Use the complete qualified finite metric table=[[0, 7, 15, 24], [7, 0, 8, 17], [15, 8, 0, 9], [24, 17, 9, 0]] on Fin(4)={0,…,3}. Preserve the convergence quantifiers: for every positive epsilon there exists N such that every n≥N satisfies d(x_n,candidate)<epsilon. A finite recorded tail alone does not certify this infinite statement. prefix=[2, 3]; every later term equals tail=0, candidate=1. This is an explicitly specified infinite sequence, not an inference from a finite pattern. Give (prefix,tail,candidate,prefix length,converges?,witness). For equality use for any epsilon>0 choose N=prefix length; otherwise use tail-value-differs-from-candidate.
G001 — Use the complete qualified finite metric table=[[0, 7, 15, 24], [7, 0, 8, 17], [15, 8, 0, 9], [24, 17, 9, 0]] on Fin(4)={0,…,3}. Preserve the convergence quantifiers: for every positive epsilon there exists N such that every n≥N satisfies d(x_n,candidate)<epsilon. A finite recorded tail alone does not certify this infinite statement. prefix=[2, 3]; every later term equals tail=0, candidate=1. This is an explicitly specified infinite sequence, not an inference from a finite pattern. Give (prefix,tail,candidate,prefix length,converges?,witness). For equality use for any epsilon>0 choose N=prefix length; otherwise use tail-value-differs-from-candidate.
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.
Qualify the whole metric table,not only queried cells.Labels are not distances:diagonal zero,positive separation,symmetry and every triangle condition are premises for later reasoning.
Step identifiers and dependencies
Rule: whole_qualified_finite_metric · Uses: G001
S002 · implies
Before
phase=candidate;prefix=[2, 3];all later terms explicitly equal=0
After
candidate=1;infinite constant tail begins at=2
Conditions: none
The problem explicitly specifies every later term as tail;this is not guessing an infinite rule from a finite prefix.The prefix does not change the tail limit.
for any proposed N choose n=max(N,2);distance=7>=epsilon
Conditions: none
Metric separation gives positive distance.Choose half of it as epsilon;after every proposed N a tail term retains that distance and violates the full-tail convergence condition.
S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S003 · premise_and_result_audit — Recompute candidate separating epsilon with exact rational arithmetic, not rounded decimals.
S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
Result
([2, 3], 0, 1, 2, No, tail-value-differs-from-candidate)
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.