Question 901: Empty Domains, Partial Lists, Ties and Local-Only Witnesses Block Different Claims

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

Step-by-step derivation

mode=empty_feasible

identify the exact missing optimization obligation

feasible set=∅

no x belongs to the feasible set

optimizer implies an existing feasible point

there is no optimizer

existence requirement fails

optimizer exists=false

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.