# Question 901: 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.

P1000-0901 · q-20261015-pulse-20261119-r07-0076



## Original question and givens

Use mode=empty_feasible, complete model={}. Give (mode, feasible_set, optimizer_exists_claim, boundary).


- G001 — Use mode=empty_feasible, complete model={}. Give (mode, feasible_set, optimizer_exists_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=empty_feasible

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

feasible set=∅

After

no x belongs to the feasible set

Conditions: none

An optimizer must first be feasible; the empty domain has no selectable element.

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





S003 · impliesBefore

optimizer implies an existing feasible point

After

there is no optimizer

Conditions: none

This denies existence, not assigns an invented value0 or a fictitious point.

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





S004 · impliesBefore

existence requirement fails

After

optimizer exists=false

Conditions: none

This task does not introduce an extended-real infimum convention for the empty set.

Step identifiers and dependenciesRule: empty_boundary_result · 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 — Completeness, neighborhoods, attainment and uniqueness are separate obligations; local data do not license a global conclusion.
- 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

(empty_feasible, [], No, empty_feasible_set_has_no_optimizer)

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: 580f8292e474bc568998de90d0b491b3369208ce5f120bcf39145bae72196c78
- candidate record SHA-256: 04ec5f2dd29133e50240d9ce8795feef54c29ae75f54543474a974d281fd854b
- 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-r07-0076.json)