Question 10529: Finite-Tail Success Does Not Certify General Convergence

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. recorded finite sequence=[3, 2, 1, 1], candidate=1, positive epsilon=4/1, cutoff N=1 may equal the recorded length. Give (sequence,candidate,epsilon,N,every [index,state,distance,status] row,all recorded indices at least N are inside?,general convergence certified?,boundary label). Status is prefix below N, otherwise inside or outside using the strict bound; general convergence is always No and the boundary label is finite-tail-insufficient-for-general-convergence.

• 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. recorded finite sequence=[3, 2, 1, 1], candidate=1, positive epsilon=4/1, cutoff N=1 may equal the recorded length. Give (sequence,candidate,epsilon,N,every [index,state,distance,status] row,all recorded indices at least N are inside?,general convergence certified?,boundary label). Status is prefix below N, otherwise inside or outside using the strict bound; general convergence is always No and the boundary label is finite-tail-insufficient-for-general-convergence.

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

original index=0;state=3;candidate=1;distance=11;cutoff=1

recorded status=prefix

recorded index1 distance times epsilon denominator := mul(5, 1)

recorded index1 distance times epsilon denominator = 5

recorded index1:distance=5;epsilon=4;positive denominator=1

cross product=5;strictly below numerator=No

original index=1;state=2;candidate=1;distance=5;cutoff=1

recorded status=outside

recorded index2 distance times epsilon denominator := mul(0, 1)

recorded index2 distance times epsilon denominator = 0

recorded index2:distance=0;epsilon=4;positive denominator=1

cross product=0;strictly below numerator=Yes

original index=2;state=1;candidate=1;distance=0;cutoff=1

recorded status=inside

recorded index3 distance times epsilon denominator := mul(0, 1)

recorded index3 distance times epsilon denominator = 0

recorded index3:distance=0;epsilon=4;positive denominator=1

cross product=0;strictly below numerator=Yes

original index=3;state=1;candidate=1;distance=0;cutoff=1

recorded status=inside

recorded rows=[[0, 3, 11, prefix], [1, 2, 5, outside], [2, 1, 0, inside], [3, 1, 0, inside]];recorded tail starts at=1

bounded recorded tail passes=No;general convergence certified=No

Reverse checks

• S013 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S012 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• 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 — Recompute recorded index3 distance times epsilon denominator with exact rational arithmetic, not rounded decimals.
• S009 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• 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 recorded index2 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 — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S004 · premise_and_result_audit — Recompute recorded index1 distance times epsilon denominator with exact rational arithmetic, not rounded decimals.
• 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, 2, 1, 1], 1, 4, 1, [[0, 3, 11, prefix], [1, 2, 5, outside], [2, 1, 0, inside], [3, 1, 0, inside]], No, No, finite-tail-insufficient-for-general-convergence)

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.