# Complete Finite Neighborhoods, Improvement Witnesses and Attained Bounds

This batch has passed Schema, deduplication, source-snapshot, and independent-answer recomputation checks; it is still not marked as human-reviewed.

This question bank has corresponding bilingual step-by-step derivations. Source candidate and English ai_candidate states are retained. AI review, finite computation and Lean formal proof are labelled separately.

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](https://ocme.evemisslab.com/data/i18n/question-output-protocols.json)



## 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](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r07-0001/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r07-0002/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r07-0003/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r07-0004/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r07-0005/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r07-0006/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r07-0007/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r07-0008/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r07-0009/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r07-0010/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r07-0011/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r07-0012/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r07-0013/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r07-0014/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r07-0015/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r07-0016/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r07-0017/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r07-0018/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r07-0019/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r07-0020/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r07-0021/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r07-0022/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r07-0023/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r07-0024/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r07-0025/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r07-0026/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r07-0027/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r07-0028/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r07-0029/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r07-0030/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r07-0031/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r07-0032/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r07-0033/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r07-0034/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r07-0035/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r07-0036/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r07-0037/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r07-0038/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r07-0039/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r07-0040/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r07-0041/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r07-0042/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r07-0043/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r07-0044/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r07-0045/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r07-0046/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r07-0047/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r07-0048/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r07-0049/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r07-0050/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r07-0051/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r07-0052/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r07-0053/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r07-0054/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r07-0055/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r07-0056/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r07-0057/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r07-0058/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r07-0059/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r07-0060/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r07-0061/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r07-0062/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r07-0063/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r07-0064/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r07-0065/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r07-0066/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r07-0067/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r07-0068/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r07-0069/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r07-0070/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r07-0071/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r07-0072/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r07-0073/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r07-0074/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r07-0075/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r07-0076/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r07-0077/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r07-0078/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r07-0079/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r07-0080/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r07-0081/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r07-0082/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r07-0083/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r07-0084/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r07-0085/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r07-0086/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r07-0087/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r07-0088/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r07-0089/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r07-0090/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r07-0091/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r07-0092/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r07-0093/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r07-0094/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r07-0095/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r07-0096/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r07-0097/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r07-0098/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r07-0099/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r07-0100/)

Optimization problem · qf-mod50-optdeep-boundary · candidate



[Canonical question JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20261015-advance-20261119-r07-optimization-problem-foundations/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20261015-advance-20261119-r07-optimization-problem-foundations.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/batches/batch-20261015-advance-20261119-r07-optimization-problem-foundations/validation-report.json)