Strict Finite-Metric Tail Audits versus Infinite Limit Witnesses
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batch-20261013-advance-20261023-r04-metric-convergence-foundations · 2026-10-13 · 100 questions
This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.
1. A Strict Open Ball in a Complete Finite Metric
Use the complete qualified finite metric table=[[0, 1, 3, 6], [1, 0, 2, 5], [3, 2, 0, 3], [6, 5, 3, 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=0, 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.
View answer and explanation
(4, 0, 1, [0])
Compare exact distances strictly; a point at distance equal to epsilon is excluded. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (4, 0, 1, [0]).
Reviewed step-by-step derivation
Convergence in a metric space · qf-mod23-metricconv-ball-inventory · candidate
2. A Strict Open Ball in a Complete Finite Metric
Use the complete qualified finite metric table=[[0, 2, 5, 9], [2, 0, 3, 7], [5, 3, 0, 4], [9, 7, 4, 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=2/1. 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.
View answer and explanation
(4, 1, 2, [1])
Compare exact distances strictly; a point at distance equal to epsilon is excluded. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (4, 1, 2, [1]).
Reviewed step-by-step derivation
Convergence in a metric space · qf-mod23-metricconv-ball-inventory · candidate
3. A Strict Open Ball in a Complete Finite Metric
Use the complete qualified finite metric table=[[0, 3, 7, 12], [3, 0, 4, 9], [7, 4, 0, 5], [12, 9, 5, 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=2, positive epsilon=3/1. 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.
View answer and explanation
(4, 2, 3, [2])
Compare exact distances strictly; a point at distance equal to epsilon is excluded. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (4, 2, 3, [2]).
Reviewed step-by-step derivation
Convergence in a metric space · qf-mod23-metricconv-ball-inventory · candidate
4. A Strict Open Ball in a Complete Finite Metric
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.
View answer and explanation
(4, 3, 4, [3])
Compare exact distances strictly; a point at distance equal to epsilon is excluded. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (4, 3, 4, [3]).
Reviewed step-by-step derivation
Convergence in a metric space · qf-mod23-metricconv-ball-inventory · candidate
5. A Strict Open Ball in a Complete Finite Metric
Use the complete qualified finite metric table=[[0, 5, 11, 18], [5, 0, 6, 13], [11, 6, 0, 7], [18, 13, 7, 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=0, positive epsilon=5/1. 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.
View answer and explanation
(4, 0, 5, [0])
Compare exact distances strictly; a point at distance equal to epsilon is excluded. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (4, 0, 5, [0]).
Reviewed step-by-step derivation
Convergence in a metric space · qf-mod23-metricconv-ball-inventory · candidate
6. A Strict Open Ball in a Complete Finite Metric
Use the complete qualified finite metric table=[[0, 6, 13, 21], [6, 0, 7, 15], [13, 7, 0, 8], [21, 15, 8, 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=6/1. 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.
View answer and explanation
(4, 1, 6, [1])
Compare exact distances strictly; a point at distance equal to epsilon is excluded. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (4, 1, 6, [1]).
Reviewed step-by-step derivation
Convergence in a metric space · qf-mod23-metricconv-ball-inventory · candidate
7. A Strict Open Ball in a Complete Finite Metric
Use the complete qualified finite metric table=[[0, 7, 15, 24], [7, 0, 8, 17], [15, 8, 0, 9], [24, 17, 9, 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=2, positive epsilon=7/1. 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.
View answer and explanation
(4, 2, 7, [2])
Compare exact distances strictly; a point at distance equal to epsilon is excluded. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (4, 2, 7, [2]).
Reviewed step-by-step derivation
Convergence in a metric space · qf-mod23-metricconv-ball-inventory · candidate
8. A Strict Open Ball in a Complete Finite Metric
Use the complete qualified finite metric table=[[0, 8, 17, 27], [8, 0, 9, 19], [17, 9, 0, 10], [27, 19, 10, 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=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.
View answer and explanation
(4, 3, 1, [3])
Compare exact distances strictly; a point at distance equal to epsilon is excluded. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (4, 3, 1, [3]).
Reviewed step-by-step derivation
Convergence in a metric space · qf-mod23-metricconv-ball-inventory · candidate
9. A Strict Open Ball in a Complete Finite Metric
Use the complete qualified finite metric table=[[0, 9, 19, 30], [9, 0, 10, 21], [19, 10, 0, 11], [30, 21, 11, 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=0, positive epsilon=2/1. 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.
View answer and explanation
(4, 0, 2, [0])
Compare exact distances strictly; a point at distance equal to epsilon is excluded. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (4, 0, 2, [0]).
Reviewed step-by-step derivation
Convergence in a metric space · qf-mod23-metricconv-ball-inventory · candidate
10. A Strict Open Ball in a Complete Finite Metric
Use the complete qualified finite metric table=[[0, 10, 21, 33], [10, 0, 11, 23], [21, 11, 0, 12], [33, 23, 12, 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=3/1. 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.
View answer and explanation
(4, 1, 3, [1])
Compare exact distances strictly; a point at distance equal to epsilon is excluded. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (4, 1, 3, [1]).
Reviewed step-by-step derivation
Convergence in a metric space · qf-mod23-metricconv-ball-inventory · candidate
11. A Strict Open Ball in a Complete Finite Metric
Use the complete qualified finite metric table=[[0, 11, 23, 36], [11, 0, 12, 25], [23, 12, 0, 13], [36, 25, 13, 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=2, 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.
View answer and explanation
(4, 2, 4, [2])
Compare exact distances strictly; a point at distance equal to epsilon is excluded. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (4, 2, 4, [2]).
Reviewed step-by-step derivation
Convergence in a metric space · qf-mod23-metricconv-ball-inventory · candidate
12. A Strict Open Ball in a Complete Finite Metric
Use the complete qualified finite metric table=[[0, 12, 25, 39], [12, 0, 13, 27], [25, 13, 0, 14], [39, 27, 14, 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=5/1. 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.
View answer and explanation
(4, 3, 5, [3])
Compare exact distances strictly; a point at distance equal to epsilon is excluded. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (4, 3, 5, [3]).
Reviewed step-by-step derivation
Convergence in a metric space · qf-mod23-metricconv-ball-inventory · candidate
13. A Strict Open Ball in a Complete Finite Metric
Use the complete qualified finite metric table=[[0, 13, 27, 42], [13, 0, 14, 29], [27, 14, 0, 15], [42, 29, 15, 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=0, positive epsilon=6/1. 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.
View answer and explanation
(4, 0, 6, [0])
Compare exact distances strictly; a point at distance equal to epsilon is excluded. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (4, 0, 6, [0]).
Reviewed step-by-step derivation
Convergence in a metric space · qf-mod23-metricconv-ball-inventory · candidate
14. A Strict Open Ball in a Complete Finite Metric
Use the complete qualified finite metric table=[[0, 14, 29, 45], [14, 0, 15, 31], [29, 15, 0, 16], [45, 31, 16, 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=7/1. 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.
View answer and explanation
(4, 1, 7, [1])
Compare exact distances strictly; a point at distance equal to epsilon is excluded. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (4, 1, 7, [1]).
Reviewed step-by-step derivation
Convergence in a metric space · qf-mod23-metricconv-ball-inventory · candidate
15. A Strict Open Ball in a Complete Finite Metric
Use the complete qualified finite metric table=[[0, 15, 31, 48], [15, 0, 16, 33], [31, 16, 0, 17], [48, 33, 17, 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=2, 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.
View answer and explanation
(4, 2, 1, [2])
Compare exact distances strictly; a point at distance equal to epsilon is excluded. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (4, 2, 1, [2]).
Reviewed step-by-step derivation
Convergence in a metric space · qf-mod23-metricconv-ball-inventory · candidate
16. A Strict Open Ball in a Complete Finite Metric
Use the complete qualified finite metric table=[[0, 16, 33, 51], [16, 0, 17, 35], [33, 17, 0, 18], [51, 35, 18, 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=2/1. 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.
View answer and explanation
(4, 3, 2, [3])
Compare exact distances strictly; a point at distance equal to epsilon is excluded. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (4, 3, 2, [3]).
Reviewed step-by-step derivation
Convergence in a metric space · qf-mod23-metricconv-ball-inventory · candidate
17. A Strict Open Ball in a Complete Finite Metric
Use the complete qualified finite metric table=[[0, 17, 35, 54], [17, 0, 18, 37], [35, 18, 0, 19], [54, 37, 19, 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=0, positive epsilon=3/1. 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.
View answer and explanation
(4, 0, 3, [0])
Compare exact distances strictly; a point at distance equal to epsilon is excluded. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (4, 0, 3, [0]).
Reviewed step-by-step derivation
Convergence in a metric space · qf-mod23-metricconv-ball-inventory · candidate
18. A Strict Open Ball in a Complete Finite Metric
Use the complete qualified finite metric table=[[0, 18, 37, 57], [18, 0, 19, 39], [37, 19, 0, 20], [57, 39, 20, 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=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.
View answer and explanation
(4, 1, 4, [1])
Compare exact distances strictly; a point at distance equal to epsilon is excluded. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (4, 1, 4, [1]).
Reviewed step-by-step derivation
Convergence in a metric space · qf-mod23-metricconv-ball-inventory · candidate
19. A Strict Open Ball in a Complete Finite Metric
Use the complete qualified finite metric table=[[0, 19, 39, 60], [19, 0, 20, 41], [39, 20, 0, 21], [60, 41, 21, 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=2, positive epsilon=5/1. 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.
View answer and explanation
(4, 2, 5, [2])
Compare exact distances strictly; a point at distance equal to epsilon is excluded. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (4, 2, 5, [2]).
Reviewed step-by-step derivation
Convergence in a metric space · qf-mod23-metricconv-ball-inventory · candidate
20. A Strict Open Ball in a Complete Finite Metric
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. center=3, positive epsilon=6/1. 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.
View answer and explanation
(4, 3, 6, [3])
Compare exact distances strictly; a point at distance equal to epsilon is excluded. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (4, 3, 6, [3]).
Reviewed step-by-step derivation
Convergence in a metric space · qf-mod23-metricconv-ball-inventory · candidate
21. A Strict Open Ball in a Complete Finite Metric
Use the complete qualified finite metric table=[[0, 21, 43, 66], [21, 0, 22, 45], [43, 22, 0, 23], [66, 45, 23, 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=0, positive epsilon=7/1. 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.
View answer and explanation
(4, 0, 7, [0])
Compare exact distances strictly; a point at distance equal to epsilon is excluded. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (4, 0, 7, [0]).
Reviewed step-by-step derivation
Convergence in a metric space · qf-mod23-metricconv-ball-inventory · candidate
22. A Strict Open Ball in a Complete Finite Metric
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.
View answer and explanation
(4, 1, 1, [1])
Compare exact distances strictly; a point at distance equal to epsilon is excluded. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (4, 1, 1, [1]).
Reviewed step-by-step derivation
Convergence in a metric space · qf-mod23-metricconv-ball-inventory · candidate
23. A Strict Open Ball in a Complete Finite Metric
Use the complete qualified finite metric table=[[0, 23, 47, 72], [23, 0, 24, 49], [47, 24, 0, 25], [72, 49, 25, 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=2, positive epsilon=2/1. 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.
View answer and explanation
(4, 2, 2, [2])
Compare exact distances strictly; a point at distance equal to epsilon is excluded. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (4, 2, 2, [2]).
Reviewed step-by-step derivation
Convergence in a metric space · qf-mod23-metricconv-ball-inventory · candidate
24. A Strict Open Ball in a Complete Finite Metric
Use the complete qualified finite metric table=[[0, 24, 49, 75], [24, 0, 25, 51], [49, 25, 0, 26], [75, 51, 26, 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=3/1. 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.
View answer and explanation
(4, 3, 3, [3])
Compare exact distances strictly; a point at distance equal to epsilon is excluded. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (4, 3, 3, [3]).
Reviewed step-by-step derivation
Convergence in a metric space · qf-mod23-metricconv-ball-inventory · candidate
25. A Strict Open Ball in a Complete Finite Metric
Use the complete qualified finite metric table=[[0, 25, 51, 78], [25, 0, 26, 53], [51, 26, 0, 27], [78, 53, 27, 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=0, 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.
View answer and explanation
(4, 0, 4, [0])
Compare exact distances strictly; a point at distance equal to epsilon is excluded. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (4, 0, 4, [0]).
Reviewed step-by-step derivation
Convergence in a metric space · qf-mod23-metricconv-ball-inventory · candidate
26. Finite-Tail Success Does Not Certify General Convergence
Use the complete qualified finite metric table=[[0, 1, 3, 6], [1, 0, 2, 5], [3, 2, 0, 3], [6, 5, 3, 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=1/1, 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.
View answer and explanation
([3, 2, 1, 1], 1, 1, 1, [[0, 3, 5, prefix], [1, 2, 2, outside], [2, 1, 0, inside], [3, 1, 0, inside]], No, No, finite-tail-insufficient-for-general-convergence)
Inspect only recorded indices. An empty recorded tail passes its bounded check vacuously but still proves no general convergence. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([3, 2, 1, 1], 1, 1, 1, [[0, 3, 5, prefix], [1, 2, 2, outside], [2, 1, 0, inside], [3, 1, 0, inside]], No, No, finite-tail-insufficient-for-general-convergence).
Reviewed step-by-step derivation
Convergence in a metric space · qf-mod23-metricconv-finite-tail-audit · candidate
27. Finite-Tail Success Does Not Certify General Convergence
Use the complete qualified finite metric table=[[0, 2, 5, 9], [2, 0, 3, 7], [5, 3, 0, 4], [9, 7, 4, 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=2/1, cutoff N=2 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.
View answer and explanation
([3, 2, 1, 1], 1, 2, 2, [[0, 3, 7, prefix], [1, 2, 3, prefix], [2, 1, 0, inside], [3, 1, 0, inside]], Yes, No, finite-tail-insufficient-for-general-convergence)
Inspect only recorded indices. An empty recorded tail passes its bounded check vacuously but still proves no general convergence. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([3, 2, 1, 1], 1, 2, 2, [[0, 3, 7, prefix], [1, 2, 3, prefix], [2, 1, 0, inside], [3, 1, 0, inside]], Yes, No, finite-tail-insufficient-for-general-convergence).
Reviewed step-by-step derivation
Convergence in a metric space · qf-mod23-metricconv-finite-tail-audit · candidate
28. Finite-Tail Success Does Not Certify General Convergence
Use the complete qualified finite metric table=[[0, 3, 7, 12], [3, 0, 4, 9], [7, 4, 0, 5], [12, 9, 5, 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=3/1, cutoff N=3 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.
View answer and explanation
([3, 2, 1, 1], 1, 3, 3, [[0, 3, 9, prefix], [1, 2, 4, prefix], [2, 1, 0, prefix], [3, 1, 0, inside]], Yes, No, finite-tail-insufficient-for-general-convergence)
Inspect only recorded indices. An empty recorded tail passes its bounded check vacuously but still proves no general convergence. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([3, 2, 1, 1], 1, 3, 3, [[0, 3, 9, prefix], [1, 2, 4, prefix], [2, 1, 0, prefix], [3, 1, 0, inside]], Yes, No, finite-tail-insufficient-for-general-convergence).
Reviewed step-by-step derivation
Convergence in a metric space · qf-mod23-metricconv-finite-tail-audit · candidate
29. Finite-Tail Success Does Not Certify General Convergence
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.
View answer and explanation
([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)
Inspect only recorded indices. An empty recorded tail passes its bounded check vacuously but still proves no general convergence. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([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).
Reviewed step-by-step derivation
Convergence in a metric space · qf-mod23-metricconv-finite-tail-audit · candidate
30. Finite-Tail Success Does Not Certify General Convergence
Use the complete qualified finite metric table=[[0, 5, 11, 18], [5, 0, 6, 13], [11, 6, 0, 7], [18, 13, 7, 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=5/1, cutoff N=2 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.
View answer and explanation
([3, 2, 1, 1], 1, 5, 2, [[0, 3, 13, prefix], [1, 2, 6, prefix], [2, 1, 0, inside], [3, 1, 0, inside]], Yes, No, finite-tail-insufficient-for-general-convergence)
Inspect only recorded indices. An empty recorded tail passes its bounded check vacuously but still proves no general convergence. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([3, 2, 1, 1], 1, 5, 2, [[0, 3, 13, prefix], [1, 2, 6, prefix], [2, 1, 0, inside], [3, 1, 0, inside]], Yes, No, finite-tail-insufficient-for-general-convergence).
Reviewed step-by-step derivation
Convergence in a metric space · qf-mod23-metricconv-finite-tail-audit · candidate
31. Finite-Tail Success Does Not Certify General Convergence
Use the complete qualified finite metric table=[[0, 6, 13, 21], [6, 0, 7, 15], [13, 7, 0, 8], [21, 15, 8, 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=1/1, cutoff N=3 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.
View answer and explanation
([3, 2, 1, 1], 1, 1, 3, [[0, 3, 15, prefix], [1, 2, 7, prefix], [2, 1, 0, prefix], [3, 1, 0, inside]], Yes, No, finite-tail-insufficient-for-general-convergence)
Inspect only recorded indices. An empty recorded tail passes its bounded check vacuously but still proves no general convergence. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([3, 2, 1, 1], 1, 1, 3, [[0, 3, 15, prefix], [1, 2, 7, prefix], [2, 1, 0, prefix], [3, 1, 0, inside]], Yes, No, finite-tail-insufficient-for-general-convergence).
Reviewed step-by-step derivation
Convergence in a metric space · qf-mod23-metricconv-finite-tail-audit · candidate
32. Finite-Tail Success Does Not Certify General Convergence
Use the complete qualified finite metric table=[[0, 7, 15, 24], [7, 0, 8, 17], [15, 8, 0, 9], [24, 17, 9, 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=2/1, 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.
View answer and explanation
([3, 2, 1, 1], 1, 2, 1, [[0, 3, 17, prefix], [1, 2, 8, outside], [2, 1, 0, inside], [3, 1, 0, inside]], No, No, finite-tail-insufficient-for-general-convergence)
Inspect only recorded indices. An empty recorded tail passes its bounded check vacuously but still proves no general convergence. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([3, 2, 1, 1], 1, 2, 1, [[0, 3, 17, prefix], [1, 2, 8, outside], [2, 1, 0, inside], [3, 1, 0, inside]], No, No, finite-tail-insufficient-for-general-convergence).
Reviewed step-by-step derivation
Convergence in a metric space · qf-mod23-metricconv-finite-tail-audit · candidate
33. Finite-Tail Success Does Not Certify General Convergence
Use the complete qualified finite metric table=[[0, 8, 17, 27], [8, 0, 9, 19], [17, 9, 0, 10], [27, 19, 10, 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=3/1, cutoff N=2 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.
View answer and explanation
([3, 2, 1, 1], 1, 3, 2, [[0, 3, 19, prefix], [1, 2, 9, prefix], [2, 1, 0, inside], [3, 1, 0, inside]], Yes, No, finite-tail-insufficient-for-general-convergence)
Inspect only recorded indices. An empty recorded tail passes its bounded check vacuously but still proves no general convergence. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([3, 2, 1, 1], 1, 3, 2, [[0, 3, 19, prefix], [1, 2, 9, prefix], [2, 1, 0, inside], [3, 1, 0, inside]], Yes, No, finite-tail-insufficient-for-general-convergence).
Reviewed step-by-step derivation
Convergence in a metric space · qf-mod23-metricconv-finite-tail-audit · candidate
34. Finite-Tail Success Does Not Certify General Convergence
Use the complete qualified finite metric table=[[0, 9, 19, 30], [9, 0, 10, 21], [19, 10, 0, 11], [30, 21, 11, 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=3 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.
View answer and explanation
([3, 2, 1, 1], 1, 4, 3, [[0, 3, 21, prefix], [1, 2, 10, prefix], [2, 1, 0, prefix], [3, 1, 0, inside]], Yes, No, finite-tail-insufficient-for-general-convergence)
Inspect only recorded indices. An empty recorded tail passes its bounded check vacuously but still proves no general convergence. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([3, 2, 1, 1], 1, 4, 3, [[0, 3, 21, prefix], [1, 2, 10, prefix], [2, 1, 0, prefix], [3, 1, 0, inside]], Yes, No, finite-tail-insufficient-for-general-convergence).
Reviewed step-by-step derivation
Convergence in a metric space · qf-mod23-metricconv-finite-tail-audit · candidate
35. Finite-Tail Success Does Not Certify General Convergence
Use the complete qualified finite metric table=[[0, 10, 21, 33], [10, 0, 11, 23], [21, 11, 0, 12], [33, 23, 12, 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=5/1, 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.
View answer and explanation
([3, 2, 1, 1], 1, 5, 1, [[0, 3, 23, prefix], [1, 2, 11, outside], [2, 1, 0, inside], [3, 1, 0, inside]], No, No, finite-tail-insufficient-for-general-convergence)
Inspect only recorded indices. An empty recorded tail passes its bounded check vacuously but still proves no general convergence. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([3, 2, 1, 1], 1, 5, 1, [[0, 3, 23, prefix], [1, 2, 11, outside], [2, 1, 0, inside], [3, 1, 0, inside]], No, No, finite-tail-insufficient-for-general-convergence).
Reviewed step-by-step derivation
Convergence in a metric space · qf-mod23-metricconv-finite-tail-audit · candidate
36. Finite-Tail Success Does Not Certify General Convergence
Use the complete qualified finite metric table=[[0, 11, 23, 36], [11, 0, 12, 25], [23, 12, 0, 13], [36, 25, 13, 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=1/1, cutoff N=2 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.
View answer and explanation
([3, 2, 1, 1], 1, 1, 2, [[0, 3, 25, prefix], [1, 2, 12, prefix], [2, 1, 0, inside], [3, 1, 0, inside]], Yes, No, finite-tail-insufficient-for-general-convergence)
Inspect only recorded indices. An empty recorded tail passes its bounded check vacuously but still proves no general convergence. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([3, 2, 1, 1], 1, 1, 2, [[0, 3, 25, prefix], [1, 2, 12, prefix], [2, 1, 0, inside], [3, 1, 0, inside]], Yes, No, finite-tail-insufficient-for-general-convergence).
Reviewed step-by-step derivation
Convergence in a metric space · qf-mod23-metricconv-finite-tail-audit · candidate
37. Finite-Tail Success Does Not Certify General Convergence
Use the complete qualified finite metric table=[[0, 12, 25, 39], [12, 0, 13, 27], [25, 13, 0, 14], [39, 27, 14, 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=2/1, cutoff N=3 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.
View answer and explanation
([3, 2, 1, 1], 1, 2, 3, [[0, 3, 27, prefix], [1, 2, 13, prefix], [2, 1, 0, prefix], [3, 1, 0, inside]], Yes, No, finite-tail-insufficient-for-general-convergence)
Inspect only recorded indices. An empty recorded tail passes its bounded check vacuously but still proves no general convergence. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([3, 2, 1, 1], 1, 2, 3, [[0, 3, 27, prefix], [1, 2, 13, prefix], [2, 1, 0, prefix], [3, 1, 0, inside]], Yes, No, finite-tail-insufficient-for-general-convergence).
Reviewed step-by-step derivation
Convergence in a metric space · qf-mod23-metricconv-finite-tail-audit · candidate
38. Finite-Tail Success Does Not Certify General Convergence
Use the complete qualified finite metric table=[[0, 13, 27, 42], [13, 0, 14, 29], [27, 14, 0, 15], [42, 29, 15, 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=3/1, 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.
View answer and explanation
([3, 2, 1, 1], 1, 3, 1, [[0, 3, 29, prefix], [1, 2, 14, outside], [2, 1, 0, inside], [3, 1, 0, inside]], No, No, finite-tail-insufficient-for-general-convergence)
Inspect only recorded indices. An empty recorded tail passes its bounded check vacuously but still proves no general convergence. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([3, 2, 1, 1], 1, 3, 1, [[0, 3, 29, prefix], [1, 2, 14, outside], [2, 1, 0, inside], [3, 1, 0, inside]], No, No, finite-tail-insufficient-for-general-convergence).
Reviewed step-by-step derivation
Convergence in a metric space · qf-mod23-metricconv-finite-tail-audit · candidate
39. Finite-Tail Success Does Not Certify General Convergence
Use the complete qualified finite metric table=[[0, 14, 29, 45], [14, 0, 15, 31], [29, 15, 0, 16], [45, 31, 16, 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=2 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.
View answer and explanation
([3, 2, 1, 1], 1, 4, 2, [[0, 3, 31, prefix], [1, 2, 15, prefix], [2, 1, 0, inside], [3, 1, 0, inside]], Yes, No, finite-tail-insufficient-for-general-convergence)
Inspect only recorded indices. An empty recorded tail passes its bounded check vacuously but still proves no general convergence. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([3, 2, 1, 1], 1, 4, 2, [[0, 3, 31, prefix], [1, 2, 15, prefix], [2, 1, 0, inside], [3, 1, 0, inside]], Yes, No, finite-tail-insufficient-for-general-convergence).
Reviewed step-by-step derivation
Convergence in a metric space · qf-mod23-metricconv-finite-tail-audit · candidate
40. Finite-Tail Success Does Not Certify General Convergence
Use the complete qualified finite metric table=[[0, 15, 31, 48], [15, 0, 16, 33], [31, 16, 0, 17], [48, 33, 17, 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=5/1, cutoff N=3 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.
View answer and explanation
([3, 2, 1, 1], 1, 5, 3, [[0, 3, 33, prefix], [1, 2, 16, prefix], [2, 1, 0, prefix], [3, 1, 0, inside]], Yes, No, finite-tail-insufficient-for-general-convergence)
Inspect only recorded indices. An empty recorded tail passes its bounded check vacuously but still proves no general convergence. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([3, 2, 1, 1], 1, 5, 3, [[0, 3, 33, prefix], [1, 2, 16, prefix], [2, 1, 0, prefix], [3, 1, 0, inside]], Yes, No, finite-tail-insufficient-for-general-convergence).
Reviewed step-by-step derivation
Convergence in a metric space · qf-mod23-metricconv-finite-tail-audit · candidate
41. Finite-Tail Success Does Not Certify General Convergence
Use the complete qualified finite metric table=[[0, 16, 33, 51], [16, 0, 17, 35], [33, 17, 0, 18], [51, 35, 18, 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=1/1, 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.
View answer and explanation
([3, 2, 1, 1], 1, 1, 1, [[0, 3, 35, prefix], [1, 2, 17, outside], [2, 1, 0, inside], [3, 1, 0, inside]], No, No, finite-tail-insufficient-for-general-convergence)
Inspect only recorded indices. An empty recorded tail passes its bounded check vacuously but still proves no general convergence. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([3, 2, 1, 1], 1, 1, 1, [[0, 3, 35, prefix], [1, 2, 17, outside], [2, 1, 0, inside], [3, 1, 0, inside]], No, No, finite-tail-insufficient-for-general-convergence).
Reviewed step-by-step derivation
Convergence in a metric space · qf-mod23-metricconv-finite-tail-audit · candidate
42. Finite-Tail Success Does Not Certify General Convergence
Use the complete qualified finite metric table=[[0, 17, 35, 54], [17, 0, 18, 37], [35, 18, 0, 19], [54, 37, 19, 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=2/1, cutoff N=2 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.
View answer and explanation
([3, 2, 1, 1], 1, 2, 2, [[0, 3, 37, prefix], [1, 2, 18, prefix], [2, 1, 0, inside], [3, 1, 0, inside]], Yes, No, finite-tail-insufficient-for-general-convergence)
Inspect only recorded indices. An empty recorded tail passes its bounded check vacuously but still proves no general convergence. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([3, 2, 1, 1], 1, 2, 2, [[0, 3, 37, prefix], [1, 2, 18, prefix], [2, 1, 0, inside], [3, 1, 0, inside]], Yes, No, finite-tail-insufficient-for-general-convergence).
Reviewed step-by-step derivation
Convergence in a metric space · qf-mod23-metricconv-finite-tail-audit · candidate
43. Finite-Tail Success Does Not Certify General Convergence
Use the complete qualified finite metric table=[[0, 18, 37, 57], [18, 0, 19, 39], [37, 19, 0, 20], [57, 39, 20, 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=3/1, cutoff N=3 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.
View answer and explanation
([3, 2, 1, 1], 1, 3, 3, [[0, 3, 39, prefix], [1, 2, 19, prefix], [2, 1, 0, prefix], [3, 1, 0, inside]], Yes, No, finite-tail-insufficient-for-general-convergence)
Inspect only recorded indices. An empty recorded tail passes its bounded check vacuously but still proves no general convergence. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([3, 2, 1, 1], 1, 3, 3, [[0, 3, 39, prefix], [1, 2, 19, prefix], [2, 1, 0, prefix], [3, 1, 0, inside]], Yes, No, finite-tail-insufficient-for-general-convergence).
Reviewed step-by-step derivation
Convergence in a metric space · qf-mod23-metricconv-finite-tail-audit · candidate
44. Finite-Tail Success Does Not Certify General Convergence
Use the complete qualified finite metric table=[[0, 19, 39, 60], [19, 0, 20, 41], [39, 20, 0, 21], [60, 41, 21, 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.
View answer and explanation
([3, 2, 1, 1], 1, 4, 1, [[0, 3, 41, prefix], [1, 2, 20, outside], [2, 1, 0, inside], [3, 1, 0, inside]], No, No, finite-tail-insufficient-for-general-convergence)
Inspect only recorded indices. An empty recorded tail passes its bounded check vacuously but still proves no general convergence. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([3, 2, 1, 1], 1, 4, 1, [[0, 3, 41, prefix], [1, 2, 20, outside], [2, 1, 0, inside], [3, 1, 0, inside]], No, No, finite-tail-insufficient-for-general-convergence).
Reviewed step-by-step derivation
Convergence in a metric space · qf-mod23-metricconv-finite-tail-audit · candidate
45. Finite-Tail Success Does Not Certify General Convergence
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. recorded finite sequence=[3, 2, 1, 1], candidate=1, positive epsilon=5/1, cutoff N=2 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.
View answer and explanation
([3, 2, 1, 1], 1, 5, 2, [[0, 3, 43, prefix], [1, 2, 21, prefix], [2, 1, 0, inside], [3, 1, 0, inside]], Yes, No, finite-tail-insufficient-for-general-convergence)
Inspect only recorded indices. An empty recorded tail passes its bounded check vacuously but still proves no general convergence. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([3, 2, 1, 1], 1, 5, 2, [[0, 3, 43, prefix], [1, 2, 21, prefix], [2, 1, 0, inside], [3, 1, 0, inside]], Yes, No, finite-tail-insufficient-for-general-convergence).
Reviewed step-by-step derivation
Convergence in a metric space · qf-mod23-metricconv-finite-tail-audit · candidate
46. Finite-Tail Success Does Not Certify General Convergence
Use the complete qualified finite metric table=[[0, 21, 43, 66], [21, 0, 22, 45], [43, 22, 0, 23], [66, 45, 23, 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=1/1, cutoff N=3 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.
View answer and explanation
([3, 2, 1, 1], 1, 1, 3, [[0, 3, 45, prefix], [1, 2, 22, prefix], [2, 1, 0, prefix], [3, 1, 0, inside]], Yes, No, finite-tail-insufficient-for-general-convergence)
Inspect only recorded indices. An empty recorded tail passes its bounded check vacuously but still proves no general convergence. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([3, 2, 1, 1], 1, 1, 3, [[0, 3, 45, prefix], [1, 2, 22, prefix], [2, 1, 0, prefix], [3, 1, 0, inside]], Yes, No, finite-tail-insufficient-for-general-convergence).
Reviewed step-by-step derivation
Convergence in a metric space · qf-mod23-metricconv-finite-tail-audit · candidate
47. Finite-Tail Success Does Not Certify General Convergence
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. recorded finite sequence=[3, 2, 1, 1], candidate=1, positive epsilon=2/1, 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.
View answer and explanation
([3, 2, 1, 1], 1, 2, 1, [[0, 3, 47, prefix], [1, 2, 23, outside], [2, 1, 0, inside], [3, 1, 0, inside]], No, No, finite-tail-insufficient-for-general-convergence)
Inspect only recorded indices. An empty recorded tail passes its bounded check vacuously but still proves no general convergence. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([3, 2, 1, 1], 1, 2, 1, [[0, 3, 47, prefix], [1, 2, 23, outside], [2, 1, 0, inside], [3, 1, 0, inside]], No, No, finite-tail-insufficient-for-general-convergence).
Reviewed step-by-step derivation
Convergence in a metric space · qf-mod23-metricconv-finite-tail-audit · candidate
48. Finite-Tail Success Does Not Certify General Convergence
Use the complete qualified finite metric table=[[0, 23, 47, 72], [23, 0, 24, 49], [47, 24, 0, 25], [72, 49, 25, 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=3/1, cutoff N=2 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.
View answer and explanation
([3, 2, 1, 1], 1, 3, 2, [[0, 3, 49, prefix], [1, 2, 24, prefix], [2, 1, 0, inside], [3, 1, 0, inside]], Yes, No, finite-tail-insufficient-for-general-convergence)
Inspect only recorded indices. An empty recorded tail passes its bounded check vacuously but still proves no general convergence. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([3, 2, 1, 1], 1, 3, 2, [[0, 3, 49, prefix], [1, 2, 24, prefix], [2, 1, 0, inside], [3, 1, 0, inside]], Yes, No, finite-tail-insufficient-for-general-convergence).
Reviewed step-by-step derivation
Convergence in a metric space · qf-mod23-metricconv-finite-tail-audit · candidate
49. Finite-Tail Success Does Not Certify General Convergence
Use the complete qualified finite metric table=[[0, 24, 49, 75], [24, 0, 25, 51], [49, 25, 0, 26], [75, 51, 26, 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=3 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.
View answer and explanation
([3, 2, 1, 1], 1, 4, 3, [[0, 3, 51, prefix], [1, 2, 25, prefix], [2, 1, 0, prefix], [3, 1, 0, inside]], Yes, No, finite-tail-insufficient-for-general-convergence)
Inspect only recorded indices. An empty recorded tail passes its bounded check vacuously but still proves no general convergence. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([3, 2, 1, 1], 1, 4, 3, [[0, 3, 51, prefix], [1, 2, 25, prefix], [2, 1, 0, prefix], [3, 1, 0, inside]], Yes, No, finite-tail-insufficient-for-general-convergence).
Reviewed step-by-step derivation
Convergence in a metric space · qf-mod23-metricconv-finite-tail-audit · candidate
50. Finite-Tail Success Does Not Certify General Convergence
Use the complete qualified finite metric table=[[0, 25, 51, 78], [25, 0, 26, 53], [51, 26, 0, 27], [78, 53, 27, 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=5/1, 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.
View answer and explanation
([3, 2, 1, 1], 1, 5, 1, [[0, 3, 53, prefix], [1, 2, 26, outside], [2, 1, 0, inside], [3, 1, 0, inside]], No, No, finite-tail-insufficient-for-general-convergence)
Inspect only recorded indices. An empty recorded tail passes its bounded check vacuously but still proves no general convergence. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([3, 2, 1, 1], 1, 5, 1, [[0, 3, 53, prefix], [1, 2, 26, outside], [2, 1, 0, inside], [3, 1, 0, inside]], No, No, finite-tail-insufficient-for-general-convergence).
Reviewed step-by-step derivation
Convergence in a metric space · qf-mod23-metricconv-finite-tail-audit · candidate
51. An Explicit Infinite Constant Tail Supplies a Quantified Witness
Use the complete qualified finite metric table=[[0, 1, 3, 6], [1, 0, 2, 5], [3, 2, 0, 3], [6, 5, 3, 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. prefix=[0, 1]; every later term equals tail=2, candidate=3. This is an explicitly specified infinite sequence, not an inference from a finite pattern. Give (prefix,tail,candidate,prefix length,converges?,witness). For equality use for any epsilon>0 choose N=prefix length; otherwise use tail-value-differs-from-candidate.
View answer and explanation
([0, 1], 2, 3, 2, No, tail-value-differs-from-candidate)
When the candidate equals the tail, every later distance is zero for every positive epsilon. Otherwise metric separation gives a positive tail distance, so the candidate is not a limit. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([0, 1], 2, 3, 2, No, tail-value-differs-from-candidate).
Reviewed step-by-step derivation
Convergence in a metric space · qf-mod23-metricconv-ultimately-constant-limit · candidate
52. An Explicit Infinite Constant Tail Supplies a Quantified Witness
Use the complete qualified finite metric table=[[0, 2, 5, 9], [2, 0, 3, 7], [5, 3, 0, 4], [9, 7, 4, 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. prefix=[1, 2]; every later term equals tail=3, candidate=3. This is an explicitly specified infinite sequence, not an inference from a finite pattern. Give (prefix,tail,candidate,prefix length,converges?,witness). For equality use for any epsilon>0 choose N=prefix length; otherwise use tail-value-differs-from-candidate.
View answer and explanation
([1, 2], 3, 3, 2, Yes, for any epsilon>0 choose N=prefix length)
When the candidate equals the tail, every later distance is zero for every positive epsilon. Otherwise metric separation gives a positive tail distance, so the candidate is not a limit. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([1, 2], 3, 3, 2, Yes, for any epsilon>0 choose N=prefix length).
Reviewed step-by-step derivation
Convergence in a metric space · qf-mod23-metricconv-ultimately-constant-limit · candidate
53. An Explicit Infinite Constant Tail Supplies a Quantified Witness
Use the complete qualified finite metric table=[[0, 3, 7, 12], [3, 0, 4, 9], [7, 4, 0, 5], [12, 9, 5, 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. prefix=[2, 3]; every later term equals tail=0, candidate=0. This is an explicitly specified infinite sequence, not an inference from a finite pattern. Give (prefix,tail,candidate,prefix length,converges?,witness). For equality use for any epsilon>0 choose N=prefix length; otherwise use tail-value-differs-from-candidate.
View answer and explanation
([2, 3], 0, 0, 2, Yes, for any epsilon>0 choose N=prefix length)
When the candidate equals the tail, every later distance is zero for every positive epsilon. Otherwise metric separation gives a positive tail distance, so the candidate is not a limit. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([2, 3], 0, 0, 2, Yes, for any epsilon>0 choose N=prefix length).
Reviewed step-by-step derivation
Convergence in a metric space · qf-mod23-metricconv-ultimately-constant-limit · candidate
54. An Explicit Infinite Constant Tail Supplies a Quantified Witness
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. prefix=[3, 0]; every later term equals tail=1, candidate=2. This is an explicitly specified infinite sequence, not an inference from a finite pattern. Give (prefix,tail,candidate,prefix length,converges?,witness). For equality use for any epsilon>0 choose N=prefix length; otherwise use tail-value-differs-from-candidate.
View answer and explanation
([3, 0], 1, 2, 2, No, tail-value-differs-from-candidate)
When the candidate equals the tail, every later distance is zero for every positive epsilon. Otherwise metric separation gives a positive tail distance, so the candidate is not a limit. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([3, 0], 1, 2, 2, No, tail-value-differs-from-candidate).
Reviewed step-by-step derivation
Convergence in a metric space · qf-mod23-metricconv-ultimately-constant-limit · candidate
55. An Explicit Infinite Constant Tail Supplies a Quantified Witness
Use the complete qualified finite metric table=[[0, 5, 11, 18], [5, 0, 6, 13], [11, 6, 0, 7], [18, 13, 7, 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. prefix=[0, 1]; every later term equals tail=2, candidate=2. This is an explicitly specified infinite sequence, not an inference from a finite pattern. Give (prefix,tail,candidate,prefix length,converges?,witness). For equality use for any epsilon>0 choose N=prefix length; otherwise use tail-value-differs-from-candidate.
View answer and explanation
([0, 1], 2, 2, 2, Yes, for any epsilon>0 choose N=prefix length)
When the candidate equals the tail, every later distance is zero for every positive epsilon. Otherwise metric separation gives a positive tail distance, so the candidate is not a limit. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([0, 1], 2, 2, 2, Yes, for any epsilon>0 choose N=prefix length).
Reviewed step-by-step derivation
Convergence in a metric space · qf-mod23-metricconv-ultimately-constant-limit · candidate
56. An Explicit Infinite Constant Tail Supplies a Quantified Witness
Use the complete qualified finite metric table=[[0, 6, 13, 21], [6, 0, 7, 15], [13, 7, 0, 8], [21, 15, 8, 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. prefix=[1, 2]; every later term equals tail=3, candidate=3. This is an explicitly specified infinite sequence, not an inference from a finite pattern. Give (prefix,tail,candidate,prefix length,converges?,witness). For equality use for any epsilon>0 choose N=prefix length; otherwise use tail-value-differs-from-candidate.
View answer and explanation
([1, 2], 3, 3, 2, Yes, for any epsilon>0 choose N=prefix length)
When the candidate equals the tail, every later distance is zero for every positive epsilon. Otherwise metric separation gives a positive tail distance, so the candidate is not a limit. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([1, 2], 3, 3, 2, Yes, for any epsilon>0 choose N=prefix length).
Reviewed step-by-step derivation
Convergence in a metric space · qf-mod23-metricconv-ultimately-constant-limit · candidate
57. An Explicit Infinite Constant Tail Supplies a Quantified Witness
Use the complete qualified finite metric table=[[0, 7, 15, 24], [7, 0, 8, 17], [15, 8, 0, 9], [24, 17, 9, 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. prefix=[2, 3]; every later term equals tail=0, candidate=1. This is an explicitly specified infinite sequence, not an inference from a finite pattern. Give (prefix,tail,candidate,prefix length,converges?,witness). For equality use for any epsilon>0 choose N=prefix length; otherwise use tail-value-differs-from-candidate.
View answer and explanation
([2, 3], 0, 1, 2, No, tail-value-differs-from-candidate)
When the candidate equals the tail, every later distance is zero for every positive epsilon. Otherwise metric separation gives a positive tail distance, so the candidate is not a limit. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([2, 3], 0, 1, 2, No, tail-value-differs-from-candidate).
Reviewed step-by-step derivation
Convergence in a metric space · qf-mod23-metricconv-ultimately-constant-limit · candidate
58. An Explicit Infinite Constant Tail Supplies a Quantified Witness
Use the complete qualified finite metric table=[[0, 8, 17, 27], [8, 0, 9, 19], [17, 9, 0, 10], [27, 19, 10, 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. prefix=[3, 0]; every later term equals tail=1, candidate=1. This is an explicitly specified infinite sequence, not an inference from a finite pattern. Give (prefix,tail,candidate,prefix length,converges?,witness). For equality use for any epsilon>0 choose N=prefix length; otherwise use tail-value-differs-from-candidate.
View answer and explanation
([3, 0], 1, 1, 2, Yes, for any epsilon>0 choose N=prefix length)
When the candidate equals the tail, every later distance is zero for every positive epsilon. Otherwise metric separation gives a positive tail distance, so the candidate is not a limit. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([3, 0], 1, 1, 2, Yes, for any epsilon>0 choose N=prefix length).
Reviewed step-by-step derivation
Convergence in a metric space · qf-mod23-metricconv-ultimately-constant-limit · candidate
59. An Explicit Infinite Constant Tail Supplies a Quantified Witness
Use the complete qualified finite metric table=[[0, 9, 19, 30], [9, 0, 10, 21], [19, 10, 0, 11], [30, 21, 11, 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. prefix=[0, 1]; every later term equals tail=2, candidate=2. This is an explicitly specified infinite sequence, not an inference from a finite pattern. Give (prefix,tail,candidate,prefix length,converges?,witness). For equality use for any epsilon>0 choose N=prefix length; otherwise use tail-value-differs-from-candidate.
View answer and explanation
([0, 1], 2, 2, 2, Yes, for any epsilon>0 choose N=prefix length)
When the candidate equals the tail, every later distance is zero for every positive epsilon. Otherwise metric separation gives a positive tail distance, so the candidate is not a limit. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([0, 1], 2, 2, 2, Yes, for any epsilon>0 choose N=prefix length).
Reviewed step-by-step derivation
Convergence in a metric space · qf-mod23-metricconv-ultimately-constant-limit · candidate
60. An Explicit Infinite Constant Tail Supplies a Quantified Witness
Use the complete qualified finite metric table=[[0, 10, 21, 33], [10, 0, 11, 23], [21, 11, 0, 12], [33, 23, 12, 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. prefix=[1, 2]; every later term equals tail=3, candidate=0. This is an explicitly specified infinite sequence, not an inference from a finite pattern. Give (prefix,tail,candidate,prefix length,converges?,witness). For equality use for any epsilon>0 choose N=prefix length; otherwise use tail-value-differs-from-candidate.
View answer and explanation
([1, 2], 3, 0, 2, No, tail-value-differs-from-candidate)
When the candidate equals the tail, every later distance is zero for every positive epsilon. Otherwise metric separation gives a positive tail distance, so the candidate is not a limit. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([1, 2], 3, 0, 2, No, tail-value-differs-from-candidate).
Reviewed step-by-step derivation
Convergence in a metric space · qf-mod23-metricconv-ultimately-constant-limit · candidate
61. An Explicit Infinite Constant Tail Supplies a Quantified Witness
Use the complete qualified finite metric table=[[0, 11, 23, 36], [11, 0, 12, 25], [23, 12, 0, 13], [36, 25, 13, 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. prefix=[2, 3]; every later term equals tail=0, candidate=0. This is an explicitly specified infinite sequence, not an inference from a finite pattern. Give (prefix,tail,candidate,prefix length,converges?,witness). For equality use for any epsilon>0 choose N=prefix length; otherwise use tail-value-differs-from-candidate.
View answer and explanation
([2, 3], 0, 0, 2, Yes, for any epsilon>0 choose N=prefix length)
When the candidate equals the tail, every later distance is zero for every positive epsilon. Otherwise metric separation gives a positive tail distance, so the candidate is not a limit. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([2, 3], 0, 0, 2, Yes, for any epsilon>0 choose N=prefix length).
Reviewed step-by-step derivation
Convergence in a metric space · qf-mod23-metricconv-ultimately-constant-limit · candidate
62. An Explicit Infinite Constant Tail Supplies a Quantified Witness
Use the complete qualified finite metric table=[[0, 12, 25, 39], [12, 0, 13, 27], [25, 13, 0, 14], [39, 27, 14, 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. prefix=[3, 0]; every later term equals tail=1, candidate=1. This is an explicitly specified infinite sequence, not an inference from a finite pattern. Give (prefix,tail,candidate,prefix length,converges?,witness). For equality use for any epsilon>0 choose N=prefix length; otherwise use tail-value-differs-from-candidate.
View answer and explanation
([3, 0], 1, 1, 2, Yes, for any epsilon>0 choose N=prefix length)
When the candidate equals the tail, every later distance is zero for every positive epsilon. Otherwise metric separation gives a positive tail distance, so the candidate is not a limit. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([3, 0], 1, 1, 2, Yes, for any epsilon>0 choose N=prefix length).
Reviewed step-by-step derivation
Convergence in a metric space · qf-mod23-metricconv-ultimately-constant-limit · candidate
63. An Explicit Infinite Constant Tail Supplies a Quantified Witness
Use the complete qualified finite metric table=[[0, 13, 27, 42], [13, 0, 14, 29], [27, 14, 0, 15], [42, 29, 15, 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. prefix=[0, 1]; every later term equals tail=2, candidate=3. This is an explicitly specified infinite sequence, not an inference from a finite pattern. Give (prefix,tail,candidate,prefix length,converges?,witness). For equality use for any epsilon>0 choose N=prefix length; otherwise use tail-value-differs-from-candidate.
View answer and explanation
([0, 1], 2, 3, 2, No, tail-value-differs-from-candidate)
When the candidate equals the tail, every later distance is zero for every positive epsilon. Otherwise metric separation gives a positive tail distance, so the candidate is not a limit. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([0, 1], 2, 3, 2, No, tail-value-differs-from-candidate).
Reviewed step-by-step derivation
Convergence in a metric space · qf-mod23-metricconv-ultimately-constant-limit · candidate
64. An Explicit Infinite Constant Tail Supplies a Quantified Witness
Use the complete qualified finite metric table=[[0, 14, 29, 45], [14, 0, 15, 31], [29, 15, 0, 16], [45, 31, 16, 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. prefix=[1, 2]; every later term equals tail=3, candidate=3. This is an explicitly specified infinite sequence, not an inference from a finite pattern. Give (prefix,tail,candidate,prefix length,converges?,witness). For equality use for any epsilon>0 choose N=prefix length; otherwise use tail-value-differs-from-candidate.
View answer and explanation
([1, 2], 3, 3, 2, Yes, for any epsilon>0 choose N=prefix length)
When the candidate equals the tail, every later distance is zero for every positive epsilon. Otherwise metric separation gives a positive tail distance, so the candidate is not a limit. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([1, 2], 3, 3, 2, Yes, for any epsilon>0 choose N=prefix length).
Reviewed step-by-step derivation
Convergence in a metric space · qf-mod23-metricconv-ultimately-constant-limit · candidate
65. An Explicit Infinite Constant Tail Supplies a Quantified Witness
Use the complete qualified finite metric table=[[0, 15, 31, 48], [15, 0, 16, 33], [31, 16, 0, 17], [48, 33, 17, 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. prefix=[2, 3]; every later term equals tail=0, candidate=0. This is an explicitly specified infinite sequence, not an inference from a finite pattern. Give (prefix,tail,candidate,prefix length,converges?,witness). For equality use for any epsilon>0 choose N=prefix length; otherwise use tail-value-differs-from-candidate.
View answer and explanation
([2, 3], 0, 0, 2, Yes, for any epsilon>0 choose N=prefix length)
When the candidate equals the tail, every later distance is zero for every positive epsilon. Otherwise metric separation gives a positive tail distance, so the candidate is not a limit. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([2, 3], 0, 0, 2, Yes, for any epsilon>0 choose N=prefix length).
Reviewed step-by-step derivation
Convergence in a metric space · qf-mod23-metricconv-ultimately-constant-limit · candidate
66. An Explicit Infinite Constant Tail Supplies a Quantified Witness
Use the complete qualified finite metric table=[[0, 16, 33, 51], [16, 0, 17, 35], [33, 17, 0, 18], [51, 35, 18, 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. prefix=[3, 0]; every later term equals tail=1, candidate=2. This is an explicitly specified infinite sequence, not an inference from a finite pattern. Give (prefix,tail,candidate,prefix length,converges?,witness). For equality use for any epsilon>0 choose N=prefix length; otherwise use tail-value-differs-from-candidate.
View answer and explanation
([3, 0], 1, 2, 2, No, tail-value-differs-from-candidate)
When the candidate equals the tail, every later distance is zero for every positive epsilon. Otherwise metric separation gives a positive tail distance, so the candidate is not a limit. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([3, 0], 1, 2, 2, No, tail-value-differs-from-candidate).
Reviewed step-by-step derivation
Convergence in a metric space · qf-mod23-metricconv-ultimately-constant-limit · candidate
67. An Explicit Infinite Constant Tail Supplies a Quantified Witness
Use the complete qualified finite metric table=[[0, 17, 35, 54], [17, 0, 18, 37], [35, 18, 0, 19], [54, 37, 19, 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. prefix=[0, 1]; every later term equals tail=2, candidate=2. This is an explicitly specified infinite sequence, not an inference from a finite pattern. Give (prefix,tail,candidate,prefix length,converges?,witness). For equality use for any epsilon>0 choose N=prefix length; otherwise use tail-value-differs-from-candidate.
View answer and explanation
([0, 1], 2, 2, 2, Yes, for any epsilon>0 choose N=prefix length)
When the candidate equals the tail, every later distance is zero for every positive epsilon. Otherwise metric separation gives a positive tail distance, so the candidate is not a limit. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([0, 1], 2, 2, 2, Yes, for any epsilon>0 choose N=prefix length).
Reviewed step-by-step derivation
Convergence in a metric space · qf-mod23-metricconv-ultimately-constant-limit · candidate
68. An Explicit Infinite Constant Tail Supplies a Quantified Witness
Use the complete qualified finite metric table=[[0, 18, 37, 57], [18, 0, 19, 39], [37, 19, 0, 20], [57, 39, 20, 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. prefix=[1, 2]; every later term equals tail=3, candidate=3. This is an explicitly specified infinite sequence, not an inference from a finite pattern. Give (prefix,tail,candidate,prefix length,converges?,witness). For equality use for any epsilon>0 choose N=prefix length; otherwise use tail-value-differs-from-candidate.
View answer and explanation
([1, 2], 3, 3, 2, Yes, for any epsilon>0 choose N=prefix length)
When the candidate equals the tail, every later distance is zero for every positive epsilon. Otherwise metric separation gives a positive tail distance, so the candidate is not a limit. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([1, 2], 3, 3, 2, Yes, for any epsilon>0 choose N=prefix length).
Reviewed step-by-step derivation
Convergence in a metric space · qf-mod23-metricconv-ultimately-constant-limit · candidate
69. An Explicit Infinite Constant Tail Supplies a Quantified Witness
Use the complete qualified finite metric table=[[0, 19, 39, 60], [19, 0, 20, 41], [39, 20, 0, 21], [60, 41, 21, 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. prefix=[2, 3]; every later term equals tail=0, candidate=1. This is an explicitly specified infinite sequence, not an inference from a finite pattern. Give (prefix,tail,candidate,prefix length,converges?,witness). For equality use for any epsilon>0 choose N=prefix length; otherwise use tail-value-differs-from-candidate.
View answer and explanation
([2, 3], 0, 1, 2, No, tail-value-differs-from-candidate)
When the candidate equals the tail, every later distance is zero for every positive epsilon. Otherwise metric separation gives a positive tail distance, so the candidate is not a limit. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([2, 3], 0, 1, 2, No, tail-value-differs-from-candidate).
Reviewed step-by-step derivation
Convergence in a metric space · qf-mod23-metricconv-ultimately-constant-limit · candidate
70. An Explicit Infinite Constant Tail Supplies a Quantified Witness
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. prefix=[3, 0]; every later term equals tail=1, candidate=1. This is an explicitly specified infinite sequence, not an inference from a finite pattern. Give (prefix,tail,candidate,prefix length,converges?,witness). For equality use for any epsilon>0 choose N=prefix length; otherwise use tail-value-differs-from-candidate.
View answer and explanation
([3, 0], 1, 1, 2, Yes, for any epsilon>0 choose N=prefix length)
When the candidate equals the tail, every later distance is zero for every positive epsilon. Otherwise metric separation gives a positive tail distance, so the candidate is not a limit. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([3, 0], 1, 1, 2, Yes, for any epsilon>0 choose N=prefix length).
Reviewed step-by-step derivation
Convergence in a metric space · qf-mod23-metricconv-ultimately-constant-limit · candidate
71. An Explicit Infinite Constant Tail Supplies a Quantified Witness
Use the complete qualified finite metric table=[[0, 21, 43, 66], [21, 0, 22, 45], [43, 22, 0, 23], [66, 45, 23, 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. prefix=[0, 1]; every later term equals tail=2, candidate=2. This is an explicitly specified infinite sequence, not an inference from a finite pattern. Give (prefix,tail,candidate,prefix length,converges?,witness). For equality use for any epsilon>0 choose N=prefix length; otherwise use tail-value-differs-from-candidate.
View answer and explanation
([0, 1], 2, 2, 2, Yes, for any epsilon>0 choose N=prefix length)
When the candidate equals the tail, every later distance is zero for every positive epsilon. Otherwise metric separation gives a positive tail distance, so the candidate is not a limit. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([0, 1], 2, 2, 2, Yes, for any epsilon>0 choose N=prefix length).
Reviewed step-by-step derivation
Convergence in a metric space · qf-mod23-metricconv-ultimately-constant-limit · candidate
72. An Explicit Infinite Constant Tail Supplies a Quantified Witness
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. prefix=[1, 2]; every later term equals tail=3, candidate=0. This is an explicitly specified infinite sequence, not an inference from a finite pattern. Give (prefix,tail,candidate,prefix length,converges?,witness). For equality use for any epsilon>0 choose N=prefix length; otherwise use tail-value-differs-from-candidate.
View answer and explanation
([1, 2], 3, 0, 2, No, tail-value-differs-from-candidate)
When the candidate equals the tail, every later distance is zero for every positive epsilon. Otherwise metric separation gives a positive tail distance, so the candidate is not a limit. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([1, 2], 3, 0, 2, No, tail-value-differs-from-candidate).
Reviewed step-by-step derivation
Convergence in a metric space · qf-mod23-metricconv-ultimately-constant-limit · candidate
73. An Explicit Infinite Constant Tail Supplies a Quantified Witness
Use the complete qualified finite metric table=[[0, 23, 47, 72], [23, 0, 24, 49], [47, 24, 0, 25], [72, 49, 25, 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. prefix=[2, 3]; every later term equals tail=0, candidate=0. This is an explicitly specified infinite sequence, not an inference from a finite pattern. Give (prefix,tail,candidate,prefix length,converges?,witness). For equality use for any epsilon>0 choose N=prefix length; otherwise use tail-value-differs-from-candidate.
View answer and explanation
([2, 3], 0, 0, 2, Yes, for any epsilon>0 choose N=prefix length)
When the candidate equals the tail, every later distance is zero for every positive epsilon. Otherwise metric separation gives a positive tail distance, so the candidate is not a limit. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([2, 3], 0, 0, 2, Yes, for any epsilon>0 choose N=prefix length).
Reviewed step-by-step derivation
Convergence in a metric space · qf-mod23-metricconv-ultimately-constant-limit · candidate
74. An Explicit Infinite Constant Tail Supplies a Quantified Witness
Use the complete qualified finite metric table=[[0, 24, 49, 75], [24, 0, 25, 51], [49, 25, 0, 26], [75, 51, 26, 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. prefix=[3, 0]; every later term equals tail=1, candidate=1. This is an explicitly specified infinite sequence, not an inference from a finite pattern. Give (prefix,tail,candidate,prefix length,converges?,witness). For equality use for any epsilon>0 choose N=prefix length; otherwise use tail-value-differs-from-candidate.
View answer and explanation
([3, 0], 1, 1, 2, Yes, for any epsilon>0 choose N=prefix length)
When the candidate equals the tail, every later distance is zero for every positive epsilon. Otherwise metric separation gives a positive tail distance, so the candidate is not a limit. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([3, 0], 1, 1, 2, Yes, for any epsilon>0 choose N=prefix length).
Reviewed step-by-step derivation
Convergence in a metric space · qf-mod23-metricconv-ultimately-constant-limit · candidate
75. An Explicit Infinite Constant Tail Supplies a Quantified Witness
Use the complete qualified finite metric table=[[0, 25, 51, 78], [25, 0, 26, 53], [51, 26, 0, 27], [78, 53, 27, 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. prefix=[0, 1]; every later term equals tail=2, candidate=3. This is an explicitly specified infinite sequence, not an inference from a finite pattern. Give (prefix,tail,candidate,prefix length,converges?,witness). For equality use for any epsilon>0 choose N=prefix length; otherwise use tail-value-differs-from-candidate.
View answer and explanation
([0, 1], 2, 3, 2, No, tail-value-differs-from-candidate)
When the candidate equals the tail, every later distance is zero for every positive epsilon. Otherwise metric separation gives a positive tail distance, so the candidate is not a limit. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([0, 1], 2, 3, 2, No, tail-value-differs-from-candidate).
Reviewed step-by-step derivation
Convergence in a metric space · qf-mod23-metricconv-ultimately-constant-limit · candidate
76. A Conditional Uniqueness Check Is Not Two Convergence Claims
Use the complete qualified finite metric table=[[0, 1, 3, 6], [1, 0, 2, 5], [3, 2, 0, 3], [6, 5, 3, 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=[0, 1] and constant tail=2; candidates a=2, 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?).
View answer and explanation
([0, 1], 2, 2, 2, 0, Yes, Yes, Yes)
Check each candidate against the declared constant tail. The final implication may be true when its antecedent is false; it does not assert both candidates converge. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([0, 1], 2, 2, 2, 0, Yes, Yes, Yes).
Reviewed step-by-step derivation
Convergence in a metric space · qf-mod23-metricconv-limit-uniqueness-audit · candidate
77. A Conditional Uniqueness Check Is Not Two Convergence Claims
Use the complete qualified finite metric table=[[0, 2, 5, 9], [2, 0, 3, 7], [5, 3, 0, 4], [9, 7, 4, 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=[1, 2] and constant tail=3; candidates a=3, b=0. Give (prefix,tail,a,b,d(a,b),a is a limit?,b is a limit?,if both are limits then a=b?).
View answer and explanation
([1, 2], 3, 3, 0, 9, Yes, No, Yes)
Check each candidate against the declared constant tail. The final implication may be true when its antecedent is false; it does not assert both candidates converge. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([1, 2], 3, 3, 0, 9, Yes, No, Yes).
Reviewed step-by-step derivation
Convergence in a metric space · qf-mod23-metricconv-limit-uniqueness-audit · candidate
78. A Conditional Uniqueness Check Is Not Two Convergence Claims
Use the complete qualified finite metric table=[[0, 3, 7, 12], [3, 0, 4, 9], [7, 4, 0, 5], [12, 9, 5, 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=[2, 3] and constant tail=0; candidates a=0, b=1. Give (prefix,tail,a,b,d(a,b),a is a limit?,b is a limit?,if both are limits then a=b?).
View answer and explanation
([2, 3], 0, 0, 1, 3, Yes, No, Yes)
Check each candidate against the declared constant tail. The final implication may be true when its antecedent is false; it does not assert both candidates converge. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([2, 3], 0, 0, 1, 3, Yes, No, Yes).
Reviewed step-by-step derivation
Convergence in a metric space · qf-mod23-metricconv-limit-uniqueness-audit · candidate
79. A Conditional Uniqueness Check Is Not Two Convergence Claims
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. infinite sequence has prefix=[3, 0] and constant tail=1; candidates a=1, b=1. Give (prefix,tail,a,b,d(a,b),a is a limit?,b is a limit?,if both are limits then a=b?).
View answer and explanation
([3, 0], 1, 1, 1, 0, Yes, Yes, Yes)
Check each candidate against the declared constant tail. The final implication may be true when its antecedent is false; it does not assert both candidates converge. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([3, 0], 1, 1, 1, 0, Yes, Yes, Yes).
Reviewed step-by-step derivation
Convergence in a metric space · qf-mod23-metricconv-limit-uniqueness-audit · candidate
80. A Conditional Uniqueness Check Is Not Two Convergence Claims
Use the complete qualified finite metric table=[[0, 5, 11, 18], [5, 0, 6, 13], [11, 6, 0, 7], [18, 13, 7, 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=[0, 1] and constant tail=2; candidates a=2, b=3. Give (prefix,tail,a,b,d(a,b),a is a limit?,b is a limit?,if both are limits then a=b?).
View answer and explanation
([0, 1], 2, 2, 3, 7, Yes, No, Yes)
Check each candidate against the declared constant tail. The final implication may be true when its antecedent is false; it does not assert both candidates converge. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([0, 1], 2, 2, 3, 7, Yes, No, Yes).
Reviewed step-by-step derivation
Convergence in a metric space · qf-mod23-metricconv-limit-uniqueness-audit · candidate
81. A Conditional Uniqueness Check Is Not Two Convergence Claims
Use the complete qualified finite metric table=[[0, 6, 13, 21], [6, 0, 7, 15], [13, 7, 0, 8], [21, 15, 8, 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=[1, 2] and constant tail=3; candidates a=3, b=0. Give (prefix,tail,a,b,d(a,b),a is a limit?,b is a limit?,if both are limits then a=b?).
View answer and explanation
([1, 2], 3, 3, 0, 21, Yes, No, Yes)
Check each candidate against the declared constant tail. The final implication may be true when its antecedent is false; it does not assert both candidates converge. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([1, 2], 3, 3, 0, 21, Yes, No, Yes).
Reviewed step-by-step derivation
Convergence in a metric space · qf-mod23-metricconv-limit-uniqueness-audit · candidate
82. A Conditional Uniqueness Check Is Not Two Convergence Claims
Use the complete qualified finite metric table=[[0, 7, 15, 24], [7, 0, 8, 17], [15, 8, 0, 9], [24, 17, 9, 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=[2, 3] and constant tail=0; candidates a=0, b=0. Give (prefix,tail,a,b,d(a,b),a is a limit?,b is a limit?,if both are limits then a=b?).
View answer and explanation
([2, 3], 0, 0, 0, 0, Yes, Yes, Yes)
Check each candidate against the declared constant tail. The final implication may be true when its antecedent is false; it does not assert both candidates converge. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([2, 3], 0, 0, 0, 0, Yes, Yes, Yes).
Reviewed step-by-step derivation
Convergence in a metric space · qf-mod23-metricconv-limit-uniqueness-audit · candidate
83. A Conditional Uniqueness Check Is Not Two Convergence Claims
Use the complete qualified finite metric table=[[0, 8, 17, 27], [8, 0, 9, 19], [17, 9, 0, 10], [27, 19, 10, 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?).
View answer and explanation
([3, 0], 1, 1, 2, 9, Yes, No, Yes)
Check each candidate against the declared constant tail. The final implication may be true when its antecedent is false; it does not assert both candidates converge. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([3, 0], 1, 1, 2, 9, Yes, No, Yes).
Reviewed step-by-step derivation
Convergence in a metric space · qf-mod23-metricconv-limit-uniqueness-audit · candidate
84. A Conditional Uniqueness Check Is Not Two Convergence Claims
Use the complete qualified finite metric table=[[0, 9, 19, 30], [9, 0, 10, 21], [19, 10, 0, 11], [30, 21, 11, 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=[0, 1] and constant tail=2; candidates a=2, b=3. Give (prefix,tail,a,b,d(a,b),a is a limit?,b is a limit?,if both are limits then a=b?).
View answer and explanation
([0, 1], 2, 2, 3, 11, Yes, No, Yes)
Check each candidate against the declared constant tail. The final implication may be true when its antecedent is false; it does not assert both candidates converge. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([0, 1], 2, 2, 3, 11, Yes, No, Yes).
Reviewed step-by-step derivation
Convergence in a metric space · qf-mod23-metricconv-limit-uniqueness-audit · candidate
85. A Conditional Uniqueness Check Is Not Two Convergence Claims
Use the complete qualified finite metric table=[[0, 10, 21, 33], [10, 0, 11, 23], [21, 11, 0, 12], [33, 23, 12, 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=[1, 2] and constant tail=3; candidates a=3, b=3. Give (prefix,tail,a,b,d(a,b),a is a limit?,b is a limit?,if both are limits then a=b?).
View answer and explanation
([1, 2], 3, 3, 3, 0, Yes, Yes, Yes)
Check each candidate against the declared constant tail. The final implication may be true when its antecedent is false; it does not assert both candidates converge. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([1, 2], 3, 3, 3, 0, Yes, Yes, Yes).
Reviewed step-by-step derivation
Convergence in a metric space · qf-mod23-metricconv-limit-uniqueness-audit · candidate
86. A Conditional Uniqueness Check Is Not Two Convergence Claims
Use the complete qualified finite metric table=[[0, 11, 23, 36], [11, 0, 12, 25], [23, 12, 0, 13], [36, 25, 13, 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=[2, 3] and constant tail=0; candidates a=0, b=1. Give (prefix,tail,a,b,d(a,b),a is a limit?,b is a limit?,if both are limits then a=b?).
View answer and explanation
([2, 3], 0, 0, 1, 11, Yes, No, Yes)
Check each candidate against the declared constant tail. The final implication may be true when its antecedent is false; it does not assert both candidates converge. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([2, 3], 0, 0, 1, 11, Yes, No, Yes).
Reviewed step-by-step derivation
Convergence in a metric space · qf-mod23-metricconv-limit-uniqueness-audit · candidate
87. A Conditional Uniqueness Check Is Not Two Convergence Claims
Use the complete qualified finite metric table=[[0, 12, 25, 39], [12, 0, 13, 27], [25, 13, 0, 14], [39, 27, 14, 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?).
View answer and explanation
([3, 0], 1, 1, 2, 13, Yes, No, Yes)
Check each candidate against the declared constant tail. The final implication may be true when its antecedent is false; it does not assert both candidates converge. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([3, 0], 1, 1, 2, 13, Yes, No, Yes).
Reviewed step-by-step derivation
Convergence in a metric space · qf-mod23-metricconv-limit-uniqueness-audit · candidate
88. A Conditional Uniqueness Check Is Not Two Convergence Claims
Use the complete qualified finite metric table=[[0, 13, 27, 42], [13, 0, 14, 29], [27, 14, 0, 15], [42, 29, 15, 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=[0, 1] and constant tail=2; candidates a=2, 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?).
View answer and explanation
([0, 1], 2, 2, 2, 0, Yes, Yes, Yes)
Check each candidate against the declared constant tail. The final implication may be true when its antecedent is false; it does not assert both candidates converge. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([0, 1], 2, 2, 2, 0, Yes, Yes, Yes).
Reviewed step-by-step derivation
Convergence in a metric space · qf-mod23-metricconv-limit-uniqueness-audit · candidate
89. A Conditional Uniqueness Check Is Not Two Convergence Claims
Use the complete qualified finite metric table=[[0, 14, 29, 45], [14, 0, 15, 31], [29, 15, 0, 16], [45, 31, 16, 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=[1, 2] and constant tail=3; candidates a=3, b=0. Give (prefix,tail,a,b,d(a,b),a is a limit?,b is a limit?,if both are limits then a=b?).
View answer and explanation
([1, 2], 3, 3, 0, 45, Yes, No, Yes)
Check each candidate against the declared constant tail. The final implication may be true when its antecedent is false; it does not assert both candidates converge. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([1, 2], 3, 3, 0, 45, Yes, No, Yes).
Reviewed step-by-step derivation
Convergence in a metric space · qf-mod23-metricconv-limit-uniqueness-audit · candidate
90. A Conditional Uniqueness Check Is Not Two Convergence Claims
Use the complete qualified finite metric table=[[0, 15, 31, 48], [15, 0, 16, 33], [31, 16, 0, 17], [48, 33, 17, 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=[2, 3] and constant tail=0; candidates a=0, b=1. Give (prefix,tail,a,b,d(a,b),a is a limit?,b is a limit?,if both are limits then a=b?).
View answer and explanation
([2, 3], 0, 0, 1, 15, Yes, No, Yes)
Check each candidate against the declared constant tail. The final implication may be true when its antecedent is false; it does not assert both candidates converge. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([2, 3], 0, 0, 1, 15, Yes, No, Yes).
Reviewed step-by-step derivation
Convergence in a metric space · qf-mod23-metricconv-limit-uniqueness-audit · candidate
91. A Conditional Uniqueness Check Is Not Two Convergence Claims
Use the complete qualified finite metric table=[[0, 16, 33, 51], [16, 0, 17, 35], [33, 17, 0, 18], [51, 35, 18, 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=1. Give (prefix,tail,a,b,d(a,b),a is a limit?,b is a limit?,if both are limits then a=b?).
View answer and explanation
([3, 0], 1, 1, 1, 0, Yes, Yes, Yes)
Check each candidate against the declared constant tail. The final implication may be true when its antecedent is false; it does not assert both candidates converge. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([3, 0], 1, 1, 1, 0, Yes, Yes, Yes).
Reviewed step-by-step derivation
Convergence in a metric space · qf-mod23-metricconv-limit-uniqueness-audit · candidate
92. A Conditional Uniqueness Check Is Not Two Convergence Claims
Use the complete qualified finite metric table=[[0, 17, 35, 54], [17, 0, 18, 37], [35, 18, 0, 19], [54, 37, 19, 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=[0, 1] and constant tail=2; candidates a=2, b=3. Give (prefix,tail,a,b,d(a,b),a is a limit?,b is a limit?,if both are limits then a=b?).
View answer and explanation
([0, 1], 2, 2, 3, 19, Yes, No, Yes)
Check each candidate against the declared constant tail. The final implication may be true when its antecedent is false; it does not assert both candidates converge. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([0, 1], 2, 2, 3, 19, Yes, No, Yes).
Reviewed step-by-step derivation
Convergence in a metric space · qf-mod23-metricconv-limit-uniqueness-audit · candidate
93. A Conditional Uniqueness Check Is Not Two Convergence Claims
Use the complete qualified finite metric table=[[0, 18, 37, 57], [18, 0, 19, 39], [37, 19, 0, 20], [57, 39, 20, 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=[1, 2] and constant tail=3; candidates a=3, b=0. Give (prefix,tail,a,b,d(a,b),a is a limit?,b is a limit?,if both are limits then a=b?).
View answer and explanation
([1, 2], 3, 3, 0, 57, Yes, No, Yes)
Check each candidate against the declared constant tail. The final implication may be true when its antecedent is false; it does not assert both candidates converge. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([1, 2], 3, 3, 0, 57, Yes, No, Yes).
Reviewed step-by-step derivation
Convergence in a metric space · qf-mod23-metricconv-limit-uniqueness-audit · candidate
94. A Conditional Uniqueness Check Is Not Two Convergence Claims
Use the complete qualified finite metric table=[[0, 19, 39, 60], [19, 0, 20, 41], [39, 20, 0, 21], [60, 41, 21, 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=[2, 3] and constant tail=0; candidates a=0, b=0. Give (prefix,tail,a,b,d(a,b),a is a limit?,b is a limit?,if both are limits then a=b?).
View answer and explanation
([2, 3], 0, 0, 0, 0, Yes, Yes, Yes)
Check each candidate against the declared constant tail. The final implication may be true when its antecedent is false; it does not assert both candidates converge. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([2, 3], 0, 0, 0, 0, Yes, Yes, Yes).
Reviewed step-by-step derivation
Convergence in a metric space · qf-mod23-metricconv-limit-uniqueness-audit · candidate
95. A Conditional Uniqueness Check Is Not Two Convergence Claims
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?).
View answer and explanation
([3, 0], 1, 1, 2, 21, Yes, No, Yes)
Check each candidate against the declared constant tail. The final implication may be true when its antecedent is false; it does not assert both candidates converge. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([3, 0], 1, 1, 2, 21, Yes, No, Yes).
Reviewed step-by-step derivation
Convergence in a metric space · qf-mod23-metricconv-limit-uniqueness-audit · candidate
96. A Conditional Uniqueness Check Is Not Two Convergence Claims
Use the complete qualified finite metric table=[[0, 21, 43, 66], [21, 0, 22, 45], [43, 22, 0, 23], [66, 45, 23, 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=[0, 1] and constant tail=2; candidates a=2, b=3. Give (prefix,tail,a,b,d(a,b),a is a limit?,b is a limit?,if both are limits then a=b?).
View answer and explanation
([0, 1], 2, 2, 3, 23, Yes, No, Yes)
Check each candidate against the declared constant tail. The final implication may be true when its antecedent is false; it does not assert both candidates converge. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([0, 1], 2, 2, 3, 23, Yes, No, Yes).
Reviewed step-by-step derivation
Convergence in a metric space · qf-mod23-metricconv-limit-uniqueness-audit · candidate
97. A Conditional Uniqueness Check Is Not Two Convergence Claims
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. infinite sequence has prefix=[1, 2] and constant tail=3; candidates a=3, b=3. Give (prefix,tail,a,b,d(a,b),a is a limit?,b is a limit?,if both are limits then a=b?).
View answer and explanation
([1, 2], 3, 3, 3, 0, Yes, Yes, Yes)
Check each candidate against the declared constant tail. The final implication may be true when its antecedent is false; it does not assert both candidates converge. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([1, 2], 3, 3, 3, 0, Yes, Yes, Yes).
Reviewed step-by-step derivation
Convergence in a metric space · qf-mod23-metricconv-limit-uniqueness-audit · candidate
98. A Conditional Uniqueness Check Is Not Two Convergence Claims
Use the complete qualified finite metric table=[[0, 23, 47, 72], [23, 0, 24, 49], [47, 24, 0, 25], [72, 49, 25, 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=[2, 3] and constant tail=0; candidates a=0, b=1. Give (prefix,tail,a,b,d(a,b),a is a limit?,b is a limit?,if both are limits then a=b?).
View answer and explanation
([2, 3], 0, 0, 1, 23, Yes, No, Yes)
Check each candidate against the declared constant tail. The final implication may be true when its antecedent is false; it does not assert both candidates converge. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([2, 3], 0, 0, 1, 23, Yes, No, Yes).
Reviewed step-by-step derivation
Convergence in a metric space · qf-mod23-metricconv-limit-uniqueness-audit · candidate
99. A Conditional Uniqueness Check Is Not Two Convergence Claims
Use the complete qualified finite metric table=[[0, 24, 49, 75], [24, 0, 25, 51], [49, 25, 0, 26], [75, 51, 26, 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?).
View answer and explanation
([3, 0], 1, 1, 2, 25, Yes, No, Yes)
Check each candidate against the declared constant tail. The final implication may be true when its antecedent is false; it does not assert both candidates converge. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([3, 0], 1, 1, 2, 25, Yes, No, Yes).
Reviewed step-by-step derivation
Convergence in a metric space · qf-mod23-metricconv-limit-uniqueness-audit · candidate
100. A Conditional Uniqueness Check Is Not Two Convergence Claims
Use the complete qualified finite metric table=[[0, 25, 51, 78], [25, 0, 26, 53], [51, 26, 0, 27], [78, 53, 27, 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=[0, 1] and constant tail=2; candidates a=2, 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?).
View answer and explanation
([0, 1], 2, 2, 2, 0, Yes, Yes, Yes)
Check each candidate against the declared constant tail. The final implication may be true when its antecedent is false; it does not assert both candidates converge. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([0, 1], 2, 2, 2, 0, Yes, Yes, Yes).
Reviewed step-by-step derivation
Convergence in a metric space · qf-mod23-metricconv-limit-uniqueness-audit · candidate
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