Question 909: Empty Domains, Partial Lists, Ties and Local-Only Witnesses Block Different Claims

Original question and givens

Use mode=tie_not_unique, complete model={"points":["A","B","C"],"values":[9,9,27]}. 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":[9,9,27]}. 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

Step-by-step derivation

mode=tie_not_unique

identify the exact missing optimization obligation

complete finite objective table

[[A, 9], [B, 9], [C, 27]]

f(A)−minimum := sub(9, 9)

f(A)−minimum = 0

f(B)−minimum := sub(9, 9)

f(B)−minimum = 0

f(C)−minimum := sub(27, 9)

f(C)−minimum = 18

minimum=9; all minimizers=[A, B]

exists=true; unique=No

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, 9, [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.