Question 10504: A Strict Open Ball in a Complete Finite Metric

Original question and givens

Use the complete qualified finite metric table=[[0, 4, 9, 15], [4, 0, 5, 11], [9, 5, 0, 6], [15, 11, 6, 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=3, positive epsilon=4/1. 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, 4, 9, 15], [4, 0, 5, 11], [9, 5, 0, 6], [15, 11, 6, 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=3, positive epsilon=4/1. 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.

• mko-metric-convergence

Step-by-step derivation

carrier Fin(4);complete distance table=[[0, 4, 9, 15], [4, 0, 5, 11], [9, 5, 0, 6], [15, 11, 6, 0]]

nonnegative typed entries=Yes;zero iff equal=Yes;symmetric=Yes;all triangle tests=Yes

raw epsilon=[4, 1]

normalized positive epsilon=4

ball state0 distance times epsilon denominator := mul(15, 1)

ball state0 distance times epsilon denominator = 15

ball state0:distance=15;epsilon=4;positive denominator=1

cross product=15;strictly below numerator=No

ball state1 distance times epsilon denominator := mul(11, 1)

ball state1 distance times epsilon denominator = 11

ball state1:distance=11;epsilon=4;positive denominator=1

cross product=11;strictly below numerator=No

ball state2 distance times epsilon denominator := mul(6, 1)

ball state2 distance times epsilon denominator = 6

ball state2:distance=6;epsilon=4;positive denominator=1

cross product=6;strictly below numerator=No

ball state3 distance times epsilon denominator := mul(0, 1)

ball state3 distance times epsilon denominator = 0

ball state3:distance=0;epsilon=4;positive denominator=1

cross product=0;strictly below numerator=Yes

center=3;strict radius=4;all carrier labels=[0, 1, 2, 3]

open ball=[3]

Reverse checks

• 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, 3, 4, [3])

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.