Use the complete qualified finite metric table=[[0, 22, 45, 69], [22, 0, 23, 47], [45, 23, 0, 24], [69, 47, 24, 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. center=1, positive epsilon=1/1. Give (carrier size,center,normalized epsilon,every label with distance from center strictly less than epsilon in increasing order). Use exact reduced rational outputs with positive denominators, omitting /1 for integers.
G001 — Use the complete qualified finite metric table=[[0, 22, 45, 69], [22, 0, 23, 47], [45, 23, 0, 24], [69, 47, 24, 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. center=1, positive epsilon=1/1. Give (carrier size,center,normalized epsilon,every label with distance from center strictly less than epsilon in increasing order). Use exact reduced rational outputs with positive denominators, omitting /1 for integers.
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
raw epsilon=[1, 1]
After
normalized positive epsilon=1
Conditions: none
The original denominator is nonzero and normalized epsilon must be positive.A negative original denominator may normalize legitimately;do not bypass the guard by totalized machine division.
S011 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S010 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S009 · premise_and_result_audit — Recompute ball state3 distance times epsilon denominator with exact rational arithmetic, not rounded decimals.
S008 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S007 · premise_and_result_audit — Recompute ball state2 distance times epsilon denominator with exact rational arithmetic, not rounded decimals.
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 ball state1 distance times epsilon denominator 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 — Recompute ball state0 distance times epsilon denominator 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
(4, 1, 1, [1])
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.