Complete Finite Neighborhoods, Improvement Witnesses and Attained Bounds
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batch-20261015-advance-20261119-r07-optimization-problem-foundations · 2026-10-15 · 100 questions
This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.
Field order uses a source-bound presentation supplement; canonical JSONL and mathematical answers are unchanged.
Field-order and source-binding data
1. Declared-Neighborhood Locality and Whole-Domain Globality Are Separate
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).
Locality means the complete declared graph neighborhood, not an unspecified continuous topology. Globality compares the complete finite feasible table.
View answer and explanation
([[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)
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).
Reviewed step-by-step derivation
Optimization problem · qf-mod50-optdeep-local-global · candidate
2. Declared-Neighborhood Locality and Whole-Domain Globality Are Separate
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).
Locality means the complete declared graph neighborhood, not an unspecified continuous topology. Globality compares the complete finite feasible table.
View answer and explanation
([[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)
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).
Reviewed step-by-step derivation
Optimization problem · qf-mod50-optdeep-local-global · candidate
3. Declared-Neighborhood Locality and Whole-Domain Globality Are Separate
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).
Locality means the complete declared graph neighborhood, not an unspecified continuous topology. Globality compares the complete finite feasible table.
View answer and explanation
([[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)
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).
Reviewed step-by-step derivation
Optimization problem · qf-mod50-optdeep-local-global · candidate
4. Declared-Neighborhood Locality and Whole-Domain Globality Are Separate
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).
Locality means the complete declared graph neighborhood, not an unspecified continuous topology. Globality compares the complete finite feasible table.
View answer and explanation
([[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)
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).
Reviewed step-by-step derivation
Optimization problem · qf-mod50-optdeep-local-global · candidate
5. Declared-Neighborhood Locality and Whole-Domain Globality Are Separate
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).
Locality means the complete declared graph neighborhood, not an unspecified continuous topology. Globality compares the complete finite feasible table.
View answer and explanation
([[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)
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).
Reviewed step-by-step derivation
Optimization problem · qf-mod50-optdeep-local-global · candidate
6. Declared-Neighborhood Locality and Whole-Domain Globality Are Separate
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).
Locality means the complete declared graph neighborhood, not an unspecified continuous topology. Globality compares the complete finite feasible table.
View answer and explanation
([[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)
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).
Reviewed step-by-step derivation
Optimization problem · qf-mod50-optdeep-local-global · candidate
7. Declared-Neighborhood Locality and Whole-Domain Globality Are Separate
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).
Locality means the complete declared graph neighborhood, not an unspecified continuous topology. Globality compares the complete finite feasible table.
View answer and explanation
([[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)
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).
Reviewed step-by-step derivation
Optimization problem · qf-mod50-optdeep-local-global · candidate
8. Declared-Neighborhood Locality and Whole-Domain Globality Are Separate
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).
Locality means the complete declared graph neighborhood, not an unspecified continuous topology. Globality compares the complete finite feasible table.
View answer and explanation
([[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)
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).
Reviewed step-by-step derivation
Optimization problem · qf-mod50-optdeep-local-global · candidate
9. Declared-Neighborhood Locality and Whole-Domain Globality Are Separate
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).
Locality means the complete declared graph neighborhood, not an unspecified continuous topology. Globality compares the complete finite feasible table.
View answer and explanation
([[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)
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).
Reviewed step-by-step derivation
Optimization problem · qf-mod50-optdeep-local-global · candidate
10. Declared-Neighborhood Locality and Whole-Domain Globality Are Separate
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).
Locality means the complete declared graph neighborhood, not an unspecified continuous topology. Globality compares the complete finite feasible table.
View answer and explanation
([[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)
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).
Reviewed step-by-step derivation
Optimization problem · qf-mod50-optdeep-local-global · candidate
11. Declared-Neighborhood Locality and Whole-Domain Globality Are Separate
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).
Locality means the complete declared graph neighborhood, not an unspecified continuous topology. Globality compares the complete finite feasible table.
View answer and explanation
([[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)
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).
Reviewed step-by-step derivation
Optimization problem · qf-mod50-optdeep-local-global · candidate
12. Declared-Neighborhood Locality and Whole-Domain Globality Are Separate
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).
Locality means the complete declared graph neighborhood, not an unspecified continuous topology. Globality compares the complete finite feasible table.
View answer and explanation
([[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)
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).
Reviewed step-by-step derivation
Optimization problem · qf-mod50-optdeep-local-global · candidate
13. Declared-Neighborhood Locality and Whole-Domain Globality Are Separate
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).
Locality means the complete declared graph neighborhood, not an unspecified continuous topology. Globality compares the complete finite feasible table.
View answer and explanation
([[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)
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).
Reviewed step-by-step derivation
Optimization problem · qf-mod50-optdeep-local-global · candidate
14. Declared-Neighborhood Locality and Whole-Domain Globality Are Separate
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).
Locality means the complete declared graph neighborhood, not an unspecified continuous topology. Globality compares the complete finite feasible table.
View answer and explanation
([[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)
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).
Reviewed step-by-step derivation
Optimization problem · qf-mod50-optdeep-local-global · candidate
15. Declared-Neighborhood Locality and Whole-Domain Globality Are Separate
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).
Locality means the complete declared graph neighborhood, not an unspecified continuous topology. Globality compares the complete finite feasible table.
View answer and explanation
([[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)
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).
Reviewed step-by-step derivation
Optimization problem · qf-mod50-optdeep-local-global · candidate
16. Declared-Neighborhood Locality and Whole-Domain Globality Are Separate
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).
Locality means the complete declared graph neighborhood, not an unspecified continuous topology. Globality compares the complete finite feasible table.
View answer and explanation
([[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)
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).
Reviewed step-by-step derivation
Optimization problem · qf-mod50-optdeep-local-global · candidate
17. Declared-Neighborhood Locality and Whole-Domain Globality Are Separate
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).
Locality means the complete declared graph neighborhood, not an unspecified continuous topology. Globality compares the complete finite feasible table.
View answer and explanation
([[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)
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).
Reviewed step-by-step derivation
Optimization problem · qf-mod50-optdeep-local-global · candidate
18. Declared-Neighborhood Locality and Whole-Domain Globality Are Separate
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).
Locality means the complete declared graph neighborhood, not an unspecified continuous topology. Globality compares the complete finite feasible table.
View answer and explanation
([[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)
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).
Reviewed step-by-step derivation
Optimization problem · qf-mod50-optdeep-local-global · candidate
19. Declared-Neighborhood Locality and Whole-Domain Globality Are Separate
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).
Locality means the complete declared graph neighborhood, not an unspecified continuous topology. Globality compares the complete finite feasible table.
View answer and explanation
([[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)
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).
Reviewed step-by-step derivation
Optimization problem · qf-mod50-optdeep-local-global · candidate
20. Declared-Neighborhood Locality and Whole-Domain Globality Are Separate
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).
Locality means the complete declared graph neighborhood, not an unspecified continuous topology. Globality compares the complete finite feasible table.
View answer and explanation
([[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)
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).
Reviewed step-by-step derivation
Optimization problem · qf-mod50-optdeep-local-global · candidate
21. Declared-Neighborhood Locality and Whole-Domain Globality Are Separate
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).
Locality means the complete declared graph neighborhood, not an unspecified continuous topology. Globality compares the complete finite feasible table.
View answer and explanation
([[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)
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).
Reviewed step-by-step derivation
Optimization problem · qf-mod50-optdeep-local-global · candidate
22. Declared-Neighborhood Locality and Whole-Domain Globality Are Separate
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).
Locality means the complete declared graph neighborhood, not an unspecified continuous topology. Globality compares the complete finite feasible table.
View answer and explanation
([[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)
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).
Reviewed step-by-step derivation
Optimization problem · qf-mod50-optdeep-local-global · candidate
23. Declared-Neighborhood Locality and Whole-Domain Globality Are Separate
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).
Locality means the complete declared graph neighborhood, not an unspecified continuous topology. Globality compares the complete finite feasible table.
View answer and explanation
([[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)
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).
Reviewed step-by-step derivation
Optimization problem · qf-mod50-optdeep-local-global · candidate
24. Declared-Neighborhood Locality and Whole-Domain Globality Are Separate
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).
Locality means the complete declared graph neighborhood, not an unspecified continuous topology. Globality compares the complete finite feasible table.
View answer and explanation
([[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)
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).
Reviewed step-by-step derivation
Optimization problem · qf-mod50-optdeep-local-global · candidate
25. Declared-Neighborhood Locality and Whole-Domain Globality Are Separate
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).
Locality means the complete declared graph neighborhood, not an unspecified continuous topology. Globality compares the complete finite feasible table.
View answer and explanation
([[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)
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).
Reviewed step-by-step derivation
Optimization problem · qf-mod50-optdeep-local-global · candidate
26. One Strictly Better Feasible Point Refutes Candidate Global Optimality
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).
View answer and explanation
([[A, 3], [B, 1], [C, 4], [D, 5]], A, 3, B, 1, Yes, No, one_strictly_better_feasible_point_refutes_candidate_global_optimality)
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).
Reviewed step-by-step derivation
Optimization problem · qf-mod50-optdeep-improvement-witness · candidate
27. One Strictly Better Feasible Point Refutes Candidate Global Optimality
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).
View answer and explanation
([[A, 6], [B, 2], [C, 8], [D, 10]], A, 6, B, 2, Yes, No, one_strictly_better_feasible_point_refutes_candidate_global_optimality)
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).
Reviewed step-by-step derivation
Optimization problem · qf-mod50-optdeep-improvement-witness · candidate
28. One Strictly Better Feasible Point Refutes Candidate Global Optimality
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).
View answer and explanation
([[A, 9], [B, 3], [C, 12], [D, 15]], A, 9, B, 3, Yes, No, one_strictly_better_feasible_point_refutes_candidate_global_optimality)
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).
Reviewed step-by-step derivation
Optimization problem · qf-mod50-optdeep-improvement-witness · candidate
29. One Strictly Better Feasible Point Refutes Candidate Global Optimality
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).
View answer and explanation
([[A, 12], [B, 4], [C, 16], [D, 20]], A, 12, B, 4, Yes, No, one_strictly_better_feasible_point_refutes_candidate_global_optimality)
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).
Reviewed step-by-step derivation
Optimization problem · qf-mod50-optdeep-improvement-witness · candidate
30. One Strictly Better Feasible Point Refutes Candidate Global Optimality
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).
View answer and explanation
([[A, 15], [B, 5], [C, 20], [D, 25]], A, 15, B, 5, Yes, No, one_strictly_better_feasible_point_refutes_candidate_global_optimality)
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).
Reviewed step-by-step derivation
Optimization problem · qf-mod50-optdeep-improvement-witness · candidate
31. One Strictly Better Feasible Point Refutes Candidate Global Optimality
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).
View answer and explanation
([[A, 18], [B, 6], [C, 24], [D, 30]], A, 18, B, 6, Yes, No, one_strictly_better_feasible_point_refutes_candidate_global_optimality)
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).
Reviewed step-by-step derivation
Optimization problem · qf-mod50-optdeep-improvement-witness · candidate
32. One Strictly Better Feasible Point Refutes Candidate Global Optimality
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).
View answer and explanation
([[A, 21], [B, 7], [C, 28], [D, 35]], A, 21, B, 7, Yes, No, one_strictly_better_feasible_point_refutes_candidate_global_optimality)
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).
Reviewed step-by-step derivation
Optimization problem · qf-mod50-optdeep-improvement-witness · candidate
33. One Strictly Better Feasible Point Refutes Candidate Global Optimality
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).
View answer and explanation
([[A, 24], [B, 8], [C, 32], [D, 40]], A, 24, B, 8, Yes, No, one_strictly_better_feasible_point_refutes_candidate_global_optimality)
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).
Reviewed step-by-step derivation
Optimization problem · qf-mod50-optdeep-improvement-witness · candidate
34. One Strictly Better Feasible Point Refutes Candidate Global Optimality
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).
View answer and explanation
([[A, 27], [B, 9], [C, 36], [D, 45]], A, 27, B, 9, Yes, No, one_strictly_better_feasible_point_refutes_candidate_global_optimality)
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).
Reviewed step-by-step derivation
Optimization problem · qf-mod50-optdeep-improvement-witness · candidate
35. One Strictly Better Feasible Point Refutes Candidate Global Optimality
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).
View answer and explanation
([[A, 30], [B, 10], [C, 40], [D, 50]], A, 30, B, 10, Yes, No, one_strictly_better_feasible_point_refutes_candidate_global_optimality)
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).
Reviewed step-by-step derivation
Optimization problem · qf-mod50-optdeep-improvement-witness · candidate
36. One Strictly Better Feasible Point Refutes Candidate Global Optimality
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).
View answer and explanation
([[A, 33], [B, 11], [C, 44], [D, 55]], A, 33, B, 11, Yes, No, one_strictly_better_feasible_point_refutes_candidate_global_optimality)
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).
Reviewed step-by-step derivation
Optimization problem · qf-mod50-optdeep-improvement-witness · candidate
37. One Strictly Better Feasible Point Refutes Candidate Global Optimality
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).
View answer and explanation
([[A, 36], [B, 12], [C, 48], [D, 60]], A, 36, B, 12, Yes, No, one_strictly_better_feasible_point_refutes_candidate_global_optimality)
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).
Reviewed step-by-step derivation
Optimization problem · qf-mod50-optdeep-improvement-witness · candidate
38. One Strictly Better Feasible Point Refutes Candidate Global Optimality
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).
View answer and explanation
([[A, 39], [B, 13], [C, 52], [D, 65]], A, 39, B, 13, Yes, No, one_strictly_better_feasible_point_refutes_candidate_global_optimality)
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).
Reviewed step-by-step derivation
Optimization problem · qf-mod50-optdeep-improvement-witness · candidate
39. One Strictly Better Feasible Point Refutes Candidate Global Optimality
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).
View answer and explanation
([[A, 42], [B, 14], [C, 56], [D, 70]], A, 42, B, 14, Yes, No, one_strictly_better_feasible_point_refutes_candidate_global_optimality)
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).
Reviewed step-by-step derivation
Optimization problem · qf-mod50-optdeep-improvement-witness · candidate
40. One Strictly Better Feasible Point Refutes Candidate Global Optimality
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).
View answer and explanation
([[A, 45], [B, 15], [C, 60], [D, 75]], A, 45, B, 15, Yes, No, one_strictly_better_feasible_point_refutes_candidate_global_optimality)
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).
Reviewed step-by-step derivation
Optimization problem · qf-mod50-optdeep-improvement-witness · candidate
41. One Strictly Better Feasible Point Refutes Candidate Global Optimality
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).
View answer and explanation
([[A, 48], [B, 16], [C, 64], [D, 80]], A, 48, B, 16, Yes, No, one_strictly_better_feasible_point_refutes_candidate_global_optimality)
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).
Reviewed step-by-step derivation
Optimization problem · qf-mod50-optdeep-improvement-witness · candidate
42. One Strictly Better Feasible Point Refutes Candidate Global Optimality
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).
View answer and explanation
([[A, 51], [B, 17], [C, 68], [D, 85]], A, 51, B, 17, Yes, No, one_strictly_better_feasible_point_refutes_candidate_global_optimality)
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).
Reviewed step-by-step derivation
Optimization problem · qf-mod50-optdeep-improvement-witness · candidate
43. One Strictly Better Feasible Point Refutes Candidate Global Optimality
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).
View answer and explanation
([[A, 54], [B, 18], [C, 72], [D, 90]], A, 54, B, 18, Yes, No, one_strictly_better_feasible_point_refutes_candidate_global_optimality)
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).
Reviewed step-by-step derivation
Optimization problem · qf-mod50-optdeep-improvement-witness · candidate
44. One Strictly Better Feasible Point Refutes Candidate Global Optimality
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).
View answer and explanation
([[A, 57], [B, 19], [C, 76], [D, 95]], A, 57, B, 19, Yes, No, one_strictly_better_feasible_point_refutes_candidate_global_optimality)
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).
Reviewed step-by-step derivation
Optimization problem · qf-mod50-optdeep-improvement-witness · candidate
45. One Strictly Better Feasible Point Refutes Candidate Global Optimality
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).
View answer and explanation
([[A, 60], [B, 20], [C, 80], [D, 100]], A, 60, B, 20, Yes, No, one_strictly_better_feasible_point_refutes_candidate_global_optimality)
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).
Reviewed step-by-step derivation
Optimization problem · qf-mod50-optdeep-improvement-witness · candidate
46. One Strictly Better Feasible Point Refutes Candidate Global Optimality
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).
View answer and explanation
([[A, 63], [B, 21], [C, 84], [D, 105]], A, 63, B, 21, Yes, No, one_strictly_better_feasible_point_refutes_candidate_global_optimality)
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).
Reviewed step-by-step derivation
Optimization problem · qf-mod50-optdeep-improvement-witness · candidate
47. One Strictly Better Feasible Point Refutes Candidate Global Optimality
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).
View answer and explanation
([[A, 66], [B, 22], [C, 88], [D, 110]], A, 66, B, 22, Yes, No, one_strictly_better_feasible_point_refutes_candidate_global_optimality)
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).
Reviewed step-by-step derivation
Optimization problem · qf-mod50-optdeep-improvement-witness · candidate
48. One Strictly Better Feasible Point Refutes Candidate Global Optimality
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).
View answer and explanation
([[A, 69], [B, 23], [C, 92], [D, 115]], A, 69, B, 23, Yes, No, one_strictly_better_feasible_point_refutes_candidate_global_optimality)
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).
Reviewed step-by-step derivation
Optimization problem · qf-mod50-optdeep-improvement-witness · candidate
49. One Strictly Better Feasible Point Refutes Candidate Global Optimality
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).
View answer and explanation
([[A, 72], [B, 24], [C, 96], [D, 120]], A, 72, B, 24, Yes, No, one_strictly_better_feasible_point_refutes_candidate_global_optimality)
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).
Reviewed step-by-step derivation
Optimization problem · qf-mod50-optdeep-improvement-witness · candidate
50. One Strictly Better Feasible Point Refutes Candidate Global Optimality
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).
View answer and explanation
([[A, 75], [B, 25], [C, 100], [D, 125]], A, 75, B, 25, Yes, No, one_strictly_better_feasible_point_refutes_candidate_global_optimality)
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).
Reviewed step-by-step derivation
Optimization problem · qf-mod50-optdeep-improvement-witness · candidate
51. A Complete Attained Lower Bound Certifies Globality but Not Automatic Uniqueness
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).
View answer and explanation
([[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)
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).
Reviewed step-by-step derivation
Optimization problem · qf-mod50-optdeep-lower-bound-certificate · candidate
52. A Complete Attained Lower Bound Certifies Globality but Not Automatic Uniqueness
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).
View answer and explanation
([[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)
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).
Reviewed step-by-step derivation
Optimization problem · qf-mod50-optdeep-lower-bound-certificate · candidate
53. A Complete Attained Lower Bound Certifies Globality but Not Automatic Uniqueness
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).
View answer and explanation
([[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)
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).
Reviewed step-by-step derivation
Optimization problem · qf-mod50-optdeep-lower-bound-certificate · candidate
54. A Complete Attained Lower Bound Certifies Globality but Not Automatic Uniqueness
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).
View answer and explanation
([[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)
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).
Reviewed step-by-step derivation
Optimization problem · qf-mod50-optdeep-lower-bound-certificate · candidate
55. A Complete Attained Lower Bound Certifies Globality but Not Automatic Uniqueness
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).
View answer and explanation
([[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)
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).
Reviewed step-by-step derivation
Optimization problem · qf-mod50-optdeep-lower-bound-certificate · candidate
56. A Complete Attained Lower Bound Certifies Globality but Not Automatic Uniqueness
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).
View answer and explanation
([[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)
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).
Reviewed step-by-step derivation
Optimization problem · qf-mod50-optdeep-lower-bound-certificate · candidate
57. A Complete Attained Lower Bound Certifies Globality but Not Automatic Uniqueness
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).
View answer and explanation
([[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)
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).
Reviewed step-by-step derivation
Optimization problem · qf-mod50-optdeep-lower-bound-certificate · candidate
58. A Complete Attained Lower Bound Certifies Globality but Not Automatic Uniqueness
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).
View answer and explanation
([[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)
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).
Reviewed step-by-step derivation
Optimization problem · qf-mod50-optdeep-lower-bound-certificate · candidate
59. A Complete Attained Lower Bound Certifies Globality but Not Automatic Uniqueness
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).
View answer and explanation
([[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)
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).
Reviewed step-by-step derivation
Optimization problem · qf-mod50-optdeep-lower-bound-certificate · candidate
60. A Complete Attained Lower Bound Certifies Globality but Not Automatic Uniqueness
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).
View answer and explanation
([[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)
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).
Reviewed step-by-step derivation
Optimization problem · qf-mod50-optdeep-lower-bound-certificate · candidate
61. A Complete Attained Lower Bound Certifies Globality but Not Automatic Uniqueness
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).
View answer and explanation
([[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)
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).
Reviewed step-by-step derivation
Optimization problem · qf-mod50-optdeep-lower-bound-certificate · candidate
62. A Complete Attained Lower Bound Certifies Globality but Not Automatic Uniqueness
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).
View answer and explanation
([[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)
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).
Reviewed step-by-step derivation
Optimization problem · qf-mod50-optdeep-lower-bound-certificate · candidate
63. A Complete Attained Lower Bound Certifies Globality but Not Automatic Uniqueness
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).
View answer and explanation
([[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)
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).
Reviewed step-by-step derivation
Optimization problem · qf-mod50-optdeep-lower-bound-certificate · candidate
64. A Complete Attained Lower Bound Certifies Globality but Not Automatic Uniqueness
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).
View answer and explanation
([[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)
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).
Reviewed step-by-step derivation
Optimization problem · qf-mod50-optdeep-lower-bound-certificate · candidate
65. A Complete Attained Lower Bound Certifies Globality but Not Automatic Uniqueness
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).
View answer and explanation
([[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)
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).
Reviewed step-by-step derivation
Optimization problem · qf-mod50-optdeep-lower-bound-certificate · candidate
66. A Complete Attained Lower Bound Certifies Globality but Not Automatic Uniqueness
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).
View answer and explanation
([[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)
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).
Reviewed step-by-step derivation
Optimization problem · qf-mod50-optdeep-lower-bound-certificate · candidate
67. A Complete Attained Lower Bound Certifies Globality but Not Automatic Uniqueness
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).
View answer and explanation
([[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)
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).
Reviewed step-by-step derivation
Optimization problem · qf-mod50-optdeep-lower-bound-certificate · candidate
68. A Complete Attained Lower Bound Certifies Globality but Not Automatic Uniqueness
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).
View answer and explanation
([[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)
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).
Reviewed step-by-step derivation
Optimization problem · qf-mod50-optdeep-lower-bound-certificate · candidate
69. A Complete Attained Lower Bound Certifies Globality but Not Automatic Uniqueness
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).
View answer and explanation
([[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)
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).
Reviewed step-by-step derivation
Optimization problem · qf-mod50-optdeep-lower-bound-certificate · candidate
70. A Complete Attained Lower Bound Certifies Globality but Not Automatic Uniqueness
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).
View answer and explanation
([[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)
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).
Reviewed step-by-step derivation
Optimization problem · qf-mod50-optdeep-lower-bound-certificate · candidate
71. A Complete Attained Lower Bound Certifies Globality but Not Automatic Uniqueness
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).
View answer and explanation
([[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)
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).
Reviewed step-by-step derivation
Optimization problem · qf-mod50-optdeep-lower-bound-certificate · candidate
72. A Complete Attained Lower Bound Certifies Globality but Not Automatic Uniqueness
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).
View answer and explanation
([[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)
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).
Reviewed step-by-step derivation
Optimization problem · qf-mod50-optdeep-lower-bound-certificate · candidate
73. A Complete Attained Lower Bound Certifies Globality but Not Automatic Uniqueness
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).
View answer and explanation
([[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)
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).
Reviewed step-by-step derivation
Optimization problem · qf-mod50-optdeep-lower-bound-certificate · candidate
74. A Complete Attained Lower Bound Certifies Globality but Not Automatic Uniqueness
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).
View answer and explanation
([[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)
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).
Reviewed step-by-step derivation
Optimization problem · qf-mod50-optdeep-lower-bound-certificate · candidate
75. A Complete Attained Lower Bound Certifies Globality but Not Automatic Uniqueness
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).
View answer and explanation
([[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)
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).
Reviewed step-by-step derivation
Optimization problem · qf-mod50-optdeep-lower-bound-certificate · candidate
76. Empty Domains, Partial Lists, Ties and Local-Only Witnesses Block Different Claims
Use mode=empty_feasible, complete model={}. Give (mode, feasible_set, optimizer_exists_claim, boundary).
The four modes have4/4/6/7 fields. A partial list cannot certify globality even if its listed minimum is computable. Retain every tied argmin; one strict improvement refutes candidate globality only.
View answer and explanation
(empty_feasible, [], No, empty_feasible_set_has_no_optimizer)
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).
Reviewed step-by-step derivation
Optimization problem · qf-mod50-optdeep-boundary · candidate
77. Empty Domains, Partial Lists, Ties and Local-Only Witnesses Block Different Claims
Use mode=incomplete_domain, complete model={"listed_points":["A","B"],"values":[2,4]}. Give (mode, listed_objective_rows, global_certificate_claim, boundary).
The four modes have4/4/6/7 fields. A partial list cannot certify globality even if its listed minimum is computable. Retain every tied argmin; one strict improvement refutes candidate globality only.
View answer and explanation
(incomplete_domain, [[A, 2], [B, 4]], No, an_unverified_partial_list_cannot_support_a_global_claim)
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).
Reviewed step-by-step derivation
Optimization problem · qf-mod50-optdeep-boundary · candidate
78. Empty Domains, Partial Lists, Ties and Local-Only Witnesses Block Different Claims
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).
The four modes have4/4/6/7 fields. A partial list cannot certify globality even if its listed minimum is computable. Retain every tied argmin; one strict improvement refutes candidate globality only.
View answer and explanation
(tie_not_unique, 3, [A, B], Yes, No, global_optimum_need_not_be_unique)
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).
Reviewed step-by-step derivation
Optimization problem · qf-mod50-optdeep-boundary · candidate
79. Empty Domains, Partial Lists, Ties and Local-Only Witnesses Block Different Claims
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).
The four modes have4/4/6/7 fields. A partial list cannot certify globality even if its listed minimum is computable. Retain every tied argmin; one strict improvement refutes candidate globality only.
View answer and explanation
(local_not_global, A, 8, [B], Yes, No, local_minimum_does_not_imply_global_minimum)
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).
Reviewed step-by-step derivation
Optimization problem · qf-mod50-optdeep-boundary · candidate
80. Empty Domains, Partial Lists, Ties and Local-Only Witnesses Block Different Claims
Use mode=incomplete_domain, complete model={"listed_points":["A","B"],"values":[5,10]}. Give (mode, listed_objective_rows, global_certificate_claim, boundary).
The four modes have4/4/6/7 fields. A partial list cannot certify globality even if its listed minimum is computable. Retain every tied argmin; one strict improvement refutes candidate globality only.
View answer and explanation
(incomplete_domain, [[A, 5], [B, 10]], No, an_unverified_partial_list_cannot_support_a_global_claim)
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).
Reviewed step-by-step derivation
Optimization problem · qf-mod50-optdeep-boundary · candidate
81. Empty Domains, Partial Lists, Ties and Local-Only Witnesses Block Different Claims
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).
The four modes have4/4/6/7 fields. A partial list cannot certify globality even if its listed minimum is computable. Retain every tied argmin; one strict improvement refutes candidate globality only.
View answer and explanation
(tie_not_unique, 6, [A, B], Yes, No, global_optimum_need_not_be_unique)
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).
Reviewed step-by-step derivation
Optimization problem · qf-mod50-optdeep-boundary · candidate
82. Empty Domains, Partial Lists, Ties and Local-Only Witnesses Block Different Claims
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).
The four modes have4/4/6/7 fields. A partial list cannot certify globality even if its listed minimum is computable. Retain every tied argmin; one strict improvement refutes candidate globality only.
View answer and explanation
(local_not_global, A, 14, [B], Yes, No, local_minimum_does_not_imply_global_minimum)
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).
Reviewed step-by-step derivation
Optimization problem · qf-mod50-optdeep-boundary · candidate
83. Empty Domains, Partial Lists, Ties and Local-Only Witnesses Block Different Claims
Use mode=incomplete_domain, complete model={"listed_points":["A","B"],"values":[8,16]}. Give (mode, listed_objective_rows, global_certificate_claim, boundary).
The four modes have4/4/6/7 fields. A partial list cannot certify globality even if its listed minimum is computable. Retain every tied argmin; one strict improvement refutes candidate globality only.
View answer and explanation
(incomplete_domain, [[A, 8], [B, 16]], No, an_unverified_partial_list_cannot_support_a_global_claim)
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).
Reviewed step-by-step derivation
Optimization problem · qf-mod50-optdeep-boundary · candidate
84. Empty Domains, Partial Lists, Ties and Local-Only Witnesses Block Different Claims
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).
The four modes have4/4/6/7 fields. A partial list cannot certify globality even if its listed minimum is computable. Retain every tied argmin; one strict improvement refutes candidate globality only.
View answer and explanation
(tie_not_unique, 9, [A, B], Yes, No, global_optimum_need_not_be_unique)
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).
Reviewed step-by-step derivation
Optimization problem · qf-mod50-optdeep-boundary · candidate
85. Empty Domains, Partial Lists, Ties and Local-Only Witnesses Block Different Claims
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).
The four modes have4/4/6/7 fields. A partial list cannot certify globality even if its listed minimum is computable. Retain every tied argmin; one strict improvement refutes candidate globality only.
View answer and explanation
(local_not_global, A, 20, [B], Yes, No, local_minimum_does_not_imply_global_minimum)
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).
Reviewed step-by-step derivation
Optimization problem · qf-mod50-optdeep-boundary · candidate
86. Empty Domains, Partial Lists, Ties and Local-Only Witnesses Block Different Claims
Use mode=incomplete_domain, complete model={"listed_points":["A","B"],"values":[11,22]}. Give (mode, listed_objective_rows, global_certificate_claim, boundary).
The four modes have4/4/6/7 fields. A partial list cannot certify globality even if its listed minimum is computable. Retain every tied argmin; one strict improvement refutes candidate globality only.
View answer and explanation
(incomplete_domain, [[A, 11], [B, 22]], No, an_unverified_partial_list_cannot_support_a_global_claim)
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).
Reviewed step-by-step derivation
Optimization problem · qf-mod50-optdeep-boundary · candidate
87. Empty Domains, Partial Lists, Ties and Local-Only Witnesses Block Different Claims
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).
The four modes have4/4/6/7 fields. A partial list cannot certify globality even if its listed minimum is computable. Retain every tied argmin; one strict improvement refutes candidate globality only.
View answer and explanation
(tie_not_unique, 12, [A, B], Yes, No, global_optimum_need_not_be_unique)
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).
Reviewed step-by-step derivation
Optimization problem · qf-mod50-optdeep-boundary · candidate
88. Empty Domains, Partial Lists, Ties and Local-Only Witnesses Block Different Claims
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).
The four modes have4/4/6/7 fields. A partial list cannot certify globality even if its listed minimum is computable. Retain every tied argmin; one strict improvement refutes candidate globality only.
View answer and explanation
(local_not_global, A, 26, [B], Yes, No, local_minimum_does_not_imply_global_minimum)
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).
Reviewed step-by-step derivation
Optimization problem · qf-mod50-optdeep-boundary · candidate
89. Empty Domains, Partial Lists, Ties and Local-Only Witnesses Block Different Claims
Use mode=incomplete_domain, complete model={"listed_points":["A","B"],"values":[14,28]}. Give (mode, listed_objective_rows, global_certificate_claim, boundary).
The four modes have4/4/6/7 fields. A partial list cannot certify globality even if its listed minimum is computable. Retain every tied argmin; one strict improvement refutes candidate globality only.
View answer and explanation
(incomplete_domain, [[A, 14], [B, 28]], No, an_unverified_partial_list_cannot_support_a_global_claim)
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).
Reviewed step-by-step derivation
Optimization problem · qf-mod50-optdeep-boundary · candidate
90. Empty Domains, Partial Lists, Ties and Local-Only Witnesses Block Different Claims
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).
The four modes have4/4/6/7 fields. A partial list cannot certify globality even if its listed minimum is computable. Retain every tied argmin; one strict improvement refutes candidate globality only.
View answer and explanation
(tie_not_unique, 15, [A, B], Yes, No, global_optimum_need_not_be_unique)
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).
Reviewed step-by-step derivation
Optimization problem · qf-mod50-optdeep-boundary · candidate
91. Empty Domains, Partial Lists, Ties and Local-Only Witnesses Block Different Claims
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).
The four modes have4/4/6/7 fields. A partial list cannot certify globality even if its listed minimum is computable. Retain every tied argmin; one strict improvement refutes candidate globality only.
View answer and explanation
(local_not_global, A, 32, [B], Yes, No, local_minimum_does_not_imply_global_minimum)
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).
Reviewed step-by-step derivation
Optimization problem · qf-mod50-optdeep-boundary · candidate
92. Empty Domains, Partial Lists, Ties and Local-Only Witnesses Block Different Claims
Use mode=incomplete_domain, complete model={"listed_points":["A","B"],"values":[17,34]}. Give (mode, listed_objective_rows, global_certificate_claim, boundary).
The four modes have4/4/6/7 fields. A partial list cannot certify globality even if its listed minimum is computable. Retain every tied argmin; one strict improvement refutes candidate globality only.
View answer and explanation
(incomplete_domain, [[A, 17], [B, 34]], No, an_unverified_partial_list_cannot_support_a_global_claim)
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).
Reviewed step-by-step derivation
Optimization problem · qf-mod50-optdeep-boundary · candidate
93. Empty Domains, Partial Lists, Ties and Local-Only Witnesses Block Different Claims
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).
The four modes have4/4/6/7 fields. A partial list cannot certify globality even if its listed minimum is computable. Retain every tied argmin; one strict improvement refutes candidate globality only.
View answer and explanation
(tie_not_unique, 18, [A, B], Yes, No, global_optimum_need_not_be_unique)
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).
Reviewed step-by-step derivation
Optimization problem · qf-mod50-optdeep-boundary · candidate
94. Empty Domains, Partial Lists, Ties and Local-Only Witnesses Block Different Claims
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).
The four modes have4/4/6/7 fields. A partial list cannot certify globality even if its listed minimum is computable. Retain every tied argmin; one strict improvement refutes candidate globality only.
View answer and explanation
(local_not_global, A, 38, [B], Yes, No, local_minimum_does_not_imply_global_minimum)
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).
Reviewed step-by-step derivation
Optimization problem · qf-mod50-optdeep-boundary · candidate
95. Empty Domains, Partial Lists, Ties and Local-Only Witnesses Block Different Claims
Use mode=incomplete_domain, complete model={"listed_points":["A","B"],"values":[20,40]}. Give (mode, listed_objective_rows, global_certificate_claim, boundary).
The four modes have4/4/6/7 fields. A partial list cannot certify globality even if its listed minimum is computable. Retain every tied argmin; one strict improvement refutes candidate globality only.
View answer and explanation
(incomplete_domain, [[A, 20], [B, 40]], No, an_unverified_partial_list_cannot_support_a_global_claim)
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).
Reviewed step-by-step derivation
Optimization problem · qf-mod50-optdeep-boundary · candidate
96. Empty Domains, Partial Lists, Ties and Local-Only Witnesses Block Different Claims
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).
The four modes have4/4/6/7 fields. A partial list cannot certify globality even if its listed minimum is computable. Retain every tied argmin; one strict improvement refutes candidate globality only.
View answer and explanation
(tie_not_unique, 21, [A, B], Yes, No, global_optimum_need_not_be_unique)
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).
Reviewed step-by-step derivation
Optimization problem · qf-mod50-optdeep-boundary · candidate
97. Empty Domains, Partial Lists, Ties and Local-Only Witnesses Block Different Claims
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).
The four modes have4/4/6/7 fields. A partial list cannot certify globality even if its listed minimum is computable. Retain every tied argmin; one strict improvement refutes candidate globality only.
View answer and explanation
(local_not_global, A, 44, [B], Yes, No, local_minimum_does_not_imply_global_minimum)
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).
Reviewed step-by-step derivation
Optimization problem · qf-mod50-optdeep-boundary · candidate
98. Empty Domains, Partial Lists, Ties and Local-Only Witnesses Block Different Claims
Use mode=incomplete_domain, complete model={"listed_points":["A","B"],"values":[23,46]}. Give (mode, listed_objective_rows, global_certificate_claim, boundary).
The four modes have4/4/6/7 fields. A partial list cannot certify globality even if its listed minimum is computable. Retain every tied argmin; one strict improvement refutes candidate globality only.
View answer and explanation
(incomplete_domain, [[A, 23], [B, 46]], No, an_unverified_partial_list_cannot_support_a_global_claim)
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).
Reviewed step-by-step derivation
Optimization problem · qf-mod50-optdeep-boundary · candidate
99. Empty Domains, Partial Lists, Ties and Local-Only Witnesses Block Different Claims
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).
The four modes have4/4/6/7 fields. A partial list cannot certify globality even if its listed minimum is computable. Retain every tied argmin; one strict improvement refutes candidate globality only.
View answer and explanation
(tie_not_unique, 24, [A, B], Yes, No, global_optimum_need_not_be_unique)
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).
Reviewed step-by-step derivation
Optimization problem · qf-mod50-optdeep-boundary · candidate
100. Empty Domains, Partial Lists, Ties and Local-Only Witnesses Block Different Claims
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).
The four modes have4/4/6/7 fields. A partial list cannot certify globality even if its listed minimum is computable. Retain every tied argmin; one strict improvement refutes candidate globality only.
View answer and explanation
(local_not_global, A, 50, [B], Yes, No, local_minimum_does_not_imply_global_minimum)
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).
Reviewed step-by-step derivation
Optimization problem · qf-mod50-optdeep-boundary · candidate
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