The explicit finite domain is Fin(0)=∅ and the complete Boolean predicate table is truth=[] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Append the explicitly given Boolean value false at new label n=0. Give (original failures, extended indexed truth table, later failures, original universal truth?, later universal truth?, new label).
G001 — The explicit finite domain is Fin(0)=∅ and the complete Boolean predicate table is truth=[] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Append the explicitly given Boolean value false at new label n=0. Give (original failures, extended indexed truth table, later failures, original universal truth?, later universal truth?, new label).
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 source interface:distance uses ONLY its complete coordinate frame;other families use a domain.Discrete operations use local hyphens,legacy banks underscores.Retain family-specific ordered output types and original bilingual data/status.
AFTER traversing the whole GIVEN finite domain,no failures means its universal statement is true,including the empty domain.This assigned finite model does not prove arbitrary infinite statements.
The NEW label is n and its truth value is explicitly GIVEN,not inferred.Raw n=8 may derive a9-row table without recapping;an empty original universal may become false after appending False.
S003 · premise_and_result_audit — Reverse-check the ORIGINAL inputs,types and premises;the conclusion need not uniquely recover its premises.This source-bound candidate is not independent review or formal proof.
S002 · premise_and_result_audit — Reverse-check the ORIGINAL inputs,types and premises;the conclusion need not uniquely recover its premises.This source-bound candidate is not independent review or formal proof.
S001 · premise_and_result_audit — Reverse-check the ORIGINAL inputs,types and premises;the conclusion need not uniquely recover its premises.This source-bound candidate is not independent review or formal proof.
Result
([], [[0, No]], [0], Yes, No, 0)
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.