# Question 856: One Strictly Better Feasible Point Refutes Candidate Global Optimality

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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.

P1000-0856 · q-20261015-pulse-20261119-r07-0031



## Original question and givens

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).


- G001 — 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).





## Definition and scope

Minimization over a complete finite feasible set; one strictly smaller feasible point refutes the candidate's global optimality.


- [mko-optimization-problem](https://ocme.evemisslab.com/en/mko/mko-optimization-problem/)





## Step-by-step derivation

S001 · impliesBefore

complete feasible domain

After

[[A, 18], [B, 6], [C, 24], [D, 30]]

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 · impliesBefore

candidate=A; challenger=B

After

both feasible; f(candidate)=18, f(challenger)=6

Conditions: none

First check that the challenger is a distinct feasible point.

Step identifiers and dependenciesRule: feasible_challenger · Uses: G001, S001





S003 · calculationBefore

challenger minus candidate := sub(6, 18)

After

challenger minus candidate = -12

Conditions: none

Subtract exactly; a strictly negative difference certifies improvement.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002





S004 · impliesBefore

difference=-12<0

After

candidate cannot minimize over all feasible points

Conditions: none

Global minimality requires a value no greater than every feasible value; one strictly better point refutes it.

Step identifiers and dependenciesRule: strict_improvement_obstruction · Uses: G001, S003









## Reverse checks


- S004 · premise_and_result_audit — Completeness, neighborhoods, attainment and uniqueness are separate obligations; local data do not license a global conclusion.
- S003 · premise_and_result_audit — Recompute challenger minus candidate with exact rational arithmetic, not rounded decimals.
- S002 · premise_and_result_audit — Completeness, neighborhoods, attainment and uniqueness are separate obligations; local data do not license a global conclusion.
- S001 · premise_and_result_audit — Completeness, neighborhoods, attainment and uniqueness are separate obligations; local data do not license a global conclusion.





## Result

([[A, 18], [B, 6], [C, 24], [D, 30]], A, 18, B, 6, Yes, No, one_strictly_better_feasible_point_refutes_candidate_global_optimality)

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: 56ff5cf1f674a9748a8a9eed9d1d1263a71cddc373dd36482d1f26ddf765c4b3
- candidate record SHA-256: 23146bd4fc12cb1ec503fb72646041cc32901cb4444c667e753849b6157d5f78
- 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-r07-0031.json)