# Question 888: A Complete Attained Lower Bound Certifies Globality but Not Automatic Uniqueness [Return to catalogue page 5](https://ocme.evemisslab.com/en/questions/page/5/) [← 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-r07-0063) 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. P1000-0888 · q-20261015-pulse-20261119-r07-0063 ## Original question and givens 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). - G001 — 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). ## 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](https://ocme.evemisslab.com/en/mko/mko-optimization-problem/) ## Step-by-step derivation S001 · impliesBefore complete feasible domain After [[A, 13], [B, 26], [C, 39], [D, 52]] Conditions: none Fix the complete table and minimization direction; sampling a few points is not a global certificate. Step identifiers and dependenciesRule: complete_objective_data · Uses: G001 S002 · calculationBefore f(A)−L := sub(13, 13) After f(A)−L = 0 Conditions: none Check pointwise that this bounds the entire feasible domain below. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001 S003 · calculationBefore f(B)−L := sub(26, 13) After f(B)−L = 13 Conditions: none Check pointwise that this bounds the entire feasible domain below. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · calculationBefore f(C)−L := sub(39, 13) After f(C)−L = 26 Conditions: none Check pointwise that this bounds the entire feasible domain below. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · calculationBefore f(D)−L := sub(52, 13) After f(D)−L = 39 Conditions: none Check pointwise that this bounds the entire feasible domain below. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · impliesBefore all f(x)≥L=13 After f(A)=13=L; attained=Yes Conditions: none A lower bound alone does not guarantee attainment; verify the designated candidate's value separately. Step identifiers and dependenciesRule: lower_bound_attainment · Uses: G001, S005 S007 · impliesBefore all attaining points=[A] After global=true; unique=Yes Conditions: none A complete lower bound plus attainment certifies global minimality; uniqueness separately depends on the complete attaining set. Step identifiers and dependenciesRule: global_certificate_and_uniqueness · Uses: G001, S006 ## 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, 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) 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. Source hashes and review receipts - input commit: 0f7be61e2c22317a4075561dc822c87d881a50db - original question SHA-256: 98a1b03add2013b312bebe738f5df90220e00eecab16f7f8299270cfdb1e6357 - candidate record SHA-256: ca70644b88408b1732522548829570738aec624a78a09323ac0911db851bb334 - semantic review receipt SHA-256: 6c2c6d46aded7582a11ac41a8c0b3bdde2669a53bdb6e796eea2039994a02772 - question-first witness SHA-256: 77e43898df9efdbf8094ad81fe4ab91cf4b4bd82f481eb9b4699b73cb4da612b - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261015-pulse-20261119-r07-0063.json)