The domain is explicitly the integers. Consider the universal proposition “Every integer is even.” and the stated witness n=9. Does this witness refute that universal proposition?
G001 — The domain is explicitly the integers. Consider the universal proposition “Every integer is even.” and the stated witness n=9. Does this witness refute that universal proposition?
Definition and scope
Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.
Check THIS bank’s keys/operation/domain:legacy table underscores,MOD50 local hyphens;permissive BASE data acquires no unknown-field restriction.Keep Number.isInteger and SAFE source guards distinct,while preserving ORIGINAL JSON number representations.
FIRST match the stated predicate/bound and INTEGER domain.THEN substitute THIS integer witness and show failure.Use an exact nonzero remainder for even/multiple and retain strict/non-strict inequalities elsewhere.Do not use a metadata flag as an answer oracle.
derived = [exists integer n with not predicate(n), not forall integers predicate(n), Yes]
Conditions: none
ONE actual counterexample IN the domain proves∃n ¬P(n),therefore refuting THIS∀n P(n).It does not prove every n fails,refute other propositions or make finite successful samples a universal proof.
S003 · premise_and_result_audit — Reverse-audit THIS step against the ORIGINAL roles,complete premises and field order.Distinguish implication from equivalence;preserve originals,English,MKOs,ASTs and review states.Finite checks are not independent review or general formal proof.
S002 · premise_and_result_audit — Reverse-audit THIS step against the ORIGINAL roles,complete premises and field order.Distinguish implication from equivalence;preserve originals,English,MKOs,ASTs and review states.Finite checks are not independent review or general formal proof.
S001 · premise_and_result_audit — Reverse-audit THIS step against the ORIGINAL roles,complete premises and field order.Distinguish implication from equivalence;preserve originals,English,MKOs,ASTs and review states.Finite checks are not independent review or general formal proof.
Result
Yes
The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.
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.