{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r03-complete-metric-space-0001","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2b5a96b612e0226cf716de2e8a8b26d05893f8612064af57e1ba544e989ddd51","family_id":"qf-mod27-complete-finite-certificate","seed":202610273001,"semantic_template":"qf-mod27-complete-finite-certificate/{\"metric\":[[0,3,4,5],[3,0,5,6],[4,5,0,7],[5,6,7,0]],\"n\":4}"},"fields":{"title":"Minimum Positive Separation Proves Finite-Space Completeness","stem":"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?).","answer":{"kind":"expression","value":"(4, 3, epsilon=min_positive_distance forces every Cauchy tail to be constant, Yes)","explanation":"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)."},"skills":["Minimum Positive Separation Proves Finite-Space Completeness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r03-complete-metric-space-0002","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"47a257d492025d3dd4892f76aea04f183438afba61619dce45915a912aab900e","family_id":"qf-mod27-complete-finite-certificate","seed":202610273002,"semantic_template":"qf-mod27-complete-finite-certificate/{\"metric\":[[0,4,7,9],[4,0,7,9],[7,7,0,12],[9,9,12,0]],\"n\":4}"},"fields":{"title":"Minimum Positive Separation Proves Finite-Space Completeness","stem":"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?).","answer":{"kind":"expression","value":"(4, 4, epsilon=min_positive_distance forces every Cauchy tail to be constant, Yes)","explanation":"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)."},"skills":["Minimum Positive Separation Proves Finite-Space Completeness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r03-complete-metric-space-0003","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cece341fb41c1cf587226221bd3100c71fa4d275a51e9b8b9b899360e87bd1d5","family_id":"qf-mod27-complete-finite-certificate","seed":202610273003,"semantic_template":"qf-mod27-complete-finite-certificate/{\"metric\":[[0,5,10,13],[5,0,9,12],[10,9,0,17],[13,12,17,0]],\"n\":4}"},"fields":{"title":"Minimum Positive Separation Proves Finite-Space Completeness","stem":"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?).","answer":{"kind":"expression","value":"(4, 5, epsilon=min_positive_distance forces every Cauchy tail to be constant, Yes)","explanation":"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)."},"skills":["Minimum Positive Separation Proves Finite-Space Completeness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r03-complete-metric-space-0004","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6c100480226dc13556f3ba99fb657d2c76b6b307e167f77fcfde876e04ac53cf","family_id":"qf-mod27-complete-finite-certificate","seed":202610273004,"semantic_template":"qf-mod27-complete-finite-certificate/{\"metric\":[[0,6,13,17],[6,0,11,15],[13,11,0,22],[17,15,22,0]],\"n\":4}"},"fields":{"title":"Minimum Positive Separation Proves Finite-Space Completeness","stem":"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?).","answer":{"kind":"expression","value":"(4, 6, epsilon=min_positive_distance forces every Cauchy tail to be constant, Yes)","explanation":"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)."},"skills":["Minimum Positive Separation Proves Finite-Space Completeness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r03-complete-metric-space-0005","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1fbe9429aabea2c7db73222175496ac0d5665014f71d40196f0736731768a5d6","family_id":"qf-mod27-complete-finite-certificate","seed":202610273005,"semantic_template":"qf-mod27-complete-finite-certificate/{\"metric\":[[0,7,9,10],[7,0,6,7],[9,6,0,9],[10,7,9,0]],\"n\":4}"},"fields":{"title":"Minimum Positive Separation Proves Finite-Space Completeness","stem":"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?).","answer":{"kind":"expression","value":"(4, 6, epsilon=min_positive_distance forces every Cauchy tail to be constant, Yes)","explanation":"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)."},"skills":["Minimum Positive Separation Proves Finite-Space Completeness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r03-complete-metric-space-0006","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6ed26d0315d836034ff2a09ba12e468ecb9de3d6bb43595955e53e8af823c20f","family_id":"qf-mod27-complete-finite-certificate","seed":202610273006,"semantic_template":"qf-mod27-complete-finite-certificate/{\"metric\":[[0,4,7,9],[4,0,9,11],[7,9,0,14],[9,11,14,0]],\"n\":4}"},"fields":{"title":"Minimum Positive Separation Proves Finite-Space Completeness","stem":"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?).","answer":{"kind":"expression","value":"(4, 4, epsilon=min_positive_distance forces every Cauchy tail to be constant, Yes)","explanation":"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)."},"skills":["Minimum Positive Separation Proves Finite-Space Completeness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r03-complete-metric-space-0007","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b1bd68483d9193b7cb40d2b5908f969af11fc7f5b74482d8e08d5ec872d3489f","family_id":"qf-mod27-complete-finite-certificate","seed":202610273007,"semantic_template":"qf-mod27-complete-finite-certificate/{\"metric\":[[0,5,10,13],[5,0,11,14],[10,11,0,19],[13,14,19,0]],\"n\":4}"},"fields":{"title":"Minimum Positive Separation Proves Finite-Space Completeness","stem":"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?).","answer":{"kind":"expression","value":"(4, 5, epsilon=min_positive_distance forces every Cauchy tail to be constant, Yes)","explanation":"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)."},"skills":["Minimum Positive Separation Proves Finite-Space Completeness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r03-complete-metric-space-0008","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fbbe961208718e36cd230a5969ed5efcc40fcbd6e8f831f95d8db21af16ebe3f","family_id":"qf-mod27-complete-finite-certificate","seed":202610273008,"semantic_template":"qf-mod27-complete-finite-certificate/{\"metric\":[[0,6,6,17],[6,0,6,17],[6,6,0,17],[17,17,17,0]],\"n\":4}"},"fields":{"title":"Minimum Positive Separation Proves Finite-Space Completeness","stem":"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?).","answer":{"kind":"expression","value":"(4, 6, epsilon=min_positive_distance forces every Cauchy tail to be constant, Yes)","explanation":"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)."},"skills":["Minimum Positive Separation Proves Finite-Space Completeness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r03-complete-metric-space-0009","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"74c8328050240799cc6d8fda1f5ff550c6467f9e1bd12c2dc9732265c5789955","family_id":"qf-mod27-complete-finite-certificate","seed":202610273009,"semantic_template":"qf-mod27-complete-finite-certificate/{\"metric\":[[0,7,9,10],[7,0,8,9],[9,8,0,11],[10,9,11,0]],\"n\":4}"},"fields":{"title":"Minimum Positive Separation Proves Finite-Space Completeness","stem":"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?).","answer":{"kind":"expression","value":"(4, 7, epsilon=min_positive_distance forces every Cauchy tail to be constant, Yes)","explanation":"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)."},"skills":["Minimum Positive Separation Proves Finite-Space Completeness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r03-complete-metric-space-0010","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"89a3b1f93744114d38e2d27533c68cc59458e95ad7839957a4134f27e9650991","family_id":"qf-mod27-complete-finite-certificate","seed":202610273010,"semantic_template":"qf-mod27-complete-finite-certificate/{\"metric\":[[0,8,12,14],[8,0,10,12],[12,10,0,16],[14,12,16,0]],\"n\":4}"},"fields":{"title":"Minimum Positive Separation Proves Finite-Space Completeness","stem":"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?).","answer":{"kind":"expression","value":"(4, 8, epsilon=min_positive_distance forces every Cauchy tail to be constant, Yes)","explanation":"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)."},"skills":["Minimum Positive Separation Proves Finite-Space Completeness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r03-complete-metric-space-0011","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9fe3456ef4b2fc447e1a7e77c80439f1d7beb790269d33f8c5f8c1180c92aed4","family_id":"qf-mod27-complete-finite-certificate","seed":202610273011,"semantic_template":"qf-mod27-complete-finite-certificate/{\"metric\":[[0,5,10,13],[5,0,13,16],[10,13,0,21],[13,16,21,0]],\"n\":4}"},"fields":{"title":"Minimum Positive Separation Proves Finite-Space Completeness","stem":"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?).","answer":{"kind":"expression","value":"(4, 5, epsilon=min_positive_distance forces every Cauchy tail to be constant, Yes)","explanation":"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)."},"skills":["Minimum Positive Separation Proves Finite-Space Completeness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r03-complete-metric-space-0012","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"89b2206b5b9a58930bfe13606af5fd1b7628c3a7857c13277dfd375d7c3c7f48","family_id":"qf-mod27-complete-finite-certificate","seed":202610273012,"semantic_template":"qf-mod27-complete-finite-certificate/{\"metric\":[[0,6,6,6],[6,0,8,8],[6,8,0,8],[6,8,8,0]],\"n\":4}"},"fields":{"title":"Minimum Positive Separation Proves Finite-Space Completeness","stem":"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?).","answer":{"kind":"expression","value":"(4, 6, epsilon=min_positive_distance forces every Cauchy tail to be constant, Yes)","explanation":"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)."},"skills":["Minimum Positive Separation Proves Finite-Space Completeness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r03-complete-metric-space-0013","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"47d9c9a3a317b060d5bcc89540a01bdac27de4c52bb289c2edad2dc4bc741cf9","family_id":"qf-mod27-complete-finite-certificate","seed":202610273013,"semantic_template":"qf-mod27-complete-finite-certificate/{\"metric\":[[0,7,9,10],[7,0,10,11],[9,10,0,13],[10,11,13,0]],\"n\":4}"},"fields":{"title":"Minimum Positive Separation Proves Finite-Space Completeness","stem":"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?).","answer":{"kind":"expression","value":"(4, 7, epsilon=min_positive_distance forces every Cauchy tail to be constant, Yes)","explanation":"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)."},"skills":["Minimum Positive Separation Proves Finite-Space Completeness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r03-complete-metric-space-0014","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5fb5b75a4837c8a8ac0edc7a195e8b5ce0c2477e7d713d487ed31a120e925c26","family_id":"qf-mod27-complete-finite-certificate","seed":202610273014,"semantic_template":"qf-mod27-complete-finite-certificate/{\"metric\":[[0,8,12,14],[8,0,12,14],[12,12,0,18],[14,14,18,0]],\"n\":4}"},"fields":{"title":"Minimum Positive Separation Proves Finite-Space Completeness","stem":"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?).","answer":{"kind":"expression","value":"(4, 8, epsilon=min_positive_distance forces every Cauchy tail to be constant, Yes)","explanation":"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)."},"skills":["Minimum Positive Separation Proves Finite-Space Completeness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r03-complete-metric-space-0015","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"59865550b18330061dda33ceff4404d590ebb23312418d8e37deee1251c81b66","family_id":"qf-mod27-complete-finite-certificate","seed":202610273015,"semantic_template":"qf-mod27-complete-finite-certificate/{\"metric\":[[0,9,8,18],[9,0,7,17],[8,7,0,16],[18,17,16,0]],\"n\":4}"},"fields":{"title":"Minimum Positive Separation Proves Finite-Space Completeness","stem":"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?).","answer":{"kind":"expression","value":"(4, 7, epsilon=min_positive_distance forces every Cauchy tail to be constant, Yes)","explanation":"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)."},"skills":["Minimum Positive Separation Proves Finite-Space Completeness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r03-complete-metric-space-0016","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"165c3a3c726b5d5168c08357c5c215b20af772c5a4efaa039e27c348d462df71","family_id":"qf-mod27-complete-finite-certificate","seed":202610273016,"semantic_template":"qf-mod27-complete-finite-certificate/{\"metric\":[[0,6,6,6],[6,0,10,10],[6,10,0,10],[6,10,10,0]],\"n\":4}"},"fields":{"title":"Minimum Positive Separation Proves Finite-Space Completeness","stem":"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?).","answer":{"kind":"expression","value":"(4, 6, epsilon=min_positive_distance forces every Cauchy tail to be constant, Yes)","explanation":"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)."},"skills":["Minimum Positive Separation Proves Finite-Space Completeness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r03-complete-metric-space-0017","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c810ff393a6198d1c7e05104ccaec54c3c616c073e6d2c3ff02042fa3c3fa567","family_id":"qf-mod27-complete-finite-certificate","seed":202610273017,"semantic_template":"qf-mod27-complete-finite-certificate/{\"metric\":[[0,7,9,10],[7,0,12,13],[9,12,0,15],[10,13,15,0]],\"n\":4}"},"fields":{"title":"Minimum Positive Separation Proves Finite-Space Completeness","stem":"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?).","answer":{"kind":"expression","value":"(4, 7, epsilon=min_positive_distance forces every Cauchy tail to be constant, Yes)","explanation":"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)."},"skills":["Minimum Positive Separation Proves Finite-Space Completeness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r03-complete-metric-space-0018","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a3b820cf38bb69ab99a6db742a36a09cc9501e8bc5f1f98f27666c94bb186fda","family_id":"qf-mod27-complete-finite-certificate","seed":202610273018,"semantic_template":"qf-mod27-complete-finite-certificate/{\"metric\":[[0,8,12,14],[8,0,14,16],[12,14,0,20],[14,16,20,0]],\"n\":4}"},"fields":{"title":"Minimum Positive Separation Proves Finite-Space Completeness","stem":"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?).","answer":{"kind":"expression","value":"(4, 8, epsilon=min_positive_distance forces every Cauchy tail to be constant, Yes)","explanation":"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)."},"skills":["Minimum Positive Separation Proves Finite-Space Completeness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r03-complete-metric-space-0019","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4661cf7b8fac6970e820ac3d17248256518573508f4ac83f9bd0ee9c7a081371","family_id":"qf-mod27-complete-finite-certificate","seed":202610273019,"semantic_template":"qf-mod27-complete-finite-certificate/{\"metric\":[[0,9,8,18],[9,0,9,19],[8,9,0,18],[18,19,18,0]],\"n\":4}"},"fields":{"title":"Minimum Positive Separation Proves Finite-Space Completeness","stem":"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?).","answer":{"kind":"expression","value":"(4, 8, epsilon=min_positive_distance forces every Cauchy tail to be constant, Yes)","explanation":"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)."},"skills":["Minimum Positive Separation Proves Finite-Space Completeness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r03-complete-metric-space-0020","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6bbe6415f320a32f641764405599d150dd9a7d12b5bd144fcf5088d3e2d8d493","family_id":"qf-mod27-complete-finite-certificate","seed":202610273020,"semantic_template":"qf-mod27-complete-finite-certificate/{\"metric\":[[0,10,11,11],[10,0,11,11],[11,11,0,12],[11,11,12,0]],\"n\":4}"},"fields":{"title":"Minimum Positive Separation Proves Finite-Space Completeness","stem":"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?).","answer":{"kind":"expression","value":"(4, 10, epsilon=min_positive_distance forces every Cauchy tail to be constant, Yes)","explanation":"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)."},"skills":["Minimum Positive Separation Proves Finite-Space Completeness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r03-complete-metric-space-0021","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dc9e6615f386246033fdfd7df7052b9eaae8b6d548520f24b56b3332cf90423a","family_id":"qf-mod27-complete-finite-certificate","seed":202610273021,"semantic_template":"qf-mod27-complete-finite-certificate/{\"metric\":[[0,7,9,10],[7,0,14,15],[9,14,0,17],[10,15,17,0]],\"n\":4}"},"fields":{"title":"Minimum Positive Separation Proves Finite-Space Completeness","stem":"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?).","answer":{"kind":"expression","value":"(4, 7, epsilon=min_positive_distance forces every Cauchy tail to be constant, Yes)","explanation":"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)."},"skills":["Minimum Positive Separation Proves Finite-Space Completeness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r03-complete-metric-space-0022","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"23261e8c1dd432b4e4154b5c5c7b28fd160ba4dbb6e4bb499e053378cb0dbc0a","family_id":"qf-mod27-complete-finite-certificate","seed":202610273022,"semantic_template":"qf-mod27-complete-finite-certificate/{\"metric\":[[0,8,5,14],[8,0,9,18],[5,9,0,15],[14,18,15,0]],\"n\":4}"},"fields":{"title":"Minimum Positive Separation Proves Finite-Space Completeness","stem":"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?).","answer":{"kind":"expression","value":"(4, 5, epsilon=min_positive_distance forces every Cauchy tail to be constant, Yes)","explanation":"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)."},"skills":["Minimum Positive Separation Proves Finite-Space Completeness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r03-complete-metric-space-0023","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a9938d0588b5222cd9be3f71e841416406e7b245d884f6207ee859714593c296","family_id":"qf-mod27-complete-finite-certificate","seed":202610273023,"semantic_template":"qf-mod27-complete-finite-certificate/{\"metric\":[[0,9,8,7],[9,0,11,10],[8,11,0,9],[7,10,9,0]],\"n\":4}"},"fields":{"title":"Minimum Positive Separation Proves Finite-Space Completeness","stem":"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?).","answer":{"kind":"expression","value":"(4, 7, epsilon=min_positive_distance forces every Cauchy tail to be constant, Yes)","explanation":"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)."},"skills":["Minimum Positive Separation Proves Finite-Space Completeness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r03-complete-metric-space-0024","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fe5bfb2f0a45447b2ae567b25017cb1bd2f11105db8ece40691053df6bae7800","family_id":"qf-mod27-complete-finite-certificate","seed":202610273024,"semantic_template":"qf-mod27-complete-finite-certificate/{\"metric\":[[0,10,11,11],[10,0,13,13],[11,13,0,14],[11,13,14,0]],\"n\":4}"},"fields":{"title":"Minimum Positive Separation Proves Finite-Space Completeness","stem":"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?).","answer":{"kind":"expression","value":"(4, 10, epsilon=min_positive_distance forces every Cauchy tail to be constant, Yes)","explanation":"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)."},"skills":["Minimum Positive Separation Proves Finite-Space Completeness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r03-complete-metric-space-0025","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5d12a67e88259882a2ab3fc9b7543cb7d68e870c136a00e6e4efc23389e02692","family_id":"qf-mod27-complete-finite-certificate","seed":202610273025,"semantic_template":"qf-mod27-complete-finite-certificate/{\"metric\":[[0,11,14,15],[11,0,15,16],[14,15,0,19],[15,16,19,0]],\"n\":4}"},"fields":{"title":"Minimum Positive Separation Proves Finite-Space Completeness","stem":"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?).","answer":{"kind":"expression","value":"(4, 11, epsilon=min_positive_distance forces every Cauchy tail to be constant, Yes)","explanation":"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)."},"skills":["Minimum Positive Separation Proves Finite-Space Completeness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r03-complete-metric-space-0026","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8d9647e5b04d9b763e7ce02fadbb38adcb586cba116f99c19bd802e39c834ca7","family_id":"qf-mod27-complete-ultimately-constant-limit","seed":202610273026,"semantic_template":"qf-mod27-complete-ultimately-constant-limit/{\"metric\":[[0,3,4,5],[3,0,5,6],[4,5,0,7],[5,6,7,0]],\"n\":4,\"prefix\":[0,1,2],\"tail\":3}"},"fields":{"title":"An Explicit Infinite Tail Converges within Its Declared Space","stem":"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?).","answer":{"kind":"expression","value":"([0, 1, 2], 3, 3, 3, Yes, Yes)","explanation":"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)."},"skills":["An Explicit Infinite Tail Converges within Its Declared Space"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r03-complete-metric-space-0027","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dbb914c702d38bbae2c661116170a179dc23c7078e003b6596e25fa9457f27d2","family_id":"qf-mod27-complete-ultimately-constant-limit","seed":202610273027,"semantic_template":"qf-mod27-complete-ultimately-constant-limit/{\"metric\":[[0,4,7,9],[4,0,7,9],[7,7,0,12],[9,9,12,0]],\"n\":4,\"prefix\":[1,2,3],\"tail\":0}"},"fields":{"title":"An Explicit Infinite Tail Converges within Its Declared Space","stem":"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?).","answer":{"kind":"expression","value":"([1, 2, 3], 0, 3, 0, Yes, Yes)","explanation":"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)."},"skills":["An Explicit Infinite Tail Converges within Its Declared Space"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r03-complete-metric-space-0028","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0ffec678200b24c1e3c7ebb5b6688d8629d87ce94af975e83acb67c3f0b86ea5","family_id":"qf-mod27-complete-ultimately-constant-limit","seed":202610273028,"semantic_template":"qf-mod27-complete-ultimately-constant-limit/{\"metric\":[[0,5,10,13],[5,0,9,12],[10,9,0,17],[13,12,17,0]],\"n\":4,\"prefix\":[2,3,0],\"tail\":1}"},"fields":{"title":"An Explicit Infinite Tail Converges within Its Declared Space","stem":"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?).","answer":{"kind":"expression","value":"([2, 3, 0], 1, 3, 1, Yes, Yes)","explanation":"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)."},"skills":["An Explicit Infinite Tail Converges within Its Declared Space"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r03-complete-metric-space-0029","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b9ed1baf53656d5bd55d4b10aee9dda786c354b2909a870bb6ef7e54c587d7b9","family_id":"qf-mod27-complete-ultimately-constant-limit","seed":202610273029,"semantic_template":"qf-mod27-complete-ultimately-constant-limit/{\"metric\":[[0,6,13,17],[6,0,11,15],[13,11,0,22],[17,15,22,0]],\"n\":4,\"prefix\":[3,0,1],\"tail\":2}"},"fields":{"title":"An Explicit Infinite Tail Converges within Its Declared Space","stem":"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?).","answer":{"kind":"expression","value":"([3, 0, 1], 2, 3, 2, Yes, Yes)","explanation":"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)."},"skills":["An Explicit Infinite Tail Converges within Its Declared Space"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r03-complete-metric-space-0030","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dc2df8c0f96ccae0428cc205312ea61b2bc246592b8ec95ca4bc451f7b885c20","family_id":"qf-mod27-complete-ultimately-constant-limit","seed":202610273030,"semantic_template":"qf-mod27-complete-ultimately-constant-limit/{\"metric\":[[0,7,9,10],[7,0,6,7],[9,6,0,9],[10,7,9,0]],\"n\":4,\"prefix\":[0,1,2],\"tail\":3}"},"fields":{"title":"An Explicit Infinite Tail Converges within Its Declared Space","stem":"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?).","answer":{"kind":"expression","value":"([0, 1, 2], 3, 3, 3, Yes, Yes)","explanation":"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)."},"skills":["An Explicit Infinite Tail Converges within Its Declared Space"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r03-complete-metric-space-0031","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2d92c4fbc156f47a9fc8add909c7b2030f2db37743c5040ece7476e8deaee0ec","family_id":"qf-mod27-complete-ultimately-constant-limit","seed":202610273031,"semantic_template":"qf-mod27-complete-ultimately-constant-limit/{\"metric\":[[0,4,7,9],[4,0,9,11],[7,9,0,14],[9,11,14,0]],\"n\":4,\"prefix\":[1,2,3],\"tail\":0}"},"fields":{"title":"An Explicit Infinite Tail Converges within Its Declared Space","stem":"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?).","answer":{"kind":"expression","value":"([1, 2, 3], 0, 3, 0, Yes, Yes)","explanation":"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)."},"skills":["An Explicit Infinite Tail Converges within Its Declared Space"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r03-complete-metric-space-0032","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2a3ee0867e6b3ad9fd68966718c6a16cd91f524cacd3c490ecd2a5a4e9ca7d5a","family_id":"qf-mod27-complete-ultimately-constant-limit","seed":202610273032,"semantic_template":"qf-mod27-complete-ultimately-constant-limit/{\"metric\":[[0,5,10,13],[5,0,11,14],[10,11,0,19],[13,14,19,0]],\"n\":4,\"prefix\":[2,3,0],\"tail\":1}"},"fields":{"title":"An Explicit Infinite Tail Converges within Its Declared Space","stem":"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?).","answer":{"kind":"expression","value":"([2, 3, 0], 1, 3, 1, Yes, Yes)","explanation":"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)."},"skills":["An Explicit Infinite Tail Converges within Its Declared Space"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r03-complete-metric-space-0033","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"eb102cbaa5d53bed847e65d93701f3d4b308a7bf7bdc26f3f14962048dcac189","family_id":"qf-mod27-complete-ultimately-constant-limit","seed":202610273033,"semantic_template":"qf-mod27-complete-ultimately-constant-limit/{\"metric\":[[0,6,6,17],[6,0,6,17],[6,6,0,17],[17,17,17,0]],\"n\":4,\"prefix\":[3,0,1],\"tail\":2}"},"fields":{"title":"An Explicit Infinite Tail Converges within Its Declared Space","stem":"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?).","answer":{"kind":"expression","value":"([3, 0, 1], 2, 3, 2, Yes, Yes)","explanation":"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)."},"skills":["An Explicit Infinite Tail Converges within Its Declared Space"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r03-complete-metric-space-0034","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c1cd3fd699e4459f7c76a7c56b00f46a78999b4675f38fb9c1e1562b19eddb98","family_id":"qf-mod27-complete-ultimately-constant-limit","seed":202610273034,"semantic_template":"qf-mod27-complete-ultimately-constant-limit/{\"metric\":[[0,7,9,10],[7,0,8,9],[9,8,0,11],[10,9,11,0]],\"n\":4,\"prefix\":[0,1,2],\"tail\":3}"},"fields":{"title":"An Explicit Infinite Tail Converges within Its Declared Space","stem":"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?).","answer":{"kind":"expression","value":"([0, 1, 2], 3, 3, 3, Yes, Yes)","explanation":"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)."},"skills":["An Explicit Infinite Tail Converges within Its Declared Space"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r03-complete-metric-space-0035","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c067db6a6267f1d5c1d7020edf2f74e987ad966f85c0ced42edd6311e20619dc","family_id":"qf-mod27-complete-ultimately-constant-limit","seed":202610273035,"semantic_template":"qf-mod27-complete-ultimately-constant-limit/{\"metric\":[[0,8,12,14],[8,0,10,12],[12,10,0,16],[14,12,16,0]],\"n\":4,\"prefix\":[1,2,3],\"tail\":0}"},"fields":{"title":"An Explicit Infinite Tail Converges within Its Declared Space","stem":"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?).","answer":{"kind":"expression","value":"([1, 2, 3], 0, 3, 0, Yes, Yes)","explanation":"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)."},"skills":["An Explicit Infinite Tail Converges within Its Declared Space"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r03-complete-metric-space-0036","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"15be6233f2aad75fee511008432f6a4a02438e7bf5853c2e4bd47d218c408dd3","family_id":"qf-mod27-complete-ultimately-constant-limit","seed":202610273036,"semantic_template":"qf-mod27-complete-ultimately-constant-limit/{\"metric\":[[0,5,10,13],[5,0,13,16],[10,13,0,21],[13,16,21,0]],\"n\":4,\"prefix\":[2,3,0],\"tail\":1}"},"fields":{"title":"An Explicit Infinite Tail Converges within Its Declared Space","stem":"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?).","answer":{"kind":"expression","value":"([2, 3, 0], 1, 3, 1, Yes, Yes)","explanation":"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)."},"skills":["An Explicit Infinite Tail Converges within Its Declared Space"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r03-complete-metric-space-0037","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ac2a2fca6203d04de9cde9f62b0c21871037bf9f80a26c0b2423473a32fcca14","family_id":"qf-mod27-complete-ultimately-constant-limit","seed":202610273037,"semantic_template":"qf-mod27-complete-ultimately-constant-limit/{\"metric\":[[0,6,6,6],[6,0,8,8],[6,8,0,8],[6,8,8,0]],\"n\":4,\"prefix\":[3,0,1],\"tail\":2}"},"fields":{"title":"An Explicit Infinite Tail Converges within Its Declared Space","stem":"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?).","answer":{"kind":"expression","value":"([3, 0, 1], 2, 3, 2, Yes, Yes)","explanation":"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)."},"skills":["An Explicit Infinite Tail Converges within Its Declared Space"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r03-complete-metric-space-0038","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"127adf81628980b0087acb1e89b505fb4e226ce1d656d9feb8cd9ea43f8bc061","family_id":"qf-mod27-complete-ultimately-constant-limit","seed":202610273038,"semantic_template":"qf-mod27-complete-ultimately-constant-limit/{\"metric\":[[0,7,9,10],[7,0,10,11],[9,10,0,13],[10,11,13,0]],\"n\":4,\"prefix\":[0,1,2],\"tail\":3}"},"fields":{"title":"An Explicit Infinite Tail Converges within Its Declared Space","stem":"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?).","answer":{"kind":"expression","value":"([0, 1, 2], 3, 3, 3, Yes, Yes)","explanation":"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)."},"skills":["An Explicit Infinite Tail Converges within Its Declared Space"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r03-complete-metric-space-0039","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"12a548e6665d89e2d218e3327bcdc2dabc3b68279edcbfc4022b6557bc6d8aa8","family_id":"qf-mod27-complete-ultimately-constant-limit","seed":202610273039,"semantic_template":"qf-mod27-complete-ultimately-constant-limit/{\"metric\":[[0,8,12,14],[8,0,12,14],[12,12,0,18],[14,14,18,0]],\"n\":4,\"prefix\":[1,2,3],\"tail\":0}"},"fields":{"title":"An Explicit Infinite Tail Converges within Its Declared Space","stem":"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?).","answer":{"kind":"expression","value":"([1, 2, 3], 0, 3, 0, Yes, Yes)","explanation":"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)."},"skills":["An Explicit Infinite Tail Converges within Its Declared Space"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r03-complete-metric-space-0040","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fb52ba9eb6b0443a78853eccc127c5fa2ce8e1b9b76ed28697450afcb26b26d3","family_id":"qf-mod27-complete-ultimately-constant-limit","seed":202610273040,"semantic_template":"qf-mod27-complete-ultimately-constant-limit/{\"metric\":[[0,9,8,18],[9,0,7,17],[8,7,0,16],[18,17,16,0]],\"n\":4,\"prefix\":[2,3,0],\"tail\":1}"},"fields":{"title":"An Explicit Infinite Tail Converges within Its Declared Space","stem":"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?).","answer":{"kind":"expression","value":"([2, 3, 0], 1, 3, 1, Yes, Yes)","explanation":"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)."},"skills":["An Explicit Infinite Tail Converges within Its Declared Space"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r03-complete-metric-space-0041","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1d7c0fc8ab86e49ee08b8bdb4c0a509682e476f94e837804c1d8f17c14133a4a","family_id":"qf-mod27-complete-ultimately-constant-limit","seed":202610273041,"semantic_template":"qf-mod27-complete-ultimately-constant-limit/{\"metric\":[[0,6,6,6],[6,0,10,10],[6,10,0,10],[6,10,10,0]],\"n\":4,\"prefix\":[3,0,1],\"tail\":2}"},"fields":{"title":"An Explicit Infinite Tail Converges within Its Declared Space","stem":"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?).","answer":{"kind":"expression","value":"([3, 0, 1], 2, 3, 2, Yes, Yes)","explanation":"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)."},"skills":["An Explicit Infinite Tail Converges within Its Declared Space"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r03-complete-metric-space-0042","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"18789f088b3aec2287a92b8da97a5fe2d21767e040c184522de8307b91aa0c75","family_id":"qf-mod27-complete-ultimately-constant-limit","seed":202610273042,"semantic_template":"qf-mod27-complete-ultimately-constant-limit/{\"metric\":[[0,7,9,10],[7,0,12,13],[9,12,0,15],[10,13,15,0]],\"n\":4,\"prefix\":[0,1,2],\"tail\":3}"},"fields":{"title":"An Explicit Infinite Tail Converges within Its Declared Space","stem":"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?).","answer":{"kind":"expression","value":"([0, 1, 2], 3, 3, 3, Yes, Yes)","explanation":"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)."},"skills":["An Explicit Infinite Tail Converges within Its Declared Space"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r03-complete-metric-space-0043","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"049eac1856f25397ab4d986cbb5f956737066f2ba8b1bde190cbfa794a5f7b03","family_id":"qf-mod27-complete-ultimately-constant-limit","seed":202610273043,"semantic_template":"qf-mod27-complete-ultimately-constant-limit/{\"metric\":[[0,8,12,14],[8,0,14,16],[12,14,0,20],[14,16,20,0]],\"n\":4,\"prefix\":[1,2,3],\"tail\":0}"},"fields":{"title":"An Explicit Infinite Tail Converges within Its Declared Space","stem":"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?).","answer":{"kind":"expression","value":"([1, 2, 3], 0, 3, 0, Yes, Yes)","explanation":"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)."},"skills":["An Explicit Infinite Tail Converges within Its Declared Space"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r03-complete-metric-space-0044","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0167016ee284a18c4c0af5df2b86439b9a95076e8a85ab1ca824560015689556","family_id":"qf-mod27-complete-ultimately-constant-limit","seed":202610273044,"semantic_template":"qf-mod27-complete-ultimately-constant-limit/{\"metric\":[[0,9,8,18],[9,0,9,19],[8,9,0,18],[18,19,18,0]],\"n\":4,\"prefix\":[2,3,0],\"tail\":1}"},"fields":{"title":"An Explicit Infinite Tail Converges within Its Declared Space","stem":"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?).","answer":{"kind":"expression","value":"([2, 3, 0], 1, 3, 1, Yes, Yes)","explanation":"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)."},"skills":["An Explicit Infinite Tail Converges within Its Declared Space"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r03-complete-metric-space-0045","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b4cf43a5fe3d69af27a1b75697c766f2e34288cdb71d2237253e1e09caec5ead","family_id":"qf-mod27-complete-ultimately-constant-limit","seed":202610273045,"semantic_template":"qf-mod27-complete-ultimately-constant-limit/{\"metric\":[[0,10,11,11],[10,0,11,11],[11,11,0,12],[11,11,12,0]],\"n\":4,\"prefix\":[3,0,1],\"tail\":2}"},"fields":{"title":"An Explicit Infinite Tail Converges within Its Declared Space","stem":"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?).","answer":{"kind":"expression","value":"([3, 0, 1], 2, 3, 2, Yes, Yes)","explanation":"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)."},"skills":["An Explicit Infinite Tail Converges within Its Declared Space"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r03-complete-metric-space-0046","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"eec08492c59d0460ea84da23c51c1861d6d76ff067d6ba700cad973689e76b2c","family_id":"qf-mod27-complete-ultimately-constant-limit","seed":202610273046,"semantic_template":"qf-mod27-complete-ultimately-constant-limit/{\"metric\":[[0,7,9,10],[7,0,14,15],[9,14,0,17],[10,15,17,0]],\"n\":4,\"prefix\":[0,1,2],\"tail\":3}"},"fields":{"title":"An Explicit Infinite Tail Converges within Its Declared Space","stem":"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?).","answer":{"kind":"expression","value":"([0, 1, 2], 3, 3, 3, Yes, Yes)","explanation":"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)."},"skills":["An Explicit Infinite Tail Converges within Its Declared Space"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r03-complete-metric-space-0047","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8c1715750d7fd53b2420582c2954e130959eab9b63c82d899f8bd562bbef5d82","family_id":"qf-mod27-complete-ultimately-constant-limit","seed":202610273047,"semantic_template":"qf-mod27-complete-ultimately-constant-limit/{\"metric\":[[0,8,5,14],[8,0,9,18],[5,9,0,15],[14,18,15,0]],\"n\":4,\"prefix\":[1,2,3],\"tail\":0}"},"fields":{"title":"An Explicit Infinite Tail Converges within Its Declared Space","stem":"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?).","answer":{"kind":"expression","value":"([1, 2, 3], 0, 3, 0, Yes, Yes)","explanation":"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)."},"skills":["An Explicit Infinite Tail Converges within Its Declared Space"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r03-complete-metric-space-0048","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7cceb1a1b30dbbefdb375ac82394ed40e905d1c6b8b489d6efaf94d24ac7e1ad","family_id":"qf-mod27-complete-ultimately-constant-limit","seed":202610273048,"semantic_template":"qf-mod27-complete-ultimately-constant-limit/{\"metric\":[[0,9,8,7],[9,0,11,10],[8,11,0,9],[7,10,9,0]],\"n\":4,\"prefix\":[2,3,0],\"tail\":1}"},"fields":{"title":"An Explicit Infinite Tail Converges within Its Declared Space","stem":"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?).","answer":{"kind":"expression","value":"([2, 3, 0], 1, 3, 1, Yes, Yes)","explanation":"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)."},"skills":["An Explicit Infinite Tail Converges within Its Declared Space"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r03-complete-metric-space-0049","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f6072d9a5d77f8ee8be725ff95533fc699544169d0eef94b77ebd31f334108fc","family_id":"qf-mod27-complete-ultimately-constant-limit","seed":202610273049,"semantic_template":"qf-mod27-complete-ultimately-constant-limit/{\"metric\":[[0,10,11,11],[10,0,13,13],[11,13,0,14],[11,13,14,0]],\"n\":4,\"prefix\":[3,0,1],\"tail\":2}"},"fields":{"title":"An Explicit Infinite Tail Converges within Its Declared Space","stem":"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?).","answer":{"kind":"expression","value":"([3, 0, 1], 2, 3, 2, Yes, Yes)","explanation":"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)."},"skills":["An Explicit Infinite Tail Converges within Its Declared Space"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r03-complete-metric-space-0050","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"70b010b03a8e923bca33415657ae0db7dfaefb189578bfa57a774bd702086925","family_id":"qf-mod27-complete-ultimately-constant-limit","seed":202610273050,"semantic_template":"qf-mod27-complete-ultimately-constant-limit/{\"metric\":[[0,11,14,15],[11,0,15,16],[14,15,0,19],[15,16,19,0]],\"n\":4,\"prefix\":[0,1,2],\"tail\":3}"},"fields":{"title":"An Explicit Infinite Tail Converges within Its Declared Space","stem":"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?).","answer":{"kind":"expression","value":"([0, 1, 2], 3, 3, 3, Yes, Yes)","explanation":"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)."},"skills":["An Explicit Infinite Tail Converges within Its Declared Space"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r03-complete-metric-space-0051","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"935cfff2567390d0d2d459b7bbabeb80d29e4d988c6ba8fcf0b7ad853ee24fa5","family_id":"qf-mod27-complete-alternating-noncauchy","seed":202610273051,"semantic_template":"qf-mod27-complete-alternating-noncauchy/{\"N\":1,\"left\":0,\"metric\":[[0,3,4,5],[3,0,5,6],[4,5,0,7],[5,6,7,0]],\"n\":4,\"right\":1}"},"fields":{"title":"A Non-Cauchy Sequence Does Not Contradict Space Completeness","stem":"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.","answer":{"kind":"expression","value":"(0, 1, 1, 2, 3, 3, Yes, No)","explanation":"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)."},"skills":["A Non-Cauchy Sequence Does Not Contradict Space Completeness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r03-complete-metric-space-0052","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"02fe7f536ea5d1520ec1625722de3625376d074e79be6aff75e571a023ba57e5","family_id":"qf-mod27-complete-alternating-noncauchy","seed":202610273052,"semantic_template":"qf-mod27-complete-alternating-noncauchy/{\"N\":3,\"left\":1,\"metric\":[[0,4,7,9],[4,0,7,9],[7,7,0,12],[9,9,12,0]],\"n\":4,\"right\":2}"},"fields":{"title":"A Non-Cauchy Sequence Does Not Contradict Space Completeness","stem":"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.","answer":{"kind":"expression","value":"(1, 2, 3, 4, 7, 7, Yes, No)","explanation":"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)."},"skills":["A Non-Cauchy Sequence Does Not Contradict Space Completeness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r03-complete-metric-space-0053","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"74ded7a985a7c0da7b865c440ba07f7f45ae718787c5a8eaed32fa36afde067c","family_id":"qf-mod27-complete-alternating-noncauchy","seed":202610273053,"semantic_template":"qf-mod27-complete-alternating-noncauchy/{\"N\":5,\"left\":2,\"metric\":[[0,5,10,13],[5,0,9,12],[10,9,0,17],[13,12,17,0]],\"n\":4,\"right\":3}"},"fields":{"title":"A Non-Cauchy Sequence Does Not Contradict Space Completeness","stem":"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.","answer":{"kind":"expression","value":"(2, 3, 5, 6, 17, 17, Yes, No)","explanation":"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)."},"skills":["A Non-Cauchy Sequence Does Not Contradict Space Completeness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r03-complete-metric-space-0054","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b36e62f8eb21e77f30c302f764e6f6038e510f749cd79db90a1cdcc558a941ca","family_id":"qf-mod27-complete-alternating-noncauchy","seed":202610273054,"semantic_template":"qf-mod27-complete-alternating-noncauchy/{\"N\":7,\"left\":3,\"metric\":[[0,6,13,17],[6,0,11,15],[13,11,0,22],[17,15,22,0]],\"n\":4,\"right\":0}"},"fields":{"title":"A Non-Cauchy Sequence Does Not Contradict Space Completeness","stem":"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.","answer":{"kind":"expression","value":"(3, 0, 7, 8, 17, 17, Yes, No)","explanation":"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)."},"skills":["A Non-Cauchy Sequence Does Not Contradict Space Completeness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r03-complete-metric-space-0055","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b7b72e25b5a28b76cbf465786a7006fbef5e39b96cd1593b670da574cd3e6649","family_id":"qf-mod27-complete-alternating-noncauchy","seed":202610273055,"semantic_template":"qf-mod27-complete-alternating-noncauchy/{\"N\":9,\"left\":0,\"metric\":[[0,7,9,10],[7,0,6,7],[9,6,0,9],[10,7,9,0]],\"n\":4,\"right\":1}"},"fields":{"title":"A Non-Cauchy Sequence Does Not Contradict Space Completeness","stem":"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.","answer":{"kind":"expression","value":"(0, 1, 9, 10, 7, 7, Yes, No)","explanation":"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)."},"skills":["A Non-Cauchy Sequence Does Not Contradict Space Completeness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r03-complete-metric-space-0056","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"077ac709f7304b1f3bf8829398a7bea669af07163357672e8c945e818a3af805","family_id":"qf-mod27-complete-alternating-noncauchy","seed":202610273056,"semantic_template":"qf-mod27-complete-alternating-noncauchy/{\"N\":11,\"left\":1,\"metric\":[[0,4,7,9],[4,0,9,11],[7,9,0,14],[9,11,14,0]],\"n\":4,\"right\":2}"},"fields":{"title":"A Non-Cauchy Sequence Does Not Contradict Space Completeness","stem":"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.","answer":{"kind":"expression","value":"(1, 2, 11, 12, 9, 9, Yes, No)","explanation":"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)."},"skills":["A Non-Cauchy Sequence Does Not Contradict Space Completeness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r03-complete-metric-space-0057","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"84cfbf57063a7bb8f564bbde939b95aec7acd69f0faa74c4babded2085092467","family_id":"qf-mod27-complete-alternating-noncauchy","seed":202610273057,"semantic_template":"qf-mod27-complete-alternating-noncauchy/{\"N\":13,\"left\":2,\"metric\":[[0,5,10,13],[5,0,11,14],[10,11,0,19],[13,14,19,0]],\"n\":4,\"right\":3}"},"fields":{"title":"A Non-Cauchy Sequence Does Not Contradict Space Completeness","stem":"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.","answer":{"kind":"expression","value":"(2, 3, 13, 14, 19, 19, Yes, No)","explanation":"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)."},"skills":["A Non-Cauchy Sequence Does Not Contradict Space Completeness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r03-complete-metric-space-0058","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cda8d63a25770ef3cb0cbd130a3988a5daf2f265cd2e05f8c8ca0b1d7f419897","family_id":"qf-mod27-complete-alternating-noncauchy","seed":202610273058,"semantic_template":"qf-mod27-complete-alternating-noncauchy/{\"N\":15,\"left\":3,\"metric\":[[0,6,6,17],[6,0,6,17],[6,6,0,17],[17,17,17,0]],\"n\":4,\"right\":0}"},"fields":{"title":"A Non-Cauchy Sequence Does Not Contradict Space Completeness","stem":"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.","answer":{"kind":"expression","value":"(3, 0, 15, 16, 17, 17, Yes, No)","explanation":"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)."},"skills":["A Non-Cauchy Sequence Does Not Contradict Space Completeness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r03-complete-metric-space-0059","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9c923ed5566ac76ab59ce11d228489718dbe0b6b0369fa7ebe41c3c54f557a98","family_id":"qf-mod27-complete-alternating-noncauchy","seed":202610273059,"semantic_template":"qf-mod27-complete-alternating-noncauchy/{\"N\":17,\"left\":0,\"metric\":[[0,7,9,10],[7,0,8,9],[9,8,0,11],[10,9,11,0]],\"n\":4,\"right\":1}"},"fields":{"title":"A Non-Cauchy Sequence Does Not Contradict Space Completeness","stem":"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.","answer":{"kind":"expression","value":"(0, 1, 17, 18, 7, 7, Yes, No)","explanation":"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)."},"skills":["A Non-Cauchy Sequence Does Not Contradict Space Completeness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r03-complete-metric-space-0060","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ebabb5fa162fc0e980080f53d3bdf96b014336ac35ed8404b33f9834139b1dd5","family_id":"qf-mod27-complete-alternating-noncauchy","seed":202610273060,"semantic_template":"qf-mod27-complete-alternating-noncauchy/{\"N\":19,\"left\":1,\"metric\":[[0,8,12,14],[8,0,10,12],[12,10,0,16],[14,12,16,0]],\"n\":4,\"right\":2}"},"fields":{"title":"A Non-Cauchy Sequence Does Not Contradict Space Completeness","stem":"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.","answer":{"kind":"expression","value":"(1, 2, 19, 20, 10, 10, Yes, No)","explanation":"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)."},"skills":["A Non-Cauchy Sequence Does Not Contradict Space Completeness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r03-complete-metric-space-0061","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"901a0d31f533f720c0ef2501fabb4696c98f884ddb789f00b3bb6469222e16d6","family_id":"qf-mod27-complete-alternating-noncauchy","seed":202610273061,"semantic_template":"qf-mod27-complete-alternating-noncauchy/{\"N\":21,\"left\":2,\"metric\":[[0,5,10,13],[5,0,13,16],[10,13,0,21],[13,16,21,0]],\"n\":4,\"right\":3}"},"fields":{"title":"A Non-Cauchy Sequence Does Not Contradict Space Completeness","stem":"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.","answer":{"kind":"expression","value":"(2, 3, 21, 22, 21, 21, Yes, No)","explanation":"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)."},"skills":["A Non-Cauchy Sequence Does Not Contradict Space Completeness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r03-complete-metric-space-0062","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"02185abb10f797dcdf2b4a25d38addbd4c03ae9e0c58396f027fb1231060bad8","family_id":"qf-mod27-complete-alternating-noncauchy","seed":202610273062,"semantic_template":"qf-mod27-complete-alternating-noncauchy/{\"N\":23,\"left\":3,\"metric\":[[0,6,6,6],[6,0,8,8],[6,8,0,8],[6,8,8,0]],\"n\":4,\"right\":0}"},"fields":{"title":"A Non-Cauchy Sequence Does Not Contradict Space Completeness","stem":"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.","answer":{"kind":"expression","value":"(3, 0, 23, 24, 6, 6, Yes, No)","explanation":"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)."},"skills":["A Non-Cauchy Sequence Does Not Contradict Space Completeness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r03-complete-metric-space-0063","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b4a9bab623598c720e3f6faa0ee1610fc4c059ce02bf6bd29daa15f611214f0e","family_id":"qf-mod27-complete-alternating-noncauchy","seed":202610273063,"semantic_template":"qf-mod27-complete-alternating-noncauchy/{\"N\":25,\"left\":0,\"metric\":[[0,7,9,10],[7,0,10,11],[9,10,0,13],[10,11,13,0]],\"n\":4,\"right\":1}"},"fields":{"title":"A Non-Cauchy Sequence Does Not Contradict Space Completeness","stem":"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.","answer":{"kind":"expression","value":"(0, 1, 25, 26, 7, 7, Yes, No)","explanation":"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)."},"skills":["A Non-Cauchy Sequence Does Not Contradict Space Completeness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r03-complete-metric-space-0064","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e18076f620b95ede2f7fe71026a48537144f0e0005dab3430ae184146d809e9d","family_id":"qf-mod27-complete-alternating-noncauchy","seed":202610273064,"semantic_template":"qf-mod27-complete-alternating-noncauchy/{\"N\":27,\"left\":1,\"metric\":[[0,8,12,14],[8,0,12,14],[12,12,0,18],[14,14,18,0]],\"n\":4,\"right\":2}"},"fields":{"title":"A Non-Cauchy Sequence Does Not Contradict Space Completeness","stem":"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.","answer":{"kind":"expression","value":"(1, 2, 27, 28, 12, 12, Yes, No)","explanation":"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)."},"skills":["A Non-Cauchy Sequence Does Not Contradict Space Completeness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r03-complete-metric-space-0065","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"30e04a4a3dad2e5facb801bf5c3a22355adb11626fc6ea4bbb47501860aba5f6","family_id":"qf-mod27-complete-alternating-noncauchy","seed":202610273065,"semantic_template":"qf-mod27-complete-alternating-noncauchy/{\"N\":29,\"left\":2,\"metric\":[[0,9,8,18],[9,0,7,17],[8,7,0,16],[18,17,16,0]],\"n\":4,\"right\":3}"},"fields":{"title":"A Non-Cauchy Sequence Does Not Contradict Space Completeness","stem":"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.","answer":{"kind":"expression","value":"(2, 3, 29, 30, 16, 16, Yes, No)","explanation":"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)."},"skills":["A Non-Cauchy Sequence Does Not Contradict Space Completeness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r03-complete-metric-space-0066","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4cd51ce99e65930514e64ebe950ccd44343c32ee832eca78b274bb63fff163dc","family_id":"qf-mod27-complete-alternating-noncauchy","seed":202610273066,"semantic_template":"qf-mod27-complete-alternating-noncauchy/{\"N\":31,\"left\":3,\"metric\":[[0,6,6,6],[6,0,10,10],[6,10,0,10],[6,10,10,0]],\"n\":4,\"right\":0}"},"fields":{"title":"A Non-Cauchy Sequence Does Not Contradict Space Completeness","stem":"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.","answer":{"kind":"expression","value":"(3, 0, 31, 32, 6, 6, Yes, No)","explanation":"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)."},"skills":["A Non-Cauchy Sequence Does Not Contradict Space Completeness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r03-complete-metric-space-0067","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2a4d191782914cab91428219012e316106937f966591bcc858a3a9f7ed1774e0","family_id":"qf-mod27-complete-alternating-noncauchy","seed":202610273067,"semantic_template":"qf-mod27-complete-alternating-noncauchy/{\"N\":33,\"left\":0,\"metric\":[[0,7,9,10],[7,0,12,13],[9,12,0,15],[10,13,15,0]],\"n\":4,\"right\":1}"},"fields":{"title":"A Non-Cauchy Sequence Does Not Contradict Space Completeness","stem":"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.","answer":{"kind":"expression","value":"(0, 1, 33, 34, 7, 7, Yes, No)","explanation":"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)."},"skills":["A Non-Cauchy Sequence Does Not Contradict Space Completeness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r03-complete-metric-space-0068","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7ee0e714b7486965faac10b41bdc9bc015cda7e73e9340590363b42702ac0a0f","family_id":"qf-mod27-complete-alternating-noncauchy","seed":202610273068,"semantic_template":"qf-mod27-complete-alternating-noncauchy/{\"N\":35,\"left\":1,\"metric\":[[0,8,12,14],[8,0,14,16],[12,14,0,20],[14,16,20,0]],\"n\":4,\"right\":2}"},"fields":{"title":"A Non-Cauchy Sequence Does Not Contradict Space Completeness","stem":"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.","answer":{"kind":"expression","value":"(1, 2, 35, 36, 14, 14, Yes, No)","explanation":"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)."},"skills":["A Non-Cauchy Sequence Does Not Contradict Space Completeness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r03-complete-metric-space-0069","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"230fa583f7118d11bd9fbc04f811e9e0502423ab58fc5e506f7bf9a10885d9cc","family_id":"qf-mod27-complete-alternating-noncauchy","seed":202610273069,"semantic_template":"qf-mod27-complete-alternating-noncauchy/{\"N\":37,\"left\":2,\"metric\":[[0,9,8,18],[9,0,9,19],[8,9,0,18],[18,19,18,0]],\"n\":4,\"right\":3}"},"fields":{"title":"A Non-Cauchy Sequence Does Not Contradict Space Completeness","stem":"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.","answer":{"kind":"expression","value":"(2, 3, 37, 38, 18, 18, Yes, No)","explanation":"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)."},"skills":["A Non-Cauchy Sequence Does Not Contradict Space Completeness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r03-complete-metric-space-0070","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9f53bfe545a92eeb65aea3664b03125a5427d8b01e60befda51ee1bde1663c97","family_id":"qf-mod27-complete-alternating-noncauchy","seed":202610273070,"semantic_template":"qf-mod27-complete-alternating-noncauchy/{\"N\":39,\"left\":3,\"metric\":[[0,10,11,11],[10,0,11,11],[11,11,0,12],[11,11,12,0]],\"n\":4,\"right\":0}"},"fields":{"title":"A Non-Cauchy Sequence Does Not Contradict Space Completeness","stem":"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.","answer":{"kind":"expression","value":"(3, 0, 39, 40, 11, 11, Yes, No)","explanation":"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)."},"skills":["A Non-Cauchy Sequence Does Not Contradict Space Completeness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r03-complete-metric-space-0071","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6aa35478079adae9492ecbd8584a0952a7d1a4e4c981acf31fc974aab911305e","family_id":"qf-mod27-complete-alternating-noncauchy","seed":202610273071,"semantic_template":"qf-mod27-complete-alternating-noncauchy/{\"N\":41,\"left\":0,\"metric\":[[0,7,9,10],[7,0,14,15],[9,14,0,17],[10,15,17,0]],\"n\":4,\"right\":1}"},"fields":{"title":"A Non-Cauchy Sequence Does Not Contradict Space Completeness","stem":"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.","answer":{"kind":"expression","value":"(0, 1, 41, 42, 7, 7, Yes, No)","explanation":"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)."},"skills":["A Non-Cauchy Sequence Does Not Contradict Space Completeness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r03-complete-metric-space-0072","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d065d8e9d2f387ca347cf98e0f4e2394491ebfb943073c6ac20ffe363ecd58d2","family_id":"qf-mod27-complete-alternating-noncauchy","seed":202610273072,"semantic_template":"qf-mod27-complete-alternating-noncauchy/{\"N\":43,\"left\":1,\"metric\":[[0,8,5,14],[8,0,9,18],[5,9,0,15],[14,18,15,0]],\"n\":4,\"right\":2}"},"fields":{"title":"A Non-Cauchy Sequence Does Not Contradict Space Completeness","stem":"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.","answer":{"kind":"expression","value":"(1, 2, 43, 44, 9, 9, Yes, No)","explanation":"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)."},"skills":["A Non-Cauchy Sequence Does Not Contradict Space Completeness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r03-complete-metric-space-0073","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"070763c36ab0fea3a9b9cd2d8c993fef3e912153ce6c157e71000d28ae158b2e","family_id":"qf-mod27-complete-alternating-noncauchy","seed":202610273073,"semantic_template":"qf-mod27-complete-alternating-noncauchy/{\"N\":45,\"left\":2,\"metric\":[[0,9,8,7],[9,0,11,10],[8,11,0,9],[7,10,9,0]],\"n\":4,\"right\":3}"},"fields":{"title":"A Non-Cauchy Sequence Does Not Contradict Space Completeness","stem":"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.","answer":{"kind":"expression","value":"(2, 3, 45, 46, 9, 9, Yes, No)","explanation":"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)."},"skills":["A Non-Cauchy Sequence Does Not Contradict Space Completeness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r03-complete-metric-space-0074","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2bdb9254c4e62f63807d2f65c0a1410a5ad0f3746a16cc94cb6f9f3e83a2a0dc","family_id":"qf-mod27-complete-alternating-noncauchy","seed":202610273074,"semantic_template":"qf-mod27-complete-alternating-noncauchy/{\"N\":47,\"left\":3,\"metric\":[[0,10,11,11],[10,0,13,13],[11,13,0,14],[11,13,14,0]],\"n\":4,\"right\":0}"},"fields":{"title":"A Non-Cauchy Sequence Does Not Contradict Space Completeness","stem":"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.","answer":{"kind":"expression","value":"(3, 0, 47, 48, 11, 11, Yes, No)","explanation":"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)."},"skills":["A Non-Cauchy Sequence Does Not Contradict Space Completeness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r03-complete-metric-space-0075","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"41b28b76e03c6789265f7a60fa55af6843fa3a83dc9e078d4cb366f15ba77226","family_id":"qf-mod27-complete-alternating-noncauchy","seed":202610273075,"semantic_template":"qf-mod27-complete-alternating-noncauchy/{\"N\":49,\"left\":0,\"metric\":[[0,11,14,15],[11,0,15,16],[14,15,0,19],[15,16,19,0]],\"n\":4,\"right\":1}"},"fields":{"title":"A Non-Cauchy Sequence Does Not Contradict Space Completeness","stem":"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.","answer":{"kind":"expression","value":"(0, 1, 49, 50, 11, 11, Yes, No)","explanation":"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)."},"skills":["A Non-Cauchy Sequence Does Not Contradict Space Completeness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r03-complete-metric-space-0076","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0f2498d5b43127d1649e6abfcb0015608eed139e8328362ce1a042267468a827","family_id":"qf-mod27-complete-evidence-mode-audit","seed":202610273076,"semantic_template":"qf-mod27-complete-evidence-mode-audit/{\"evidence_mode\":\"sampled_sequences\",\"metric\":[[0,3,4,5],[3,0,5,6],[4,5,0,7],[5,6,7,0]],\"n\":4}"},"fields":{"title":"A Space-Wide Certificate Is Not a List of Successful Sequences","stem":"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.","answer":{"kind":"expression","value":"(4, 3, sampled_sequences, No, sampled_sequences_insufficient)","explanation":"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)."},"skills":["A Space-Wide Certificate Is Not a List of Successful Sequences"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r03-complete-metric-space-0077","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fc6723f58cb3c9c1c8570f7d007d464c17ff8a6ca1a8ce9b5e578dfea893c047","family_id":"qf-mod27-complete-evidence-mode-audit","seed":202610273077,"semantic_template":"qf-mod27-complete-evidence-mode-audit/{\"evidence_mode\":\"finite_minimum_separation\",\"metric\":[[0,4,7,9],[4,0,7,9],[7,7,0,12],[9,9,12,0]],\"n\":4}"},"fields":{"title":"A Space-Wide Certificate Is Not a List of Successful Sequences","stem":"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.","answer":{"kind":"expression","value":"(4, 4, finite_minimum_separation, Yes, universal_finite_certificate)","explanation":"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)."},"skills":["A Space-Wide Certificate Is Not a List of Successful Sequences"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r03-complete-metric-space-0078","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7e5718f303cbc100b625d05ec0e4af491554ccafbc7b2bedfa41eadecbf2c62c","family_id":"qf-mod27-complete-evidence-mode-audit","seed":202610273078,"semantic_template":"qf-mod27-complete-evidence-mode-audit/{\"evidence_mode\":\"finite_minimum_separation\",\"metric\":[[0,5,10,13],[5,0,9,12],[10,9,0,17],[13,12,17,0]],\"n\":4}"},"fields":{"title":"A Space-Wide Certificate Is Not a List of Successful Sequences","stem":"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.","answer":{"kind":"expression","value":"(4, 5, finite_minimum_separation, Yes, universal_finite_certificate)","explanation":"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)."},"skills":["A Space-Wide Certificate Is Not a List of Successful Sequences"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r03-complete-metric-space-0079","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b0cff7ecae3c2cbe6a7c625bf974ebe89a40353fb681fa33fbbed8a411aa3456","family_id":"qf-mod27-complete-evidence-mode-audit","seed":202610273079,"semantic_template":"qf-mod27-complete-evidence-mode-audit/{\"evidence_mode\":\"finite_minimum_separation\",\"metric\":[[0,6,13,17],[6,0,11,15],[13,11,0,22],[17,15,22,0]],\"n\":4}"},"fields":{"title":"A Space-Wide Certificate Is Not a List of Successful Sequences","stem":"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.","answer":{"kind":"expression","value":"(4, 6, finite_minimum_separation, Yes, universal_finite_certificate)","explanation":"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)."},"skills":["A Space-Wide Certificate Is Not a List of Successful Sequences"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r03-complete-metric-space-0080","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"37890826d4fc3d7b83fd4e0debf589c3e24805f5dd879440857b6062bddbb571","family_id":"qf-mod27-complete-evidence-mode-audit","seed":202610273080,"semantic_template":"qf-mod27-complete-evidence-mode-audit/{\"evidence_mode\":\"sampled_sequences\",\"metric\":[[0,7,9,10],[7,0,6,7],[9,6,0,9],[10,7,9,0]],\"n\":4}"},"fields":{"title":"A Space-Wide Certificate Is Not a List of Successful Sequences","stem":"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.","answer":{"kind":"expression","value":"(4, 6, sampled_sequences, No, sampled_sequences_insufficient)","explanation":"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)."},"skills":["A Space-Wide Certificate Is Not a List of Successful Sequences"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r03-complete-metric-space-0081","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3007b4e1800264380c4b5b5eb0ae580a862f319e6213f2079c3608687c3f610a","family_id":"qf-mod27-complete-evidence-mode-audit","seed":202610273081,"semantic_template":"qf-mod27-complete-evidence-mode-audit/{\"evidence_mode\":\"finite_minimum_separation\",\"metric\":[[0,4,7,9],[4,0,9,11],[7,9,0,14],[9,11,14,0]],\"n\":4}"},"fields":{"title":"A Space-Wide Certificate Is Not a List of Successful Sequences","stem":"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.","answer":{"kind":"expression","value":"(4, 4, finite_minimum_separation, Yes, universal_finite_certificate)","explanation":"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)."},"skills":["A Space-Wide Certificate Is Not a List of Successful Sequences"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r03-complete-metric-space-0082","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ac24c1267928f70cce74d1823a367cb3866a39742134230114d3da63ca512034","family_id":"qf-mod27-complete-evidence-mode-audit","seed":202610273082,"semantic_template":"qf-mod27-complete-evidence-mode-audit/{\"evidence_mode\":\"finite_minimum_separation\",\"metric\":[[0,5,10,13],[5,0,11,14],[10,11,0,19],[13,14,19,0]],\"n\":4}"},"fields":{"title":"A Space-Wide Certificate Is Not a List of Successful Sequences","stem":"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.","answer":{"kind":"expression","value":"(4, 5, finite_minimum_separation, Yes, universal_finite_certificate)","explanation":"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)."},"skills":["A Space-Wide Certificate Is Not a List of Successful Sequences"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r03-complete-metric-space-0083","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"be3af7b099abd00e83d86698aa62b7c3ac9047ba44c6fb70fb16ac4ccc6dff85","family_id":"qf-mod27-complete-evidence-mode-audit","seed":202610273083,"semantic_template":"qf-mod27-complete-evidence-mode-audit/{\"evidence_mode\":\"finite_minimum_separation\",\"metric\":[[0,6,6,17],[6,0,6,17],[6,6,0,17],[17,17,17,0]],\"n\":4}"},"fields":{"title":"A Space-Wide Certificate Is Not a List of Successful Sequences","stem":"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.","answer":{"kind":"expression","value":"(4, 6, finite_minimum_separation, Yes, universal_finite_certificate)","explanation":"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)."},"skills":["A Space-Wide Certificate Is Not a List of Successful Sequences"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r03-complete-metric-space-0084","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"60b5bd69785d9e2499aed5899cf7319287cb660be4666952f354fc266faa4dd1","family_id":"qf-mod27-complete-evidence-mode-audit","seed":202610273084,"semantic_template":"qf-mod27-complete-evidence-mode-audit/{\"evidence_mode\":\"sampled_sequences\",\"metric\":[[0,7,9,10],[7,0,8,9],[9,8,0,11],[10,9,11,0]],\"n\":4}"},"fields":{"title":"A Space-Wide Certificate Is Not a List of Successful Sequences","stem":"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.","answer":{"kind":"expression","value":"(4, 7, sampled_sequences, No, sampled_sequences_insufficient)","explanation":"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)."},"skills":["A Space-Wide Certificate Is Not a List of Successful Sequences"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r03-complete-metric-space-0085","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3c0d357908a77dea5c7ed3faaaa26983d6fc9798b3fcd6af800c52dd56ce577f","family_id":"qf-mod27-complete-evidence-mode-audit","seed":202610273085,"semantic_template":"qf-mod27-complete-evidence-mode-audit/{\"evidence_mode\":\"finite_minimum_separation\",\"metric\":[[0,8,12,14],[8,0,10,12],[12,10,0,16],[14,12,16,0]],\"n\":4}"},"fields":{"title":"A Space-Wide Certificate Is Not a List of Successful Sequences","stem":"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.","answer":{"kind":"expression","value":"(4, 8, finite_minimum_separation, Yes, universal_finite_certificate)","explanation":"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)."},"skills":["A Space-Wide Certificate Is Not a List of Successful Sequences"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r03-complete-metric-space-0086","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"70c9545613085e7245e9254f8c7a1423b8414b9c2cab05e6aaf21cf744e81925","family_id":"qf-mod27-complete-evidence-mode-audit","seed":202610273086,"semantic_template":"qf-mod27-complete-evidence-mode-audit/{\"evidence_mode\":\"finite_minimum_separation\",\"metric\":[[0,5,10,13],[5,0,13,16],[10,13,0,21],[13,16,21,0]],\"n\":4}"},"fields":{"title":"A Space-Wide Certificate Is Not a List of Successful Sequences","stem":"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.","answer":{"kind":"expression","value":"(4, 5, finite_minimum_separation, Yes, universal_finite_certificate)","explanation":"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)."},"skills":["A Space-Wide Certificate Is Not a List of Successful Sequences"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r03-complete-metric-space-0087","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3578487c6134458954f186888a46aa4b6de0783c3e3c7839e2e41189397b0870","family_id":"qf-mod27-complete-evidence-mode-audit","seed":202610273087,"semantic_template":"qf-mod27-complete-evidence-mode-audit/{\"evidence_mode\":\"finite_minimum_separation\",\"metric\":[[0,6,6,6],[6,0,8,8],[6,8,0,8],[6,8,8,0]],\"n\":4}"},"fields":{"title":"A Space-Wide Certificate Is Not a List of Successful Sequences","stem":"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.","answer":{"kind":"expression","value":"(4, 6, finite_minimum_separation, Yes, universal_finite_certificate)","explanation":"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)."},"skills":["A Space-Wide Certificate Is Not a List of Successful Sequences"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r03-complete-metric-space-0088","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0f21fec174270b0da92e021402bd4db850676bd4141381cb5ab223db185bf1f3","family_id":"qf-mod27-complete-evidence-mode-audit","seed":202610273088,"semantic_template":"qf-mod27-complete-evidence-mode-audit/{\"evidence_mode\":\"sampled_sequences\",\"metric\":[[0,7,9,10],[7,0,10,11],[9,10,0,13],[10,11,13,0]],\"n\":4}"},"fields":{"title":"A Space-Wide Certificate Is Not a List of Successful Sequences","stem":"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.","answer":{"kind":"expression","value":"(4, 7, sampled_sequences, No, sampled_sequences_insufficient)","explanation":"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)."},"skills":["A Space-Wide Certificate Is Not a List of Successful Sequences"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r03-complete-metric-space-0089","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"770c8bb6383cfc7b53f0983eff679b9794afdb8a9de8148964a640054b1a1295","family_id":"qf-mod27-complete-evidence-mode-audit","seed":202610273089,"semantic_template":"qf-mod27-complete-evidence-mode-audit/{\"evidence_mode\":\"finite_minimum_separation\",\"metric\":[[0,8,12,14],[8,0,12,14],[12,12,0,18],[14,14,18,0]],\"n\":4}"},"fields":{"title":"A Space-Wide Certificate Is Not a List of Successful Sequences","stem":"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.","answer":{"kind":"expression","value":"(4, 8, finite_minimum_separation, Yes, universal_finite_certificate)","explanation":"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)."},"skills":["A Space-Wide Certificate Is Not a List of Successful Sequences"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r03-complete-metric-space-0090","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4962966ecdb7a6f82f0f3480ceb12d781a8753679427ba0fb5cbc5794ffdc09f","family_id":"qf-mod27-complete-evidence-mode-audit","seed":202610273090,"semantic_template":"qf-mod27-complete-evidence-mode-audit/{\"evidence_mode\":\"finite_minimum_separation\",\"metric\":[[0,9,8,18],[9,0,7,17],[8,7,0,16],[18,17,16,0]],\"n\":4}"},"fields":{"title":"A Space-Wide Certificate Is Not a List of Successful Sequences","stem":"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.","answer":{"kind":"expression","value":"(4, 7, finite_minimum_separation, Yes, universal_finite_certificate)","explanation":"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)."},"skills":["A Space-Wide Certificate Is Not a List of Successful Sequences"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r03-complete-metric-space-0091","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e0f6681bc4816d4db252e94f259c870e32ab19c9ee21931a1533d0bde840dba5","family_id":"qf-mod27-complete-evidence-mode-audit","seed":202610273091,"semantic_template":"qf-mod27-complete-evidence-mode-audit/{\"evidence_mode\":\"finite_minimum_separation\",\"metric\":[[0,6,6,6],[6,0,10,10],[6,10,0,10],[6,10,10,0]],\"n\":4}"},"fields":{"title":"A Space-Wide Certificate Is Not a List of Successful Sequences","stem":"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.","answer":{"kind":"expression","value":"(4, 6, finite_minimum_separation, Yes, universal_finite_certificate)","explanation":"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)."},"skills":["A Space-Wide Certificate Is Not a List of Successful Sequences"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r03-complete-metric-space-0092","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"83e60b00673f3b0182342c55510b0bbcf9dee52a488c49cfe961e9d2d3561503","family_id":"qf-mod27-complete-evidence-mode-audit","seed":202610273092,"semantic_template":"qf-mod27-complete-evidence-mode-audit/{\"evidence_mode\":\"sampled_sequences\",\"metric\":[[0,7,9,10],[7,0,12,13],[9,12,0,15],[10,13,15,0]],\"n\":4}"},"fields":{"title":"A Space-Wide Certificate Is Not a List of Successful Sequences","stem":"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.","answer":{"kind":"expression","value":"(4, 7, sampled_sequences, No, sampled_sequences_insufficient)","explanation":"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)."},"skills":["A Space-Wide Certificate Is Not a List of Successful Sequences"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r03-complete-metric-space-0093","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4834a24d3d17fb37b3a841fdedfb6185bb813fcfc41abcf83ea8d4b15f55c76f","family_id":"qf-mod27-complete-evidence-mode-audit","seed":202610273093,"semantic_template":"qf-mod27-complete-evidence-mode-audit/{\"evidence_mode\":\"finite_minimum_separation\",\"metric\":[[0,8,12,14],[8,0,14,16],[12,14,0,20],[14,16,20,0]],\"n\":4}"},"fields":{"title":"A Space-Wide Certificate Is Not a List of Successful Sequences","stem":"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.","answer":{"kind":"expression","value":"(4, 8, finite_minimum_separation, Yes, universal_finite_certificate)","explanation":"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)."},"skills":["A Space-Wide Certificate Is Not a List of Successful Sequences"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r03-complete-metric-space-0094","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a24ab5a2f5cde40c5386225f669f91f13a676da38eac79b83b931f7867d08d3b","family_id":"qf-mod27-complete-evidence-mode-audit","seed":202610273094,"semantic_template":"qf-mod27-complete-evidence-mode-audit/{\"evidence_mode\":\"finite_minimum_separation\",\"metric\":[[0,9,8,18],[9,0,9,19],[8,9,0,18],[18,19,18,0]],\"n\":4}"},"fields":{"title":"A Space-Wide Certificate Is Not a List of Successful Sequences","stem":"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.","answer":{"kind":"expression","value":"(4, 8, finite_minimum_separation, Yes, universal_finite_certificate)","explanation":"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)."},"skills":["A Space-Wide Certificate Is Not a List of Successful Sequences"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r03-complete-metric-space-0095","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"adf84b9d81f3fb0e363cf0d7aa654e930c6de76af1d0747914568f50824a050c","family_id":"qf-mod27-complete-evidence-mode-audit","seed":202610273095,"semantic_template":"qf-mod27-complete-evidence-mode-audit/{\"evidence_mode\":\"finite_minimum_separation\",\"metric\":[[0,10,11,11],[10,0,11,11],[11,11,0,12],[11,11,12,0]],\"n\":4}"},"fields":{"title":"A Space-Wide Certificate Is Not a List of Successful Sequences","stem":"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.","answer":{"kind":"expression","value":"(4, 10, finite_minimum_separation, Yes, universal_finite_certificate)","explanation":"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)."},"skills":["A Space-Wide Certificate Is Not a List of Successful Sequences"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r03-complete-metric-space-0096","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"691bc7d7287ae07b8743e9c87b9230a520c9d982bd4041c3aa6d132192694e62","family_id":"qf-mod27-complete-evidence-mode-audit","seed":202610273096,"semantic_template":"qf-mod27-complete-evidence-mode-audit/{\"evidence_mode\":\"sampled_sequences\",\"metric\":[[0,7,9,10],[7,0,14,15],[9,14,0,17],[10,15,17,0]],\"n\":4}"},"fields":{"title":"A Space-Wide Certificate Is Not a List of Successful Sequences","stem":"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.","answer":{"kind":"expression","value":"(4, 7, sampled_sequences, No, sampled_sequences_insufficient)","explanation":"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)."},"skills":["A Space-Wide Certificate Is Not a List of Successful Sequences"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r03-complete-metric-space-0097","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ee402d913d49ac82244e63d0ade99fc1b16575992a0903f0dedecf7d0468b6d4","family_id":"qf-mod27-complete-evidence-mode-audit","seed":202610273097,"semantic_template":"qf-mod27-complete-evidence-mode-audit/{\"evidence_mode\":\"finite_minimum_separation\",\"metric\":[[0,8,5,14],[8,0,9,18],[5,9,0,15],[14,18,15,0]],\"n\":4}"},"fields":{"title":"A Space-Wide Certificate Is Not a List of Successful Sequences","stem":"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.","answer":{"kind":"expression","value":"(4, 5, finite_minimum_separation, Yes, universal_finite_certificate)","explanation":"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)."},"skills":["A Space-Wide Certificate Is Not a List of Successful Sequences"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r03-complete-metric-space-0098","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bc6aacd9c27ebe6cae7bd5811684d16cb0623392016bb05ab99bfdc2e450f43f","family_id":"qf-mod27-complete-evidence-mode-audit","seed":202610273098,"semantic_template":"qf-mod27-complete-evidence-mode-audit/{\"evidence_mode\":\"finite_minimum_separation\",\"metric\":[[0,9,8,7],[9,0,11,10],[8,11,0,9],[7,10,9,0]],\"n\":4}"},"fields":{"title":"A Space-Wide Certificate Is Not a List of Successful Sequences","stem":"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.","answer":{"kind":"expression","value":"(4, 7, finite_minimum_separation, Yes, universal_finite_certificate)","explanation":"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)."},"skills":["A Space-Wide Certificate Is Not a List of Successful Sequences"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r03-complete-metric-space-0099","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2d352776eeffdaa495940a5421db61203ad7cdb98c9f3e238bb91b6c3ef3a6c0","family_id":"qf-mod27-complete-evidence-mode-audit","seed":202610273099,"semantic_template":"qf-mod27-complete-evidence-mode-audit/{\"evidence_mode\":\"finite_minimum_separation\",\"metric\":[[0,10,11,11],[10,0,13,13],[11,13,0,14],[11,13,14,0]],\"n\":4}"},"fields":{"title":"A Space-Wide Certificate Is Not a List of Successful Sequences","stem":"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.","answer":{"kind":"expression","value":"(4, 10, finite_minimum_separation, Yes, universal_finite_certificate)","explanation":"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)."},"skills":["A Space-Wide Certificate Is Not a List of Successful Sequences"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r03-complete-metric-space-0100","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"caf43723e65cbbf7a01b5aba36bbb9bd756b20390365210e5d9532db11d55b66","family_id":"qf-mod27-complete-evidence-mode-audit","seed":202610273100,"semantic_template":"qf-mod27-complete-evidence-mode-audit/{\"evidence_mode\":\"sampled_sequences\",\"metric\":[[0,11,14,15],[11,0,15,16],[14,15,0,19],[15,16,19,0]],\"n\":4}"},"fields":{"title":"A Space-Wide Certificate Is Not a List of Successful Sequences","stem":"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.","answer":{"kind":"expression","value":"(4, 11, sampled_sequences, No, sampled_sequences_insufficient)","explanation":"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)."},"skills":["A Space-Wide Certificate Is Not a List of Successful Sequences"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}