Question 910: Empty Domains, Partial Lists, Ties and Local-Only Witnesses Block Different Claims

Original question and givens

Use mode=local_not_global, complete model={"points":["A","B","C","D"],"values":[20,30,10,40],"edges":[["A","B"]],"candidate":"A"}. Give (mode, candidate, candidate_value, declared_neighbors, neighborhood_local_minimum, global_minimum, boundary).

• G001 — Use mode=local_not_global, complete model={"points":["A","B","C","D"],"values":[20,30,10,40],"edges":[["A","B"]],"candidate":"A"}. Give (mode, candidate, candidate_value, declared_neighbors, neighborhood_local_minimum, global_minimum, 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=local_not_global

identify the exact missing optimization obligation

complete finite objective table

[[A, 20], [B, 30], [C, 10], [D, 40]]

neighbor f(B)−f(A) := sub(30, 20)

neighbor f(B)−f(A) = 10

neighbors=[B]; better feasible points=[C]

local=true; global=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 — Recompute neighbor f(B)−f(A) 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

(local_not_global, A, 20, [B], Yes, No, local_minimum_does_not_imply_global_minimum)

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.