Use the complete qualified finite metric table=[[0, 20, 41, 63], [20, 0, 21, 43], [41, 21, 0, 22], [63, 43, 22, 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. infinite sequence has prefix=[3, 0] and constant tail=1; candidates a=1, b=2. Give (prefix,tail,a,b,d(a,b),a is a limit?,b is a limit?,if both are limits then a=b?).
G001 — Use the complete qualified finite metric table=[[0, 20, 41, 63], [20, 0, 21, 43], [41, 21, 0, 22], [63, 43, 22, 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. infinite sequence has prefix=[3, 0] and constant tail=1; candidates a=1, b=2. Give (prefix,tail,a,b,d(a,b),a is a limit?,b is a limit?,if both are limits then a=b?).
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 a;prefix=[3, 0];all later terms explicitly equal=1
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.
phase=candidate b;prefix=[3, 0];all later terms explicitly equal=1
After
candidate=2;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=21>=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.
Audit both limit qualifications separately.The last field is conditional implication;it can be true with a false antecedent and does not assert both candidates converge.
S007 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S005 · premise_and_result_audit — Recompute candidate b separating epsilon with exact rational arithmetic, not rounded decimals.
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 — 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
([3, 0], 1, 1, 2, 21, Yes, No, 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.