# Question 918: Empty Domains, Partial Lists, Ties and Local-Only Witnesses Block Different Claims

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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.

P1000-0918 · q-20261015-pulse-20261119-r07-0093



## Original question and givens

Use mode=tie_not_unique, complete model={"points":["A","B","C"],"values":[18,18,54]}. Give (mode, minimum_value, all_global_minimizers, global_optimum_exists, unique_claim, boundary).


- G001 — Use mode=tie_not_unique, complete model={"points":["A","B","C"],"values":[18,18,54]}. Give (mode, minimum_value, all_global_minimizers, global_optimum_exists, unique_claim, boundary).





## Definition and scope

An empty set has no optimizer; an unverified partial list supports no global claim; ties are not unique; a local minimum need not be global.


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





## Step-by-step derivation

S001 · impliesBefore

mode=tie_not_unique

After

identify the exact missing optimization obligation

Conditions: none

Distinguish nonexistence, unverified completeness, nonuniqueness and merely local optimality.

Step identifiers and dependenciesRule: optimization_scope · Uses: G001





S002 · impliesBefore

complete finite objective table

After

[[A, 18], [B, 18], [C, 54]]

Conditions: none

This branch declares the complete domain, licensing pointwise global comparison.

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





S003 · calculationBefore

f(A)−minimum := sub(18, 18)

After

f(A)−minimum = 0

Conditions: none

Check whether each point attains the minimum.

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





S004 · calculationBefore

f(B)−minimum := sub(18, 18)

After

f(B)−minimum = 0

Conditions: none

Check whether each point attains the minimum.

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





S005 · calculationBefore

f(C)−minimum := sub(54, 18)

After

f(C)−minimum = 36

Conditions: none

Check whether each point attains the minimum.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004





S006 · impliesBefore

minimum=18; all minimizers=[A, B]

After

exists=true; unique=No

Conditions: none

A global optimum need not be unique; retain every tied attaining point.

Step identifiers and dependenciesRule: tie_not_uniqueness · Uses: G001, S005









## Reverse checks


- 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(C)−minimum with exact rational arithmetic, not rounded decimals.
- S004 · premise_and_result_audit — Recompute f(B)−minimum with exact rational arithmetic, not rounded decimals.
- S003 · premise_and_result_audit — Recompute f(A)−minimum 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

(tie_not_unique, 18, [A, B], Yes, No, global_optimum_need_not_be_unique)

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: db601a9779a3ce892044ffa0a260f394533e765d078f9f0f65e512b56b547ee9
- candidate record SHA-256: 1879bf61a5edf286a9b474b29ce3b1baecbdcfe7c82770805a31562a94639f80
- 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-r07-0093.json)