["Empty Domains, Partial Lists, Ties and Local-Only Witnesses Block Different Claims"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r07-0090","batch_id":"batch-20261015-advance-20261119-r07-optimization-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7baefe228530279e71b1b8e33ccb52ec46d91b0b1a93fc96f00ed4ed1ddc116c","family_id":"qf-mod50-optdeep-boundary","seed":202611197090,"semantic_template":"optdeep-boundary|{\"mode\":\"tie_not_unique\",\"model\":{\"points\":[\"A\",\"B\",\"C\"],\"values\":[[15,1],[15,1],[45,1]]}}"},"fields":{"title":"Empty Domains, Partial Lists, Ties and Local-Only Witnesses Block Different Claims","stem":"Use mode=tie_not_unique, complete model={\"points\":[\"A\",\"B\",\"C\"],\"values\":[15,15,45]}. Give (mode, minimum_value, all_global_minimizers, global_optimum_exists, unique_claim, boundary).","answer":{"kind":"expression","value":"(tie_not_unique, 15, [A, B], Yes, No, global_optimum_need_not_be_unique)","explanation":"The empty feasible set has no optimizer. A list not established as complete cannot certify globality. Tied minima establish existence but not uniqueness. A graph-local minimum can fail globally; use the actual complete neighborhood and feasible objective table rather than conflating the two comparison scopes. The exact result, retaining the requested order, is (tie_not_unique, 15, [A, B], Yes, No, global_optimum_need_not_be_unique)."},"skills":["Empty Domains, Partial Lists, Ties and Local-Only Witnesses Block Different Claims"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r07-0091","batch_id":"batch-20261015-advance-20261119-r07-optimization-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bd878a15f4e11a396633b3092a12821ea10db563911fdaa82168583c91ad3b66","family_id":"qf-mod50-optdeep-boundary","seed":202611197091,"semantic_template":"optdeep-boundary|{\"mode\":\"local_not_global\",\"model\":{\"candidate\":\"A\",\"edges\":[[\"A\",\"B\"]],\"points\":[\"A\",\"B\",\"C\",\"D\"],\"values\":[[32,1],[48,1],[16,1],[64,1]]}}"},"fields":{"title":"Empty Domains, Partial Lists, Ties and Local-Only Witnesses Block Different Claims","stem":"Use mode=local_not_global, complete model={\"points\":[\"A\",\"B\",\"C\",\"D\"],\"values\":[32,48,16,64],\"edges\":[[\"A\",\"B\"]],\"candidate\":\"A\"}. Give (mode, candidate, candidate_value, declared_neighbors, neighborhood_local_minimum, global_minimum, boundary).","answer":{"kind":"expression","value":"(local_not_global, A, 32, [B], Yes, No, local_minimum_does_not_imply_global_minimum)","explanation":"The empty feasible set has no optimizer. A list not established as complete cannot certify globality. Tied minima establish existence but not uniqueness. A graph-local minimum can fail globally; use the actual complete neighborhood and feasible objective table rather than conflating the two comparison scopes. The exact result, retaining the requested order, is (local_not_global, A, 32, [B], Yes, No, local_minimum_does_not_imply_global_minimum)."},"skills":["Empty Domains, Partial Lists, Ties and Local-Only Witnesses Block Different Claims"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r07-0092","batch_id":"batch-20261015-advance-20261119-r07-optimization-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5ddc9e3f6683511548d252068ce4499f6d94b7d361a3dad83e392bc28b101931","family_id":"qf-mod50-optdeep-boundary","seed":202611197092,"semantic_template":"optdeep-boundary|{\"mode\":\"incomplete_domain\",\"model\":{\"listed_points\":[\"A\",\"B\"],\"values\":[[17,1],[34,1]]}}"},"fields":{"title":"Empty Domains, Partial Lists, Ties and Local-Only Witnesses Block Different Claims","stem":"Use mode=incomplete_domain, complete model={\"listed_points\":[\"A\",\"B\"],\"values\":[17,34]}. Give (mode, listed_objective_rows, global_certificate_claim, boundary).","answer":{"kind":"expression","value":"(incomplete_domain, [[A, 17], [B, 34]], No, an_unverified_partial_list_cannot_support_a_global_claim)","explanation":"The empty feasible set has no optimizer. A list not established as complete cannot certify globality. Tied minima establish existence but not uniqueness. A graph-local minimum can fail globally; use the actual complete neighborhood and feasible objective table rather than conflating the two comparison scopes. The exact result, retaining the requested order, is (incomplete_domain, [[A, 17], [B, 34]], No, an_unverified_partial_list_cannot_support_a_global_claim)."},"skills":["Empty Domains, Partial Lists, Ties and Local-Only Witnesses Block Different Claims"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r07-0093","batch_id":"batch-20261015-advance-20261119-r07-optimization-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"db601a9779a3ce892044ffa0a260f394533e765d078f9f0f65e512b56b547ee9","family_id":"qf-mod50-optdeep-boundary","seed":202611197093,"semantic_template":"optdeep-boundary|{\"mode\":\"tie_not_unique\",\"model\":{\"points\":[\"A\",\"B\",\"C\"],\"values\":[[18,1],[18,1],[54,1]]}}"},"fields":{"title":"Empty Domains, Partial Lists, Ties and Local-Only Witnesses Block Different Claims","stem":"Use mode=tie_not_unique, complete model={\"points\":[\"A\",\"B\",\"C\"],\"values\":[18,18,54]}. Give (mode, minimum_value, all_global_minimizers, global_optimum_exists, unique_claim, boundary).","answer":{"kind":"expression","value":"(tie_not_unique, 18, [A, B], Yes, No, global_optimum_need_not_be_unique)","explanation":"The empty feasible set has no optimizer. A list not established as complete cannot certify globality. Tied minima establish existence but not uniqueness. A graph-local minimum can fail globally; use the actual complete neighborhood and feasible objective table rather than conflating the two comparison scopes. The exact result, retaining the requested order, is (tie_not_unique, 18, [A, B], Yes, No, global_optimum_need_not_be_unique)."},"skills":["Empty Domains, Partial Lists, Ties and Local-Only Witnesses Block Different Claims"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r07-0094","batch_id":"batch-20261015-advance-20261119-r07-optimization-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2c788c99393dc11018b89c335f25fe0af5f1dbc75797a7d5b96e8c099062e73f","family_id":"qf-mod50-optdeep-boundary","seed":202611197094,"semantic_template":"optdeep-boundary|{\"mode\":\"local_not_global\",\"model\":{\"candidate\":\"A\",\"edges\":[[\"A\",\"B\"]],\"points\":[\"A\",\"B\",\"C\",\"D\"],\"values\":[[38,1],[57,1],[19,1],[76,1]]}}"},"fields":{"title":"Empty Domains, Partial Lists, Ties and Local-Only Witnesses Block Different Claims","stem":"Use mode=local_not_global, complete model={\"points\":[\"A\",\"B\",\"C\",\"D\"],\"values\":[38,57,19,76],\"edges\":[[\"A\",\"B\"]],\"candidate\":\"A\"}. Give (mode, candidate, candidate_value, declared_neighbors, neighborhood_local_minimum, global_minimum, boundary).","answer":{"kind":"expression","value":"(local_not_global, A, 38, [B], Yes, No, local_minimum_does_not_imply_global_minimum)","explanation":"The empty feasible set has no optimizer. A list not established as complete cannot certify globality. Tied minima establish existence but not uniqueness. A graph-local minimum can fail globally; use the actual complete neighborhood and feasible objective table rather than conflating the two comparison scopes. The exact result, retaining the requested order, is (local_not_global, A, 38, [B], Yes, No, local_minimum_does_not_imply_global_minimum)."},"skills":["Empty Domains, Partial Lists, Ties and Local-Only Witnesses Block Different Claims"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r07-0095","batch_id":"batch-20261015-advance-20261119-r07-optimization-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e7591d8584969404f9a0caceaeda140f6d3a269a9f0b1c0dc92b04f340057a38","family_id":"qf-mod50-optdeep-boundary","seed":202611197095,"semantic_template":"optdeep-boundary|{\"mode\":\"incomplete_domain\",\"model\":{\"listed_points\":[\"A\",\"B\"],\"values\":[[20,1],[40,1]]}}"},"fields":{"title":"Empty Domains, Partial Lists, Ties and Local-Only Witnesses Block Different Claims","stem":"Use mode=incomplete_domain, complete model={\"listed_points\":[\"A\",\"B\"],\"values\":[20,40]}. Give (mode, listed_objective_rows, global_certificate_claim, boundary).","answer":{"kind":"expression","value":"(incomplete_domain, [[A, 20], [B, 40]], No, an_unverified_partial_list_cannot_support_a_global_claim)","explanation":"The empty feasible set has no optimizer. A list not established as complete cannot certify globality. Tied minima establish existence but not uniqueness. A graph-local minimum can fail globally; use the actual complete neighborhood and feasible objective table rather than conflating the two comparison scopes. The exact result, retaining the requested order, is (incomplete_domain, [[A, 20], [B, 40]], No, an_unverified_partial_list_cannot_support_a_global_claim)."},"skills":["Empty Domains, Partial Lists, Ties and Local-Only Witnesses Block Different Claims"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r07-0096","batch_id":"batch-20261015-advance-20261119-r07-optimization-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ad71ed879a823ff7ce6f11ce3d9b0ed47de3b0eb9962ef4af8860dc6bba5d6fb","family_id":"qf-mod50-optdeep-boundary","seed":202611197096,"semantic_template":"optdeep-boundary|{\"mode\":\"tie_not_unique\",\"model\":{\"points\":[\"A\",\"B\",\"C\"],\"values\":[[21,1],[21,1],[63,1]]}}"},"fields":{"title":"Empty Domains, Partial Lists, Ties and Local-Only Witnesses Block Different Claims","stem":"Use mode=tie_not_unique, complete model={\"points\":[\"A\",\"B\",\"C\"],\"values\":[21,21,63]}. Give (mode, minimum_value, all_global_minimizers, global_optimum_exists, unique_claim, boundary).","answer":{"kind":"expression","value":"(tie_not_unique, 21, [A, B], Yes, No, global_optimum_need_not_be_unique)","explanation":"The empty feasible set has no optimizer. A list not established as complete cannot certify globality. Tied minima establish existence but not uniqueness. A graph-local minimum can fail globally; use the actual complete neighborhood and feasible objective table rather than conflating the two comparison scopes. The exact result, retaining the requested order, is (tie_not_unique, 21, [A, B], Yes, No, global_optimum_need_not_be_unique)."},"skills":["Empty Domains, Partial Lists, Ties and Local-Only Witnesses Block Different Claims"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r07-0097","batch_id":"batch-20261015-advance-20261119-r07-optimization-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2af99d761422b8b94c307dafa59a4c7c7e5aba89c72a24a5a4c317eb3d40170a","family_id":"qf-mod50-optdeep-boundary","seed":202611197097,"semantic_template":"optdeep-boundary|{\"mode\":\"local_not_global\",\"model\":{\"candidate\":\"A\",\"edges\":[[\"A\",\"B\"]],\"points\":[\"A\",\"B\",\"C\",\"D\"],\"values\":[[44,1],[66,1],[22,1],[88,1]]}}"},"fields":{"title":"Empty Domains, Partial Lists, Ties and Local-Only Witnesses Block Different Claims","stem":"Use mode=local_not_global, complete model={\"points\":[\"A\",\"B\",\"C\",\"D\"],\"values\":[44,66,22,88],\"edges\":[[\"A\",\"B\"]],\"candidate\":\"A\"}. Give (mode, candidate, candidate_value, declared_neighbors, neighborhood_local_minimum, global_minimum, boundary).","answer":{"kind":"expression","value":"(local_not_global, A, 44, [B], Yes, No, local_minimum_does_not_imply_global_minimum)","explanation":"The empty feasible set has no optimizer. A list not established as complete cannot certify globality. Tied minima establish existence but not uniqueness. A graph-local minimum can fail globally; use the actual complete neighborhood and feasible objective table rather than conflating the two comparison scopes. The exact result, retaining the requested order, is (local_not_global, A, 44, [B], Yes, No, local_minimum_does_not_imply_global_minimum)."},"skills":["Empty Domains, Partial Lists, Ties and Local-Only Witnesses Block Different Claims"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r07-0098","batch_id":"batch-20261015-advance-20261119-r07-optimization-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2c5d5bda5fccd0cac9ebdacce3c5ec2a72010ffa56d4eba33df3256b6ec63187","family_id":"qf-mod50-optdeep-boundary","seed":202611197098,"semantic_template":"optdeep-boundary|{\"mode\":\"incomplete_domain\",\"model\":{\"listed_points\":[\"A\",\"B\"],\"values\":[[23,1],[46,1]]}}"},"fields":{"title":"Empty Domains, Partial Lists, Ties and Local-Only Witnesses Block Different Claims","stem":"Use mode=incomplete_domain, complete model={\"listed_points\":[\"A\",\"B\"],\"values\":[23,46]}. Give (mode, listed_objective_rows, global_certificate_claim, boundary).","answer":{"kind":"expression","value":"(incomplete_domain, [[A, 23], [B, 46]], No, an_unverified_partial_list_cannot_support_a_global_claim)","explanation":"The empty feasible set has no optimizer. A list not established as complete cannot certify globality. Tied minima establish existence but not uniqueness. A graph-local minimum can fail globally; use the actual complete neighborhood and feasible objective table rather than conflating the two comparison scopes. The exact result, retaining the requested order, is (incomplete_domain, [[A, 23], [B, 46]], No, an_unverified_partial_list_cannot_support_a_global_claim)."},"skills":["Empty Domains, Partial Lists, Ties and Local-Only Witnesses Block Different Claims"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r07-0099","batch_id":"batch-20261015-advance-20261119-r07-optimization-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4b7fe0a96acc0308e5ed685f99380647b52431a000e7510c24ca94df25632eaf","family_id":"qf-mod50-optdeep-boundary","seed":202611197099,"semantic_template":"optdeep-boundary|{\"mode\":\"tie_not_unique\",\"model\":{\"points\":[\"A\",\"B\",\"C\"],\"values\":[[24,1],[24,1],[72,1]]}}"},"fields":{"title":"Empty Domains, Partial Lists, Ties and Local-Only Witnesses Block Different Claims","stem":"Use mode=tie_not_unique, complete model={\"points\":[\"A\",\"B\",\"C\"],\"values\":[24,24,72]}. Give (mode, minimum_value, all_global_minimizers, global_optimum_exists, unique_claim, boundary).","answer":{"kind":"expression","value":"(tie_not_unique, 24, [A, B], Yes, No, global_optimum_need_not_be_unique)","explanation":"The empty feasible set has no optimizer. A list not established as complete cannot certify globality. Tied minima establish existence but not uniqueness. A graph-local minimum can fail globally; use the actual complete neighborhood and feasible objective table rather than conflating the two comparison scopes. The exact result, retaining the requested order, is (tie_not_unique, 24, [A, B], Yes, No, global_optimum_need_not_be_unique)."},"skills":["Empty Domains, Partial Lists, Ties and Local-Only Witnesses Block Different Claims"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r07-0100","batch_id":"batch-20261015-advance-20261119-r07-optimization-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"96ef4305d0187781c068e0406d72604159ae6e8a2781d1dd485f1b95707164fb","family_id":"qf-mod50-optdeep-boundary","seed":202611197100,"semantic_template":"optdeep-boundary|{\"mode\":\"local_not_global\",\"model\":{\"candidate\":\"A\",\"edges\":[[\"A\",\"B\"]],\"points\":[\"A\",\"B\",\"C\",\"D\"],\"values\":[[50,1],[75,1],[25,1],[100,1]]}}"},"fields":{"title":"Empty Domains, Partial Lists, Ties and Local-Only Witnesses Block Different Claims","stem":"Use mode=local_not_global, complete model={\"points\":[\"A\",\"B\",\"C\",\"D\"],\"values\":[50,75,25,100],\"edges\":[[\"A\",\"B\"]],\"candidate\":\"A\"}. Give (mode, candidate, candidate_value, declared_neighbors, neighborhood_local_minimum, global_minimum, boundary).","answer":{"kind":"expression","value":"(local_not_global, A, 50, [B], Yes, No, local_minimum_does_not_imply_global_minimum)","explanation":"The empty feasible set has no optimizer. A list not established as complete cannot certify globality. Tied minima establish existence but not uniqueness. A graph-local minimum can fail globally; use the actual complete neighborhood and feasible objective table rather than conflating the two comparison scopes. The exact result, retaining the requested order, is (local_not_global, A, 50, [B], Yes, No, local_minimum_does_not_imply_global_minimum)."},"skills":["Empty Domains, Partial Lists, Ties and Local-Only Witnesses Block Different Claims"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}