# Question 246: Declared-Neighborhood Locality and Whole-Domain Globality Are Separate [Return to catalogue page 2](https://ocme.evemisslab.com/en/questions/page/2/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261015-advance-20261119-r07-optimization-problem-foundations/#q-20261015-pulse-20261119-r07-0021) This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth. P250-0246 · q-20261015-pulse-20261119-r07-0021 ## Original question and givens 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). - G001 — 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). ## Definition and scope Use the complete finite feasible carrier and declared undirected neighborhood. Local checks cover every neighbor; global checks cover every feasible point. - [mko-optimization-problem](https://ocme.evemisslab.com/en/mko/mko-optimization-problem/) ## Step-by-step derivation S001 · impliesBefore feasible points=[A, B, C, D] After [[A, 21], [B, 42], [C, 63], [D, 84]] Conditions: none Fix the complete feasible domain and each objective value; this task is minimization. Step identifiers and dependenciesRule: complete_objective_table · Uses: G001 S002 · impliesBefore candidate=A; edges=[[A, B], [A, C], [B, D], [C, D]] After neighbors(A)=[B, C]; f(A)=21 Conditions: none List every neighbor of the candidate from the declared undirected relation; not every feasible point is automatically a neighbor. Step identifiers and dependenciesRule: declared_neighborhood · Uses: G001, S001 S003 · calculationBefore f(B)−f(A) := sub(42, 21) After f(B)−f(A) = 21 Conditions: none Compare the objective difference at every neighbor. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · calculationBefore f(C)−f(A) := sub(63, 21) After f(C)−f(A) = 42 Conditions: none Compare the objective difference at every neighbor. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · impliesBefore [[B, 21, Yes], [C, 42, Yes]] After local minimum=Yes Conditions: none If all neighbor differences are nonnegative, no neighbor is strictly better; equal values are allowed for a non-strict local minimum. Step identifiers and dependenciesRule: local_minimum_definition · Uses: G001, S004 S006 · calculationBefore global f(A)−f(A) := sub(21, 21) After global f(A)−f(A) = 0 Conditions: none For global optimality compare every feasible point, including the candidate itself. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · calculationBefore global f(B)−f(A) := sub(42, 21) After global f(B)−f(A) = 21 Conditions: none For global optimality compare every feasible point, including the candidate itself. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006 S008 · calculationBefore global f(C)−f(A) := sub(63, 21) After global f(C)−f(A) = 42 Conditions: none For global optimality compare every feasible point, including the candidate itself. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S007 S009 · calculationBefore global f(D)−f(A) := sub(84, 21) After global f(D)−f(A) = 63 Conditions: none For global optimality compare every feasible point, including the candidate itself. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S008 S010 · impliesBefore [[A, 0, Yes], [B, 21, Yes], [C, 42, Yes], [D, 63, Yes]] After global minimum=Yes; minimum value=21; all minimizers=[A] Conditions: none The pointwise differences determine global optimality; retain every point attaining the minimum, including ties. Step identifiers and dependenciesRule: global_minimum_definition · Uses: G001, S009 ## Reverse checks - S010 · premise_and_result_audit — One strictly better feasible point refutes global optimality, while a local pass alone does not prove global optimality. - S009 · premise_and_result_audit — Recompute global f(D)−f(A) with exact rational arithmetic, not rounded decimals. - S008 · premise_and_result_audit — Recompute global f(C)−f(A) with exact rational arithmetic, not rounded decimals. - S007 · premise_and_result_audit — Recompute global f(B)−f(A) with exact rational arithmetic, not rounded decimals. - S006 · premise_and_result_audit — Recompute global f(A)−f(A) with exact rational arithmetic, not rounded decimals. - S005 · premise_and_result_audit — This test alone has not compared nonneighbors. - S004 · premise_and_result_audit — Recompute f(C)−f(A) with exact rational arithmetic, not rounded decimals. - S003 · premise_and_result_audit — Recompute f(B)−f(A) with exact rational arithmetic, not rounded decimals. - S002 · premise_and_result_audit — Local uses the neighborhood; global uses the entire feasible domain. - S001 · premise_and_result_audit — Completeness of the finite table is a premise here, not a statement about unlisted points. ## Result ([[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) Candidate A: local=Yes, global=Yes; all minimizers=[A]. ## This question review and source binding Review date: 2026-10-24 Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof. Source hashes and review receipts - input commit: 0f7be61e2c22317a4075561dc822c87d881a50db - original question SHA-256: 0022cba58de3b8b6724a6b2d1ec7a872d19c5336afb81672a59ec7a06650dd4f - candidate record SHA-256: 139d539b8ba8668e9497773b90a2f5ff6994f773bea084656137e4a214b37952 - semantic review receipt SHA-256: 4727d1bd957a9b7f6cadb7972ddb5ab80a31f5259dc5394c1dff2b71944ed36b - question-first witness SHA-256: 51e18649133238ac992229f6e16542a5f3af53ca73b2262ab0524508ea937dee - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261015-pulse-20261119-r07-0021.json)