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

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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.

R0020-10504 · q-20261013-advance-20261023-r04-metric-convergence-0004



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





## Step-by-step derivation

S001 · impliesBefore

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

raw epsilon=[4, 1]

After

normalized positive epsilon=4

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.

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





S003 · calculationBefore

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

After

ball state0 distance times epsilon denominator = 15

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, S002





S004 · impliesBefore

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

After

cross product=15;strictly below numerator=No

Conditions: none

Cross-multiply by the positive denominator and retain strict inequality.Equality with epsilon is outside the open ball.

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





S005 · calculationBefore

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

After

ball state1 distance times epsilon denominator = 11

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

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

After

cross product=11;strictly below numerator=No

Conditions: none

Cross-multiply by the positive denominator and retain strict inequality.Equality with epsilon is outside the open ball.

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





S007 · calculationBefore

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

After

ball state2 distance times epsilon denominator = 6

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, S006





S008 · impliesBefore

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

After

cross product=6;strictly below numerator=No

Conditions: none

Cross-multiply by the positive denominator and retain strict inequality.Equality with epsilon is outside the open ball.

Step identifiers and dependenciesRule: strict_metric_distance_comparison · Uses: G001, S007





S009 · calculationBefore

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

After

ball state3 distance times epsilon denominator = 0

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, S008





S010 · impliesBefore

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

After

cross product=0;strictly below numerator=Yes

Conditions: none

Cross-multiply by the positive denominator and retain strict inequality.Equality with epsilon is outside the open ball.

Step identifiers and dependenciesRule: strict_metric_distance_comparison · Uses: G001, S009





S011 · impliesBefore

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

After

open ball=[3]

Conditions: none

Inspect the complete carrier,including the center and excluding the boundary.Do not infer the ball from label sizes or a drawing.

Step identifiers and dependenciesRule: complete_strict_ball_inventory · Uses: G001, S010









## 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.

Source hashes and review receipts
- input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
- original question SHA-256: 0b536d78b7b89780c7c781594333fbf68b585fd6a45f0bdddce0ec88236eb38e
- candidate record SHA-256: 1fc41f84816d6b721ebaf997327fc48257bb10149d74f23d3dbdf325309c3d55
- semantic review receipt SHA-256: b4c47938fd38085d44967226cfa8337d61334455472e7307c60d3a488cef598f
- question-first witness SHA-256: 3c23baa97a5933bc07b44f2257017ef4e845bf1429bd6f5381410681c43b97e4
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261013-advance-20261023-r04-metric-convergence-0004.json)