# Question 10570: A Conditional Uniqueness Check Is Not Two Convergence Claims

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This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth.

R0020-10570 · q-20261013-advance-20261023-r04-metric-convergence-0095



## Original question and givens

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.


- [mko-metric-convergence](https://ocme.evemisslab.com/en/mko/mko-metric-convergence/)





## Step-by-step derivation

S001 · impliesBefore

carrier Fin(4);complete distance table=[[0, 20, 41, 63], [20, 0, 21, 43], [41, 21, 0, 22], [63, 43, 22, 0]]

After

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

Conditions: none

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 dependenciesRule: whole_qualified_finite_metric · Uses: G001





S002 · impliesBefore

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.

Step identifiers and dependenciesRule: declared_infinite_tail_not_pattern_guess · Uses: G001, S001





S003 · impliesBefore

phase=candidate a;candidate=tail=1;distance=0

After

for every epsilon>0 choose N=2;every n>=N has distance0<epsilon

Conditions: none

Retain the order for every epsilon>0 there exists N such that every n≥N works.N equal to prefix length works for every positive epsilon.

Step identifiers and dependenciesRule: constant_tail_full_quantified_witness · Uses: G001, S002





S004 · impliesBefore

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.

Step identifiers and dependenciesRule: declared_infinite_tail_not_pattern_guess · Uses: G001, S003





S005 · calculationBefore

candidate b separating epsilon := div(21, 2)

After

candidate b separating epsilon = 21/2

Conditions: none

Compute exactly with this step's original roles and operands,without floating approximation or reordering source data.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004





S006 · impliesBefore

phase=candidate b;tail distance=21;epsilon=21/2

After

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.

Step identifiers and dependenciesRule: constant_tail_nonlimit_counterwitness · Uses: G001, S005





S007 · impliesBefore

a limit=Yes;b limit=No;distance(a,b)=21

After

if both are limits then a=b=Yes

Conditions: none

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.

Step identifiers and dependenciesRule: limit_uniqueness_conditional_not_two_claims · Uses: G001, S006









## Reverse checks


- 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.

Source hashes and review receipts
- input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
- original question SHA-256: 2fc85fea837498029d076cb7f306dc8175323f7a681ad0bf5d29dcfc3725b5e2
- candidate record SHA-256: 86840711203ebffce5fe708e9c1b79e85e0c6041f58fc5e75a2866f5afd71d8b
- semantic review receipt SHA-256: b4c47938fd38085d44967226cfa8337d61334455472e7307c60d3a488cef598f
- question-first witness SHA-256: 3c23baa97a5933bc07b44f2257017ef4e845bf1429bd6f5381410681c43b97e4
- Lean: NOT_RUN · human review: NOT_PERFORMED

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