Question 876: A Complete Attained Lower Bound Certifies Globality but Not Automatic Uniqueness

Original question and givens

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).

• G001 — 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).

Definition and scope

A complete lower bound attained at the candidate proves global minimality; uniqueness is decided separately by the attaining set.

• mko-optimization-problem

Step-by-step derivation

complete feasible domain

[[A, 1], [B, 2], [C, 3], [D, 4]]

f(A)−L := sub(1, 1)

f(A)−L = 0

f(B)−L := sub(2, 1)

f(B)−L = 1

f(C)−L := sub(3, 1)

f(C)−L = 2

f(D)−L := sub(4, 1)

f(D)−L = 3

all f(x)≥L=1

f(A)=1=L; attained=Yes

all attaining points=[A]

global=true; unique=Yes

Reverse checks

• S007 · premise_and_result_audit — Completeness, neighborhoods, attainment and uniqueness are separate obligations; local data do not license a global conclusion.
• S006 · premise_and_result_audit — Completeness, neighborhoods, attainment and uniqueness are separate obligations; local data do not license a global conclusion.
• S005 · premise_and_result_audit — Recompute f(D)−L with exact rational arithmetic, not rounded decimals.
• S004 · premise_and_result_audit — Recompute f(C)−L with exact rational arithmetic, not rounded decimals.
• S003 · premise_and_result_audit — Recompute f(B)−L with exact rational arithmetic, not rounded decimals.
• S002 · premise_and_result_audit — Recompute f(A)−L with exact rational arithmetic, not rounded decimals.
• S001 · premise_and_result_audit — Completeness, neighborhoods, attainment and uniqueness are separate obligations; local data do not license a global conclusion.

Result

([[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)

The full substitutions and premise audits recover the original answer, retaining its field order and verdict meanings.

This question review and source binding

Review date: 2026-10-26

Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.