Question 250: Declared-Neighborhood Locality and Whole-Domain Globality Are Separate

Original question and givens

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).

• G001 — 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).

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

Step-by-step derivation

feasible points=[A, B, C, D]

[[A, 25], [B, 50], [C, 75], [D, 100]]

candidate=A; edges=[[A, B], [A, C], [B, D], [C, D]]

neighbors(A)=[B, C]; f(A)=25

f(B)−f(A) := sub(50, 25)

f(B)−f(A) = 25

f(C)−f(A) := sub(75, 25)

f(C)−f(A) = 50

[[B, 25, Yes], [C, 50, Yes]]

local minimum=Yes

global f(A)−f(A) := sub(25, 25)

global f(A)−f(A) = 0

global f(B)−f(A) := sub(50, 25)

global f(B)−f(A) = 25

global f(C)−f(A) := sub(75, 25)

global f(C)−f(A) = 50

global f(D)−f(A) := sub(100, 25)

global f(D)−f(A) = 75

[[A, 0, Yes], [B, 25, Yes], [C, 50, Yes], [D, 75, Yes]]

global minimum=Yes; minimum value=25; all minimizers=[A]

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, 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)

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.