Finite-Space Completeness, Explicit Cauchy Limits and Correctly Labeled Non-Cauchy Counterexamples
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This question bank has corresponding bilingual step-by-step derivations. Source candidate and English ai_candidate states are retained. AI review, finite computation and Lean formal proof are labelled separately.
batch-20261014-advance-20261027-r03-complete-metric-space-foundations · 2026-10-14 · 100 questions
This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.
1. Minimum Positive Separation Proves Finite-Space Completeness
Use the complete qualified finite metric table=[[0, 3, 4, 5], [3, 0, 5, 6], [4, 5, 0, 7], [5, 6, 7, 0]] on Fin(4)={0,…,3}, with at least two points for this minimum-positive-distance protocol. Completeness quantifies over every Cauchy sequence and requires a limit in this same space, not merely an ambient space. Testing finitely many sequences or terms is insufficient. Give (n,min_positive_distance,epsilon=min_positive_distance forces every Cauchy tail to be constant,complete?).
View answer and explanation
(4, 3, epsilon=min_positive_distance forces every Cauchy tail to be constant, Yes)
For any Cauchy sequence choose epsilon equal to the least positive distance. All sufficiently late pairwise distances are strictly smaller, so those terms must coincide. Their common value lies in the finite space and is the limit. This is universal over Cauchy sequences, not sampling. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (4, 3, epsilon=min_positive_distance forces every Cauchy tail to be constant, Yes).
Reviewed step-by-step derivation
Complete metric space · qf-mod27-complete-finite-certificate · candidate
2. Minimum Positive Separation Proves Finite-Space Completeness
Use the complete qualified finite metric table=[[0, 4, 7, 9], [4, 0, 7, 9], [7, 7, 0, 12], [9, 9, 12, 0]] on Fin(4)={0,…,3}, with at least two points for this minimum-positive-distance protocol. Completeness quantifies over every Cauchy sequence and requires a limit in this same space, not merely an ambient space. Testing finitely many sequences or terms is insufficient. Give (n,min_positive_distance,epsilon=min_positive_distance forces every Cauchy tail to be constant,complete?).
View answer and explanation
(4, 4, epsilon=min_positive_distance forces every Cauchy tail to be constant, Yes)
For any Cauchy sequence choose epsilon equal to the least positive distance. All sufficiently late pairwise distances are strictly smaller, so those terms must coincide. Their common value lies in the finite space and is the limit. This is universal over Cauchy sequences, not sampling. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (4, 4, epsilon=min_positive_distance forces every Cauchy tail to be constant, Yes).
Reviewed step-by-step derivation
Complete metric space · qf-mod27-complete-finite-certificate · candidate
3. Minimum Positive Separation Proves Finite-Space Completeness
Use the complete qualified finite metric table=[[0, 5, 10, 13], [5, 0, 9, 12], [10, 9, 0, 17], [13, 12, 17, 0]] on Fin(4)={0,…,3}, with at least two points for this minimum-positive-distance protocol. Completeness quantifies over every Cauchy sequence and requires a limit in this same space, not merely an ambient space. Testing finitely many sequences or terms is insufficient. Give (n,min_positive_distance,epsilon=min_positive_distance forces every Cauchy tail to be constant,complete?).
View answer and explanation
(4, 5, epsilon=min_positive_distance forces every Cauchy tail to be constant, Yes)
For any Cauchy sequence choose epsilon equal to the least positive distance. All sufficiently late pairwise distances are strictly smaller, so those terms must coincide. Their common value lies in the finite space and is the limit. This is universal over Cauchy sequences, not sampling. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (4, 5, epsilon=min_positive_distance forces every Cauchy tail to be constant, Yes).
Reviewed step-by-step derivation
Complete metric space · qf-mod27-complete-finite-certificate · candidate
4. Minimum Positive Separation Proves Finite-Space Completeness
Use the complete qualified finite metric table=[[0, 6, 13, 17], [6, 0, 11, 15], [13, 11, 0, 22], [17, 15, 22, 0]] on Fin(4)={0,…,3}, with at least two points for this minimum-positive-distance protocol. Completeness quantifies over every Cauchy sequence and requires a limit in this same space, not merely an ambient space. Testing finitely many sequences or terms is insufficient. Give (n,min_positive_distance,epsilon=min_positive_distance forces every Cauchy tail to be constant,complete?).
View answer and explanation
(4, 6, epsilon=min_positive_distance forces every Cauchy tail to be constant, Yes)
For any Cauchy sequence choose epsilon equal to the least positive distance. All sufficiently late pairwise distances are strictly smaller, so those terms must coincide. Their common value lies in the finite space and is the limit. This is universal over Cauchy sequences, not sampling. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (4, 6, epsilon=min_positive_distance forces every Cauchy tail to be constant, Yes).
Reviewed step-by-step derivation
Complete metric space · qf-mod27-complete-finite-certificate · candidate
5. Minimum Positive Separation Proves Finite-Space Completeness
Use the complete qualified finite metric table=[[0, 7, 9, 10], [7, 0, 6, 7], [9, 6, 0, 9], [10, 7, 9, 0]] on Fin(4)={0,…,3}, with at least two points for this minimum-positive-distance protocol. Completeness quantifies over every Cauchy sequence and requires a limit in this same space, not merely an ambient space. Testing finitely many sequences or terms is insufficient. Give (n,min_positive_distance,epsilon=min_positive_distance forces every Cauchy tail to be constant,complete?).
View answer and explanation
(4, 6, epsilon=min_positive_distance forces every Cauchy tail to be constant, Yes)
For any Cauchy sequence choose epsilon equal to the least positive distance. All sufficiently late pairwise distances are strictly smaller, so those terms must coincide. Their common value lies in the finite space and is the limit. This is universal over Cauchy sequences, not sampling. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (4, 6, epsilon=min_positive_distance forces every Cauchy tail to be constant, Yes).
Reviewed step-by-step derivation
Complete metric space · qf-mod27-complete-finite-certificate · candidate
6. Minimum Positive Separation Proves Finite-Space Completeness
Use the complete qualified finite metric table=[[0, 4, 7, 9], [4, 0, 9, 11], [7, 9, 0, 14], [9, 11, 14, 0]] on Fin(4)={0,…,3}, with at least two points for this minimum-positive-distance protocol. Completeness quantifies over every Cauchy sequence and requires a limit in this same space, not merely an ambient space. Testing finitely many sequences or terms is insufficient. Give (n,min_positive_distance,epsilon=min_positive_distance forces every Cauchy tail to be constant,complete?).
View answer and explanation
(4, 4, epsilon=min_positive_distance forces every Cauchy tail to be constant, Yes)
For any Cauchy sequence choose epsilon equal to the least positive distance. All sufficiently late pairwise distances are strictly smaller, so those terms must coincide. Their common value lies in the finite space and is the limit. This is universal over Cauchy sequences, not sampling. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (4, 4, epsilon=min_positive_distance forces every Cauchy tail to be constant, Yes).
Reviewed step-by-step derivation
Complete metric space · qf-mod27-complete-finite-certificate · candidate
7. Minimum Positive Separation Proves Finite-Space Completeness
Use the complete qualified finite metric table=[[0, 5, 10, 13], [5, 0, 11, 14], [10, 11, 0, 19], [13, 14, 19, 0]] on Fin(4)={0,…,3}, with at least two points for this minimum-positive-distance protocol. Completeness quantifies over every Cauchy sequence and requires a limit in this same space, not merely an ambient space. Testing finitely many sequences or terms is insufficient. Give (n,min_positive_distance,epsilon=min_positive_distance forces every Cauchy tail to be constant,complete?).
View answer and explanation
(4, 5, epsilon=min_positive_distance forces every Cauchy tail to be constant, Yes)
For any Cauchy sequence choose epsilon equal to the least positive distance. All sufficiently late pairwise distances are strictly smaller, so those terms must coincide. Their common value lies in the finite space and is the limit. This is universal over Cauchy sequences, not sampling. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (4, 5, epsilon=min_positive_distance forces every Cauchy tail to be constant, Yes).
Reviewed step-by-step derivation
Complete metric space · qf-mod27-complete-finite-certificate · candidate
8. Minimum Positive Separation Proves Finite-Space Completeness
Use the complete qualified finite metric table=[[0, 6, 6, 17], [6, 0, 6, 17], [6, 6, 0, 17], [17, 17, 17, 0]] on Fin(4)={0,…,3}, with at least two points for this minimum-positive-distance protocol. Completeness quantifies over every Cauchy sequence and requires a limit in this same space, not merely an ambient space. Testing finitely many sequences or terms is insufficient. Give (n,min_positive_distance,epsilon=min_positive_distance forces every Cauchy tail to be constant,complete?).
View answer and explanation
(4, 6, epsilon=min_positive_distance forces every Cauchy tail to be constant, Yes)
For any Cauchy sequence choose epsilon equal to the least positive distance. All sufficiently late pairwise distances are strictly smaller, so those terms must coincide. Their common value lies in the finite space and is the limit. This is universal over Cauchy sequences, not sampling. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (4, 6, epsilon=min_positive_distance forces every Cauchy tail to be constant, Yes).
Reviewed step-by-step derivation
Complete metric space · qf-mod27-complete-finite-certificate · candidate
9. Minimum Positive Separation Proves Finite-Space Completeness
Use the complete qualified finite metric table=[[0, 7, 9, 10], [7, 0, 8, 9], [9, 8, 0, 11], [10, 9, 11, 0]] on Fin(4)={0,…,3}, with at least two points for this minimum-positive-distance protocol. Completeness quantifies over every Cauchy sequence and requires a limit in this same space, not merely an ambient space. Testing finitely many sequences or terms is insufficient. Give (n,min_positive_distance,epsilon=min_positive_distance forces every Cauchy tail to be constant,complete?).
View answer and explanation
(4, 7, epsilon=min_positive_distance forces every Cauchy tail to be constant, Yes)
For any Cauchy sequence choose epsilon equal to the least positive distance. All sufficiently late pairwise distances are strictly smaller, so those terms must coincide. Their common value lies in the finite space and is the limit. This is universal over Cauchy sequences, not sampling. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (4, 7, epsilon=min_positive_distance forces every Cauchy tail to be constant, Yes).
Reviewed step-by-step derivation
Complete metric space · qf-mod27-complete-finite-certificate · candidate
10. Minimum Positive Separation Proves Finite-Space Completeness
Use the complete qualified finite metric table=[[0, 8, 12, 14], [8, 0, 10, 12], [12, 10, 0, 16], [14, 12, 16, 0]] on Fin(4)={0,…,3}, with at least two points for this minimum-positive-distance protocol. Completeness quantifies over every Cauchy sequence and requires a limit in this same space, not merely an ambient space. Testing finitely many sequences or terms is insufficient. Give (n,min_positive_distance,epsilon=min_positive_distance forces every Cauchy tail to be constant,complete?).
View answer and explanation
(4, 8, epsilon=min_positive_distance forces every Cauchy tail to be constant, Yes)
For any Cauchy sequence choose epsilon equal to the least positive distance. All sufficiently late pairwise distances are strictly smaller, so those terms must coincide. Their common value lies in the finite space and is the limit. This is universal over Cauchy sequences, not sampling. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (4, 8, epsilon=min_positive_distance forces every Cauchy tail to be constant, Yes).
Reviewed step-by-step derivation
Complete metric space · qf-mod27-complete-finite-certificate · candidate
11. Minimum Positive Separation Proves Finite-Space Completeness
Use the complete qualified finite metric table=[[0, 5, 10, 13], [5, 0, 13, 16], [10, 13, 0, 21], [13, 16, 21, 0]] on Fin(4)={0,…,3}, with at least two points for this minimum-positive-distance protocol. Completeness quantifies over every Cauchy sequence and requires a limit in this same space, not merely an ambient space. Testing finitely many sequences or terms is insufficient. Give (n,min_positive_distance,epsilon=min_positive_distance forces every Cauchy tail to be constant,complete?).
View answer and explanation
(4, 5, epsilon=min_positive_distance forces every Cauchy tail to be constant, Yes)
For any Cauchy sequence choose epsilon equal to the least positive distance. All sufficiently late pairwise distances are strictly smaller, so those terms must coincide. Their common value lies in the finite space and is the limit. This is universal over Cauchy sequences, not sampling. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (4, 5, epsilon=min_positive_distance forces every Cauchy tail to be constant, Yes).
Reviewed step-by-step derivation
Complete metric space · qf-mod27-complete-finite-certificate · candidate
12. Minimum Positive Separation Proves Finite-Space Completeness
Use the complete qualified finite metric table=[[0, 6, 6, 6], [6, 0, 8, 8], [6, 8, 0, 8], [6, 8, 8, 0]] on Fin(4)={0,…,3}, with at least two points for this minimum-positive-distance protocol. Completeness quantifies over every Cauchy sequence and requires a limit in this same space, not merely an ambient space. Testing finitely many sequences or terms is insufficient. Give (n,min_positive_distance,epsilon=min_positive_distance forces every Cauchy tail to be constant,complete?).
View answer and explanation
(4, 6, epsilon=min_positive_distance forces every Cauchy tail to be constant, Yes)
For any Cauchy sequence choose epsilon equal to the least positive distance. All sufficiently late pairwise distances are strictly smaller, so those terms must coincide. Their common value lies in the finite space and is the limit. This is universal over Cauchy sequences, not sampling. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (4, 6, epsilon=min_positive_distance forces every Cauchy tail to be constant, Yes).
Reviewed step-by-step derivation
Complete metric space · qf-mod27-complete-finite-certificate · candidate
13. Minimum Positive Separation Proves Finite-Space Completeness
Use the complete qualified finite metric table=[[0, 7, 9, 10], [7, 0, 10, 11], [9, 10, 0, 13], [10, 11, 13, 0]] on Fin(4)={0,…,3}, with at least two points for this minimum-positive-distance protocol. Completeness quantifies over every Cauchy sequence and requires a limit in this same space, not merely an ambient space. Testing finitely many sequences or terms is insufficient. Give (n,min_positive_distance,epsilon=min_positive_distance forces every Cauchy tail to be constant,complete?).
View answer and explanation
(4, 7, epsilon=min_positive_distance forces every Cauchy tail to be constant, Yes)
For any Cauchy sequence choose epsilon equal to the least positive distance. All sufficiently late pairwise distances are strictly smaller, so those terms must coincide. Their common value lies in the finite space and is the limit. This is universal over Cauchy sequences, not sampling. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (4, 7, epsilon=min_positive_distance forces every Cauchy tail to be constant, Yes).
Reviewed step-by-step derivation
Complete metric space · qf-mod27-complete-finite-certificate · candidate
14. Minimum Positive Separation Proves Finite-Space Completeness
Use the complete qualified finite metric table=[[0, 8, 12, 14], [8, 0, 12, 14], [12, 12, 0, 18], [14, 14, 18, 0]] on Fin(4)={0,…,3}, with at least two points for this minimum-positive-distance protocol. Completeness quantifies over every Cauchy sequence and requires a limit in this same space, not merely an ambient space. Testing finitely many sequences or terms is insufficient. Give (n,min_positive_distance,epsilon=min_positive_distance forces every Cauchy tail to be constant,complete?).
View answer and explanation
(4, 8, epsilon=min_positive_distance forces every Cauchy tail to be constant, Yes)
For any Cauchy sequence choose epsilon equal to the least positive distance. All sufficiently late pairwise distances are strictly smaller, so those terms must coincide. Their common value lies in the finite space and is the limit. This is universal over Cauchy sequences, not sampling. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (4, 8, epsilon=min_positive_distance forces every Cauchy tail to be constant, Yes).
Reviewed step-by-step derivation
Complete metric space · qf-mod27-complete-finite-certificate · candidate
15. Minimum Positive Separation Proves Finite-Space Completeness
Use the complete qualified finite metric table=[[0, 9, 8, 18], [9, 0, 7, 17], [8, 7, 0, 16], [18, 17, 16, 0]] on Fin(4)={0,…,3}, with at least two points for this minimum-positive-distance protocol. Completeness quantifies over every Cauchy sequence and requires a limit in this same space, not merely an ambient space. Testing finitely many sequences or terms is insufficient. Give (n,min_positive_distance,epsilon=min_positive_distance forces every Cauchy tail to be constant,complete?).
View answer and explanation
(4, 7, epsilon=min_positive_distance forces every Cauchy tail to be constant, Yes)
For any Cauchy sequence choose epsilon equal to the least positive distance. All sufficiently late pairwise distances are strictly smaller, so those terms must coincide. Their common value lies in the finite space and is the limit. This is universal over Cauchy sequences, not sampling. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (4, 7, epsilon=min_positive_distance forces every Cauchy tail to be constant, Yes).
Reviewed step-by-step derivation
Complete metric space · qf-mod27-complete-finite-certificate · candidate
16. Minimum Positive Separation Proves Finite-Space Completeness
Use the complete qualified finite metric table=[[0, 6, 6, 6], [6, 0, 10, 10], [6, 10, 0, 10], [6, 10, 10, 0]] on Fin(4)={0,…,3}, with at least two points for this minimum-positive-distance protocol. Completeness quantifies over every Cauchy sequence and requires a limit in this same space, not merely an ambient space. Testing finitely many sequences or terms is insufficient. Give (n,min_positive_distance,epsilon=min_positive_distance forces every Cauchy tail to be constant,complete?).
View answer and explanation
(4, 6, epsilon=min_positive_distance forces every Cauchy tail to be constant, Yes)
For any Cauchy sequence choose epsilon equal to the least positive distance. All sufficiently late pairwise distances are strictly smaller, so those terms must coincide. Their common value lies in the finite space and is the limit. This is universal over Cauchy sequences, not sampling. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (4, 6, epsilon=min_positive_distance forces every Cauchy tail to be constant, Yes).
Reviewed step-by-step derivation
Complete metric space · qf-mod27-complete-finite-certificate · candidate
17. Minimum Positive Separation Proves Finite-Space Completeness
Use the complete qualified finite metric table=[[0, 7, 9, 10], [7, 0, 12, 13], [9, 12, 0, 15], [10, 13, 15, 0]] on Fin(4)={0,…,3}, with at least two points for this minimum-positive-distance protocol. Completeness quantifies over every Cauchy sequence and requires a limit in this same space, not merely an ambient space. Testing finitely many sequences or terms is insufficient. Give (n,min_positive_distance,epsilon=min_positive_distance forces every Cauchy tail to be constant,complete?).
View answer and explanation
(4, 7, epsilon=min_positive_distance forces every Cauchy tail to be constant, Yes)
For any Cauchy sequence choose epsilon equal to the least positive distance. All sufficiently late pairwise distances are strictly smaller, so those terms must coincide. Their common value lies in the finite space and is the limit. This is universal over Cauchy sequences, not sampling. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (4, 7, epsilon=min_positive_distance forces every Cauchy tail to be constant, Yes).
Reviewed step-by-step derivation
Complete metric space · qf-mod27-complete-finite-certificate · candidate
18. Minimum Positive Separation Proves Finite-Space Completeness
Use the complete qualified finite metric table=[[0, 8, 12, 14], [8, 0, 14, 16], [12, 14, 0, 20], [14, 16, 20, 0]] on Fin(4)={0,…,3}, with at least two points for this minimum-positive-distance protocol. Completeness quantifies over every Cauchy sequence and requires a limit in this same space, not merely an ambient space. Testing finitely many sequences or terms is insufficient. Give (n,min_positive_distance,epsilon=min_positive_distance forces every Cauchy tail to be constant,complete?).
View answer and explanation
(4, 8, epsilon=min_positive_distance forces every Cauchy tail to be constant, Yes)
For any Cauchy sequence choose epsilon equal to the least positive distance. All sufficiently late pairwise distances are strictly smaller, so those terms must coincide. Their common value lies in the finite space and is the limit. This is universal over Cauchy sequences, not sampling. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (4, 8, epsilon=min_positive_distance forces every Cauchy tail to be constant, Yes).
Reviewed step-by-step derivation
Complete metric space · qf-mod27-complete-finite-certificate · candidate
19. Minimum Positive Separation Proves Finite-Space Completeness
Use the complete qualified finite metric table=[[0, 9, 8, 18], [9, 0, 9, 19], [8, 9, 0, 18], [18, 19, 18, 0]] on Fin(4)={0,…,3}, with at least two points for this minimum-positive-distance protocol. Completeness quantifies over every Cauchy sequence and requires a limit in this same space, not merely an ambient space. Testing finitely many sequences or terms is insufficient. Give (n,min_positive_distance,epsilon=min_positive_distance forces every Cauchy tail to be constant,complete?).
View answer and explanation
(4, 8, epsilon=min_positive_distance forces every Cauchy tail to be constant, Yes)
For any Cauchy sequence choose epsilon equal to the least positive distance. All sufficiently late pairwise distances are strictly smaller, so those terms must coincide. Their common value lies in the finite space and is the limit. This is universal over Cauchy sequences, not sampling. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (4, 8, epsilon=min_positive_distance forces every Cauchy tail to be constant, Yes).
Reviewed step-by-step derivation
Complete metric space · qf-mod27-complete-finite-certificate · candidate
20. Minimum Positive Separation Proves Finite-Space Completeness
Use the complete qualified finite metric table=[[0, 10, 11, 11], [10, 0, 11, 11], [11, 11, 0, 12], [11, 11, 12, 0]] on Fin(4)={0,…,3}, with at least two points for this minimum-positive-distance protocol. Completeness quantifies over every Cauchy sequence and requires a limit in this same space, not merely an ambient space. Testing finitely many sequences or terms is insufficient. Give (n,min_positive_distance,epsilon=min_positive_distance forces every Cauchy tail to be constant,complete?).
View answer and explanation
(4, 10, epsilon=min_positive_distance forces every Cauchy tail to be constant, Yes)
For any Cauchy sequence choose epsilon equal to the least positive distance. All sufficiently late pairwise distances are strictly smaller, so those terms must coincide. Their common value lies in the finite space and is the limit. This is universal over Cauchy sequences, not sampling. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (4, 10, epsilon=min_positive_distance forces every Cauchy tail to be constant, Yes).
Reviewed step-by-step derivation
Complete metric space · qf-mod27-complete-finite-certificate · candidate
21. Minimum Positive Separation Proves Finite-Space Completeness
Use the complete qualified finite metric table=[[0, 7, 9, 10], [7, 0, 14, 15], [9, 14, 0, 17], [10, 15, 17, 0]] on Fin(4)={0,…,3}, with at least two points for this minimum-positive-distance protocol. Completeness quantifies over every Cauchy sequence and requires a limit in this same space, not merely an ambient space. Testing finitely many sequences or terms is insufficient. Give (n,min_positive_distance,epsilon=min_positive_distance forces every Cauchy tail to be constant,complete?).
View answer and explanation
(4, 7, epsilon=min_positive_distance forces every Cauchy tail to be constant, Yes)
For any Cauchy sequence choose epsilon equal to the least positive distance. All sufficiently late pairwise distances are strictly smaller, so those terms must coincide. Their common value lies in the finite space and is the limit. This is universal over Cauchy sequences, not sampling. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (4, 7, epsilon=min_positive_distance forces every Cauchy tail to be constant, Yes).
Reviewed step-by-step derivation
Complete metric space · qf-mod27-complete-finite-certificate · candidate
22. Minimum Positive Separation Proves Finite-Space Completeness
Use the complete qualified finite metric table=[[0, 8, 5, 14], [8, 0, 9, 18], [5, 9, 0, 15], [14, 18, 15, 0]] on Fin(4)={0,…,3}, with at least two points for this minimum-positive-distance protocol. Completeness quantifies over every Cauchy sequence and requires a limit in this same space, not merely an ambient space. Testing finitely many sequences or terms is insufficient. Give (n,min_positive_distance,epsilon=min_positive_distance forces every Cauchy tail to be constant,complete?).
View answer and explanation
(4, 5, epsilon=min_positive_distance forces every Cauchy tail to be constant, Yes)
For any Cauchy sequence choose epsilon equal to the least positive distance. All sufficiently late pairwise distances are strictly smaller, so those terms must coincide. Their common value lies in the finite space and is the limit. This is universal over Cauchy sequences, not sampling. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (4, 5, epsilon=min_positive_distance forces every Cauchy tail to be constant, Yes).
Reviewed step-by-step derivation
Complete metric space · qf-mod27-complete-finite-certificate · candidate
23. Minimum Positive Separation Proves Finite-Space Completeness
Use the complete qualified finite metric table=[[0, 9, 8, 7], [9, 0, 11, 10], [8, 11, 0, 9], [7, 10, 9, 0]] on Fin(4)={0,…,3}, with at least two points for this minimum-positive-distance protocol. Completeness quantifies over every Cauchy sequence and requires a limit in this same space, not merely an ambient space. Testing finitely many sequences or terms is insufficient. Give (n,min_positive_distance,epsilon=min_positive_distance forces every Cauchy tail to be constant,complete?).
View answer and explanation
(4, 7, epsilon=min_positive_distance forces every Cauchy tail to be constant, Yes)
For any Cauchy sequence choose epsilon equal to the least positive distance. All sufficiently late pairwise distances are strictly smaller, so those terms must coincide. Their common value lies in the finite space and is the limit. This is universal over Cauchy sequences, not sampling. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (4, 7, epsilon=min_positive_distance forces every Cauchy tail to be constant, Yes).
Reviewed step-by-step derivation
Complete metric space · qf-mod27-complete-finite-certificate · candidate
24. Minimum Positive Separation Proves Finite-Space Completeness
Use the complete qualified finite metric table=[[0, 10, 11, 11], [10, 0, 13, 13], [11, 13, 0, 14], [11, 13, 14, 0]] on Fin(4)={0,…,3}, with at least two points for this minimum-positive-distance protocol. Completeness quantifies over every Cauchy sequence and requires a limit in this same space, not merely an ambient space. Testing finitely many sequences or terms is insufficient. Give (n,min_positive_distance,epsilon=min_positive_distance forces every Cauchy tail to be constant,complete?).
View answer and explanation
(4, 10, epsilon=min_positive_distance forces every Cauchy tail to be constant, Yes)
For any Cauchy sequence choose epsilon equal to the least positive distance. All sufficiently late pairwise distances are strictly smaller, so those terms must coincide. Their common value lies in the finite space and is the limit. This is universal over Cauchy sequences, not sampling. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (4, 10, epsilon=min_positive_distance forces every Cauchy tail to be constant, Yes).
Reviewed step-by-step derivation
Complete metric space · qf-mod27-complete-finite-certificate · candidate
25. Minimum Positive Separation Proves Finite-Space Completeness
Use the complete qualified finite metric table=[[0, 11, 14, 15], [11, 0, 15, 16], [14, 15, 0, 19], [15, 16, 19, 0]] on Fin(4)={0,…,3}, with at least two points for this minimum-positive-distance protocol. Completeness quantifies over every Cauchy sequence and requires a limit in this same space, not merely an ambient space. Testing finitely many sequences or terms is insufficient. Give (n,min_positive_distance,epsilon=min_positive_distance forces every Cauchy tail to be constant,complete?).
View answer and explanation
(4, 11, epsilon=min_positive_distance forces every Cauchy tail to be constant, Yes)
For any Cauchy sequence choose epsilon equal to the least positive distance. All sufficiently late pairwise distances are strictly smaller, so those terms must coincide. Their common value lies in the finite space and is the limit. This is universal over Cauchy sequences, not sampling. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (4, 11, epsilon=min_positive_distance forces every Cauchy tail to be constant, Yes).
Reviewed step-by-step derivation
Complete metric space · qf-mod27-complete-finite-certificate · candidate
26. An Explicit Infinite Tail Converges within Its Declared Space
Use the complete qualified finite metric table=[[0, 3, 4, 5], [3, 0, 5, 6], [4, 5, 0, 7], [5, 6, 7, 0]] on Fin(4)={0,…,3}, with at least two points for this minimum-positive-distance protocol. Completeness quantifies over every Cauchy sequence and requires a limit in this same space, not merely an ambient space. Testing finitely many sequences or terms is insufficient. The infinite sequence starts with prefix=[0, 1, 2] then remains forever at tail=3. Give (prefix,tail,N,limit,cauchy?,converges_in_space?).
View answer and explanation
([0, 1, 2], 3, 3, 3, Yes, Yes)
Choose N equal to the prefix length. All later terms coincide with the specified tail point, giving both Cauchy behavior and convergence inside this space. A finite prefix alone would not supply this certificate. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([0, 1, 2], 3, 3, 3, Yes, Yes).
Reviewed step-by-step derivation
Complete metric space · qf-mod27-complete-ultimately-constant-limit · candidate
27. An Explicit Infinite Tail Converges within Its Declared Space
Use the complete qualified finite metric table=[[0, 4, 7, 9], [4, 0, 7, 9], [7, 7, 0, 12], [9, 9, 12, 0]] on Fin(4)={0,…,3}, with at least two points for this minimum-positive-distance protocol. Completeness quantifies over every Cauchy sequence and requires a limit in this same space, not merely an ambient space. Testing finitely many sequences or terms is insufficient. The infinite sequence starts with prefix=[1, 2, 3] then remains forever at tail=0. Give (prefix,tail,N,limit,cauchy?,converges_in_space?).
View answer and explanation
([1, 2, 3], 0, 3, 0, Yes, Yes)
Choose N equal to the prefix length. All later terms coincide with the specified tail point, giving both Cauchy behavior and convergence inside this space. A finite prefix alone would not supply this certificate. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([1, 2, 3], 0, 3, 0, Yes, Yes).
Reviewed step-by-step derivation
Complete metric space · qf-mod27-complete-ultimately-constant-limit · candidate
28. An Explicit Infinite Tail Converges within Its Declared Space
Use the complete qualified finite metric table=[[0, 5, 10, 13], [5, 0, 9, 12], [10, 9, 0, 17], [13, 12, 17, 0]] on Fin(4)={0,…,3}, with at least two points for this minimum-positive-distance protocol. Completeness quantifies over every Cauchy sequence and requires a limit in this same space, not merely an ambient space. Testing finitely many sequences or terms is insufficient. The infinite sequence starts with prefix=[2, 3, 0] then remains forever at tail=1. Give (prefix,tail,N,limit,cauchy?,converges_in_space?).
View answer and explanation
([2, 3, 0], 1, 3, 1, Yes, Yes)
Choose N equal to the prefix length. All later terms coincide with the specified tail point, giving both Cauchy behavior and convergence inside this space. A finite prefix alone would not supply this certificate. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([2, 3, 0], 1, 3, 1, Yes, Yes).
Reviewed step-by-step derivation
Complete metric space · qf-mod27-complete-ultimately-constant-limit · candidate
29. An Explicit Infinite Tail Converges within Its Declared Space
Use the complete qualified finite metric table=[[0, 6, 13, 17], [6, 0, 11, 15], [13, 11, 0, 22], [17, 15, 22, 0]] on Fin(4)={0,…,3}, with at least two points for this minimum-positive-distance protocol. Completeness quantifies over every Cauchy sequence and requires a limit in this same space, not merely an ambient space. Testing finitely many sequences or terms is insufficient. The infinite sequence starts with prefix=[3, 0, 1] then remains forever at tail=2. Give (prefix,tail,N,limit,cauchy?,converges_in_space?).
View answer and explanation
([3, 0, 1], 2, 3, 2, Yes, Yes)
Choose N equal to the prefix length. All later terms coincide with the specified tail point, giving both Cauchy behavior and convergence inside this space. A finite prefix alone would not supply this certificate. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([3, 0, 1], 2, 3, 2, Yes, Yes).
Reviewed step-by-step derivation
Complete metric space · qf-mod27-complete-ultimately-constant-limit · candidate
30. An Explicit Infinite Tail Converges within Its Declared Space
Use the complete qualified finite metric table=[[0, 7, 9, 10], [7, 0, 6, 7], [9, 6, 0, 9], [10, 7, 9, 0]] on Fin(4)={0,…,3}, with at least two points for this minimum-positive-distance protocol. Completeness quantifies over every Cauchy sequence and requires a limit in this same space, not merely an ambient space. Testing finitely many sequences or terms is insufficient. The infinite sequence starts with prefix=[0, 1, 2] then remains forever at tail=3. Give (prefix,tail,N,limit,cauchy?,converges_in_space?).
View answer and explanation
([0, 1, 2], 3, 3, 3, Yes, Yes)
Choose N equal to the prefix length. All later terms coincide with the specified tail point, giving both Cauchy behavior and convergence inside this space. A finite prefix alone would not supply this certificate. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([0, 1, 2], 3, 3, 3, Yes, Yes).
Reviewed step-by-step derivation
Complete metric space · qf-mod27-complete-ultimately-constant-limit · candidate
31. An Explicit Infinite Tail Converges within Its Declared Space
Use the complete qualified finite metric table=[[0, 4, 7, 9], [4, 0, 9, 11], [7, 9, 0, 14], [9, 11, 14, 0]] on Fin(4)={0,…,3}, with at least two points for this minimum-positive-distance protocol. Completeness quantifies over every Cauchy sequence and requires a limit in this same space, not merely an ambient space. Testing finitely many sequences or terms is insufficient. The infinite sequence starts with prefix=[1, 2, 3] then remains forever at tail=0. Give (prefix,tail,N,limit,cauchy?,converges_in_space?).
View answer and explanation
([1, 2, 3], 0, 3, 0, Yes, Yes)
Choose N equal to the prefix length. All later terms coincide with the specified tail point, giving both Cauchy behavior and convergence inside this space. A finite prefix alone would not supply this certificate. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([1, 2, 3], 0, 3, 0, Yes, Yes).
Reviewed step-by-step derivation
Complete metric space · qf-mod27-complete-ultimately-constant-limit · candidate
32. An Explicit Infinite Tail Converges within Its Declared Space
Use the complete qualified finite metric table=[[0, 5, 10, 13], [5, 0, 11, 14], [10, 11, 0, 19], [13, 14, 19, 0]] on Fin(4)={0,…,3}, with at least two points for this minimum-positive-distance protocol. Completeness quantifies over every Cauchy sequence and requires a limit in this same space, not merely an ambient space. Testing finitely many sequences or terms is insufficient. The infinite sequence starts with prefix=[2, 3, 0] then remains forever at tail=1. Give (prefix,tail,N,limit,cauchy?,converges_in_space?).
View answer and explanation
([2, 3, 0], 1, 3, 1, Yes, Yes)
Choose N equal to the prefix length. All later terms coincide with the specified tail point, giving both Cauchy behavior and convergence inside this space. A finite prefix alone would not supply this certificate. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([2, 3, 0], 1, 3, 1, Yes, Yes).
Reviewed step-by-step derivation
Complete metric space · qf-mod27-complete-ultimately-constant-limit · candidate
33. An Explicit Infinite Tail Converges within Its Declared Space
Use the complete qualified finite metric table=[[0, 6, 6, 17], [6, 0, 6, 17], [6, 6, 0, 17], [17, 17, 17, 0]] on Fin(4)={0,…,3}, with at least two points for this minimum-positive-distance protocol. Completeness quantifies over every Cauchy sequence and requires a limit in this same space, not merely an ambient space. Testing finitely many sequences or terms is insufficient. The infinite sequence starts with prefix=[3, 0, 1] then remains forever at tail=2. Give (prefix,tail,N,limit,cauchy?,converges_in_space?).
View answer and explanation
([3, 0, 1], 2, 3, 2, Yes, Yes)
Choose N equal to the prefix length. All later terms coincide with the specified tail point, giving both Cauchy behavior and convergence inside this space. A finite prefix alone would not supply this certificate. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([3, 0, 1], 2, 3, 2, Yes, Yes).
Reviewed step-by-step derivation
Complete metric space · qf-mod27-complete-ultimately-constant-limit · candidate
34. An Explicit Infinite Tail Converges within Its Declared Space
Use the complete qualified finite metric table=[[0, 7, 9, 10], [7, 0, 8, 9], [9, 8, 0, 11], [10, 9, 11, 0]] on Fin(4)={0,…,3}, with at least two points for this minimum-positive-distance protocol. Completeness quantifies over every Cauchy sequence and requires a limit in this same space, not merely an ambient space. Testing finitely many sequences or terms is insufficient. The infinite sequence starts with prefix=[0, 1, 2] then remains forever at tail=3. Give (prefix,tail,N,limit,cauchy?,converges_in_space?).
View answer and explanation
([0, 1, 2], 3, 3, 3, Yes, Yes)
Choose N equal to the prefix length. All later terms coincide with the specified tail point, giving both Cauchy behavior and convergence inside this space. A finite prefix alone would not supply this certificate. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([0, 1, 2], 3, 3, 3, Yes, Yes).
Reviewed step-by-step derivation
Complete metric space · qf-mod27-complete-ultimately-constant-limit · candidate
35. An Explicit Infinite Tail Converges within Its Declared Space
Use the complete qualified finite metric table=[[0, 8, 12, 14], [8, 0, 10, 12], [12, 10, 0, 16], [14, 12, 16, 0]] on Fin(4)={0,…,3}, with at least two points for this minimum-positive-distance protocol. Completeness quantifies over every Cauchy sequence and requires a limit in this same space, not merely an ambient space. Testing finitely many sequences or terms is insufficient. The infinite sequence starts with prefix=[1, 2, 3] then remains forever at tail=0. Give (prefix,tail,N,limit,cauchy?,converges_in_space?).
View answer and explanation
([1, 2, 3], 0, 3, 0, Yes, Yes)
Choose N equal to the prefix length. All later terms coincide with the specified tail point, giving both Cauchy behavior and convergence inside this space. A finite prefix alone would not supply this certificate. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([1, 2, 3], 0, 3, 0, Yes, Yes).
Reviewed step-by-step derivation
Complete metric space · qf-mod27-complete-ultimately-constant-limit · candidate
36. An Explicit Infinite Tail Converges within Its Declared Space
Use the complete qualified finite metric table=[[0, 5, 10, 13], [5, 0, 13, 16], [10, 13, 0, 21], [13, 16, 21, 0]] on Fin(4)={0,…,3}, with at least two points for this minimum-positive-distance protocol. Completeness quantifies over every Cauchy sequence and requires a limit in this same space, not merely an ambient space. Testing finitely many sequences or terms is insufficient. The infinite sequence starts with prefix=[2, 3, 0] then remains forever at tail=1. Give (prefix,tail,N,limit,cauchy?,converges_in_space?).
View answer and explanation
([2, 3, 0], 1, 3, 1, Yes, Yes)
Choose N equal to the prefix length. All later terms coincide with the specified tail point, giving both Cauchy behavior and convergence inside this space. A finite prefix alone would not supply this certificate. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([2, 3, 0], 1, 3, 1, Yes, Yes).
Reviewed step-by-step derivation
Complete metric space · qf-mod27-complete-ultimately-constant-limit · candidate
37. An Explicit Infinite Tail Converges within Its Declared Space
Use the complete qualified finite metric table=[[0, 6, 6, 6], [6, 0, 8, 8], [6, 8, 0, 8], [6, 8, 8, 0]] on Fin(4)={0,…,3}, with at least two points for this minimum-positive-distance protocol. Completeness quantifies over every Cauchy sequence and requires a limit in this same space, not merely an ambient space. Testing finitely many sequences or terms is insufficient. The infinite sequence starts with prefix=[3, 0, 1] then remains forever at tail=2. Give (prefix,tail,N,limit,cauchy?,converges_in_space?).
View answer and explanation
([3, 0, 1], 2, 3, 2, Yes, Yes)
Choose N equal to the prefix length. All later terms coincide with the specified tail point, giving both Cauchy behavior and convergence inside this space. A finite prefix alone would not supply this certificate. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([3, 0, 1], 2, 3, 2, Yes, Yes).
Reviewed step-by-step derivation
Complete metric space · qf-mod27-complete-ultimately-constant-limit · candidate
38. An Explicit Infinite Tail Converges within Its Declared Space
Use the complete qualified finite metric table=[[0, 7, 9, 10], [7, 0, 10, 11], [9, 10, 0, 13], [10, 11, 13, 0]] on Fin(4)={0,…,3}, with at least two points for this minimum-positive-distance protocol. Completeness quantifies over every Cauchy sequence and requires a limit in this same space, not merely an ambient space. Testing finitely many sequences or terms is insufficient. The infinite sequence starts with prefix=[0, 1, 2] then remains forever at tail=3. Give (prefix,tail,N,limit,cauchy?,converges_in_space?).
View answer and explanation
([0, 1, 2], 3, 3, 3, Yes, Yes)
Choose N equal to the prefix length. All later terms coincide with the specified tail point, giving both Cauchy behavior and convergence inside this space. A finite prefix alone would not supply this certificate. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([0, 1, 2], 3, 3, 3, Yes, Yes).
Reviewed step-by-step derivation
Complete metric space · qf-mod27-complete-ultimately-constant-limit · candidate
39. An Explicit Infinite Tail Converges within Its Declared Space
Use the complete qualified finite metric table=[[0, 8, 12, 14], [8, 0, 12, 14], [12, 12, 0, 18], [14, 14, 18, 0]] on Fin(4)={0,…,3}, with at least two points for this minimum-positive-distance protocol. Completeness quantifies over every Cauchy sequence and requires a limit in this same space, not merely an ambient space. Testing finitely many sequences or terms is insufficient. The infinite sequence starts with prefix=[1, 2, 3] then remains forever at tail=0. Give (prefix,tail,N,limit,cauchy?,converges_in_space?).
View answer and explanation
([1, 2, 3], 0, 3, 0, Yes, Yes)
Choose N equal to the prefix length. All later terms coincide with the specified tail point, giving both Cauchy behavior and convergence inside this space. A finite prefix alone would not supply this certificate. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([1, 2, 3], 0, 3, 0, Yes, Yes).
Reviewed step-by-step derivation
Complete metric space · qf-mod27-complete-ultimately-constant-limit · candidate
40. An Explicit Infinite Tail Converges within Its Declared Space
Use the complete qualified finite metric table=[[0, 9, 8, 18], [9, 0, 7, 17], [8, 7, 0, 16], [18, 17, 16, 0]] on Fin(4)={0,…,3}, with at least two points for this minimum-positive-distance protocol. Completeness quantifies over every Cauchy sequence and requires a limit in this same space, not merely an ambient space. Testing finitely many sequences or terms is insufficient. The infinite sequence starts with prefix=[2, 3, 0] then remains forever at tail=1. Give (prefix,tail,N,limit,cauchy?,converges_in_space?).
View answer and explanation
([2, 3, 0], 1, 3, 1, Yes, Yes)
Choose N equal to the prefix length. All later terms coincide with the specified tail point, giving both Cauchy behavior and convergence inside this space. A finite prefix alone would not supply this certificate. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([2, 3, 0], 1, 3, 1, Yes, Yes).
Reviewed step-by-step derivation
Complete metric space · qf-mod27-complete-ultimately-constant-limit · candidate
41. An Explicit Infinite Tail Converges within Its Declared Space
Use the complete qualified finite metric table=[[0, 6, 6, 6], [6, 0, 10, 10], [6, 10, 0, 10], [6, 10, 10, 0]] on Fin(4)={0,…,3}, with at least two points for this minimum-positive-distance protocol. Completeness quantifies over every Cauchy sequence and requires a limit in this same space, not merely an ambient space. Testing finitely many sequences or terms is insufficient. The infinite sequence starts with prefix=[3, 0, 1] then remains forever at tail=2. Give (prefix,tail,N,limit,cauchy?,converges_in_space?).
View answer and explanation
([3, 0, 1], 2, 3, 2, Yes, Yes)
Choose N equal to the prefix length. All later terms coincide with the specified tail point, giving both Cauchy behavior and convergence inside this space. A finite prefix alone would not supply this certificate. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([3, 0, 1], 2, 3, 2, Yes, Yes).
Reviewed step-by-step derivation
Complete metric space · qf-mod27-complete-ultimately-constant-limit · candidate
42. An Explicit Infinite Tail Converges within Its Declared Space
Use the complete qualified finite metric table=[[0, 7, 9, 10], [7, 0, 12, 13], [9, 12, 0, 15], [10, 13, 15, 0]] on Fin(4)={0,…,3}, with at least two points for this minimum-positive-distance protocol. Completeness quantifies over every Cauchy sequence and requires a limit in this same space, not merely an ambient space. Testing finitely many sequences or terms is insufficient. The infinite sequence starts with prefix=[0, 1, 2] then remains forever at tail=3. Give (prefix,tail,N,limit,cauchy?,converges_in_space?).
View answer and explanation
([0, 1, 2], 3, 3, 3, Yes, Yes)
Choose N equal to the prefix length. All later terms coincide with the specified tail point, giving both Cauchy behavior and convergence inside this space. A finite prefix alone would not supply this certificate. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([0, 1, 2], 3, 3, 3, Yes, Yes).
Reviewed step-by-step derivation
Complete metric space · qf-mod27-complete-ultimately-constant-limit · candidate
43. An Explicit Infinite Tail Converges within Its Declared Space
Use the complete qualified finite metric table=[[0, 8, 12, 14], [8, 0, 14, 16], [12, 14, 0, 20], [14, 16, 20, 0]] on Fin(4)={0,…,3}, with at least two points for this minimum-positive-distance protocol. Completeness quantifies over every Cauchy sequence and requires a limit in this same space, not merely an ambient space. Testing finitely many sequences or terms is insufficient. The infinite sequence starts with prefix=[1, 2, 3] then remains forever at tail=0. Give (prefix,tail,N,limit,cauchy?,converges_in_space?).
View answer and explanation
([1, 2, 3], 0, 3, 0, Yes, Yes)
Choose N equal to the prefix length. All later terms coincide with the specified tail point, giving both Cauchy behavior and convergence inside this space. A finite prefix alone would not supply this certificate. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([1, 2, 3], 0, 3, 0, Yes, Yes).
Reviewed step-by-step derivation
Complete metric space · qf-mod27-complete-ultimately-constant-limit · candidate
44. An Explicit Infinite Tail Converges within Its Declared Space
Use the complete qualified finite metric table=[[0, 9, 8, 18], [9, 0, 9, 19], [8, 9, 0, 18], [18, 19, 18, 0]] on Fin(4)={0,…,3}, with at least two points for this minimum-positive-distance protocol. Completeness quantifies over every Cauchy sequence and requires a limit in this same space, not merely an ambient space. Testing finitely many sequences or terms is insufficient. The infinite sequence starts with prefix=[2, 3, 0] then remains forever at tail=1. Give (prefix,tail,N,limit,cauchy?,converges_in_space?).
View answer and explanation
([2, 3, 0], 1, 3, 1, Yes, Yes)
Choose N equal to the prefix length. All later terms coincide with the specified tail point, giving both Cauchy behavior and convergence inside this space. A finite prefix alone would not supply this certificate. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([2, 3, 0], 1, 3, 1, Yes, Yes).
Reviewed step-by-step derivation
Complete metric space · qf-mod27-complete-ultimately-constant-limit · candidate
45. An Explicit Infinite Tail Converges within Its Declared Space
Use the complete qualified finite metric table=[[0, 10, 11, 11], [10, 0, 11, 11], [11, 11, 0, 12], [11, 11, 12, 0]] on Fin(4)={0,…,3}, with at least two points for this minimum-positive-distance protocol. Completeness quantifies over every Cauchy sequence and requires a limit in this same space, not merely an ambient space. Testing finitely many sequences or terms is insufficient. The infinite sequence starts with prefix=[3, 0, 1] then remains forever at tail=2. Give (prefix,tail,N,limit,cauchy?,converges_in_space?).
View answer and explanation
([3, 0, 1], 2, 3, 2, Yes, Yes)
Choose N equal to the prefix length. All later terms coincide with the specified tail point, giving both Cauchy behavior and convergence inside this space. A finite prefix alone would not supply this certificate. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([3, 0, 1], 2, 3, 2, Yes, Yes).
Reviewed step-by-step derivation
Complete metric space · qf-mod27-complete-ultimately-constant-limit · candidate
46. An Explicit Infinite Tail Converges within Its Declared Space
Use the complete qualified finite metric table=[[0, 7, 9, 10], [7, 0, 14, 15], [9, 14, 0, 17], [10, 15, 17, 0]] on Fin(4)={0,…,3}, with at least two points for this minimum-positive-distance protocol. Completeness quantifies over every Cauchy sequence and requires a limit in this same space, not merely an ambient space. Testing finitely many sequences or terms is insufficient. The infinite sequence starts with prefix=[0, 1, 2] then remains forever at tail=3. Give (prefix,tail,N,limit,cauchy?,converges_in_space?).
View answer and explanation
([0, 1, 2], 3, 3, 3, Yes, Yes)
Choose N equal to the prefix length. All later terms coincide with the specified tail point, giving both Cauchy behavior and convergence inside this space. A finite prefix alone would not supply this certificate. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([0, 1, 2], 3, 3, 3, Yes, Yes).
Reviewed step-by-step derivation
Complete metric space · qf-mod27-complete-ultimately-constant-limit · candidate
47. An Explicit Infinite Tail Converges within Its Declared Space
Use the complete qualified finite metric table=[[0, 8, 5, 14], [8, 0, 9, 18], [5, 9, 0, 15], [14, 18, 15, 0]] on Fin(4)={0,…,3}, with at least two points for this minimum-positive-distance protocol. Completeness quantifies over every Cauchy sequence and requires a limit in this same space, not merely an ambient space. Testing finitely many sequences or terms is insufficient. The infinite sequence starts with prefix=[1, 2, 3] then remains forever at tail=0. Give (prefix,tail,N,limit,cauchy?,converges_in_space?).
View answer and explanation
([1, 2, 3], 0, 3, 0, Yes, Yes)
Choose N equal to the prefix length. All later terms coincide with the specified tail point, giving both Cauchy behavior and convergence inside this space. A finite prefix alone would not supply this certificate. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([1, 2, 3], 0, 3, 0, Yes, Yes).
Reviewed step-by-step derivation
Complete metric space · qf-mod27-complete-ultimately-constant-limit · candidate
48. An Explicit Infinite Tail Converges within Its Declared Space
Use the complete qualified finite metric table=[[0, 9, 8, 7], [9, 0, 11, 10], [8, 11, 0, 9], [7, 10, 9, 0]] on Fin(4)={0,…,3}, with at least two points for this minimum-positive-distance protocol. Completeness quantifies over every Cauchy sequence and requires a limit in this same space, not merely an ambient space. Testing finitely many sequences or terms is insufficient. The infinite sequence starts with prefix=[2, 3, 0] then remains forever at tail=1. Give (prefix,tail,N,limit,cauchy?,converges_in_space?).
View answer and explanation
([2, 3, 0], 1, 3, 1, Yes, Yes)
Choose N equal to the prefix length. All later terms coincide with the specified tail point, giving both Cauchy behavior and convergence inside this space. A finite prefix alone would not supply this certificate. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([2, 3, 0], 1, 3, 1, Yes, Yes).
Reviewed step-by-step derivation
Complete metric space · qf-mod27-complete-ultimately-constant-limit · candidate
49. An Explicit Infinite Tail Converges within Its Declared Space
Use the complete qualified finite metric table=[[0, 10, 11, 11], [10, 0, 13, 13], [11, 13, 0, 14], [11, 13, 14, 0]] on Fin(4)={0,…,3}, with at least two points for this minimum-positive-distance protocol. Completeness quantifies over every Cauchy sequence and requires a limit in this same space, not merely an ambient space. Testing finitely many sequences or terms is insufficient. The infinite sequence starts with prefix=[3, 0, 1] then remains forever at tail=2. Give (prefix,tail,N,limit,cauchy?,converges_in_space?).
View answer and explanation
([3, 0, 1], 2, 3, 2, Yes, Yes)
Choose N equal to the prefix length. All later terms coincide with the specified tail point, giving both Cauchy behavior and convergence inside this space. A finite prefix alone would not supply this certificate. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([3, 0, 1], 2, 3, 2, Yes, Yes).
Reviewed step-by-step derivation
Complete metric space · qf-mod27-complete-ultimately-constant-limit · candidate
50. An Explicit Infinite Tail Converges within Its Declared Space
Use the complete qualified finite metric table=[[0, 11, 14, 15], [11, 0, 15, 16], [14, 15, 0, 19], [15, 16, 19, 0]] on Fin(4)={0,…,3}, with at least two points for this minimum-positive-distance protocol. Completeness quantifies over every Cauchy sequence and requires a limit in this same space, not merely an ambient space. Testing finitely many sequences or terms is insufficient. The infinite sequence starts with prefix=[0, 1, 2] then remains forever at tail=3. Give (prefix,tail,N,limit,cauchy?,converges_in_space?).
View answer and explanation
([0, 1, 2], 3, 3, 3, Yes, Yes)
Choose N equal to the prefix length. All later terms coincide with the specified tail point, giving both Cauchy behavior and convergence inside this space. A finite prefix alone would not supply this certificate. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([0, 1, 2], 3, 3, 3, Yes, Yes).
Reviewed step-by-step derivation
Complete metric space · qf-mod27-complete-ultimately-constant-limit · candidate
51. A Non-Cauchy Sequence Does Not Contradict Space Completeness
Use the complete qualified finite metric table=[[0, 3, 4, 5], [3, 0, 5, 6], [4, 5, 0, 7], [5, 6, 7, 0]] on Fin(4)={0,…,3}, with at least two points for this minimum-positive-distance protocol. Completeness quantifies over every Cauchy sequence and requires a limit in this same space, not merely an ambient space. Testing finitely many sequences or terms is insufficient. The infinite sequence alternates forever between distinct left=0 and right=1, with arbitrary proposed cutoff N=1. Give [left,right,N,next_index,epsilon,distance,counterexample,cauchy] as a tuple. Choose next_index=N+1 and epsilon=d(left,right)>0. Retain the strict Cauchy condition on every pair of later terms.
View answer and explanation
(0, 1, 1, 2, 3, 3, Yes, No)
The adjacent terms at N and N+1 are the two distinct points in one order, so symmetry gives distance=epsilon, not distance<epsilon. The same fixed positive epsilon defeats every cutoff: counterexample=Yes and cauchy=No. Being non-Cauchy, this sequence does not refute completeness of the space, which concerns only Cauchy sequences. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (0, 1, 1, 2, 3, 3, Yes, No).
Reviewed step-by-step derivation
Complete metric space · qf-mod27-complete-alternating-noncauchy · candidate
52. A Non-Cauchy Sequence Does Not Contradict Space Completeness
Use the complete qualified finite metric table=[[0, 4, 7, 9], [4, 0, 7, 9], [7, 7, 0, 12], [9, 9, 12, 0]] on Fin(4)={0,…,3}, with at least two points for this minimum-positive-distance protocol. Completeness quantifies over every Cauchy sequence and requires a limit in this same space, not merely an ambient space. Testing finitely many sequences or terms is insufficient. The infinite sequence alternates forever between distinct left=1 and right=2, with arbitrary proposed cutoff N=3. Give [left,right,N,next_index,epsilon,distance,counterexample,cauchy] as a tuple. Choose next_index=N+1 and epsilon=d(left,right)>0. Retain the strict Cauchy condition on every pair of later terms.
View answer and explanation
(1, 2, 3, 4, 7, 7, Yes, No)
The adjacent terms at N and N+1 are the two distinct points in one order, so symmetry gives distance=epsilon, not distance<epsilon. The same fixed positive epsilon defeats every cutoff: counterexample=Yes and cauchy=No. Being non-Cauchy, this sequence does not refute completeness of the space, which concerns only Cauchy sequences. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (1, 2, 3, 4, 7, 7, Yes, No).
Reviewed step-by-step derivation
Complete metric space · qf-mod27-complete-alternating-noncauchy · candidate
53. A Non-Cauchy Sequence Does Not Contradict Space Completeness
Use the complete qualified finite metric table=[[0, 5, 10, 13], [5, 0, 9, 12], [10, 9, 0, 17], [13, 12, 17, 0]] on Fin(4)={0,…,3}, with at least two points for this minimum-positive-distance protocol. Completeness quantifies over every Cauchy sequence and requires a limit in this same space, not merely an ambient space. Testing finitely many sequences or terms is insufficient. The infinite sequence alternates forever between distinct left=2 and right=3, with arbitrary proposed cutoff N=5. Give [left,right,N,next_index,epsilon,distance,counterexample,cauchy] as a tuple. Choose next_index=N+1 and epsilon=d(left,right)>0. Retain the strict Cauchy condition on every pair of later terms.
View answer and explanation
(2, 3, 5, 6, 17, 17, Yes, No)
The adjacent terms at N and N+1 are the two distinct points in one order, so symmetry gives distance=epsilon, not distance<epsilon. The same fixed positive epsilon defeats every cutoff: counterexample=Yes and cauchy=No. Being non-Cauchy, this sequence does not refute completeness of the space, which concerns only Cauchy sequences. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (2, 3, 5, 6, 17, 17, Yes, No).
Reviewed step-by-step derivation
Complete metric space · qf-mod27-complete-alternating-noncauchy · candidate
54. A Non-Cauchy Sequence Does Not Contradict Space Completeness
Use the complete qualified finite metric table=[[0, 6, 13, 17], [6, 0, 11, 15], [13, 11, 0, 22], [17, 15, 22, 0]] on Fin(4)={0,…,3}, with at least two points for this minimum-positive-distance protocol. Completeness quantifies over every Cauchy sequence and requires a limit in this same space, not merely an ambient space. Testing finitely many sequences or terms is insufficient. The infinite sequence alternates forever between distinct left=3 and right=0, with arbitrary proposed cutoff N=7. Give [left,right,N,next_index,epsilon,distance,counterexample,cauchy] as a tuple. Choose next_index=N+1 and epsilon=d(left,right)>0. Retain the strict Cauchy condition on every pair of later terms.
View answer and explanation
(3, 0, 7, 8, 17, 17, Yes, No)
The adjacent terms at N and N+1 are the two distinct points in one order, so symmetry gives distance=epsilon, not distance<epsilon. The same fixed positive epsilon defeats every cutoff: counterexample=Yes and cauchy=No. Being non-Cauchy, this sequence does not refute completeness of the space, which concerns only Cauchy sequences. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (3, 0, 7, 8, 17, 17, Yes, No).
Reviewed step-by-step derivation
Complete metric space · qf-mod27-complete-alternating-noncauchy · candidate
55. A Non-Cauchy Sequence Does Not Contradict Space Completeness
Use the complete qualified finite metric table=[[0, 7, 9, 10], [7, 0, 6, 7], [9, 6, 0, 9], [10, 7, 9, 0]] on Fin(4)={0,…,3}, with at least two points for this minimum-positive-distance protocol. Completeness quantifies over every Cauchy sequence and requires a limit in this same space, not merely an ambient space. Testing finitely many sequences or terms is insufficient. The infinite sequence alternates forever between distinct left=0 and right=1, with arbitrary proposed cutoff N=9. Give [left,right,N,next_index,epsilon,distance,counterexample,cauchy] as a tuple. Choose next_index=N+1 and epsilon=d(left,right)>0. Retain the strict Cauchy condition on every pair of later terms.
View answer and explanation
(0, 1, 9, 10, 7, 7, Yes, No)
The adjacent terms at N and N+1 are the two distinct points in one order, so symmetry gives distance=epsilon, not distance<epsilon. The same fixed positive epsilon defeats every cutoff: counterexample=Yes and cauchy=No. Being non-Cauchy, this sequence does not refute completeness of the space, which concerns only Cauchy sequences. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (0, 1, 9, 10, 7, 7, Yes, No).
Reviewed step-by-step derivation
Complete metric space · qf-mod27-complete-alternating-noncauchy · candidate
56. A Non-Cauchy Sequence Does Not Contradict Space Completeness
Use the complete qualified finite metric table=[[0, 4, 7, 9], [4, 0, 9, 11], [7, 9, 0, 14], [9, 11, 14, 0]] on Fin(4)={0,…,3}, with at least two points for this minimum-positive-distance protocol. Completeness quantifies over every Cauchy sequence and requires a limit in this same space, not merely an ambient space. Testing finitely many sequences or terms is insufficient. The infinite sequence alternates forever between distinct left=1 and right=2, with arbitrary proposed cutoff N=11. Give [left,right,N,next_index,epsilon,distance,counterexample,cauchy] as a tuple. Choose next_index=N+1 and epsilon=d(left,right)>0. Retain the strict Cauchy condition on every pair of later terms.
View answer and explanation
(1, 2, 11, 12, 9, 9, Yes, No)
The adjacent terms at N and N+1 are the two distinct points in one order, so symmetry gives distance=epsilon, not distance<epsilon. The same fixed positive epsilon defeats every cutoff: counterexample=Yes and cauchy=No. Being non-Cauchy, this sequence does not refute completeness of the space, which concerns only Cauchy sequences. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (1, 2, 11, 12, 9, 9, Yes, No).
Reviewed step-by-step derivation
Complete metric space · qf-mod27-complete-alternating-noncauchy · candidate
57. A Non-Cauchy Sequence Does Not Contradict Space Completeness
Use the complete qualified finite metric table=[[0, 5, 10, 13], [5, 0, 11, 14], [10, 11, 0, 19], [13, 14, 19, 0]] on Fin(4)={0,…,3}, with at least two points for this minimum-positive-distance protocol. Completeness quantifies over every Cauchy sequence and requires a limit in this same space, not merely an ambient space. Testing finitely many sequences or terms is insufficient. The infinite sequence alternates forever between distinct left=2 and right=3, with arbitrary proposed cutoff N=13. Give [left,right,N,next_index,epsilon,distance,counterexample,cauchy] as a tuple. Choose next_index=N+1 and epsilon=d(left,right)>0. Retain the strict Cauchy condition on every pair of later terms.
View answer and explanation
(2, 3, 13, 14, 19, 19, Yes, No)
The adjacent terms at N and N+1 are the two distinct points in one order, so symmetry gives distance=epsilon, not distance<epsilon. The same fixed positive epsilon defeats every cutoff: counterexample=Yes and cauchy=No. Being non-Cauchy, this sequence does not refute completeness of the space, which concerns only Cauchy sequences. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (2, 3, 13, 14, 19, 19, Yes, No).
Reviewed step-by-step derivation
Complete metric space · qf-mod27-complete-alternating-noncauchy · candidate
58. A Non-Cauchy Sequence Does Not Contradict Space Completeness
Use the complete qualified finite metric table=[[0, 6, 6, 17], [6, 0, 6, 17], [6, 6, 0, 17], [17, 17, 17, 0]] on Fin(4)={0,…,3}, with at least two points for this minimum-positive-distance protocol. Completeness quantifies over every Cauchy sequence and requires a limit in this same space, not merely an ambient space. Testing finitely many sequences or terms is insufficient. The infinite sequence alternates forever between distinct left=3 and right=0, with arbitrary proposed cutoff N=15. Give [left,right,N,next_index,epsilon,distance,counterexample,cauchy] as a tuple. Choose next_index=N+1 and epsilon=d(left,right)>0. Retain the strict Cauchy condition on every pair of later terms.
View answer and explanation
(3, 0, 15, 16, 17, 17, Yes, No)
The adjacent terms at N and N+1 are the two distinct points in one order, so symmetry gives distance=epsilon, not distance<epsilon. The same fixed positive epsilon defeats every cutoff: counterexample=Yes and cauchy=No. Being non-Cauchy, this sequence does not refute completeness of the space, which concerns only Cauchy sequences. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (3, 0, 15, 16, 17, 17, Yes, No).
Reviewed step-by-step derivation
Complete metric space · qf-mod27-complete-alternating-noncauchy · candidate
59. A Non-Cauchy Sequence Does Not Contradict Space Completeness
Use the complete qualified finite metric table=[[0, 7, 9, 10], [7, 0, 8, 9], [9, 8, 0, 11], [10, 9, 11, 0]] on Fin(4)={0,…,3}, with at least two points for this minimum-positive-distance protocol. Completeness quantifies over every Cauchy sequence and requires a limit in this same space, not merely an ambient space. Testing finitely many sequences or terms is insufficient. The infinite sequence alternates forever between distinct left=0 and right=1, with arbitrary proposed cutoff N=17. Give [left,right,N,next_index,epsilon,distance,counterexample,cauchy] as a tuple. Choose next_index=N+1 and epsilon=d(left,right)>0. Retain the strict Cauchy condition on every pair of later terms.
View answer and explanation
(0, 1, 17, 18, 7, 7, Yes, No)
The adjacent terms at N and N+1 are the two distinct points in one order, so symmetry gives distance=epsilon, not distance<epsilon. The same fixed positive epsilon defeats every cutoff: counterexample=Yes and cauchy=No. Being non-Cauchy, this sequence does not refute completeness of the space, which concerns only Cauchy sequences. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (0, 1, 17, 18, 7, 7, Yes, No).
Reviewed step-by-step derivation
Complete metric space · qf-mod27-complete-alternating-noncauchy · candidate
60. A Non-Cauchy Sequence Does Not Contradict Space Completeness
Use the complete qualified finite metric table=[[0, 8, 12, 14], [8, 0, 10, 12], [12, 10, 0, 16], [14, 12, 16, 0]] on Fin(4)={0,…,3}, with at least two points for this minimum-positive-distance protocol. Completeness quantifies over every Cauchy sequence and requires a limit in this same space, not merely an ambient space. Testing finitely many sequences or terms is insufficient. The infinite sequence alternates forever between distinct left=1 and right=2, with arbitrary proposed cutoff N=19. Give [left,right,N,next_index,epsilon,distance,counterexample,cauchy] as a tuple. Choose next_index=N+1 and epsilon=d(left,right)>0. Retain the strict Cauchy condition on every pair of later terms.
View answer and explanation
(1, 2, 19, 20, 10, 10, Yes, No)
The adjacent terms at N and N+1 are the two distinct points in one order, so symmetry gives distance=epsilon, not distance<epsilon. The same fixed positive epsilon defeats every cutoff: counterexample=Yes and cauchy=No. Being non-Cauchy, this sequence does not refute completeness of the space, which concerns only Cauchy sequences. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (1, 2, 19, 20, 10, 10, Yes, No).
Reviewed step-by-step derivation
Complete metric space · qf-mod27-complete-alternating-noncauchy · candidate
61. A Non-Cauchy Sequence Does Not Contradict Space Completeness
Use the complete qualified finite metric table=[[0, 5, 10, 13], [5, 0, 13, 16], [10, 13, 0, 21], [13, 16, 21, 0]] on Fin(4)={0,…,3}, with at least two points for this minimum-positive-distance protocol. Completeness quantifies over every Cauchy sequence and requires a limit in this same space, not merely an ambient space. Testing finitely many sequences or terms is insufficient. The infinite sequence alternates forever between distinct left=2 and right=3, with arbitrary proposed cutoff N=21. Give [left,right,N,next_index,epsilon,distance,counterexample,cauchy] as a tuple. Choose next_index=N+1 and epsilon=d(left,right)>0. Retain the strict Cauchy condition on every pair of later terms.
View answer and explanation
(2, 3, 21, 22, 21, 21, Yes, No)
The adjacent terms at N and N+1 are the two distinct points in one order, so symmetry gives distance=epsilon, not distance<epsilon. The same fixed positive epsilon defeats every cutoff: counterexample=Yes and cauchy=No. Being non-Cauchy, this sequence does not refute completeness of the space, which concerns only Cauchy sequences. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (2, 3, 21, 22, 21, 21, Yes, No).
Reviewed step-by-step derivation
Complete metric space · qf-mod27-complete-alternating-noncauchy · candidate
62. A Non-Cauchy Sequence Does Not Contradict Space Completeness
Use the complete qualified finite metric table=[[0, 6, 6, 6], [6, 0, 8, 8], [6, 8, 0, 8], [6, 8, 8, 0]] on Fin(4)={0,…,3}, with at least two points for this minimum-positive-distance protocol. Completeness quantifies over every Cauchy sequence and requires a limit in this same space, not merely an ambient space. Testing finitely many sequences or terms is insufficient. The infinite sequence alternates forever between distinct left=3 and right=0, with arbitrary proposed cutoff N=23. Give [left,right,N,next_index,epsilon,distance,counterexample,cauchy] as a tuple. Choose next_index=N+1 and epsilon=d(left,right)>0. Retain the strict Cauchy condition on every pair of later terms.
View answer and explanation
(3, 0, 23, 24, 6, 6, Yes, No)
The adjacent terms at N and N+1 are the two distinct points in one order, so symmetry gives distance=epsilon, not distance<epsilon. The same fixed positive epsilon defeats every cutoff: counterexample=Yes and cauchy=No. Being non-Cauchy, this sequence does not refute completeness of the space, which concerns only Cauchy sequences. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (3, 0, 23, 24, 6, 6, Yes, No).
Reviewed step-by-step derivation
Complete metric space · qf-mod27-complete-alternating-noncauchy · candidate
63. A Non-Cauchy Sequence Does Not Contradict Space Completeness
Use the complete qualified finite metric table=[[0, 7, 9, 10], [7, 0, 10, 11], [9, 10, 0, 13], [10, 11, 13, 0]] on Fin(4)={0,…,3}, with at least two points for this minimum-positive-distance protocol. Completeness quantifies over every Cauchy sequence and requires a limit in this same space, not merely an ambient space. Testing finitely many sequences or terms is insufficient. The infinite sequence alternates forever between distinct left=0 and right=1, with arbitrary proposed cutoff N=25. Give [left,right,N,next_index,epsilon,distance,counterexample,cauchy] as a tuple. Choose next_index=N+1 and epsilon=d(left,right)>0. Retain the strict Cauchy condition on every pair of later terms.
View answer and explanation
(0, 1, 25, 26, 7, 7, Yes, No)
The adjacent terms at N and N+1 are the two distinct points in one order, so symmetry gives distance=epsilon, not distance<epsilon. The same fixed positive epsilon defeats every cutoff: counterexample=Yes and cauchy=No. Being non-Cauchy, this sequence does not refute completeness of the space, which concerns only Cauchy sequences. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (0, 1, 25, 26, 7, 7, Yes, No).
Reviewed step-by-step derivation
Complete metric space · qf-mod27-complete-alternating-noncauchy · candidate
64. A Non-Cauchy Sequence Does Not Contradict Space Completeness
Use the complete qualified finite metric table=[[0, 8, 12, 14], [8, 0, 12, 14], [12, 12, 0, 18], [14, 14, 18, 0]] on Fin(4)={0,…,3}, with at least two points for this minimum-positive-distance protocol. Completeness quantifies over every Cauchy sequence and requires a limit in this same space, not merely an ambient space. Testing finitely many sequences or terms is insufficient. The infinite sequence alternates forever between distinct left=1 and right=2, with arbitrary proposed cutoff N=27. Give [left,right,N,next_index,epsilon,distance,counterexample,cauchy] as a tuple. Choose next_index=N+1 and epsilon=d(left,right)>0. Retain the strict Cauchy condition on every pair of later terms.
View answer and explanation
(1, 2, 27, 28, 12, 12, Yes, No)
The adjacent terms at N and N+1 are the two distinct points in one order, so symmetry gives distance=epsilon, not distance<epsilon. The same fixed positive epsilon defeats every cutoff: counterexample=Yes and cauchy=No. Being non-Cauchy, this sequence does not refute completeness of the space, which concerns only Cauchy sequences. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (1, 2, 27, 28, 12, 12, Yes, No).
Reviewed step-by-step derivation
Complete metric space · qf-mod27-complete-alternating-noncauchy · candidate
65. A Non-Cauchy Sequence Does Not Contradict Space Completeness
Use the complete qualified finite metric table=[[0, 9, 8, 18], [9, 0, 7, 17], [8, 7, 0, 16], [18, 17, 16, 0]] on Fin(4)={0,…,3}, with at least two points for this minimum-positive-distance protocol. Completeness quantifies over every Cauchy sequence and requires a limit in this same space, not merely an ambient space. Testing finitely many sequences or terms is insufficient. The infinite sequence alternates forever between distinct left=2 and right=3, with arbitrary proposed cutoff N=29. Give [left,right,N,next_index,epsilon,distance,counterexample,cauchy] as a tuple. Choose next_index=N+1 and epsilon=d(left,right)>0. Retain the strict Cauchy condition on every pair of later terms.
View answer and explanation
(2, 3, 29, 30, 16, 16, Yes, No)
The adjacent terms at N and N+1 are the two distinct points in one order, so symmetry gives distance=epsilon, not distance<epsilon. The same fixed positive epsilon defeats every cutoff: counterexample=Yes and cauchy=No. Being non-Cauchy, this sequence does not refute completeness of the space, which concerns only Cauchy sequences. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (2, 3, 29, 30, 16, 16, Yes, No).
Reviewed step-by-step derivation
Complete metric space · qf-mod27-complete-alternating-noncauchy · candidate
66. A Non-Cauchy Sequence Does Not Contradict Space Completeness
Use the complete qualified finite metric table=[[0, 6, 6, 6], [6, 0, 10, 10], [6, 10, 0, 10], [6, 10, 10, 0]] on Fin(4)={0,…,3}, with at least two points for this minimum-positive-distance protocol. Completeness quantifies over every Cauchy sequence and requires a limit in this same space, not merely an ambient space. Testing finitely many sequences or terms is insufficient. The infinite sequence alternates forever between distinct left=3 and right=0, with arbitrary proposed cutoff N=31. Give [left,right,N,next_index,epsilon,distance,counterexample,cauchy] as a tuple. Choose next_index=N+1 and epsilon=d(left,right)>0. Retain the strict Cauchy condition on every pair of later terms.
View answer and explanation
(3, 0, 31, 32, 6, 6, Yes, No)
The adjacent terms at N and N+1 are the two distinct points in one order, so symmetry gives distance=epsilon, not distance<epsilon. The same fixed positive epsilon defeats every cutoff: counterexample=Yes and cauchy=No. Being non-Cauchy, this sequence does not refute completeness of the space, which concerns only Cauchy sequences. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (3, 0, 31, 32, 6, 6, Yes, No).
Reviewed step-by-step derivation
Complete metric space · qf-mod27-complete-alternating-noncauchy · candidate
67. A Non-Cauchy Sequence Does Not Contradict Space Completeness
Use the complete qualified finite metric table=[[0, 7, 9, 10], [7, 0, 12, 13], [9, 12, 0, 15], [10, 13, 15, 0]] on Fin(4)={0,…,3}, with at least two points for this minimum-positive-distance protocol. Completeness quantifies over every Cauchy sequence and requires a limit in this same space, not merely an ambient space. Testing finitely many sequences or terms is insufficient. The infinite sequence alternates forever between distinct left=0 and right=1, with arbitrary proposed cutoff N=33. Give [left,right,N,next_index,epsilon,distance,counterexample,cauchy] as a tuple. Choose next_index=N+1 and epsilon=d(left,right)>0. Retain the strict Cauchy condition on every pair of later terms.
View answer and explanation
(0, 1, 33, 34, 7, 7, Yes, No)
The adjacent terms at N and N+1 are the two distinct points in one order, so symmetry gives distance=epsilon, not distance<epsilon. The same fixed positive epsilon defeats every cutoff: counterexample=Yes and cauchy=No. Being non-Cauchy, this sequence does not refute completeness of the space, which concerns only Cauchy sequences. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (0, 1, 33, 34, 7, 7, Yes, No).
Reviewed step-by-step derivation
Complete metric space · qf-mod27-complete-alternating-noncauchy · candidate
68. A Non-Cauchy Sequence Does Not Contradict Space Completeness
Use the complete qualified finite metric table=[[0, 8, 12, 14], [8, 0, 14, 16], [12, 14, 0, 20], [14, 16, 20, 0]] on Fin(4)={0,…,3}, with at least two points for this minimum-positive-distance protocol. Completeness quantifies over every Cauchy sequence and requires a limit in this same space, not merely an ambient space. Testing finitely many sequences or terms is insufficient. The infinite sequence alternates forever between distinct left=1 and right=2, with arbitrary proposed cutoff N=35. Give [left,right,N,next_index,epsilon,distance,counterexample,cauchy] as a tuple. Choose next_index=N+1 and epsilon=d(left,right)>0. Retain the strict Cauchy condition on every pair of later terms.
View answer and explanation
(1, 2, 35, 36, 14, 14, Yes, No)
The adjacent terms at N and N+1 are the two distinct points in one order, so symmetry gives distance=epsilon, not distance<epsilon. The same fixed positive epsilon defeats every cutoff: counterexample=Yes and cauchy=No. Being non-Cauchy, this sequence does not refute completeness of the space, which concerns only Cauchy sequences. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (1, 2, 35, 36, 14, 14, Yes, No).
Reviewed step-by-step derivation
Complete metric space · qf-mod27-complete-alternating-noncauchy · candidate
69. A Non-Cauchy Sequence Does Not Contradict Space Completeness
Use the complete qualified finite metric table=[[0, 9, 8, 18], [9, 0, 9, 19], [8, 9, 0, 18], [18, 19, 18, 0]] on Fin(4)={0,…,3}, with at least two points for this minimum-positive-distance protocol. Completeness quantifies over every Cauchy sequence and requires a limit in this same space, not merely an ambient space. Testing finitely many sequences or terms is insufficient. The infinite sequence alternates forever between distinct left=2 and right=3, with arbitrary proposed cutoff N=37. Give [left,right,N,next_index,epsilon,distance,counterexample,cauchy] as a tuple. Choose next_index=N+1 and epsilon=d(left,right)>0. Retain the strict Cauchy condition on every pair of later terms.
View answer and explanation
(2, 3, 37, 38, 18, 18, Yes, No)
The adjacent terms at N and N+1 are the two distinct points in one order, so symmetry gives distance=epsilon, not distance<epsilon. The same fixed positive epsilon defeats every cutoff: counterexample=Yes and cauchy=No. Being non-Cauchy, this sequence does not refute completeness of the space, which concerns only Cauchy sequences. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (2, 3, 37, 38, 18, 18, Yes, No).
Reviewed step-by-step derivation
Complete metric space · qf-mod27-complete-alternating-noncauchy · candidate
70. A Non-Cauchy Sequence Does Not Contradict Space Completeness
Use the complete qualified finite metric table=[[0, 10, 11, 11], [10, 0, 11, 11], [11, 11, 0, 12], [11, 11, 12, 0]] on Fin(4)={0,…,3}, with at least two points for this minimum-positive-distance protocol. Completeness quantifies over every Cauchy sequence and requires a limit in this same space, not merely an ambient space. Testing finitely many sequences or terms is insufficient. The infinite sequence alternates forever between distinct left=3 and right=0, with arbitrary proposed cutoff N=39. Give [left,right,N,next_index,epsilon,distance,counterexample,cauchy] as a tuple. Choose next_index=N+1 and epsilon=d(left,right)>0. Retain the strict Cauchy condition on every pair of later terms.
View answer and explanation
(3, 0, 39, 40, 11, 11, Yes, No)
The adjacent terms at N and N+1 are the two distinct points in one order, so symmetry gives distance=epsilon, not distance<epsilon. The same fixed positive epsilon defeats every cutoff: counterexample=Yes and cauchy=No. Being non-Cauchy, this sequence does not refute completeness of the space, which concerns only Cauchy sequences. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (3, 0, 39, 40, 11, 11, Yes, No).
Reviewed step-by-step derivation
Complete metric space · qf-mod27-complete-alternating-noncauchy · candidate
71. A Non-Cauchy Sequence Does Not Contradict Space Completeness
Use the complete qualified finite metric table=[[0, 7, 9, 10], [7, 0, 14, 15], [9, 14, 0, 17], [10, 15, 17, 0]] on Fin(4)={0,…,3}, with at least two points for this minimum-positive-distance protocol. Completeness quantifies over every Cauchy sequence and requires a limit in this same space, not merely an ambient space. Testing finitely many sequences or terms is insufficient. The infinite sequence alternates forever between distinct left=0 and right=1, with arbitrary proposed cutoff N=41. Give [left,right,N,next_index,epsilon,distance,counterexample,cauchy] as a tuple. Choose next_index=N+1 and epsilon=d(left,right)>0. Retain the strict Cauchy condition on every pair of later terms.
View answer and explanation
(0, 1, 41, 42, 7, 7, Yes, No)
The adjacent terms at N and N+1 are the two distinct points in one order, so symmetry gives distance=epsilon, not distance<epsilon. The same fixed positive epsilon defeats every cutoff: counterexample=Yes and cauchy=No. Being non-Cauchy, this sequence does not refute completeness of the space, which concerns only Cauchy sequences. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (0, 1, 41, 42, 7, 7, Yes, No).
Reviewed step-by-step derivation
Complete metric space · qf-mod27-complete-alternating-noncauchy · candidate
72. A Non-Cauchy Sequence Does Not Contradict Space Completeness
Use the complete qualified finite metric table=[[0, 8, 5, 14], [8, 0, 9, 18], [5, 9, 0, 15], [14, 18, 15, 0]] on Fin(4)={0,…,3}, with at least two points for this minimum-positive-distance protocol. Completeness quantifies over every Cauchy sequence and requires a limit in this same space, not merely an ambient space. Testing finitely many sequences or terms is insufficient. The infinite sequence alternates forever between distinct left=1 and right=2, with arbitrary proposed cutoff N=43. Give [left,right,N,next_index,epsilon,distance,counterexample,cauchy] as a tuple. Choose next_index=N+1 and epsilon=d(left,right)>0. Retain the strict Cauchy condition on every pair of later terms.
View answer and explanation
(1, 2, 43, 44, 9, 9, Yes, No)
The adjacent terms at N and N+1 are the two distinct points in one order, so symmetry gives distance=epsilon, not distance<epsilon. The same fixed positive epsilon defeats every cutoff: counterexample=Yes and cauchy=No. Being non-Cauchy, this sequence does not refute completeness of the space, which concerns only Cauchy sequences. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (1, 2, 43, 44, 9, 9, Yes, No).
Reviewed step-by-step derivation
Complete metric space · qf-mod27-complete-alternating-noncauchy · candidate
73. A Non-Cauchy Sequence Does Not Contradict Space Completeness
Use the complete qualified finite metric table=[[0, 9, 8, 7], [9, 0, 11, 10], [8, 11, 0, 9], [7, 10, 9, 0]] on Fin(4)={0,…,3}, with at least two points for this minimum-positive-distance protocol. Completeness quantifies over every Cauchy sequence and requires a limit in this same space, not merely an ambient space. Testing finitely many sequences or terms is insufficient. The infinite sequence alternates forever between distinct left=2 and right=3, with arbitrary proposed cutoff N=45. Give [left,right,N,next_index,epsilon,distance,counterexample,cauchy] as a tuple. Choose next_index=N+1 and epsilon=d(left,right)>0. Retain the strict Cauchy condition on every pair of later terms.
View answer and explanation
(2, 3, 45, 46, 9, 9, Yes, No)
The adjacent terms at N and N+1 are the two distinct points in one order, so symmetry gives distance=epsilon, not distance<epsilon. The same fixed positive epsilon defeats every cutoff: counterexample=Yes and cauchy=No. Being non-Cauchy, this sequence does not refute completeness of the space, which concerns only Cauchy sequences. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (2, 3, 45, 46, 9, 9, Yes, No).
Reviewed step-by-step derivation
Complete metric space · qf-mod27-complete-alternating-noncauchy · candidate
74. A Non-Cauchy Sequence Does Not Contradict Space Completeness
Use the complete qualified finite metric table=[[0, 10, 11, 11], [10, 0, 13, 13], [11, 13, 0, 14], [11, 13, 14, 0]] on Fin(4)={0,…,3}, with at least two points for this minimum-positive-distance protocol. Completeness quantifies over every Cauchy sequence and requires a limit in this same space, not merely an ambient space. Testing finitely many sequences or terms is insufficient. The infinite sequence alternates forever between distinct left=3 and right=0, with arbitrary proposed cutoff N=47. Give [left,right,N,next_index,epsilon,distance,counterexample,cauchy] as a tuple. Choose next_index=N+1 and epsilon=d(left,right)>0. Retain the strict Cauchy condition on every pair of later terms.
View answer and explanation
(3, 0, 47, 48, 11, 11, Yes, No)
The adjacent terms at N and N+1 are the two distinct points in one order, so symmetry gives distance=epsilon, not distance<epsilon. The same fixed positive epsilon defeats every cutoff: counterexample=Yes and cauchy=No. Being non-Cauchy, this sequence does not refute completeness of the space, which concerns only Cauchy sequences. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (3, 0, 47, 48, 11, 11, Yes, No).
Reviewed step-by-step derivation
Complete metric space · qf-mod27-complete-alternating-noncauchy · candidate
75. A Non-Cauchy Sequence Does Not Contradict Space Completeness
Use the complete qualified finite metric table=[[0, 11, 14, 15], [11, 0, 15, 16], [14, 15, 0, 19], [15, 16, 19, 0]] on Fin(4)={0,…,3}, with at least two points for this minimum-positive-distance protocol. Completeness quantifies over every Cauchy sequence and requires a limit in this same space, not merely an ambient space. Testing finitely many sequences or terms is insufficient. The infinite sequence alternates forever between distinct left=0 and right=1, with arbitrary proposed cutoff N=49. Give [left,right,N,next_index,epsilon,distance,counterexample,cauchy] as a tuple. Choose next_index=N+1 and epsilon=d(left,right)>0. Retain the strict Cauchy condition on every pair of later terms.
View answer and explanation
(0, 1, 49, 50, 11, 11, Yes, No)
The adjacent terms at N and N+1 are the two distinct points in one order, so symmetry gives distance=epsilon, not distance<epsilon. The same fixed positive epsilon defeats every cutoff: counterexample=Yes and cauchy=No. Being non-Cauchy, this sequence does not refute completeness of the space, which concerns only Cauchy sequences. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (0, 1, 49, 50, 11, 11, Yes, No).
Reviewed step-by-step derivation
Complete metric space · qf-mod27-complete-alternating-noncauchy · candidate
76. A Space-Wide Certificate Is Not a List of Successful Sequences
Use the complete qualified finite metric table=[[0, 3, 4, 5], [3, 0, 5, 6], [4, 5, 0, 7], [5, 6, 7, 0]] on Fin(4)={0,…,3}, with at least two points for this minimum-positive-distance protocol. Completeness quantifies over every Cauchy sequence and requires a limit in this same space, not merely an ambient space. Testing finitely many sequences or terms is insufficient. evidence_mode=sampled_sequences. Give (n,min_positive_distance,evidence_mode,sufficient?,boundary). finite_minimum_separation yields universal_finite_certificate; sampled_sequences yields sampled_sequences_insufficient.
View answer and explanation
(4, 3, sampled_sequences, No, sampled_sequences_insufficient)
Compute the same exact separation in both modes, but distinguish the universal argument from a proposed sample-only justification. Insufficient evidence is not a claim that the qualified finite space is incomplete. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (4, 3, sampled_sequences, No, sampled_sequences_insufficient).
Reviewed step-by-step derivation
Complete metric space · qf-mod27-complete-evidence-mode-audit · candidate
77. A Space-Wide Certificate Is Not a List of Successful Sequences
Use the complete qualified finite metric table=[[0, 4, 7, 9], [4, 0, 7, 9], [7, 7, 0, 12], [9, 9, 12, 0]] on Fin(4)={0,…,3}, with at least two points for this minimum-positive-distance protocol. Completeness quantifies over every Cauchy sequence and requires a limit in this same space, not merely an ambient space. Testing finitely many sequences or terms is insufficient. evidence_mode=finite_minimum_separation. Give (n,min_positive_distance,evidence_mode,sufficient?,boundary). finite_minimum_separation yields universal_finite_certificate; sampled_sequences yields sampled_sequences_insufficient.
View answer and explanation
(4, 4, finite_minimum_separation, Yes, universal_finite_certificate)
Compute the same exact separation in both modes, but distinguish the universal argument from a proposed sample-only justification. Insufficient evidence is not a claim that the qualified finite space is incomplete. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (4, 4, finite_minimum_separation, Yes, universal_finite_certificate).
Reviewed step-by-step derivation
Complete metric space · qf-mod27-complete-evidence-mode-audit · candidate
78. A Space-Wide Certificate Is Not a List of Successful Sequences
Use the complete qualified finite metric table=[[0, 5, 10, 13], [5, 0, 9, 12], [10, 9, 0, 17], [13, 12, 17, 0]] on Fin(4)={0,…,3}, with at least two points for this minimum-positive-distance protocol. Completeness quantifies over every Cauchy sequence and requires a limit in this same space, not merely an ambient space. Testing finitely many sequences or terms is insufficient. evidence_mode=finite_minimum_separation. Give (n,min_positive_distance,evidence_mode,sufficient?,boundary). finite_minimum_separation yields universal_finite_certificate; sampled_sequences yields sampled_sequences_insufficient.
View answer and explanation
(4, 5, finite_minimum_separation, Yes, universal_finite_certificate)
Compute the same exact separation in both modes, but distinguish the universal argument from a proposed sample-only justification. Insufficient evidence is not a claim that the qualified finite space is incomplete. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (4, 5, finite_minimum_separation, Yes, universal_finite_certificate).
Reviewed step-by-step derivation
Complete metric space · qf-mod27-complete-evidence-mode-audit · candidate
79. A Space-Wide Certificate Is Not a List of Successful Sequences
Use the complete qualified finite metric table=[[0, 6, 13, 17], [6, 0, 11, 15], [13, 11, 0, 22], [17, 15, 22, 0]] on Fin(4)={0,…,3}, with at least two points for this minimum-positive-distance protocol. Completeness quantifies over every Cauchy sequence and requires a limit in this same space, not merely an ambient space. Testing finitely many sequences or terms is insufficient. evidence_mode=finite_minimum_separation. Give (n,min_positive_distance,evidence_mode,sufficient?,boundary). finite_minimum_separation yields universal_finite_certificate; sampled_sequences yields sampled_sequences_insufficient.
View answer and explanation
(4, 6, finite_minimum_separation, Yes, universal_finite_certificate)
Compute the same exact separation in both modes, but distinguish the universal argument from a proposed sample-only justification. Insufficient evidence is not a claim that the qualified finite space is incomplete. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (4, 6, finite_minimum_separation, Yes, universal_finite_certificate).
Reviewed step-by-step derivation
Complete metric space · qf-mod27-complete-evidence-mode-audit · candidate
80. A Space-Wide Certificate Is Not a List of Successful Sequences
Use the complete qualified finite metric table=[[0, 7, 9, 10], [7, 0, 6, 7], [9, 6, 0, 9], [10, 7, 9, 0]] on Fin(4)={0,…,3}, with at least two points for this minimum-positive-distance protocol. Completeness quantifies over every Cauchy sequence and requires a limit in this same space, not merely an ambient space. Testing finitely many sequences or terms is insufficient. evidence_mode=sampled_sequences. Give (n,min_positive_distance,evidence_mode,sufficient?,boundary). finite_minimum_separation yields universal_finite_certificate; sampled_sequences yields sampled_sequences_insufficient.
View answer and explanation
(4, 6, sampled_sequences, No, sampled_sequences_insufficient)
Compute the same exact separation in both modes, but distinguish the universal argument from a proposed sample-only justification. Insufficient evidence is not a claim that the qualified finite space is incomplete. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (4, 6, sampled_sequences, No, sampled_sequences_insufficient).
Reviewed step-by-step derivation
Complete metric space · qf-mod27-complete-evidence-mode-audit · candidate
81. A Space-Wide Certificate Is Not a List of Successful Sequences
Use the complete qualified finite metric table=[[0, 4, 7, 9], [4, 0, 9, 11], [7, 9, 0, 14], [9, 11, 14, 0]] on Fin(4)={0,…,3}, with at least two points for this minimum-positive-distance protocol. Completeness quantifies over every Cauchy sequence and requires a limit in this same space, not merely an ambient space. Testing finitely many sequences or terms is insufficient. evidence_mode=finite_minimum_separation. Give (n,min_positive_distance,evidence_mode,sufficient?,boundary). finite_minimum_separation yields universal_finite_certificate; sampled_sequences yields sampled_sequences_insufficient.
View answer and explanation
(4, 4, finite_minimum_separation, Yes, universal_finite_certificate)
Compute the same exact separation in both modes, but distinguish the universal argument from a proposed sample-only justification. Insufficient evidence is not a claim that the qualified finite space is incomplete. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (4, 4, finite_minimum_separation, Yes, universal_finite_certificate).
Reviewed step-by-step derivation
Complete metric space · qf-mod27-complete-evidence-mode-audit · candidate
82. A Space-Wide Certificate Is Not a List of Successful Sequences
Use the complete qualified finite metric table=[[0, 5, 10, 13], [5, 0, 11, 14], [10, 11, 0, 19], [13, 14, 19, 0]] on Fin(4)={0,…,3}, with at least two points for this minimum-positive-distance protocol. Completeness quantifies over every Cauchy sequence and requires a limit in this same space, not merely an ambient space. Testing finitely many sequences or terms is insufficient. evidence_mode=finite_minimum_separation. Give (n,min_positive_distance,evidence_mode,sufficient?,boundary). finite_minimum_separation yields universal_finite_certificate; sampled_sequences yields sampled_sequences_insufficient.
View answer and explanation
(4, 5, finite_minimum_separation, Yes, universal_finite_certificate)
Compute the same exact separation in both modes, but distinguish the universal argument from a proposed sample-only justification. Insufficient evidence is not a claim that the qualified finite space is incomplete. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (4, 5, finite_minimum_separation, Yes, universal_finite_certificate).
Reviewed step-by-step derivation
Complete metric space · qf-mod27-complete-evidence-mode-audit · candidate
83. A Space-Wide Certificate Is Not a List of Successful Sequences
Use the complete qualified finite metric table=[[0, 6, 6, 17], [6, 0, 6, 17], [6, 6, 0, 17], [17, 17, 17, 0]] on Fin(4)={0,…,3}, with at least two points for this minimum-positive-distance protocol. Completeness quantifies over every Cauchy sequence and requires a limit in this same space, not merely an ambient space. Testing finitely many sequences or terms is insufficient. evidence_mode=finite_minimum_separation. Give (n,min_positive_distance,evidence_mode,sufficient?,boundary). finite_minimum_separation yields universal_finite_certificate; sampled_sequences yields sampled_sequences_insufficient.
View answer and explanation
(4, 6, finite_minimum_separation, Yes, universal_finite_certificate)
Compute the same exact separation in both modes, but distinguish the universal argument from a proposed sample-only justification. Insufficient evidence is not a claim that the qualified finite space is incomplete. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (4, 6, finite_minimum_separation, Yes, universal_finite_certificate).
Reviewed step-by-step derivation
Complete metric space · qf-mod27-complete-evidence-mode-audit · candidate
84. A Space-Wide Certificate Is Not a List of Successful Sequences
Use the complete qualified finite metric table=[[0, 7, 9, 10], [7, 0, 8, 9], [9, 8, 0, 11], [10, 9, 11, 0]] on Fin(4)={0,…,3}, with at least two points for this minimum-positive-distance protocol. Completeness quantifies over every Cauchy sequence and requires a limit in this same space, not merely an ambient space. Testing finitely many sequences or terms is insufficient. evidence_mode=sampled_sequences. Give (n,min_positive_distance,evidence_mode,sufficient?,boundary). finite_minimum_separation yields universal_finite_certificate; sampled_sequences yields sampled_sequences_insufficient.
View answer and explanation
(4, 7, sampled_sequences, No, sampled_sequences_insufficient)
Compute the same exact separation in both modes, but distinguish the universal argument from a proposed sample-only justification. Insufficient evidence is not a claim that the qualified finite space is incomplete. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (4, 7, sampled_sequences, No, sampled_sequences_insufficient).
Reviewed step-by-step derivation
Complete metric space · qf-mod27-complete-evidence-mode-audit · candidate
85. A Space-Wide Certificate Is Not a List of Successful Sequences
Use the complete qualified finite metric table=[[0, 8, 12, 14], [8, 0, 10, 12], [12, 10, 0, 16], [14, 12, 16, 0]] on Fin(4)={0,…,3}, with at least two points for this minimum-positive-distance protocol. Completeness quantifies over every Cauchy sequence and requires a limit in this same space, not merely an ambient space. Testing finitely many sequences or terms is insufficient. evidence_mode=finite_minimum_separation. Give (n,min_positive_distance,evidence_mode,sufficient?,boundary). finite_minimum_separation yields universal_finite_certificate; sampled_sequences yields sampled_sequences_insufficient.
View answer and explanation
(4, 8, finite_minimum_separation, Yes, universal_finite_certificate)
Compute the same exact separation in both modes, but distinguish the universal argument from a proposed sample-only justification. Insufficient evidence is not a claim that the qualified finite space is incomplete. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (4, 8, finite_minimum_separation, Yes, universal_finite_certificate).
Reviewed step-by-step derivation
Complete metric space · qf-mod27-complete-evidence-mode-audit · candidate
86. A Space-Wide Certificate Is Not a List of Successful Sequences
Use the complete qualified finite metric table=[[0, 5, 10, 13], [5, 0, 13, 16], [10, 13, 0, 21], [13, 16, 21, 0]] on Fin(4)={0,…,3}, with at least two points for this minimum-positive-distance protocol. Completeness quantifies over every Cauchy sequence and requires a limit in this same space, not merely an ambient space. Testing finitely many sequences or terms is insufficient. evidence_mode=finite_minimum_separation. Give (n,min_positive_distance,evidence_mode,sufficient?,boundary). finite_minimum_separation yields universal_finite_certificate; sampled_sequences yields sampled_sequences_insufficient.
View answer and explanation
(4, 5, finite_minimum_separation, Yes, universal_finite_certificate)
Compute the same exact separation in both modes, but distinguish the universal argument from a proposed sample-only justification. Insufficient evidence is not a claim that the qualified finite space is incomplete. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (4, 5, finite_minimum_separation, Yes, universal_finite_certificate).
Reviewed step-by-step derivation
Complete metric space · qf-mod27-complete-evidence-mode-audit · candidate
87. A Space-Wide Certificate Is Not a List of Successful Sequences
Use the complete qualified finite metric table=[[0, 6, 6, 6], [6, 0, 8, 8], [6, 8, 0, 8], [6, 8, 8, 0]] on Fin(4)={0,…,3}, with at least two points for this minimum-positive-distance protocol. Completeness quantifies over every Cauchy sequence and requires a limit in this same space, not merely an ambient space. Testing finitely many sequences or terms is insufficient. evidence_mode=finite_minimum_separation. Give (n,min_positive_distance,evidence_mode,sufficient?,boundary). finite_minimum_separation yields universal_finite_certificate; sampled_sequences yields sampled_sequences_insufficient.
View answer and explanation
(4, 6, finite_minimum_separation, Yes, universal_finite_certificate)
Compute the same exact separation in both modes, but distinguish the universal argument from a proposed sample-only justification. Insufficient evidence is not a claim that the qualified finite space is incomplete. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (4, 6, finite_minimum_separation, Yes, universal_finite_certificate).
Reviewed step-by-step derivation
Complete metric space · qf-mod27-complete-evidence-mode-audit · candidate
88. A Space-Wide Certificate Is Not a List of Successful Sequences
Use the complete qualified finite metric table=[[0, 7, 9, 10], [7, 0, 10, 11], [9, 10, 0, 13], [10, 11, 13, 0]] on Fin(4)={0,…,3}, with at least two points for this minimum-positive-distance protocol. Completeness quantifies over every Cauchy sequence and requires a limit in this same space, not merely an ambient space. Testing finitely many sequences or terms is insufficient. evidence_mode=sampled_sequences. Give (n,min_positive_distance,evidence_mode,sufficient?,boundary). finite_minimum_separation yields universal_finite_certificate; sampled_sequences yields sampled_sequences_insufficient.
View answer and explanation
(4, 7, sampled_sequences, No, sampled_sequences_insufficient)
Compute the same exact separation in both modes, but distinguish the universal argument from a proposed sample-only justification. Insufficient evidence is not a claim that the qualified finite space is incomplete. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (4, 7, sampled_sequences, No, sampled_sequences_insufficient).
Reviewed step-by-step derivation
Complete metric space · qf-mod27-complete-evidence-mode-audit · candidate
89. A Space-Wide Certificate Is Not a List of Successful Sequences
Use the complete qualified finite metric table=[[0, 8, 12, 14], [8, 0, 12, 14], [12, 12, 0, 18], [14, 14, 18, 0]] on Fin(4)={0,…,3}, with at least two points for this minimum-positive-distance protocol. Completeness quantifies over every Cauchy sequence and requires a limit in this same space, not merely an ambient space. Testing finitely many sequences or terms is insufficient. evidence_mode=finite_minimum_separation. Give (n,min_positive_distance,evidence_mode,sufficient?,boundary). finite_minimum_separation yields universal_finite_certificate; sampled_sequences yields sampled_sequences_insufficient.
View answer and explanation
(4, 8, finite_minimum_separation, Yes, universal_finite_certificate)
Compute the same exact separation in both modes, but distinguish the universal argument from a proposed sample-only justification. Insufficient evidence is not a claim that the qualified finite space is incomplete. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (4, 8, finite_minimum_separation, Yes, universal_finite_certificate).
Reviewed step-by-step derivation
Complete metric space · qf-mod27-complete-evidence-mode-audit · candidate
90. A Space-Wide Certificate Is Not a List of Successful Sequences
Use the complete qualified finite metric table=[[0, 9, 8, 18], [9, 0, 7, 17], [8, 7, 0, 16], [18, 17, 16, 0]] on Fin(4)={0,…,3}, with at least two points for this minimum-positive-distance protocol. Completeness quantifies over every Cauchy sequence and requires a limit in this same space, not merely an ambient space. Testing finitely many sequences or terms is insufficient. evidence_mode=finite_minimum_separation. Give (n,min_positive_distance,evidence_mode,sufficient?,boundary). finite_minimum_separation yields universal_finite_certificate; sampled_sequences yields sampled_sequences_insufficient.
View answer and explanation
(4, 7, finite_minimum_separation, Yes, universal_finite_certificate)
Compute the same exact separation in both modes, but distinguish the universal argument from a proposed sample-only justification. Insufficient evidence is not a claim that the qualified finite space is incomplete. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (4, 7, finite_minimum_separation, Yes, universal_finite_certificate).
Reviewed step-by-step derivation
Complete metric space · qf-mod27-complete-evidence-mode-audit · candidate
91. A Space-Wide Certificate Is Not a List of Successful Sequences
Use the complete qualified finite metric table=[[0, 6, 6, 6], [6, 0, 10, 10], [6, 10, 0, 10], [6, 10, 10, 0]] on Fin(4)={0,…,3}, with at least two points for this minimum-positive-distance protocol. Completeness quantifies over every Cauchy sequence and requires a limit in this same space, not merely an ambient space. Testing finitely many sequences or terms is insufficient. evidence_mode=finite_minimum_separation. Give (n,min_positive_distance,evidence_mode,sufficient?,boundary). finite_minimum_separation yields universal_finite_certificate; sampled_sequences yields sampled_sequences_insufficient.
View answer and explanation
(4, 6, finite_minimum_separation, Yes, universal_finite_certificate)
Compute the same exact separation in both modes, but distinguish the universal argument from a proposed sample-only justification. Insufficient evidence is not a claim that the qualified finite space is incomplete. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (4, 6, finite_minimum_separation, Yes, universal_finite_certificate).
Reviewed step-by-step derivation
Complete metric space · qf-mod27-complete-evidence-mode-audit · candidate
92. A Space-Wide Certificate Is Not a List of Successful Sequences
Use the complete qualified finite metric table=[[0, 7, 9, 10], [7, 0, 12, 13], [9, 12, 0, 15], [10, 13, 15, 0]] on Fin(4)={0,…,3}, with at least two points for this minimum-positive-distance protocol. Completeness quantifies over every Cauchy sequence and requires a limit in this same space, not merely an ambient space. Testing finitely many sequences or terms is insufficient. evidence_mode=sampled_sequences. Give (n,min_positive_distance,evidence_mode,sufficient?,boundary). finite_minimum_separation yields universal_finite_certificate; sampled_sequences yields sampled_sequences_insufficient.
View answer and explanation
(4, 7, sampled_sequences, No, sampled_sequences_insufficient)
Compute the same exact separation in both modes, but distinguish the universal argument from a proposed sample-only justification. Insufficient evidence is not a claim that the qualified finite space is incomplete. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (4, 7, sampled_sequences, No, sampled_sequences_insufficient).
Reviewed step-by-step derivation
Complete metric space · qf-mod27-complete-evidence-mode-audit · candidate
93. A Space-Wide Certificate Is Not a List of Successful Sequences
Use the complete qualified finite metric table=[[0, 8, 12, 14], [8, 0, 14, 16], [12, 14, 0, 20], [14, 16, 20, 0]] on Fin(4)={0,…,3}, with at least two points for this minimum-positive-distance protocol. Completeness quantifies over every Cauchy sequence and requires a limit in this same space, not merely an ambient space. Testing finitely many sequences or terms is insufficient. evidence_mode=finite_minimum_separation. Give (n,min_positive_distance,evidence_mode,sufficient?,boundary). finite_minimum_separation yields universal_finite_certificate; sampled_sequences yields sampled_sequences_insufficient.
View answer and explanation
(4, 8, finite_minimum_separation, Yes, universal_finite_certificate)
Compute the same exact separation in both modes, but distinguish the universal argument from a proposed sample-only justification. Insufficient evidence is not a claim that the qualified finite space is incomplete. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (4, 8, finite_minimum_separation, Yes, universal_finite_certificate).
Reviewed step-by-step derivation
Complete metric space · qf-mod27-complete-evidence-mode-audit · candidate
94. A Space-Wide Certificate Is Not a List of Successful Sequences
Use the complete qualified finite metric table=[[0, 9, 8, 18], [9, 0, 9, 19], [8, 9, 0, 18], [18, 19, 18, 0]] on Fin(4)={0,…,3}, with at least two points for this minimum-positive-distance protocol. Completeness quantifies over every Cauchy sequence and requires a limit in this same space, not merely an ambient space. Testing finitely many sequences or terms is insufficient. evidence_mode=finite_minimum_separation. Give (n,min_positive_distance,evidence_mode,sufficient?,boundary). finite_minimum_separation yields universal_finite_certificate; sampled_sequences yields sampled_sequences_insufficient.
View answer and explanation
(4, 8, finite_minimum_separation, Yes, universal_finite_certificate)
Compute the same exact separation in both modes, but distinguish the universal argument from a proposed sample-only justification. Insufficient evidence is not a claim that the qualified finite space is incomplete. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (4, 8, finite_minimum_separation, Yes, universal_finite_certificate).
Reviewed step-by-step derivation
Complete metric space · qf-mod27-complete-evidence-mode-audit · candidate
95. A Space-Wide Certificate Is Not a List of Successful Sequences
Use the complete qualified finite metric table=[[0, 10, 11, 11], [10, 0, 11, 11], [11, 11, 0, 12], [11, 11, 12, 0]] on Fin(4)={0,…,3}, with at least two points for this minimum-positive-distance protocol. Completeness quantifies over every Cauchy sequence and requires a limit in this same space, not merely an ambient space. Testing finitely many sequences or terms is insufficient. evidence_mode=finite_minimum_separation. Give (n,min_positive_distance,evidence_mode,sufficient?,boundary). finite_minimum_separation yields universal_finite_certificate; sampled_sequences yields sampled_sequences_insufficient.
View answer and explanation
(4, 10, finite_minimum_separation, Yes, universal_finite_certificate)
Compute the same exact separation in both modes, but distinguish the universal argument from a proposed sample-only justification. Insufficient evidence is not a claim that the qualified finite space is incomplete. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (4, 10, finite_minimum_separation, Yes, universal_finite_certificate).
Reviewed step-by-step derivation
Complete metric space · qf-mod27-complete-evidence-mode-audit · candidate
96. A Space-Wide Certificate Is Not a List of Successful Sequences
Use the complete qualified finite metric table=[[0, 7, 9, 10], [7, 0, 14, 15], [9, 14, 0, 17], [10, 15, 17, 0]] on Fin(4)={0,…,3}, with at least two points for this minimum-positive-distance protocol. Completeness quantifies over every Cauchy sequence and requires a limit in this same space, not merely an ambient space. Testing finitely many sequences or terms is insufficient. evidence_mode=sampled_sequences. Give (n,min_positive_distance,evidence_mode,sufficient?,boundary). finite_minimum_separation yields universal_finite_certificate; sampled_sequences yields sampled_sequences_insufficient.
View answer and explanation
(4, 7, sampled_sequences, No, sampled_sequences_insufficient)
Compute the same exact separation in both modes, but distinguish the universal argument from a proposed sample-only justification. Insufficient evidence is not a claim that the qualified finite space is incomplete. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (4, 7, sampled_sequences, No, sampled_sequences_insufficient).
Reviewed step-by-step derivation
Complete metric space · qf-mod27-complete-evidence-mode-audit · candidate
97. A Space-Wide Certificate Is Not a List of Successful Sequences
Use the complete qualified finite metric table=[[0, 8, 5, 14], [8, 0, 9, 18], [5, 9, 0, 15], [14, 18, 15, 0]] on Fin(4)={0,…,3}, with at least two points for this minimum-positive-distance protocol. Completeness quantifies over every Cauchy sequence and requires a limit in this same space, not merely an ambient space. Testing finitely many sequences or terms is insufficient. evidence_mode=finite_minimum_separation. Give (n,min_positive_distance,evidence_mode,sufficient?,boundary). finite_minimum_separation yields universal_finite_certificate; sampled_sequences yields sampled_sequences_insufficient.
View answer and explanation
(4, 5, finite_minimum_separation, Yes, universal_finite_certificate)
Compute the same exact separation in both modes, but distinguish the universal argument from a proposed sample-only justification. Insufficient evidence is not a claim that the qualified finite space is incomplete. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (4, 5, finite_minimum_separation, Yes, universal_finite_certificate).
Reviewed step-by-step derivation
Complete metric space · qf-mod27-complete-evidence-mode-audit · candidate
98. A Space-Wide Certificate Is Not a List of Successful Sequences
Use the complete qualified finite metric table=[[0, 9, 8, 7], [9, 0, 11, 10], [8, 11, 0, 9], [7, 10, 9, 0]] on Fin(4)={0,…,3}, with at least two points for this minimum-positive-distance protocol. Completeness quantifies over every Cauchy sequence and requires a limit in this same space, not merely an ambient space. Testing finitely many sequences or terms is insufficient. evidence_mode=finite_minimum_separation. Give (n,min_positive_distance,evidence_mode,sufficient?,boundary). finite_minimum_separation yields universal_finite_certificate; sampled_sequences yields sampled_sequences_insufficient.
View answer and explanation
(4, 7, finite_minimum_separation, Yes, universal_finite_certificate)
Compute the same exact separation in both modes, but distinguish the universal argument from a proposed sample-only justification. Insufficient evidence is not a claim that the qualified finite space is incomplete. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (4, 7, finite_minimum_separation, Yes, universal_finite_certificate).
Reviewed step-by-step derivation
Complete metric space · qf-mod27-complete-evidence-mode-audit · candidate
99. A Space-Wide Certificate Is Not a List of Successful Sequences
Use the complete qualified finite metric table=[[0, 10, 11, 11], [10, 0, 13, 13], [11, 13, 0, 14], [11, 13, 14, 0]] on Fin(4)={0,…,3}, with at least two points for this minimum-positive-distance protocol. Completeness quantifies over every Cauchy sequence and requires a limit in this same space, not merely an ambient space. Testing finitely many sequences or terms is insufficient. evidence_mode=finite_minimum_separation. Give (n,min_positive_distance,evidence_mode,sufficient?,boundary). finite_minimum_separation yields universal_finite_certificate; sampled_sequences yields sampled_sequences_insufficient.
View answer and explanation
(4, 10, finite_minimum_separation, Yes, universal_finite_certificate)
Compute the same exact separation in both modes, but distinguish the universal argument from a proposed sample-only justification. Insufficient evidence is not a claim that the qualified finite space is incomplete. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (4, 10, finite_minimum_separation, Yes, universal_finite_certificate).
Reviewed step-by-step derivation
Complete metric space · qf-mod27-complete-evidence-mode-audit · candidate
100. A Space-Wide Certificate Is Not a List of Successful Sequences
Use the complete qualified finite metric table=[[0, 11, 14, 15], [11, 0, 15, 16], [14, 15, 0, 19], [15, 16, 19, 0]] on Fin(4)={0,…,3}, with at least two points for this minimum-positive-distance protocol. Completeness quantifies over every Cauchy sequence and requires a limit in this same space, not merely an ambient space. Testing finitely many sequences or terms is insufficient. evidence_mode=sampled_sequences. Give (n,min_positive_distance,evidence_mode,sufficient?,boundary). finite_minimum_separation yields universal_finite_certificate; sampled_sequences yields sampled_sequences_insufficient.
View answer and explanation
(4, 11, sampled_sequences, No, sampled_sequences_insufficient)
Compute the same exact separation in both modes, but distinguish the universal argument from a proposed sample-only justification. Insufficient evidence is not a claim that the qualified finite space is incomplete. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (4, 11, sampled_sequences, No, sampled_sequences_insufficient).
Reviewed step-by-step derivation
Complete metric space · qf-mod27-complete-evidence-mode-audit · candidate
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