Use the complete qualified finite metric table=[[0, 18, 37, 57], [18, 0, 19, 39], [37, 19, 0, 20], [57, 39, 20, 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=[1, 2]; every later term equals tail=3, candidate=3. 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, 18, 37, 57], [18, 0, 19, 39], [37, 19, 0, 20], [57, 39, 20, 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=[1, 2]; every later term equals tail=3, candidate=3. 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=[1, 2];all later terms explicitly equal=3
After
candidate=3;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.
S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
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
([1, 2], 3, 3, 2, Yes, for any epsilon>0 choose N=prefix length)
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.