{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r04-metric-convergence-0001","batch_id":"batch-20261013-advance-20261023-r04-metric-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fbd99f942b57eb7012fee6c1eb025d58fe87ecade0eed131b86ea0767646bacb","family_id":"qf-mod23-metricconv-ball-inventory","seed":202610234001,"semantic_template":"qf-mod23-metricconv-ball-inventory/{\"center\":0,\"distances\":[[0,1,3,6],[1,0,2,5],[3,2,0,3],[6,5,3,0]],\"epsilon\":[1,1],\"n\":4}"},"fields":{"title":"A Strict Open Ball in a Complete Finite Metric","stem":"Use the complete qualified finite metric table=[[0, 1, 3, 6], [1, 0, 2, 5], [3, 2, 0, 3], [6, 5, 3, 0]] on Fin(4)={0,…,3}. Preserve the convergence quantifiers: for every positive epsilon there exists N such that every n≥N satisfies d(x_n,candidate)<epsilon. A finite recorded tail alone does not certify this infinite statement. center=0, positive epsilon=1/1. Give (carrier size,center,normalized epsilon,every label with distance from center strictly less than epsilon in increasing order). Use exact reduced rational outputs with positive denominators, omitting /1 for integers.","answer":{"kind":"expression","value":"(4, 0, 1, [0])","explanation":"Compare exact distances strictly; a point at distance equal to epsilon is excluded. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (4, 0, 1, [0])."},"skills":["A Strict Open Ball in a Complete Finite Metric"],"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-20261013-advance-20261023-r04-metric-convergence-0002","batch_id":"batch-20261013-advance-20261023-r04-metric-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3f019808f2d033b13aa75b50a625d0f133d3856798770cb3dc3d58baf9b52dd0","family_id":"qf-mod23-metricconv-ball-inventory","seed":202610234002,"semantic_template":"qf-mod23-metricconv-ball-inventory/{\"center\":1,\"distances\":[[0,2,5,9],[2,0,3,7],[5,3,0,4],[9,7,4,0]],\"epsilon\":[2,1],\"n\":4}"},"fields":{"title":"A Strict Open Ball in a Complete Finite Metric","stem":"Use the complete qualified finite metric table=[[0, 2, 5, 9], [2, 0, 3, 7], [5, 3, 0, 4], [9, 7, 4, 0]] on Fin(4)={0,…,3}. Preserve the convergence quantifiers: for every positive epsilon there exists N such that every n≥N satisfies d(x_n,candidate)<epsilon. A finite recorded tail alone does not certify this infinite statement. center=1, positive epsilon=2/1. Give (carrier size,center,normalized epsilon,every label with distance from center strictly less than epsilon in increasing order). Use exact reduced rational outputs with positive denominators, omitting /1 for integers.","answer":{"kind":"expression","value":"(4, 1, 2, [1])","explanation":"Compare exact distances strictly; a point at distance equal to epsilon is excluded. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (4, 1, 2, [1])."},"skills":["A Strict Open Ball in a Complete Finite Metric"],"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-20261013-advance-20261023-r04-metric-convergence-0003","batch_id":"batch-20261013-advance-20261023-r04-metric-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"63575b5d2b5606758ed7db8a2a25db402cbf67ceaddcde14f8769c5a4bf51fd5","family_id":"qf-mod23-metricconv-ball-inventory","seed":202610234003,"semantic_template":"qf-mod23-metricconv-ball-inventory/{\"center\":2,\"distances\":[[0,3,7,12],[3,0,4,9],[7,4,0,5],[12,9,5,0]],\"epsilon\":[3,1],\"n\":4}"},"fields":{"title":"A Strict Open Ball in a Complete Finite Metric","stem":"Use the complete qualified finite metric table=[[0, 3, 7, 12], [3, 0, 4, 9], [7, 4, 0, 5], [12, 9, 5, 0]] on Fin(4)={0,…,3}. Preserve the convergence quantifiers: for every positive epsilon there exists N such that every n≥N satisfies d(x_n,candidate)<epsilon. A finite recorded tail alone does not certify this infinite statement. center=2, positive epsilon=3/1. Give (carrier size,center,normalized epsilon,every label with distance from center strictly less than epsilon in increasing order). Use exact reduced rational outputs with positive denominators, omitting /1 for integers.","answer":{"kind":"expression","value":"(4, 2, 3, [2])","explanation":"Compare exact distances strictly; a point at distance equal to epsilon is excluded. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (4, 2, 3, [2])."},"skills":["A Strict Open Ball in a Complete Finite Metric"],"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-20261013-advance-20261023-r04-metric-convergence-0004","batch_id":"batch-20261013-advance-20261023-r04-metric-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0b536d78b7b89780c7c781594333fbf68b585fd6a45f0bdddce0ec88236eb38e","family_id":"qf-mod23-metricconv-ball-inventory","seed":202610234004,"semantic_template":"qf-mod23-metricconv-ball-inventory/{\"center\":3,\"distances\":[[0,4,9,15],[4,0,5,11],[9,5,0,6],[15,11,6,0]],\"epsilon\":[4,1],\"n\":4}"},"fields":{"title":"A Strict Open Ball in a Complete Finite Metric","stem":"Use the complete qualified finite metric table=[[0, 4, 9, 15], [4, 0, 5, 11], [9, 5, 0, 6], [15, 11, 6, 0]] on Fin(4)={0,…,3}. Preserve the convergence quantifiers: for every positive epsilon there exists N such that every n≥N satisfies d(x_n,candidate)<epsilon. A finite recorded tail alone does not certify this infinite statement. center=3, positive epsilon=4/1. Give (carrier size,center,normalized epsilon,every label with distance from center strictly less than epsilon in increasing order). Use exact reduced rational outputs with positive denominators, omitting /1 for integers.","answer":{"kind":"expression","value":"(4, 3, 4, [3])","explanation":"Compare exact distances strictly; a point at distance equal to epsilon is excluded. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (4, 3, 4, [3])."},"skills":["A Strict Open Ball in a Complete Finite Metric"],"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-20261013-advance-20261023-r04-metric-convergence-0005","batch_id":"batch-20261013-advance-20261023-r04-metric-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b478fa1f60152048f926cfc22c5ced93b1f193df5b93677f284ed191e390acaf","family_id":"qf-mod23-metricconv-ball-inventory","seed":202610234005,"semantic_template":"qf-mod23-metricconv-ball-inventory/{\"center\":0,\"distances\":[[0,5,11,18],[5,0,6,13],[11,6,0,7],[18,13,7,0]],\"epsilon\":[5,1],\"n\":4}"},"fields":{"title":"A Strict Open Ball in a Complete Finite Metric","stem":"Use the complete qualified finite metric table=[[0, 5, 11, 18], [5, 0, 6, 13], [11, 6, 0, 7], [18, 13, 7, 0]] on Fin(4)={0,…,3}. Preserve the convergence quantifiers: for every positive epsilon there exists N such that every n≥N satisfies d(x_n,candidate)<epsilon. A finite recorded tail alone does not certify this infinite statement. center=0, positive epsilon=5/1. Give (carrier size,center,normalized epsilon,every label with distance from center strictly less than epsilon in increasing order). Use exact reduced rational outputs with positive denominators, omitting /1 for integers.","answer":{"kind":"expression","value":"(4, 0, 5, [0])","explanation":"Compare exact distances strictly; a point at distance equal to epsilon is excluded. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (4, 0, 5, [0])."},"skills":["A Strict Open Ball in a Complete Finite Metric"],"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-20261013-advance-20261023-r04-metric-convergence-0006","batch_id":"batch-20261013-advance-20261023-r04-metric-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a5cac78b45cb6c824cb7551a78dbe9a528f4055aea8e73995e51b2a291c20c5b","family_id":"qf-mod23-metricconv-ball-inventory","seed":202610234006,"semantic_template":"qf-mod23-metricconv-ball-inventory/{\"center\":1,\"distances\":[[0,6,13,21],[6,0,7,15],[13,7,0,8],[21,15,8,0]],\"epsilon\":[6,1],\"n\":4}"},"fields":{"title":"A Strict Open Ball in a Complete Finite Metric","stem":"Use the complete qualified finite metric table=[[0, 6, 13, 21], [6, 0, 7, 15], [13, 7, 0, 8], [21, 15, 8, 0]] on Fin(4)={0,…,3}. Preserve the convergence quantifiers: for every positive epsilon there exists N such that every n≥N satisfies d(x_n,candidate)<epsilon. A finite recorded tail alone does not certify this infinite statement. center=1, positive epsilon=6/1. Give (carrier size,center,normalized epsilon,every label with distance from center strictly less than epsilon in increasing order). Use exact reduced rational outputs with positive denominators, omitting /1 for integers.","answer":{"kind":"expression","value":"(4, 1, 6, [1])","explanation":"Compare exact distances strictly; a point at distance equal to epsilon is excluded. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (4, 1, 6, [1])."},"skills":["A Strict Open Ball in a Complete Finite Metric"],"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-20261013-advance-20261023-r04-metric-convergence-0007","batch_id":"batch-20261013-advance-20261023-r04-metric-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"478b56aad8c87e7a1faf1775a56733750283861a334e07ad45d48a3d3135e44b","family_id":"qf-mod23-metricconv-ball-inventory","seed":202610234007,"semantic_template":"qf-mod23-metricconv-ball-inventory/{\"center\":2,\"distances\":[[0,7,15,24],[7,0,8,17],[15,8,0,9],[24,17,9,0]],\"epsilon\":[7,1],\"n\":4}"},"fields":{"title":"A Strict Open Ball in a Complete Finite Metric","stem":"Use the complete qualified finite metric table=[[0, 7, 15, 24], [7, 0, 8, 17], [15, 8, 0, 9], [24, 17, 9, 0]] on Fin(4)={0,…,3}. Preserve the convergence quantifiers: for every positive epsilon there exists N such that every n≥N satisfies d(x_n,candidate)<epsilon. A finite recorded tail alone does not certify this infinite statement. center=2, positive epsilon=7/1. Give (carrier size,center,normalized epsilon,every label with distance from center strictly less than epsilon in increasing order). Use exact reduced rational outputs with positive denominators, omitting /1 for integers.","answer":{"kind":"expression","value":"(4, 2, 7, [2])","explanation":"Compare exact distances strictly; a point at distance equal to epsilon is excluded. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (4, 2, 7, [2])."},"skills":["A Strict Open Ball in a Complete Finite Metric"],"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-20261013-advance-20261023-r04-metric-convergence-0008","batch_id":"batch-20261013-advance-20261023-r04-metric-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"49531cfe024e0ed8e0673f85222facfa32b6e159815a8eaae5b3368daf5b1c87","family_id":"qf-mod23-metricconv-ball-inventory","seed":202610234008,"semantic_template":"qf-mod23-metricconv-ball-inventory/{\"center\":3,\"distances\":[[0,8,17,27],[8,0,9,19],[17,9,0,10],[27,19,10,0]],\"epsilon\":[1,1],\"n\":4}"},"fields":{"title":"A Strict Open Ball in a Complete Finite Metric","stem":"Use the complete qualified finite metric table=[[0, 8, 17, 27], [8, 0, 9, 19], [17, 9, 0, 10], [27, 19, 10, 0]] on Fin(4)={0,…,3}. Preserve the convergence quantifiers: for every positive epsilon there exists N such that every n≥N satisfies d(x_n,candidate)<epsilon. A finite recorded tail alone does not certify this infinite statement. center=3, positive epsilon=1/1. Give (carrier size,center,normalized epsilon,every label with distance from center strictly less than epsilon in increasing order). Use exact reduced rational outputs with positive denominators, omitting /1 for integers.","answer":{"kind":"expression","value":"(4, 3, 1, [3])","explanation":"Compare exact distances strictly; a point at distance equal to epsilon is excluded. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (4, 3, 1, [3])."},"skills":["A Strict Open Ball in a Complete Finite Metric"],"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-20261013-advance-20261023-r04-metric-convergence-0009","batch_id":"batch-20261013-advance-20261023-r04-metric-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e9758699b17eca7cb8c992e4137d3af1889e9ed4312af00a3d92c88773e4ae2b","family_id":"qf-mod23-metricconv-ball-inventory","seed":202610234009,"semantic_template":"qf-mod23-metricconv-ball-inventory/{\"center\":0,\"distances\":[[0,9,19,30],[9,0,10,21],[19,10,0,11],[30,21,11,0]],\"epsilon\":[2,1],\"n\":4}"},"fields":{"title":"A Strict Open Ball in a Complete Finite Metric","stem":"Use the complete qualified finite metric table=[[0, 9, 19, 30], [9, 0, 10, 21], [19, 10, 0, 11], [30, 21, 11, 0]] on Fin(4)={0,…,3}. Preserve the convergence quantifiers: for every positive epsilon there exists N such that every n≥N satisfies d(x_n,candidate)<epsilon. A finite recorded tail alone does not certify this infinite statement. center=0, positive epsilon=2/1. Give (carrier size,center,normalized epsilon,every label with distance from center strictly less than epsilon in increasing order). Use exact reduced rational outputs with positive denominators, omitting /1 for integers.","answer":{"kind":"expression","value":"(4, 0, 2, [0])","explanation":"Compare exact distances strictly; a point at distance equal to epsilon is excluded. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (4, 0, 2, [0])."},"skills":["A Strict Open Ball in a Complete Finite Metric"],"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-20261013-advance-20261023-r04-metric-convergence-0010","batch_id":"batch-20261013-advance-20261023-r04-metric-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e1f11b3f6c194c2518ef59e54df8672a570c43682747620b33fabc29be427105","family_id":"qf-mod23-metricconv-ball-inventory","seed":202610234010,"semantic_template":"qf-mod23-metricconv-ball-inventory/{\"center\":1,\"distances\":[[0,10,21,33],[10,0,11,23],[21,11,0,12],[33,23,12,0]],\"epsilon\":[3,1],\"n\":4}"},"fields":{"title":"A Strict Open Ball in a Complete Finite Metric","stem":"Use the complete qualified finite metric table=[[0, 10, 21, 33], [10, 0, 11, 23], [21, 11, 0, 12], [33, 23, 12, 0]] on Fin(4)={0,…,3}. Preserve the convergence quantifiers: for every positive epsilon there exists N such that every n≥N satisfies d(x_n,candidate)<epsilon. A finite recorded tail alone does not certify this infinite statement. center=1, positive epsilon=3/1. Give (carrier size,center,normalized epsilon,every label with distance from center strictly less than epsilon in increasing order). Use exact reduced rational outputs with positive denominators, omitting /1 for integers.","answer":{"kind":"expression","value":"(4, 1, 3, [1])","explanation":"Compare exact distances strictly; a point at distance equal to epsilon is excluded. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (4, 1, 3, [1])."},"skills":["A Strict Open Ball in a Complete Finite Metric"],"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-20261013-advance-20261023-r04-metric-convergence-0011","batch_id":"batch-20261013-advance-20261023-r04-metric-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"aee8b992aec08cc59d4f99ae5bd749ff51164b4fad90f712d072c7f7b79d9ab7","family_id":"qf-mod23-metricconv-ball-inventory","seed":202610234011,"semantic_template":"qf-mod23-metricconv-ball-inventory/{\"center\":2,\"distances\":[[0,11,23,36],[11,0,12,25],[23,12,0,13],[36,25,13,0]],\"epsilon\":[4,1],\"n\":4}"},"fields":{"title":"A Strict Open Ball in a Complete Finite Metric","stem":"Use the complete qualified finite metric table=[[0, 11, 23, 36], [11, 0, 12, 25], [23, 12, 0, 13], [36, 25, 13, 0]] on Fin(4)={0,…,3}. Preserve the convergence quantifiers: for every positive epsilon there exists N such that every n≥N satisfies d(x_n,candidate)<epsilon. A finite recorded tail alone does not certify this infinite statement. center=2, positive epsilon=4/1. Give (carrier size,center,normalized epsilon,every label with distance from center strictly less than epsilon in increasing order). Use exact reduced rational outputs with positive denominators, omitting /1 for integers.","answer":{"kind":"expression","value":"(4, 2, 4, [2])","explanation":"Compare exact distances strictly; a point at distance equal to epsilon is excluded. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (4, 2, 4, [2])."},"skills":["A Strict Open Ball in a Complete Finite Metric"],"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-20261013-advance-20261023-r04-metric-convergence-0012","batch_id":"batch-20261013-advance-20261023-r04-metric-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0c793668a6c62d370e9cca208672483cf6c9ced80864c80f7ad75afbfe1ac04b","family_id":"qf-mod23-metricconv-ball-inventory","seed":202610234012,"semantic_template":"qf-mod23-metricconv-ball-inventory/{\"center\":3,\"distances\":[[0,12,25,39],[12,0,13,27],[25,13,0,14],[39,27,14,0]],\"epsilon\":[5,1],\"n\":4}"},"fields":{"title":"A Strict Open Ball in a Complete Finite Metric","stem":"Use the complete qualified finite metric table=[[0, 12, 25, 39], [12, 0, 13, 27], [25, 13, 0, 14], [39, 27, 14, 0]] on Fin(4)={0,…,3}. Preserve the convergence quantifiers: for every positive epsilon there exists N such that every n≥N satisfies d(x_n,candidate)<epsilon. A finite recorded tail alone does not certify this infinite statement. center=3, positive epsilon=5/1. Give (carrier size,center,normalized epsilon,every label with distance from center strictly less than epsilon in increasing order). Use exact reduced rational outputs with positive denominators, omitting /1 for integers.","answer":{"kind":"expression","value":"(4, 3, 5, [3])","explanation":"Compare exact distances strictly; a point at distance equal to epsilon is excluded. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (4, 3, 5, [3])."},"skills":["A Strict Open Ball in a Complete Finite Metric"],"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-20261013-advance-20261023-r04-metric-convergence-0013","batch_id":"batch-20261013-advance-20261023-r04-metric-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a9cbb7ad6f5c91ffa57f751b47a827d4cd3c6c99330ddef2e5873ebe01552b13","family_id":"qf-mod23-metricconv-ball-inventory","seed":202610234013,"semantic_template":"qf-mod23-metricconv-ball-inventory/{\"center\":0,\"distances\":[[0,13,27,42],[13,0,14,29],[27,14,0,15],[42,29,15,0]],\"epsilon\":[6,1],\"n\":4}"},"fields":{"title":"A Strict Open Ball in a Complete Finite Metric","stem":"Use the complete qualified finite metric table=[[0, 13, 27, 42], [13, 0, 14, 29], [27, 14, 0, 15], [42, 29, 15, 0]] on Fin(4)={0,…,3}. Preserve the convergence quantifiers: for every positive epsilon there exists N such that every n≥N satisfies d(x_n,candidate)<epsilon. A finite recorded tail alone does not certify this infinite statement. center=0, positive epsilon=6/1. Give (carrier size,center,normalized epsilon,every label with distance from center strictly less than epsilon in increasing order). Use exact reduced rational outputs with positive denominators, omitting /1 for integers.","answer":{"kind":"expression","value":"(4, 0, 6, [0])","explanation":"Compare exact distances strictly; a point at distance equal to epsilon is excluded. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (4, 0, 6, [0])."},"skills":["A Strict Open Ball in a Complete Finite Metric"],"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-20261013-advance-20261023-r04-metric-convergence-0014","batch_id":"batch-20261013-advance-20261023-r04-metric-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0578a6de583b98466de8fd0848350904a1e7520e19178ece5cf56fbff1a40e09","family_id":"qf-mod23-metricconv-ball-inventory","seed":202610234014,"semantic_template":"qf-mod23-metricconv-ball-inventory/{\"center\":1,\"distances\":[[0,14,29,45],[14,0,15,31],[29,15,0,16],[45,31,16,0]],\"epsilon\":[7,1],\"n\":4}"},"fields":{"title":"A Strict Open Ball in a Complete Finite Metric","stem":"Use the complete qualified finite metric table=[[0, 14, 29, 45], [14, 0, 15, 31], [29, 15, 0, 16], [45, 31, 16, 0]] on Fin(4)={0,…,3}. Preserve the convergence quantifiers: for every positive epsilon there exists N such that every n≥N satisfies d(x_n,candidate)<epsilon. A finite recorded tail alone does not certify this infinite statement. center=1, positive epsilon=7/1. Give (carrier size,center,normalized epsilon,every label with distance from center strictly less than epsilon in increasing order). Use exact reduced rational outputs with positive denominators, omitting /1 for integers.","answer":{"kind":"expression","value":"(4, 1, 7, [1])","explanation":"Compare exact distances strictly; a point at distance equal to epsilon is excluded. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (4, 1, 7, [1])."},"skills":["A Strict Open Ball in a Complete Finite Metric"],"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-20261013-advance-20261023-r04-metric-convergence-0015","batch_id":"batch-20261013-advance-20261023-r04-metric-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"92487fb354dda712905ff484ad915f07f55f7cb58d4992dd4384bde0e8b31a5d","family_id":"qf-mod23-metricconv-ball-inventory","seed":202610234015,"semantic_template":"qf-mod23-metricconv-ball-inventory/{\"center\":2,\"distances\":[[0,15,31,48],[15,0,16,33],[31,16,0,17],[48,33,17,0]],\"epsilon\":[1,1],\"n\":4}"},"fields":{"title":"A Strict Open Ball in a Complete Finite Metric","stem":"Use the complete qualified finite metric table=[[0, 15, 31, 48], [15, 0, 16, 33], [31, 16, 0, 17], [48, 33, 17, 0]] on Fin(4)={0,…,3}. Preserve the convergence quantifiers: for every positive epsilon there exists N such that every n≥N satisfies d(x_n,candidate)<epsilon. A finite recorded tail alone does not certify this infinite statement. center=2, positive epsilon=1/1. Give (carrier size,center,normalized epsilon,every label with distance from center strictly less than epsilon in increasing order). Use exact reduced rational outputs with positive denominators, omitting /1 for integers.","answer":{"kind":"expression","value":"(4, 2, 1, [2])","explanation":"Compare exact distances strictly; a point at distance equal to epsilon is excluded. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (4, 2, 1, [2])."},"skills":["A Strict Open Ball in a Complete Finite Metric"],"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-20261013-advance-20261023-r04-metric-convergence-0016","batch_id":"batch-20261013-advance-20261023-r04-metric-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e7fc4063cb372e80f569b536fe76a4eb67c10214961e68540b056a3f683307ed","family_id":"qf-mod23-metricconv-ball-inventory","seed":202610234016,"semantic_template":"qf-mod23-metricconv-ball-inventory/{\"center\":3,\"distances\":[[0,16,33,51],[16,0,17,35],[33,17,0,18],[51,35,18,0]],\"epsilon\":[2,1],\"n\":4}"},"fields":{"title":"A Strict Open Ball in a Complete Finite Metric","stem":"Use the complete qualified finite metric table=[[0, 16, 33, 51], [16, 0, 17, 35], [33, 17, 0, 18], [51, 35, 18, 0]] on Fin(4)={0,…,3}. Preserve the convergence quantifiers: for every positive epsilon there exists N such that every n≥N satisfies d(x_n,candidate)<epsilon. A finite recorded tail alone does not certify this infinite statement. center=3, positive epsilon=2/1. Give (carrier size,center,normalized epsilon,every label with distance from center strictly less than epsilon in increasing order). Use exact reduced rational outputs with positive denominators, omitting /1 for integers.","answer":{"kind":"expression","value":"(4, 3, 2, [3])","explanation":"Compare exact distances strictly; a point at distance equal to epsilon is excluded. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (4, 3, 2, [3])."},"skills":["A Strict Open Ball in a Complete Finite Metric"],"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-20261013-advance-20261023-r04-metric-convergence-0017","batch_id":"batch-20261013-advance-20261023-r04-metric-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f89207ba834a144068e3093c7026dd13a35117e22d20a7cb5af3627d4fd1e591","family_id":"qf-mod23-metricconv-ball-inventory","seed":202610234017,"semantic_template":"qf-mod23-metricconv-ball-inventory/{\"center\":0,\"distances\":[[0,17,35,54],[17,0,18,37],[35,18,0,19],[54,37,19,0]],\"epsilon\":[3,1],\"n\":4}"},"fields":{"title":"A Strict Open Ball in a Complete Finite Metric","stem":"Use the complete qualified finite metric table=[[0, 17, 35, 54], [17, 0, 18, 37], [35, 18, 0, 19], [54, 37, 19, 0]] on Fin(4)={0,…,3}. Preserve the convergence quantifiers: for every positive epsilon there exists N such that every n≥N satisfies d(x_n,candidate)<epsilon. A finite recorded tail alone does not certify this infinite statement. center=0, positive epsilon=3/1. Give (carrier size,center,normalized epsilon,every label with distance from center strictly less than epsilon in increasing order). Use exact reduced rational outputs with positive denominators, omitting /1 for integers.","answer":{"kind":"expression","value":"(4, 0, 3, [0])","explanation":"Compare exact distances strictly; a point at distance equal to epsilon is excluded. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (4, 0, 3, [0])."},"skills":["A Strict Open Ball in a Complete Finite Metric"],"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-20261013-advance-20261023-r04-metric-convergence-0018","batch_id":"batch-20261013-advance-20261023-r04-metric-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d0a61e845e7472060d8abfb10d9cf55562a3700616ca483ce7176e4dfdbc08cc","family_id":"qf-mod23-metricconv-ball-inventory","seed":202610234018,"semantic_template":"qf-mod23-metricconv-ball-inventory/{\"center\":1,\"distances\":[[0,18,37,57],[18,0,19,39],[37,19,0,20],[57,39,20,0]],\"epsilon\":[4,1],\"n\":4}"},"fields":{"title":"A Strict Open Ball in a Complete Finite Metric","stem":"Use the complete qualified finite metric table=[[0, 18, 37, 57], [18, 0, 19, 39], [37, 19, 0, 20], [57, 39, 20, 0]] on Fin(4)={0,…,3}. Preserve the convergence quantifiers: for every positive epsilon there exists N such that every n≥N satisfies d(x_n,candidate)<epsilon. A finite recorded tail alone does not certify this infinite statement. center=1, positive epsilon=4/1. Give (carrier size,center,normalized epsilon,every label with distance from center strictly less than epsilon in increasing order). Use exact reduced rational outputs with positive denominators, omitting /1 for integers.","answer":{"kind":"expression","value":"(4, 1, 4, [1])","explanation":"Compare exact distances strictly; a point at distance equal to epsilon is excluded. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (4, 1, 4, [1])."},"skills":["A Strict Open Ball in a Complete Finite Metric"],"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-20261013-advance-20261023-r04-metric-convergence-0019","batch_id":"batch-20261013-advance-20261023-r04-metric-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d4fa026e0a0163146a42ded6e26672e254597bab179627de3419ef1930067bd0","family_id":"qf-mod23-metricconv-ball-inventory","seed":202610234019,"semantic_template":"qf-mod23-metricconv-ball-inventory/{\"center\":2,\"distances\":[[0,19,39,60],[19,0,20,41],[39,20,0,21],[60,41,21,0]],\"epsilon\":[5,1],\"n\":4}"},"fields":{"title":"A Strict Open Ball in a Complete Finite Metric","stem":"Use the complete qualified finite metric table=[[0, 19, 39, 60], [19, 0, 20, 41], [39, 20, 0, 21], [60, 41, 21, 0]] on Fin(4)={0,…,3}. Preserve the convergence quantifiers: for every positive epsilon there exists N such that every n≥N satisfies d(x_n,candidate)<epsilon. A finite recorded tail alone does not certify this infinite statement. center=2, positive epsilon=5/1. Give (carrier size,center,normalized epsilon,every label with distance from center strictly less than epsilon in increasing order). Use exact reduced rational outputs with positive denominators, omitting /1 for integers.","answer":{"kind":"expression","value":"(4, 2, 5, [2])","explanation":"Compare exact distances strictly; a point at distance equal to epsilon is excluded. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (4, 2, 5, [2])."},"skills":["A Strict Open Ball in a Complete Finite Metric"],"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-20261013-advance-20261023-r04-metric-convergence-0020","batch_id":"batch-20261013-advance-20261023-r04-metric-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c1c064f8d0eea30237c6363748991fb9715d4671e53db468685669990035586f","family_id":"qf-mod23-metricconv-ball-inventory","seed":202610234020,"semantic_template":"qf-mod23-metricconv-ball-inventory/{\"center\":3,\"distances\":[[0,20,41,63],[20,0,21,43],[41,21,0,22],[63,43,22,0]],\"epsilon\":[6,1],\"n\":4}"},"fields":{"title":"A Strict Open Ball in a Complete Finite Metric","stem":"Use the complete qualified finite metric table=[[0, 20, 41, 63], [20, 0, 21, 43], [41, 21, 0, 22], [63, 43, 22, 0]] on Fin(4)={0,…,3}. Preserve the convergence quantifiers: for every positive epsilon there exists N such that every n≥N satisfies d(x_n,candidate)<epsilon. A finite recorded tail alone does not certify this infinite statement. center=3, positive epsilon=6/1. Give (carrier size,center,normalized epsilon,every label with distance from center strictly less than epsilon in increasing order). Use exact reduced rational outputs with positive denominators, omitting /1 for integers.","answer":{"kind":"expression","value":"(4, 3, 6, [3])","explanation":"Compare exact distances strictly; a point at distance equal to epsilon is excluded. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (4, 3, 6, [3])."},"skills":["A Strict Open Ball in a Complete Finite Metric"],"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-20261013-advance-20261023-r04-metric-convergence-0021","batch_id":"batch-20261013-advance-20261023-r04-metric-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1133aa354a23cccbd1bba0ffa3b537e4fccd3e11d3d767a50a7a170bdd591a67","family_id":"qf-mod23-metricconv-ball-inventory","seed":202610234021,"semantic_template":"qf-mod23-metricconv-ball-inventory/{\"center\":0,\"distances\":[[0,21,43,66],[21,0,22,45],[43,22,0,23],[66,45,23,0]],\"epsilon\":[7,1],\"n\":4}"},"fields":{"title":"A Strict Open Ball in a Complete Finite Metric","stem":"Use the complete qualified finite metric table=[[0, 21, 43, 66], [21, 0, 22, 45], [43, 22, 0, 23], [66, 45, 23, 0]] on Fin(4)={0,…,3}. Preserve the convergence quantifiers: for every positive epsilon there exists N such that every n≥N satisfies d(x_n,candidate)<epsilon. A finite recorded tail alone does not certify this infinite statement. center=0, positive epsilon=7/1. Give (carrier size,center,normalized epsilon,every label with distance from center strictly less than epsilon in increasing order). Use exact reduced rational outputs with positive denominators, omitting /1 for integers.","answer":{"kind":"expression","value":"(4, 0, 7, [0])","explanation":"Compare exact distances strictly; a point at distance equal to epsilon is excluded. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (4, 0, 7, [0])."},"skills":["A Strict Open Ball in a Complete Finite Metric"],"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-20261013-advance-20261023-r04-metric-convergence-0022","batch_id":"batch-20261013-advance-20261023-r04-metric-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fa2982ed0522e83c77673970b4b9535663a0ed2365efeb57d91961bc0655f41b","family_id":"qf-mod23-metricconv-ball-inventory","seed":202610234022,"semantic_template":"qf-mod23-metricconv-ball-inventory/{\"center\":1,\"distances\":[[0,22,45,69],[22,0,23,47],[45,23,0,24],[69,47,24,0]],\"epsilon\":[1,1],\"n\":4}"},"fields":{"title":"A Strict Open Ball in a Complete Finite Metric","stem":"Use the complete qualified finite metric table=[[0, 22, 45, 69], [22, 0, 23, 47], [45, 23, 0, 24], [69, 47, 24, 0]] on Fin(4)={0,…,3}. Preserve the convergence quantifiers: for every positive epsilon there exists N such that every n≥N satisfies d(x_n,candidate)<epsilon. A finite recorded tail alone does not certify this infinite statement. center=1, positive epsilon=1/1. Give (carrier size,center,normalized epsilon,every label with distance from center strictly less than epsilon in increasing order). Use exact reduced rational outputs with positive denominators, omitting /1 for integers.","answer":{"kind":"expression","value":"(4, 1, 1, [1])","explanation":"Compare exact distances strictly; a point at distance equal to epsilon is excluded. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (4, 1, 1, [1])."},"skills":["A Strict Open Ball in a Complete Finite Metric"],"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-20261013-advance-20261023-r04-metric-convergence-0023","batch_id":"batch-20261013-advance-20261023-r04-metric-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"504b1735fd3140e7f9cfd3d5d0ad0518908a1a123ec7cfbb83e1f448e3c19f20","family_id":"qf-mod23-metricconv-ball-inventory","seed":202610234023,"semantic_template":"qf-mod23-metricconv-ball-inventory/{\"center\":2,\"distances\":[[0,23,47,72],[23,0,24,49],[47,24,0,25],[72,49,25,0]],\"epsilon\":[2,1],\"n\":4}"},"fields":{"title":"A Strict Open Ball in a Complete Finite Metric","stem":"Use the complete qualified finite metric table=[[0, 23, 47, 72], [23, 0, 24, 49], [47, 24, 0, 25], [72, 49, 25, 0]] on Fin(4)={0,…,3}. Preserve the convergence quantifiers: for every positive epsilon there exists N such that every n≥N satisfies d(x_n,candidate)<epsilon. A finite recorded tail alone does not certify this infinite statement. center=2, positive epsilon=2/1. Give (carrier size,center,normalized epsilon,every label with distance from center strictly less than epsilon in increasing order). Use exact reduced rational outputs with positive denominators, omitting /1 for integers.","answer":{"kind":"expression","value":"(4, 2, 2, [2])","explanation":"Compare exact distances strictly; a point at distance equal to epsilon is excluded. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (4, 2, 2, [2])."},"skills":["A Strict Open Ball in a Complete Finite Metric"],"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-20261013-advance-20261023-r04-metric-convergence-0024","batch_id":"batch-20261013-advance-20261023-r04-metric-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9c857309bd08727a25c703fd950e4aa666026070c6818246840f2ef96ee4b2ad","family_id":"qf-mod23-metricconv-ball-inventory","seed":202610234024,"semantic_template":"qf-mod23-metricconv-ball-inventory/{\"center\":3,\"distances\":[[0,24,49,75],[24,0,25,51],[49,25,0,26],[75,51,26,0]],\"epsilon\":[3,1],\"n\":4}"},"fields":{"title":"A Strict Open Ball in a Complete Finite Metric","stem":"Use the complete qualified finite metric table=[[0, 24, 49, 75], [24, 0, 25, 51], [49, 25, 0, 26], [75, 51, 26, 0]] on Fin(4)={0,…,3}. Preserve the convergence quantifiers: for every positive epsilon there exists N such that every n≥N satisfies d(x_n,candidate)<epsilon. A finite recorded tail alone does not certify this infinite statement. center=3, positive epsilon=3/1. Give (carrier size,center,normalized epsilon,every label with distance from center strictly less than epsilon in increasing order). Use exact reduced rational outputs with positive denominators, omitting /1 for integers.","answer":{"kind":"expression","value":"(4, 3, 3, [3])","explanation":"Compare exact distances strictly; a point at distance equal to epsilon is excluded. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (4, 3, 3, [3])."},"skills":["A Strict Open Ball in a Complete Finite Metric"],"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-20261013-advance-20261023-r04-metric-convergence-0025","batch_id":"batch-20261013-advance-20261023-r04-metric-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f0371fdcb87fe7924a839fd6e6ffaad0426b200a9f98d9751164f9b486e436de","family_id":"qf-mod23-metricconv-ball-inventory","seed":202610234025,"semantic_template":"qf-mod23-metricconv-ball-inventory/{\"center\":0,\"distances\":[[0,25,51,78],[25,0,26,53],[51,26,0,27],[78,53,27,0]],\"epsilon\":[4,1],\"n\":4}"},"fields":{"title":"A Strict Open Ball in a Complete Finite Metric","stem":"Use the complete qualified finite metric table=[[0, 25, 51, 78], [25, 0, 26, 53], [51, 26, 0, 27], [78, 53, 27, 0]] on Fin(4)={0,…,3}. Preserve the convergence quantifiers: for every positive epsilon there exists N such that every n≥N satisfies d(x_n,candidate)<epsilon. A finite recorded tail alone does not certify this infinite statement. center=0, positive epsilon=4/1. Give (carrier size,center,normalized epsilon,every label with distance from center strictly less than epsilon in increasing order). Use exact reduced rational outputs with positive denominators, omitting /1 for integers.","answer":{"kind":"expression","value":"(4, 0, 4, [0])","explanation":"Compare exact distances strictly; a point at distance equal to epsilon is excluded. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (4, 0, 4, [0])."},"skills":["A Strict Open Ball in a Complete Finite Metric"],"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-20261013-advance-20261023-r04-metric-convergence-0026","batch_id":"batch-20261013-advance-20261023-r04-metric-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b5999fa6b7a4b68c0b4c936fbccd326e4c1ebc73801e351b6bac302d91f96fb0","family_id":"qf-mod23-metricconv-finite-tail-audit","seed":202610234026,"semantic_template":"qf-mod23-metricconv-finite-tail-audit/{\"N\":1,\"candidate\":1,\"distances\":[[0,1,3,6],[1,0,2,5],[3,2,0,3],[6,5,3,0]],\"epsilon\":[1,1],\"n\":4,\"sequence\":[3,2,1,1]}"},"fields":{"title":"Finite-Tail Success Does Not Certify General Convergence","stem":"Use the complete qualified finite metric table=[[0, 1, 3, 6], [1, 0, 2, 5], [3, 2, 0, 3], [6, 5, 3, 0]] on Fin(4)={0,…,3}. Preserve the convergence quantifiers: for every positive epsilon there exists N such that every n≥N satisfies d(x_n,candidate)<epsilon. A finite recorded tail alone does not certify this infinite statement. recorded finite sequence=[3, 2, 1, 1], candidate=1, positive epsilon=1/1, cutoff N=1 may equal the recorded length. Give (sequence,candidate,epsilon,N,every [index,state,distance,status] row,all recorded indices at least N are inside?,general convergence certified?,boundary label). Status is prefix below N, otherwise inside or outside using the strict bound; general convergence is always No and the boundary label is finite-tail-insufficient-for-general-convergence.","answer":{"kind":"expression","value":"([3, 2, 1, 1], 1, 1, 1, [[0, 3, 5, prefix], [1, 2, 2, outside], [2, 1, 0, inside], [3, 1, 0, inside]], No, No, finite-tail-insufficient-for-general-convergence)","explanation":"Inspect only recorded indices. An empty recorded tail passes its bounded check vacuously but still proves no general convergence. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([3, 2, 1, 1], 1, 1, 1, [[0, 3, 5, prefix], [1, 2, 2, outside], [2, 1, 0, inside], [3, 1, 0, inside]], No, No, finite-tail-insufficient-for-general-convergence)."},"skills":["Finite-Tail Success Does Not Certify General Convergence"],"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-20261013-advance-20261023-r04-metric-convergence-0027","batch_id":"batch-20261013-advance-20261023-r04-metric-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8212ed7092caca832fa5f60b9dea0242d72c2832776dce8a3c4afc1eb12a7acf","family_id":"qf-mod23-metricconv-finite-tail-audit","seed":202610234027,"semantic_template":"qf-mod23-metricconv-finite-tail-audit/{\"N\":2,\"candidate\":1,\"distances\":[[0,2,5,9],[2,0,3,7],[5,3,0,4],[9,7,4,0]],\"epsilon\":[2,1],\"n\":4,\"sequence\":[3,2,1,1]}"},"fields":{"title":"Finite-Tail Success Does Not Certify General Convergence","stem":"Use the complete qualified finite metric table=[[0, 2, 5, 9], [2, 0, 3, 7], [5, 3, 0, 4], [9, 7, 4, 0]] on Fin(4)={0,…,3}. Preserve the convergence quantifiers: for every positive epsilon there exists N such that every n≥N satisfies d(x_n,candidate)<epsilon. A finite recorded tail alone does not certify this infinite statement. recorded finite sequence=[3, 2, 1, 1], candidate=1, positive epsilon=2/1, cutoff N=2 may equal the recorded length. Give (sequence,candidate,epsilon,N,every [index,state,distance,status] row,all recorded indices at least N are inside?,general convergence certified?,boundary label). Status is prefix below N, otherwise inside or outside using the strict bound; general convergence is always No and the boundary label is finite-tail-insufficient-for-general-convergence.","answer":{"kind":"expression","value":"([3, 2, 1, 1], 1, 2, 2, [[0, 3, 7, prefix], [1, 2, 3, prefix], [2, 1, 0, inside], [3, 1, 0, inside]], Yes, No, finite-tail-insufficient-for-general-convergence)","explanation":"Inspect only recorded indices. An empty recorded tail passes its bounded check vacuously but still proves no general convergence. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([3, 2, 1, 1], 1, 2, 2, [[0, 3, 7, prefix], [1, 2, 3, prefix], [2, 1, 0, inside], [3, 1, 0, inside]], Yes, No, finite-tail-insufficient-for-general-convergence)."},"skills":["Finite-Tail Success Does Not Certify General Convergence"],"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-20261013-advance-20261023-r04-metric-convergence-0028","batch_id":"batch-20261013-advance-20261023-r04-metric-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"72ada61ff2e0e1863784b5258a64d680c146ce59fdc97b14911da2b4d4c2c245","family_id":"qf-mod23-metricconv-finite-tail-audit","seed":202610234028,"semantic_template":"qf-mod23-metricconv-finite-tail-audit/{\"N\":3,\"candidate\":1,\"distances\":[[0,3,7,12],[3,0,4,9],[7,4,0,5],[12,9,5,0]],\"epsilon\":[3,1],\"n\":4,\"sequence\":[3,2,1,1]}"},"fields":{"title":"Finite-Tail Success Does Not Certify General Convergence","stem":"Use the complete qualified finite metric table=[[0, 3, 7, 12], [3, 0, 4, 9], [7, 4, 0, 5], [12, 9, 5, 0]] on Fin(4)={0,…,3}. Preserve the convergence quantifiers: for every positive epsilon there exists N such that every n≥N satisfies d(x_n,candidate)<epsilon. A finite recorded tail alone does not certify this infinite statement. recorded finite sequence=[3, 2, 1, 1], candidate=1, positive epsilon=3/1, cutoff N=3 may equal the recorded length. Give (sequence,candidate,epsilon,N,every [index,state,distance,status] row,all recorded indices at least N are inside?,general convergence certified?,boundary label). Status is prefix below N, otherwise inside or outside using the strict bound; general convergence is always No and the boundary label is finite-tail-insufficient-for-general-convergence.","answer":{"kind":"expression","value":"([3, 2, 1, 1], 1, 3, 3, [[0, 3, 9, prefix], [1, 2, 4, prefix], [2, 1, 0, prefix], [3, 1, 0, inside]], Yes, No, finite-tail-insufficient-for-general-convergence)","explanation":"Inspect only recorded indices. An empty recorded tail passes its bounded check vacuously but still proves no general convergence. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([3, 2, 1, 1], 1, 3, 3, [[0, 3, 9, prefix], [1, 2, 4, prefix], [2, 1, 0, prefix], [3, 1, 0, inside]], Yes, No, finite-tail-insufficient-for-general-convergence)."},"skills":["Finite-Tail Success Does Not Certify General Convergence"],"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-20261013-advance-20261023-r04-metric-convergence-0029","batch_id":"batch-20261013-advance-20261023-r04-metric-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6d29f7f5a32cb27c8806374098f699a99ffc2c06019f2de203ae7d62d59a0819","family_id":"qf-mod23-metricconv-finite-tail-audit","seed":202610234029,"semantic_template":"qf-mod23-metricconv-finite-tail-audit/{\"N\":1,\"candidate\":1,\"distances\":[[0,4,9,15],[4,0,5,11],[9,5,0,6],[15,11,6,0]],\"epsilon\":[4,1],\"n\":4,\"sequence\":[3,2,1,1]}"},"fields":{"title":"Finite-Tail Success Does Not Certify General Convergence","stem":"Use the complete qualified finite metric table=[[0, 4, 9, 15], [4, 0, 5, 11], [9, 5, 0, 6], [15, 11, 6, 0]] on Fin(4)={0,…,3}. Preserve the convergence quantifiers: for every positive epsilon there exists N such that every n≥N satisfies d(x_n,candidate)<epsilon. A finite recorded tail alone does not certify this infinite statement. recorded finite sequence=[3, 2, 1, 1], candidate=1, positive epsilon=4/1, cutoff N=1 may equal the recorded length. Give (sequence,candidate,epsilon,N,every [index,state,distance,status] row,all recorded indices at least N are inside?,general convergence certified?,boundary label). Status is prefix below N, otherwise inside or outside using the strict bound; general convergence is always No and the boundary label is finite-tail-insufficient-for-general-convergence.","answer":{"kind":"expression","value":"([3, 2, 1, 1], 1, 4, 1, [[0, 3, 11, prefix], [1, 2, 5, outside], [2, 1, 0, inside], [3, 1, 0, inside]], No, No, finite-tail-insufficient-for-general-convergence)","explanation":"Inspect only recorded indices. An empty recorded tail passes its bounded check vacuously but still proves no general convergence. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([3, 2, 1, 1], 1, 4, 1, [[0, 3, 11, prefix], [1, 2, 5, outside], [2, 1, 0, inside], [3, 1, 0, inside]], No, No, finite-tail-insufficient-for-general-convergence)."},"skills":["Finite-Tail Success Does Not Certify General Convergence"],"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-20261013-advance-20261023-r04-metric-convergence-0030","batch_id":"batch-20261013-advance-20261023-r04-metric-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"889c36caeab0126e5706cb4ff3d2045177c4dce2fd807b636317a26dc4e1b54a","family_id":"qf-mod23-metricconv-finite-tail-audit","seed":202610234030,"semantic_template":"qf-mod23-metricconv-finite-tail-audit/{\"N\":2,\"candidate\":1,\"distances\":[[0,5,11,18],[5,0,6,13],[11,6,0,7],[18,13,7,0]],\"epsilon\":[5,1],\"n\":4,\"sequence\":[3,2,1,1]}"},"fields":{"title":"Finite-Tail Success Does Not Certify General Convergence","stem":"Use the complete qualified finite metric table=[[0, 5, 11, 18], [5, 0, 6, 13], [11, 6, 0, 7], [18, 13, 7, 0]] on Fin(4)={0,…,3}. Preserve the convergence quantifiers: for every positive epsilon there exists N such that every n≥N satisfies d(x_n,candidate)<epsilon. A finite recorded tail alone does not certify this infinite statement. recorded finite sequence=[3, 2, 1, 1], candidate=1, positive epsilon=5/1, cutoff N=2 may equal the recorded length. Give (sequence,candidate,epsilon,N,every [index,state,distance,status] row,all recorded indices at least N are inside?,general convergence certified?,boundary label). Status is prefix below N, otherwise inside or outside using the strict bound; general convergence is always No and the boundary label is finite-tail-insufficient-for-general-convergence.","answer":{"kind":"expression","value":"([3, 2, 1, 1], 1, 5, 2, [[0, 3, 13, prefix], [1, 2, 6, prefix], [2, 1, 0, inside], [3, 1, 0, inside]], Yes, No, finite-tail-insufficient-for-general-convergence)","explanation":"Inspect only recorded indices. An empty recorded tail passes its bounded check vacuously but still proves no general convergence. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([3, 2, 1, 1], 1, 5, 2, [[0, 3, 13, prefix], [1, 2, 6, prefix], [2, 1, 0, inside], [3, 1, 0, inside]], Yes, No, finite-tail-insufficient-for-general-convergence)."},"skills":["Finite-Tail Success Does Not Certify General Convergence"],"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-20261013-advance-20261023-r04-metric-convergence-0031","batch_id":"batch-20261013-advance-20261023-r04-metric-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ef8bd48ea3caad5c1abe3cce62d113a0fb18a3234734f1023dce2f72f59e3ed0","family_id":"qf-mod23-metricconv-finite-tail-audit","seed":202610234031,"semantic_template":"qf-mod23-metricconv-finite-tail-audit/{\"N\":3,\"candidate\":1,\"distances\":[[0,6,13,21],[6,0,7,15],[13,7,0,8],[21,15,8,0]],\"epsilon\":[1,1],\"n\":4,\"sequence\":[3,2,1,1]}"},"fields":{"title":"Finite-Tail Success Does Not Certify General Convergence","stem":"Use the complete qualified finite metric table=[[0, 6, 13, 21], [6, 0, 7, 15], [13, 7, 0, 8], [21, 15, 8, 0]] on Fin(4)={0,…,3}. Preserve the convergence quantifiers: for every positive epsilon there exists N such that every n≥N satisfies d(x_n,candidate)<epsilon. A finite recorded tail alone does not certify this infinite statement. recorded finite sequence=[3, 2, 1, 1], candidate=1, positive epsilon=1/1, cutoff N=3 may equal the recorded length. Give (sequence,candidate,epsilon,N,every [index,state,distance,status] row,all recorded indices at least N are inside?,general convergence certified?,boundary label). Status is prefix below N, otherwise inside or outside using the strict bound; general convergence is always No and the boundary label is finite-tail-insufficient-for-general-convergence.","answer":{"kind":"expression","value":"([3, 2, 1, 1], 1, 1, 3, [[0, 3, 15, prefix], [1, 2, 7, prefix], [2, 1, 0, prefix], [3, 1, 0, inside]], Yes, No, finite-tail-insufficient-for-general-convergence)","explanation":"Inspect only recorded indices. An empty recorded tail passes its bounded check vacuously but still proves no general convergence. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([3, 2, 1, 1], 1, 1, 3, [[0, 3, 15, prefix], [1, 2, 7, prefix], [2, 1, 0, prefix], [3, 1, 0, inside]], Yes, No, finite-tail-insufficient-for-general-convergence)."},"skills":["Finite-Tail Success Does Not Certify General Convergence"],"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-20261013-advance-20261023-r04-metric-convergence-0032","batch_id":"batch-20261013-advance-20261023-r04-metric-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6854620afc372f36d0c6be4d2d48943b38876cb5d5f7dbbc549e3ab183c652cf","family_id":"qf-mod23-metricconv-finite-tail-audit","seed":202610234032,"semantic_template":"qf-mod23-metricconv-finite-tail-audit/{\"N\":1,\"candidate\":1,\"distances\":[[0,7,15,24],[7,0,8,17],[15,8,0,9],[24,17,9,0]],\"epsilon\":[2,1],\"n\":4,\"sequence\":[3,2,1,1]}"},"fields":{"title":"Finite-Tail Success Does Not Certify General Convergence","stem":"Use the complete qualified finite metric table=[[0, 7, 15, 24], [7, 0, 8, 17], [15, 8, 0, 9], [24, 17, 9, 0]] on Fin(4)={0,…,3}. Preserve the convergence quantifiers: for every positive epsilon there exists N such that every n≥N satisfies d(x_n,candidate)<epsilon. A finite recorded tail alone does not certify this infinite statement. recorded finite sequence=[3, 2, 1, 1], candidate=1, positive epsilon=2/1, cutoff N=1 may equal the recorded length. Give (sequence,candidate,epsilon,N,every [index,state,distance,status] row,all recorded indices at least N are inside?,general convergence certified?,boundary label). Status is prefix below N, otherwise inside or outside using the strict bound; general convergence is always No and the boundary label is finite-tail-insufficient-for-general-convergence.","answer":{"kind":"expression","value":"([3, 2, 1, 1], 1, 2, 1, [[0, 3, 17, prefix], [1, 2, 8, outside], [2, 1, 0, inside], [3, 1, 0, inside]], No, No, finite-tail-insufficient-for-general-convergence)","explanation":"Inspect only recorded indices. An empty recorded tail passes its bounded check vacuously but still proves no general convergence. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([3, 2, 1, 1], 1, 2, 1, [[0, 3, 17, prefix], [1, 2, 8, outside], [2, 1, 0, inside], [3, 1, 0, inside]], No, No, finite-tail-insufficient-for-general-convergence)."},"skills":["Finite-Tail Success Does Not Certify General Convergence"],"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-20261013-advance-20261023-r04-metric-convergence-0033","batch_id":"batch-20261013-advance-20261023-r04-metric-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a6fa37abbd01c9280ecd2853c5975acd53fcd6bb471cd4991a7921a70743116f","family_id":"qf-mod23-metricconv-finite-tail-audit","seed":202610234033,"semantic_template":"qf-mod23-metricconv-finite-tail-audit/{\"N\":2,\"candidate\":1,\"distances\":[[0,8,17,27],[8,0,9,19],[17,9,0,10],[27,19,10,0]],\"epsilon\":[3,1],\"n\":4,\"sequence\":[3,2,1,1]}"},"fields":{"title":"Finite-Tail Success Does Not Certify General Convergence","stem":"Use the complete qualified finite metric table=[[0, 8, 17, 27], [8, 0, 9, 19], [17, 9, 0, 10], [27, 19, 10, 0]] on Fin(4)={0,…,3}. Preserve the convergence quantifiers: for every positive epsilon there exists N such that every n≥N satisfies d(x_n,candidate)<epsilon. A finite recorded tail alone does not certify this infinite statement. recorded finite sequence=[3, 2, 1, 1], candidate=1, positive epsilon=3/1, cutoff N=2 may equal the recorded length. Give (sequence,candidate,epsilon,N,every [index,state,distance,status] row,all recorded indices at least N are inside?,general convergence certified?,boundary label). Status is prefix below N, otherwise inside or outside using the strict bound; general convergence is always No and the boundary label is finite-tail-insufficient-for-general-convergence.","answer":{"kind":"expression","value":"([3, 2, 1, 1], 1, 3, 2, [[0, 3, 19, prefix], [1, 2, 9, prefix], [2, 1, 0, inside], [3, 1, 0, inside]], Yes, No, finite-tail-insufficient-for-general-convergence)","explanation":"Inspect only recorded indices. An empty recorded tail passes its bounded check vacuously but still proves no general convergence. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([3, 2, 1, 1], 1, 3, 2, [[0, 3, 19, prefix], [1, 2, 9, prefix], [2, 1, 0, inside], [3, 1, 0, inside]], Yes, No, finite-tail-insufficient-for-general-convergence)."},"skills":["Finite-Tail Success Does Not Certify General Convergence"],"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-20261013-advance-20261023-r04-metric-convergence-0034","batch_id":"batch-20261013-advance-20261023-r04-metric-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"86179f79ca5ecffd6916c80adddeb7bf47162777e283d0055802e5ba1fd8ad45","family_id":"qf-mod23-metricconv-finite-tail-audit","seed":202610234034,"semantic_template":"qf-mod23-metricconv-finite-tail-audit/{\"N\":3,\"candidate\":1,\"distances\":[[0,9,19,30],[9,0,10,21],[19,10,0,11],[30,21,11,0]],\"epsilon\":[4,1],\"n\":4,\"sequence\":[3,2,1,1]}"},"fields":{"title":"Finite-Tail Success Does Not Certify General Convergence","stem":"Use the complete qualified finite metric table=[[0, 9, 19, 30], [9, 0, 10, 21], [19, 10, 0, 11], [30, 21, 11, 0]] on Fin(4)={0,…,3}. Preserve the convergence quantifiers: for every positive epsilon there exists N such that every n≥N satisfies d(x_n,candidate)<epsilon. A finite recorded tail alone does not certify this infinite statement. recorded finite sequence=[3, 2, 1, 1], candidate=1, positive epsilon=4/1, cutoff N=3 may equal the recorded length. Give (sequence,candidate,epsilon,N,every [index,state,distance,status] row,all recorded indices at least N are inside?,general convergence certified?,boundary label). Status is prefix below N, otherwise inside or outside using the strict bound; general convergence is always No and the boundary label is finite-tail-insufficient-for-general-convergence.","answer":{"kind":"expression","value":"([3, 2, 1, 1], 1, 4, 3, [[0, 3, 21, prefix], [1, 2, 10, prefix], [2, 1, 0, prefix], [3, 1, 0, inside]], Yes, No, finite-tail-insufficient-for-general-convergence)","explanation":"Inspect only recorded indices. An empty recorded tail passes its bounded check vacuously but still proves no general convergence. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([3, 2, 1, 1], 1, 4, 3, [[0, 3, 21, prefix], [1, 2, 10, prefix], [2, 1, 0, prefix], [3, 1, 0, inside]], Yes, No, finite-tail-insufficient-for-general-convergence)."},"skills":["Finite-Tail Success Does Not Certify General Convergence"],"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-20261013-advance-20261023-r04-metric-convergence-0035","batch_id":"batch-20261013-advance-20261023-r04-metric-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fc8935e2af3e65b3ee9406ce773d59f4adadbc95eabd4eafa18367822050600b","family_id":"qf-mod23-metricconv-finite-tail-audit","seed":202610234035,"semantic_template":"qf-mod23-metricconv-finite-tail-audit/{\"N\":1,\"candidate\":1,\"distances\":[[0,10,21,33],[10,0,11,23],[21,11,0,12],[33,23,12,0]],\"epsilon\":[5,1],\"n\":4,\"sequence\":[3,2,1,1]}"},"fields":{"title":"Finite-Tail Success Does Not Certify General Convergence","stem":"Use the complete qualified finite metric table=[[0, 10, 21, 33], [10, 0, 11, 23], [21, 11, 0, 12], [33, 23, 12, 0]] on Fin(4)={0,…,3}. Preserve the convergence quantifiers: for every positive epsilon there exists N such that every n≥N satisfies d(x_n,candidate)<epsilon. A finite recorded tail alone does not certify this infinite statement. recorded finite sequence=[3, 2, 1, 1], candidate=1, positive epsilon=5/1, cutoff N=1 may equal the recorded length. Give (sequence,candidate,epsilon,N,every [index,state,distance,status] row,all recorded indices at least N are inside?,general convergence certified?,boundary label). Status is prefix below N, otherwise inside or outside using the strict bound; general convergence is always No and the boundary label is finite-tail-insufficient-for-general-convergence.","answer":{"kind":"expression","value":"([3, 2, 1, 1], 1, 5, 1, [[0, 3, 23, prefix], [1, 2, 11, outside], [2, 1, 0, inside], [3, 1, 0, inside]], No, No, finite-tail-insufficient-for-general-convergence)","explanation":"Inspect only recorded indices. An empty recorded tail passes its bounded check vacuously but still proves no general convergence. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([3, 2, 1, 1], 1, 5, 1, [[0, 3, 23, prefix], [1, 2, 11, outside], [2, 1, 0, inside], [3, 1, 0, inside]], No, No, finite-tail-insufficient-for-general-convergence)."},"skills":["Finite-Tail Success Does Not Certify General Convergence"],"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-20261013-advance-20261023-r04-metric-convergence-0036","batch_id":"batch-20261013-advance-20261023-r04-metric-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d2244d63dfb3af8062225c8c73160d66594e1767efbdfcd15cb25c99186c49ff","family_id":"qf-mod23-metricconv-finite-tail-audit","seed":202610234036,"semantic_template":"qf-mod23-metricconv-finite-tail-audit/{\"N\":2,\"candidate\":1,\"distances\":[[0,11,23,36],[11,0,12,25],[23,12,0,13],[36,25,13,0]],\"epsilon\":[1,1],\"n\":4,\"sequence\":[3,2,1,1]}"},"fields":{"title":"Finite-Tail Success Does Not Certify General Convergence","stem":"Use the complete qualified finite metric table=[[0, 11, 23, 36], [11, 0, 12, 25], [23, 12, 0, 13], [36, 25, 13, 0]] on Fin(4)={0,…,3}. Preserve the convergence quantifiers: for every positive epsilon there exists N such that every n≥N satisfies d(x_n,candidate)<epsilon. A finite recorded tail alone does not certify this infinite statement. recorded finite sequence=[3, 2, 1, 1], candidate=1, positive epsilon=1/1, cutoff N=2 may equal the recorded length. Give (sequence,candidate,epsilon,N,every [index,state,distance,status] row,all recorded indices at least N are inside?,general convergence certified?,boundary label). Status is prefix below N, otherwise inside or outside using the strict bound; general convergence is always No and the boundary label is finite-tail-insufficient-for-general-convergence.","answer":{"kind":"expression","value":"([3, 2, 1, 1], 1, 1, 2, [[0, 3, 25, prefix], [1, 2, 12, prefix], [2, 1, 0, inside], [3, 1, 0, inside]], Yes, No, finite-tail-insufficient-for-general-convergence)","explanation":"Inspect only recorded indices. An empty recorded tail passes its bounded check vacuously but still proves no general convergence. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([3, 2, 1, 1], 1, 1, 2, [[0, 3, 25, prefix], [1, 2, 12, prefix], [2, 1, 0, inside], [3, 1, 0, inside]], Yes, No, finite-tail-insufficient-for-general-convergence)."},"skills":["Finite-Tail Success Does Not Certify General Convergence"],"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-20261013-advance-20261023-r04-metric-convergence-0037","batch_id":"batch-20261013-advance-20261023-r04-metric-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9afb510b823e51bd625759d47ddfbb47de4cc2dafc9f1d725a85134cbed54155","family_id":"qf-mod23-metricconv-finite-tail-audit","seed":202610234037,"semantic_template":"qf-mod23-metricconv-finite-tail-audit/{\"N\":3,\"candidate\":1,\"distances\":[[0,12,25,39],[12,0,13,27],[25,13,0,14],[39,27,14,0]],\"epsilon\":[2,1],\"n\":4,\"sequence\":[3,2,1,1]}"},"fields":{"title":"Finite-Tail Success Does Not Certify General Convergence","stem":"Use the complete qualified finite metric table=[[0, 12, 25, 39], [12, 0, 13, 27], [25, 13, 0, 14], [39, 27, 14, 0]] on Fin(4)={0,…,3}. Preserve the convergence quantifiers: for every positive epsilon there exists N such that every n≥N satisfies d(x_n,candidate)<epsilon. A finite recorded tail alone does not certify this infinite statement. recorded finite sequence=[3, 2, 1, 1], candidate=1, positive epsilon=2/1, cutoff N=3 may equal the recorded length. Give (sequence,candidate,epsilon,N,every [index,state,distance,status] row,all recorded indices at least N are inside?,general convergence certified?,boundary label). Status is prefix below N, otherwise inside or outside using the strict bound; general convergence is always No and the boundary label is finite-tail-insufficient-for-general-convergence.","answer":{"kind":"expression","value":"([3, 2, 1, 1], 1, 2, 3, [[0, 3, 27, prefix], [1, 2, 13, prefix], [2, 1, 0, prefix], [3, 1, 0, inside]], Yes, No, finite-tail-insufficient-for-general-convergence)","explanation":"Inspect only recorded indices. An empty recorded tail passes its bounded check vacuously but still proves no general convergence. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([3, 2, 1, 1], 1, 2, 3, [[0, 3, 27, prefix], [1, 2, 13, prefix], [2, 1, 0, prefix], [3, 1, 0, inside]], Yes, No, finite-tail-insufficient-for-general-convergence)."},"skills":["Finite-Tail Success Does Not Certify General Convergence"],"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-20261013-advance-20261023-r04-metric-convergence-0038","batch_id":"batch-20261013-advance-20261023-r04-metric-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"95af9fa9626c284b7bb238d26f88a9e69a9e29eaadd7878a62b339a89d0cb795","family_id":"qf-mod23-metricconv-finite-tail-audit","seed":202610234038,"semantic_template":"qf-mod23-metricconv-finite-tail-audit/{\"N\":1,\"candidate\":1,\"distances\":[[0,13,27,42],[13,0,14,29],[27,14,0,15],[42,29,15,0]],\"epsilon\":[3,1],\"n\":4,\"sequence\":[3,2,1,1]}"},"fields":{"title":"Finite-Tail Success Does Not Certify General Convergence","stem":"Use the complete qualified finite metric table=[[0, 13, 27, 42], [13, 0, 14, 29], [27, 14, 0, 15], [42, 29, 15, 0]] on Fin(4)={0,…,3}. Preserve the convergence quantifiers: for every positive epsilon there exists N such that every n≥N satisfies d(x_n,candidate)<epsilon. A finite recorded tail alone does not certify this infinite statement. recorded finite sequence=[3, 2, 1, 1], candidate=1, positive epsilon=3/1, cutoff N=1 may equal the recorded length. Give (sequence,candidate,epsilon,N,every [index,state,distance,status] row,all recorded indices at least N are inside?,general convergence certified?,boundary label). Status is prefix below N, otherwise inside or outside using the strict bound; general convergence is always No and the boundary label is finite-tail-insufficient-for-general-convergence.","answer":{"kind":"expression","value":"([3, 2, 1, 1], 1, 3, 1, [[0, 3, 29, prefix], [1, 2, 14, outside], [2, 1, 0, inside], [3, 1, 0, inside]], No, No, finite-tail-insufficient-for-general-convergence)","explanation":"Inspect only recorded indices. An empty recorded tail passes its bounded check vacuously but still proves no general convergence. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([3, 2, 1, 1], 1, 3, 1, [[0, 3, 29, prefix], [1, 2, 14, outside], [2, 1, 0, inside], [3, 1, 0, inside]], No, No, finite-tail-insufficient-for-general-convergence)."},"skills":["Finite-Tail Success Does Not Certify General Convergence"],"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-20261013-advance-20261023-r04-metric-convergence-0039","batch_id":"batch-20261013-advance-20261023-r04-metric-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ce6ad240b25ed6b161b2fc515613f861d3ff259952294585c566eef98cf4234e","family_id":"qf-mod23-metricconv-finite-tail-audit","seed":202610234039,"semantic_template":"qf-mod23-metricconv-finite-tail-audit/{\"N\":2,\"candidate\":1,\"distances\":[[0,14,29,45],[14,0,15,31],[29,15,0,16],[45,31,16,0]],\"epsilon\":[4,1],\"n\":4,\"sequence\":[3,2,1,1]}"},"fields":{"title":"Finite-Tail Success Does Not Certify General Convergence","stem":"Use the complete qualified finite metric table=[[0, 14, 29, 45], [14, 0, 15, 31], [29, 15, 0, 16], [45, 31, 16, 0]] on Fin(4)={0,…,3}. Preserve the convergence quantifiers: for every positive epsilon there exists N such that every n≥N satisfies d(x_n,candidate)<epsilon. A finite recorded tail alone does not certify this infinite statement. recorded finite sequence=[3, 2, 1, 1], candidate=1, positive epsilon=4/1, cutoff N=2 may equal the recorded length. Give (sequence,candidate,epsilon,N,every [index,state,distance,status] row,all recorded indices at least N are inside?,general convergence certified?,boundary label). Status is prefix below N, otherwise inside or outside using the strict bound; general convergence is always No and the boundary label is finite-tail-insufficient-for-general-convergence.","answer":{"kind":"expression","value":"([3, 2, 1, 1], 1, 4, 2, [[0, 3, 31, prefix], [1, 2, 15, prefix], [2, 1, 0, inside], [3, 1, 0, inside]], Yes, No, finite-tail-insufficient-for-general-convergence)","explanation":"Inspect only recorded indices. An empty recorded tail passes its bounded check vacuously but still proves no general convergence. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([3, 2, 1, 1], 1, 4, 2, [[0, 3, 31, prefix], [1, 2, 15, prefix], [2, 1, 0, inside], [3, 1, 0, inside]], Yes, No, finite-tail-insufficient-for-general-convergence)."},"skills":["Finite-Tail Success Does Not Certify General Convergence"],"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-20261013-advance-20261023-r04-metric-convergence-0040","batch_id":"batch-20261013-advance-20261023-r04-metric-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b470a028737b5e039883a15349590bf38ad524a865c3135e291cd3cd9358dfcd","family_id":"qf-mod23-metricconv-finite-tail-audit","seed":202610234040,"semantic_template":"qf-mod23-metricconv-finite-tail-audit/{\"N\":3,\"candidate\":1,\"distances\":[[0,15,31,48],[15,0,16,33],[31,16,0,17],[48,33,17,0]],\"epsilon\":[5,1],\"n\":4,\"sequence\":[3,2,1,1]}"},"fields":{"title":"Finite-Tail Success Does Not Certify General Convergence","stem":"Use the complete qualified finite metric table=[[0, 15, 31, 48], [15, 0, 16, 33], [31, 16, 0, 17], [48, 33, 17, 0]] on Fin(4)={0,…,3}. Preserve the convergence quantifiers: for every positive epsilon there exists N such that every n≥N satisfies d(x_n,candidate)<epsilon. A finite recorded tail alone does not certify this infinite statement. recorded finite sequence=[3, 2, 1, 1], candidate=1, positive epsilon=5/1, cutoff N=3 may equal the recorded length. Give (sequence,candidate,epsilon,N,every [index,state,distance,status] row,all recorded indices at least N are inside?,general convergence certified?,boundary label). Status is prefix below N, otherwise inside or outside using the strict bound; general convergence is always No and the boundary label is finite-tail-insufficient-for-general-convergence.","answer":{"kind":"expression","value":"([3, 2, 1, 1], 1, 5, 3, [[0, 3, 33, prefix], [1, 2, 16, prefix], [2, 1, 0, prefix], [3, 1, 0, inside]], Yes, No, finite-tail-insufficient-for-general-convergence)","explanation":"Inspect only recorded indices. An empty recorded tail passes its bounded check vacuously but still proves no general convergence. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([3, 2, 1, 1], 1, 5, 3, [[0, 3, 33, prefix], [1, 2, 16, prefix], [2, 1, 0, prefix], [3, 1, 0, inside]], Yes, No, finite-tail-insufficient-for-general-convergence)."},"skills":["Finite-Tail Success Does Not Certify General Convergence"],"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-20261013-advance-20261023-r04-metric-convergence-0041","batch_id":"batch-20261013-advance-20261023-r04-metric-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6cf098ab93666e970fec503f6e0bd150173ad5c87009ca1142c443f78e16d655","family_id":"qf-mod23-metricconv-finite-tail-audit","seed":202610234041,"semantic_template":"qf-mod23-metricconv-finite-tail-audit/{\"N\":1,\"candidate\":1,\"distances\":[[0,16,33,51],[16,0,17,35],[33,17,0,18],[51,35,18,0]],\"epsilon\":[1,1],\"n\":4,\"sequence\":[3,2,1,1]}"},"fields":{"title":"Finite-Tail Success Does Not Certify General Convergence","stem":"Use the complete qualified finite metric table=[[0, 16, 33, 51], [16, 0, 17, 35], [33, 17, 0, 18], [51, 35, 18, 0]] on Fin(4)={0,…,3}. Preserve the convergence quantifiers: for every positive epsilon there exists N such that every n≥N satisfies d(x_n,candidate)<epsilon. A finite recorded tail alone does not certify this infinite statement. recorded finite sequence=[3, 2, 1, 1], candidate=1, positive epsilon=1/1, cutoff N=1 may equal the recorded length. Give (sequence,candidate,epsilon,N,every [index,state,distance,status] row,all recorded indices at least N are inside?,general convergence certified?,boundary label). Status is prefix below N, otherwise inside or outside using the strict bound; general convergence is always No and the boundary label is finite-tail-insufficient-for-general-convergence.","answer":{"kind":"expression","value":"([3, 2, 1, 1], 1, 1, 1, [[0, 3, 35, prefix], [1, 2, 17, outside], [2, 1, 0, inside], [3, 1, 0, inside]], No, No, finite-tail-insufficient-for-general-convergence)","explanation":"Inspect only recorded indices. An empty recorded tail passes its bounded check vacuously but still proves no general convergence. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([3, 2, 1, 1], 1, 1, 1, [[0, 3, 35, prefix], [1, 2, 17, outside], [2, 1, 0, inside], [3, 1, 0, inside]], No, No, finite-tail-insufficient-for-general-convergence)."},"skills":["Finite-Tail Success Does Not Certify General Convergence"],"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-20261013-advance-20261023-r04-metric-convergence-0042","batch_id":"batch-20261013-advance-20261023-r04-metric-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6142aa49346becd2e9a07830cf4a71adaef3cb6525f5c98cf3d25833453b46f4","family_id":"qf-mod23-metricconv-finite-tail-audit","seed":202610234042,"semantic_template":"qf-mod23-metricconv-finite-tail-audit/{\"N\":2,\"candidate\":1,\"distances\":[[0,17,35,54],[17,0,18,37],[35,18,0,19],[54,37,19,0]],\"epsilon\":[2,1],\"n\":4,\"sequence\":[3,2,1,1]}"},"fields":{"title":"Finite-Tail Success Does Not Certify General Convergence","stem":"Use the complete qualified finite metric table=[[0, 17, 35, 54], [17, 0, 18, 37], [35, 18, 0, 19], [54, 37, 19, 0]] on Fin(4)={0,…,3}. Preserve the convergence quantifiers: for every positive epsilon there exists N such that every n≥N satisfies d(x_n,candidate)<epsilon. A finite recorded tail alone does not certify this infinite statement. recorded finite sequence=[3, 2, 1, 1], candidate=1, positive epsilon=2/1, cutoff N=2 may equal the recorded length. Give (sequence,candidate,epsilon,N,every [index,state,distance,status] row,all recorded indices at least N are inside?,general convergence certified?,boundary label). Status is prefix below N, otherwise inside or outside using the strict bound; general convergence is always No and the boundary label is finite-tail-insufficient-for-general-convergence.","answer":{"kind":"expression","value":"([3, 2, 1, 1], 1, 2, 2, [[0, 3, 37, prefix], [1, 2, 18, prefix], [2, 1, 0, inside], [3, 1, 0, inside]], Yes, No, finite-tail-insufficient-for-general-convergence)","explanation":"Inspect only recorded indices. An empty recorded tail passes its bounded check vacuously but still proves no general convergence. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([3, 2, 1, 1], 1, 2, 2, [[0, 3, 37, prefix], [1, 2, 18, prefix], [2, 1, 0, inside], [3, 1, 0, inside]], Yes, No, finite-tail-insufficient-for-general-convergence)."},"skills":["Finite-Tail Success Does Not Certify General Convergence"],"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-20261013-advance-20261023-r04-metric-convergence-0043","batch_id":"batch-20261013-advance-20261023-r04-metric-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"64e62a4524ff6889ad02caa1c321ed7475aa6b9ecd712a1e61f9d727b8422a10","family_id":"qf-mod23-metricconv-finite-tail-audit","seed":202610234043,"semantic_template":"qf-mod23-metricconv-finite-tail-audit/{\"N\":3,\"candidate\":1,\"distances\":[[0,18,37,57],[18,0,19,39],[37,19,0,20],[57,39,20,0]],\"epsilon\":[3,1],\"n\":4,\"sequence\":[3,2,1,1]}"},"fields":{"title":"Finite-Tail Success Does Not Certify General Convergence","stem":"Use the complete qualified finite metric table=[[0, 18, 37, 57], [18, 0, 19, 39], [37, 19, 0, 20], [57, 39, 20, 0]] on Fin(4)={0,…,3}. Preserve the convergence quantifiers: for every positive epsilon there exists N such that every n≥N satisfies d(x_n,candidate)<epsilon. A finite recorded tail alone does not certify this infinite statement. recorded finite sequence=[3, 2, 1, 1], candidate=1, positive epsilon=3/1, cutoff N=3 may equal the recorded length. Give (sequence,candidate,epsilon,N,every [index,state,distance,status] row,all recorded indices at least N are inside?,general convergence certified?,boundary label). Status is prefix below N, otherwise inside or outside using the strict bound; general convergence is always No and the boundary label is finite-tail-insufficient-for-general-convergence.","answer":{"kind":"expression","value":"([3, 2, 1, 1], 1, 3, 3, [[0, 3, 39, prefix], [1, 2, 19, prefix], [2, 1, 0, prefix], [3, 1, 0, inside]], Yes, No, finite-tail-insufficient-for-general-convergence)","explanation":"Inspect only recorded indices. An empty recorded tail passes its bounded check vacuously but still proves no general convergence. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([3, 2, 1, 1], 1, 3, 3, [[0, 3, 39, prefix], [1, 2, 19, prefix], [2, 1, 0, prefix], [3, 1, 0, inside]], Yes, No, finite-tail-insufficient-for-general-convergence)."},"skills":["Finite-Tail Success Does Not Certify General Convergence"],"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-20261013-advance-20261023-r04-metric-convergence-0044","batch_id":"batch-20261013-advance-20261023-r04-metric-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a3359f77af0c180087423215a2b0402e78a52b8f426c3ed4906b849151c5c327","family_id":"qf-mod23-metricconv-finite-tail-audit","seed":202610234044,"semantic_template":"qf-mod23-metricconv-finite-tail-audit/{\"N\":1,\"candidate\":1,\"distances\":[[0,19,39,60],[19,0,20,41],[39,20,0,21],[60,41,21,0]],\"epsilon\":[4,1],\"n\":4,\"sequence\":[3,2,1,1]}"},"fields":{"title":"Finite-Tail Success Does Not Certify General Convergence","stem":"Use the complete qualified finite metric table=[[0, 19, 39, 60], [19, 0, 20, 41], [39, 20, 0, 21], [60, 41, 21, 0]] on Fin(4)={0,…,3}. Preserve the convergence quantifiers: for every positive epsilon there exists N such that every n≥N satisfies d(x_n,candidate)<epsilon. A finite recorded tail alone does not certify this infinite statement. recorded finite sequence=[3, 2, 1, 1], candidate=1, positive epsilon=4/1, cutoff N=1 may equal the recorded length. Give (sequence,candidate,epsilon,N,every [index,state,distance,status] row,all recorded indices at least N are inside?,general convergence certified?,boundary label). Status is prefix below N, otherwise inside or outside using the strict bound; general convergence is always No and the boundary label is finite-tail-insufficient-for-general-convergence.","answer":{"kind":"expression","value":"([3, 2, 1, 1], 1, 4, 1, [[0, 3, 41, prefix], [1, 2, 20, outside], [2, 1, 0, inside], [3, 1, 0, inside]], No, No, finite-tail-insufficient-for-general-convergence)","explanation":"Inspect only recorded indices. An empty recorded tail passes its bounded check vacuously but still proves no general convergence. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([3, 2, 1, 1], 1, 4, 1, [[0, 3, 41, prefix], [1, 2, 20, outside], [2, 1, 0, inside], [3, 1, 0, inside]], No, No, finite-tail-insufficient-for-general-convergence)."},"skills":["Finite-Tail Success Does Not Certify General Convergence"],"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-20261013-advance-20261023-r04-metric-convergence-0045","batch_id":"batch-20261013-advance-20261023-r04-metric-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"beffbdb03c0a9ffae6e126979a4406d09043e6f38e20d527d3840ca12f04664b","family_id":"qf-mod23-metricconv-finite-tail-audit","seed":202610234045,"semantic_template":"qf-mod23-metricconv-finite-tail-audit/{\"N\":2,\"candidate\":1,\"distances\":[[0,20,41,63],[20,0,21,43],[41,21,0,22],[63,43,22,0]],\"epsilon\":[5,1],\"n\":4,\"sequence\":[3,2,1,1]}"},"fields":{"title":"Finite-Tail Success Does Not Certify General Convergence","stem":"Use the complete qualified finite metric table=[[0, 20, 41, 63], [20, 0, 21, 43], [41, 21, 0, 22], [63, 43, 22, 0]] on Fin(4)={0,…,3}. Preserve the convergence quantifiers: for every positive epsilon there exists N such that every n≥N satisfies d(x_n,candidate)<epsilon. A finite recorded tail alone does not certify this infinite statement. recorded finite sequence=[3, 2, 1, 1], candidate=1, positive epsilon=5/1, cutoff N=2 may equal the recorded length. Give (sequence,candidate,epsilon,N,every [index,state,distance,status] row,all recorded indices at least N are inside?,general convergence certified?,boundary label). Status is prefix below N, otherwise inside or outside using the strict bound; general convergence is always No and the boundary label is finite-tail-insufficient-for-general-convergence.","answer":{"kind":"expression","value":"([3, 2, 1, 1], 1, 5, 2, [[0, 3, 43, prefix], [1, 2, 21, prefix], [2, 1, 0, inside], [3, 1, 0, inside]], Yes, No, finite-tail-insufficient-for-general-convergence)","explanation":"Inspect only recorded indices. An empty recorded tail passes its bounded check vacuously but still proves no general convergence. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([3, 2, 1, 1], 1, 5, 2, [[0, 3, 43, prefix], [1, 2, 21, prefix], [2, 1, 0, inside], [3, 1, 0, inside]], Yes, No, finite-tail-insufficient-for-general-convergence)."},"skills":["Finite-Tail Success Does Not Certify General Convergence"],"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-20261013-advance-20261023-r04-metric-convergence-0046","batch_id":"batch-20261013-advance-20261023-r04-metric-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5749f3c0b990a56cc76ecd8ec390a3463139e1760543418d01de6fd600a25e91","family_id":"qf-mod23-metricconv-finite-tail-audit","seed":202610234046,"semantic_template":"qf-mod23-metricconv-finite-tail-audit/{\"N\":3,\"candidate\":1,\"distances\":[[0,21,43,66],[21,0,22,45],[43,22,0,23],[66,45,23,0]],\"epsilon\":[1,1],\"n\":4,\"sequence\":[3,2,1,1]}"},"fields":{"title":"Finite-Tail Success Does Not Certify General Convergence","stem":"Use the complete qualified finite metric table=[[0, 21, 43, 66], [21, 0, 22, 45], [43, 22, 0, 23], [66, 45, 23, 0]] on Fin(4)={0,…,3}. Preserve the convergence quantifiers: for every positive epsilon there exists N such that every n≥N satisfies d(x_n,candidate)<epsilon. A finite recorded tail alone does not certify this infinite statement. recorded finite sequence=[3, 2, 1, 1], candidate=1, positive epsilon=1/1, cutoff N=3 may equal the recorded length. Give (sequence,candidate,epsilon,N,every [index,state,distance,status] row,all recorded indices at least N are inside?,general convergence certified?,boundary label). Status is prefix below N, otherwise inside or outside using the strict bound; general convergence is always No and the boundary label is finite-tail-insufficient-for-general-convergence.","answer":{"kind":"expression","value":"([3, 2, 1, 1], 1, 1, 3, [[0, 3, 45, prefix], [1, 2, 22, prefix], [2, 1, 0, prefix], [3, 1, 0, inside]], Yes, No, finite-tail-insufficient-for-general-convergence)","explanation":"Inspect only recorded indices. An empty recorded tail passes its bounded check vacuously but still proves no general convergence. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([3, 2, 1, 1], 1, 1, 3, [[0, 3, 45, prefix], [1, 2, 22, prefix], [2, 1, 0, prefix], [3, 1, 0, inside]], Yes, No, finite-tail-insufficient-for-general-convergence)."},"skills":["Finite-Tail Success Does Not Certify General Convergence"],"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-20261013-advance-20261023-r04-metric-convergence-0047","batch_id":"batch-20261013-advance-20261023-r04-metric-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bb42e9a1239fb23b90a9f652ff986d34492164d49445fe85319bf8a32981dad6","family_id":"qf-mod23-metricconv-finite-tail-audit","seed":202610234047,"semantic_template":"qf-mod23-metricconv-finite-tail-audit/{\"N\":1,\"candidate\":1,\"distances\":[[0,22,45,69],[22,0,23,47],[45,23,0,24],[69,47,24,0]],\"epsilon\":[2,1],\"n\":4,\"sequence\":[3,2,1,1]}"},"fields":{"title":"Finite-Tail Success Does Not Certify General Convergence","stem":"Use the complete qualified finite metric table=[[0, 22, 45, 69], [22, 0, 23, 47], [45, 23, 0, 24], [69, 47, 24, 0]] on Fin(4)={0,…,3}. Preserve the convergence quantifiers: for every positive epsilon there exists N such that every n≥N satisfies d(x_n,candidate)<epsilon. A finite recorded tail alone does not certify this infinite statement. recorded finite sequence=[3, 2, 1, 1], candidate=1, positive epsilon=2/1, cutoff N=1 may equal the recorded length. Give (sequence,candidate,epsilon,N,every [index,state,distance,status] row,all recorded indices at least N are inside?,general convergence certified?,boundary label). Status is prefix below N, otherwise inside or outside using the strict bound; general convergence is always No and the boundary label is finite-tail-insufficient-for-general-convergence.","answer":{"kind":"expression","value":"([3, 2, 1, 1], 1, 2, 1, [[0, 3, 47, prefix], [1, 2, 23, outside], [2, 1, 0, inside], [3, 1, 0, inside]], No, No, finite-tail-insufficient-for-general-convergence)","explanation":"Inspect only recorded indices. An empty recorded tail passes its bounded check vacuously but still proves no general convergence. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([3, 2, 1, 1], 1, 2, 1, [[0, 3, 47, prefix], [1, 2, 23, outside], [2, 1, 0, inside], [3, 1, 0, inside]], No, No, finite-tail-insufficient-for-general-convergence)."},"skills":["Finite-Tail Success Does Not Certify General Convergence"],"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-20261013-advance-20261023-r04-metric-convergence-0048","batch_id":"batch-20261013-advance-20261023-r04-metric-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"190a2c1b0b5ff405ada9a6bf637b8d3b546fa3fe628642ae7bc37d64815103a3","family_id":"qf-mod23-metricconv-finite-tail-audit","seed":202610234048,"semantic_template":"qf-mod23-metricconv-finite-tail-audit/{\"N\":2,\"candidate\":1,\"distances\":[[0,23,47,72],[23,0,24,49],[47,24,0,25],[72,49,25,0]],\"epsilon\":[3,1],\"n\":4,\"sequence\":[3,2,1,1]}"},"fields":{"title":"Finite-Tail Success Does Not Certify General Convergence","stem":"Use the complete qualified finite metric table=[[0, 23, 47, 72], [23, 0, 24, 49], [47, 24, 0, 25], [72, 49, 25, 0]] on Fin(4)={0,…,3}. Preserve the convergence quantifiers: for every positive epsilon there exists N such that every n≥N satisfies d(x_n,candidate)<epsilon. A finite recorded tail alone does not certify this infinite statement. recorded finite sequence=[3, 2, 1, 1], candidate=1, positive epsilon=3/1, cutoff N=2 may equal the recorded length. Give (sequence,candidate,epsilon,N,every [index,state,distance,status] row,all recorded indices at least N are inside?,general convergence certified?,boundary label). Status is prefix below N, otherwise inside or outside using the strict bound; general convergence is always No and the boundary label is finite-tail-insufficient-for-general-convergence.","answer":{"kind":"expression","value":"([3, 2, 1, 1], 1, 3, 2, [[0, 3, 49, prefix], [1, 2, 24, prefix], [2, 1, 0, inside], [3, 1, 0, inside]], Yes, No, finite-tail-insufficient-for-general-convergence)","explanation":"Inspect only recorded indices. An empty recorded tail passes its bounded check vacuously but still proves no general convergence. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([3, 2, 1, 1], 1, 3, 2, [[0, 3, 49, prefix], [1, 2, 24, prefix], [2, 1, 0, inside], [3, 1, 0, inside]], Yes, No, finite-tail-insufficient-for-general-convergence)."},"skills":["Finite-Tail Success Does Not Certify General Convergence"],"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-20261013-advance-20261023-r04-metric-convergence-0049","batch_id":"batch-20261013-advance-20261023-r04-metric-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f79248b243d884d6a53c6d6a12cdd2bdc31fdbbc1ac394901cd6f4fc02221f9d","family_id":"qf-mod23-metricconv-finite-tail-audit","seed":202610234049,"semantic_template":"qf-mod23-metricconv-finite-tail-audit/{\"N\":3,\"candidate\":1,\"distances\":[[0,24,49,75],[24,0,25,51],[49,25,0,26],[75,51,26,0]],\"epsilon\":[4,1],\"n\":4,\"sequence\":[3,2,1,1]}"},"fields":{"title":"Finite-Tail Success Does Not Certify General Convergence","stem":"Use the complete qualified finite metric table=[[0, 24, 49, 75], [24, 0, 25, 51], [49, 25, 0, 26], [75, 51, 26, 0]] on Fin(4)={0,…,3}. Preserve the convergence quantifiers: for every positive epsilon there exists N such that every n≥N satisfies d(x_n,candidate)<epsilon. A finite recorded tail alone does not certify this infinite statement. recorded finite sequence=[3, 2, 1, 1], candidate=1, positive epsilon=4/1, cutoff N=3 may equal the recorded length. Give (sequence,candidate,epsilon,N,every [index,state,distance,status] row,all recorded indices at least N are inside?,general convergence certified?,boundary label). Status is prefix below N, otherwise inside or outside using the strict bound; general convergence is always No and the boundary label is finite-tail-insufficient-for-general-convergence.","answer":{"kind":"expression","value":"([3, 2, 1, 1], 1, 4, 3, [[0, 3, 51, prefix], [1, 2, 25, prefix], [2, 1, 0, prefix], [3, 1, 0, inside]], Yes, No, finite-tail-insufficient-for-general-convergence)","explanation":"Inspect only recorded indices. An empty recorded tail passes its bounded check vacuously but still proves no general convergence. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([3, 2, 1, 1], 1, 4, 3, [[0, 3, 51, prefix], [1, 2, 25, prefix], [2, 1, 0, prefix], [3, 1, 0, inside]], Yes, No, finite-tail-insufficient-for-general-convergence)."},"skills":["Finite-Tail Success Does Not Certify General Convergence"],"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-20261013-advance-20261023-r04-metric-convergence-0050","batch_id":"batch-20261013-advance-20261023-r04-metric-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4c271ef24920fdff270a6c7fcb3dff853bf9e53fe9939c9993c9532718d46b91","family_id":"qf-mod23-metricconv-finite-tail-audit","seed":202610234050,"semantic_template":"qf-mod23-metricconv-finite-tail-audit/{\"N\":1,\"candidate\":1,\"distances\":[[0,25,51,78],[25,0,26,53],[51,26,0,27],[78,53,27,0]],\"epsilon\":[5,1],\"n\":4,\"sequence\":[3,2,1,1]}"},"fields":{"title":"Finite-Tail Success Does Not Certify General Convergence","stem":"Use the complete qualified finite metric table=[[0, 25, 51, 78], [25, 0, 26, 53], [51, 26, 0, 27], [78, 53, 27, 0]] on Fin(4)={0,…,3}. Preserve the convergence quantifiers: for every positive epsilon there exists N such that every n≥N satisfies d(x_n,candidate)<epsilon. A finite recorded tail alone does not certify this infinite statement. recorded finite sequence=[3, 2, 1, 1], candidate=1, positive epsilon=5/1, cutoff N=1 may equal the recorded length. Give (sequence,candidate,epsilon,N,every [index,state,distance,status] row,all recorded indices at least N are inside?,general convergence certified?,boundary label). Status is prefix below N, otherwise inside or outside using the strict bound; general convergence is always No and the boundary label is finite-tail-insufficient-for-general-convergence.","answer":{"kind":"expression","value":"([3, 2, 1, 1], 1, 5, 1, [[0, 3, 53, prefix], [1, 2, 26, outside], [2, 1, 0, inside], [3, 1, 0, inside]], No, No, finite-tail-insufficient-for-general-convergence)","explanation":"Inspect only recorded indices. An empty recorded tail passes its bounded check vacuously but still proves no general convergence. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([3, 2, 1, 1], 1, 5, 1, [[0, 3, 53, prefix], [1, 2, 26, outside], [2, 1, 0, inside], [3, 1, 0, inside]], No, No, finite-tail-insufficient-for-general-convergence)."},"skills":["Finite-Tail Success Does Not Certify General Convergence"],"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-20261013-advance-20261023-r04-metric-convergence-0051","batch_id":"batch-20261013-advance-20261023-r04-metric-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1571d52177b420e3415a20a7ae5cb58219b9cb13a8446d2e244e34221ff21257","family_id":"qf-mod23-metricconv-ultimately-constant-limit","seed":202610234051,"semantic_template":"qf-mod23-metricconv-ultimately-constant-limit/{\"candidate\":3,\"distances\":[[0,1,3,6],[1,0,2,5],[3,2,0,3],[6,5,3,0]],\"n\":4,\"prefix\":[0,1],\"tail_value\":2}"},"fields":{"title":"An Explicit Infinite Constant Tail Supplies a Quantified Witness","stem":"Use the complete qualified finite metric table=[[0, 1, 3, 6], [1, 0, 2, 5], [3, 2, 0, 3], [6, 5, 3, 0]] on Fin(4)={0,…,3}. Preserve the convergence quantifiers: for every positive epsilon there exists N such that every n≥N satisfies d(x_n,candidate)<epsilon. A finite recorded tail alone does not certify this infinite statement. prefix=[0, 1]; every later term equals tail=2, candidate=3. This is an explicitly specified infinite sequence, not an inference from a finite pattern. Give (prefix,tail,candidate,prefix length,converges?,witness). For equality use for any epsilon>0 choose N=prefix length; otherwise use tail-value-differs-from-candidate.","answer":{"kind":"expression","value":"([0, 1], 2, 3, 2, No, tail-value-differs-from-candidate)","explanation":"When the candidate equals the tail, every later distance is zero for every positive epsilon. Otherwise metric separation gives a positive tail distance, so the candidate is not a limit. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([0, 1], 2, 3, 2, No, tail-value-differs-from-candidate)."},"skills":["An Explicit Infinite Constant Tail Supplies a Quantified Witness"],"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-20261013-advance-20261023-r04-metric-convergence-0052","batch_id":"batch-20261013-advance-20261023-r04-metric-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dc2a2666abaa87b91b0bb905dadf00b32eb97adcb9bf9b3e6fcb9dc72e071c10","family_id":"qf-mod23-metricconv-ultimately-constant-limit","seed":202610234052,"semantic_template":"qf-mod23-metricconv-ultimately-constant-limit/{\"candidate\":3,\"distances\":[[0,2,5,9],[2,0,3,7],[5,3,0,4],[9,7,4,0]],\"n\":4,\"prefix\":[1,2],\"tail_value\":3}"},"fields":{"title":"An Explicit Infinite Constant Tail Supplies a Quantified Witness","stem":"Use the complete qualified finite metric table=[[0, 2, 5, 9], [2, 0, 3, 7], [5, 3, 0, 4], [9, 7, 4, 0]] on Fin(4)={0,…,3}. Preserve the convergence quantifiers: for every positive epsilon there exists N such that every n≥N satisfies d(x_n,candidate)<epsilon. A finite recorded tail alone does not certify this infinite statement. prefix=[1, 2]; every later term equals tail=3, candidate=3. This is an explicitly specified infinite sequence, not an inference from a finite pattern. Give (prefix,tail,candidate,prefix length,converges?,witness). For equality use for any epsilon>0 choose N=prefix length; otherwise use tail-value-differs-from-candidate.","answer":{"kind":"expression","value":"([1, 2], 3, 3, 2, Yes, for any epsilon>0 choose N=prefix length)","explanation":"When the candidate equals the tail, every later distance is zero for every positive epsilon. Otherwise metric separation gives a positive tail distance, so the candidate is not a limit. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([1, 2], 3, 3, 2, Yes, for any epsilon>0 choose N=prefix length)."},"skills":["An Explicit Infinite Constant Tail Supplies a Quantified Witness"],"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-20261013-advance-20261023-r04-metric-convergence-0053","batch_id":"batch-20261013-advance-20261023-r04-metric-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a698b6eb769661baaf758d16bd31dc385ed529b5b6b7bb638359ce4e16a4714d","family_id":"qf-mod23-metricconv-ultimately-constant-limit","seed":202610234053,"semantic_template":"qf-mod23-metricconv-ultimately-constant-limit/{\"candidate\":0,\"distances\":[[0,3,7,12],[3,0,4,9],[7,4,0,5],[12,9,5,0]],\"n\":4,\"prefix\":[2,3],\"tail_value\":0}"},"fields":{"title":"An Explicit Infinite Constant Tail Supplies a Quantified Witness","stem":"Use the complete qualified finite metric table=[[0, 3, 7, 12], [3, 0, 4, 9], [7, 4, 0, 5], [12, 9, 5, 0]] on Fin(4)={0,…,3}. Preserve the convergence quantifiers: for every positive epsilon there exists N such that every n≥N satisfies d(x_n,candidate)<epsilon. A finite recorded tail alone does not certify this infinite statement. prefix=[2, 3]; every later term equals tail=0, candidate=0. This is an explicitly specified infinite sequence, not an inference from a finite pattern. Give (prefix,tail,candidate,prefix length,converges?,witness). For equality use for any epsilon>0 choose N=prefix length; otherwise use tail-value-differs-from-candidate.","answer":{"kind":"expression","value":"([2, 3], 0, 0, 2, Yes, for any epsilon>0 choose N=prefix length)","explanation":"When the candidate equals the tail, every later distance is zero for every positive epsilon. Otherwise metric separation gives a positive tail distance, so the candidate is not a limit. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([2, 3], 0, 0, 2, Yes, for any epsilon>0 choose N=prefix length)."},"skills":["An Explicit Infinite Constant Tail Supplies a Quantified Witness"],"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-20261013-advance-20261023-r04-metric-convergence-0054","batch_id":"batch-20261013-advance-20261023-r04-metric-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cd04924861506f5fecce60c0550e9854283cb2c77ea1593fe027dbcbff853b9e","family_id":"qf-mod23-metricconv-ultimately-constant-limit","seed":202610234054,"semantic_template":"qf-mod23-metricconv-ultimately-constant-limit/{\"candidate\":2,\"distances\":[[0,4,9,15],[4,0,5,11],[9,5,0,6],[15,11,6,0]],\"n\":4,\"prefix\":[3,0],\"tail_value\":1}"},"fields":{"title":"An Explicit Infinite Constant Tail Supplies a Quantified Witness","stem":"Use the complete qualified finite metric table=[[0, 4, 9, 15], [4, 0, 5, 11], [9, 5, 0, 6], [15, 11, 6, 0]] on Fin(4)={0,…,3}. Preserve the convergence quantifiers: for every positive epsilon there exists N such that every n≥N satisfies d(x_n,candidate)<epsilon. A finite recorded tail alone does not certify this infinite statement. prefix=[3, 0]; every later term equals tail=1, candidate=2. This is an explicitly specified infinite sequence, not an inference from a finite pattern. Give (prefix,tail,candidate,prefix length,converges?,witness). For equality use for any epsilon>0 choose N=prefix length; otherwise use tail-value-differs-from-candidate.","answer":{"kind":"expression","value":"([3, 0], 1, 2, 2, No, tail-value-differs-from-candidate)","explanation":"When the candidate equals the tail, every later distance is zero for every positive epsilon. Otherwise metric separation gives a positive tail distance, so the candidate is not a limit. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([3, 0], 1, 2, 2, No, tail-value-differs-from-candidate)."},"skills":["An Explicit Infinite Constant Tail Supplies a Quantified Witness"],"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-20261013-advance-20261023-r04-metric-convergence-0055","batch_id":"batch-20261013-advance-20261023-r04-metric-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cf47dd2f6e8a915b6191b43029457503fb147e7c05caf056c1900758e8009d8b","family_id":"qf-mod23-metricconv-ultimately-constant-limit","seed":202610234055,"semantic_template":"qf-mod23-metricconv-ultimately-constant-limit/{\"candidate\":2,\"distances\":[[0,5,11,18],[5,0,6,13],[11,6,0,7],[18,13,7,0]],\"n\":4,\"prefix\":[0,1],\"tail_value\":2}"},"fields":{"title":"An Explicit Infinite Constant Tail Supplies a Quantified Witness","stem":"Use the complete qualified finite metric table=[[0, 5, 11, 18], [5, 0, 6, 13], [11, 6, 0, 7], [18, 13, 7, 0]] on Fin(4)={0,…,3}. Preserve the convergence quantifiers: for every positive epsilon there exists N such that every n≥N satisfies d(x_n,candidate)<epsilon. A finite recorded tail alone does not certify this infinite statement. prefix=[0, 1]; every later term equals tail=2, candidate=2. This is an explicitly specified infinite sequence, not an inference from a finite pattern. Give (prefix,tail,candidate,prefix length,converges?,witness). For equality use for any epsilon>0 choose N=prefix length; otherwise use tail-value-differs-from-candidate.","answer":{"kind":"expression","value":"([0, 1], 2, 2, 2, Yes, for any epsilon>0 choose N=prefix length)","explanation":"When the candidate equals the tail, every later distance is zero for every positive epsilon. Otherwise metric separation gives a positive tail distance, so the candidate is not a limit. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([0, 1], 2, 2, 2, Yes, for any epsilon>0 choose N=prefix length)."},"skills":["An Explicit Infinite Constant Tail Supplies a Quantified Witness"],"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-20261013-advance-20261023-r04-metric-convergence-0056","batch_id":"batch-20261013-advance-20261023-r04-metric-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0e4cde99e61d823363c32dd8004a71a198d933eb91a24ad8bc19aa175743de7d","family_id":"qf-mod23-metricconv-ultimately-constant-limit","seed":202610234056,"semantic_template":"qf-mod23-metricconv-ultimately-constant-limit/{\"candidate\":3,\"distances\":[[0,6,13,21],[6,0,7,15],[13,7,0,8],[21,15,8,0]],\"n\":4,\"prefix\":[1,2],\"tail_value\":3}"},"fields":{"title":"An Explicit Infinite Constant Tail Supplies a Quantified Witness","stem":"Use the complete qualified finite metric table=[[0, 6, 13, 21], [6, 0, 7, 15], [13, 7, 0, 8], [21, 15, 8, 0]] on Fin(4)={0,…,3}. Preserve the convergence quantifiers: for every positive epsilon there exists N such that every n≥N satisfies d(x_n,candidate)<epsilon. A finite recorded tail alone does not certify this infinite statement. prefix=[1, 2]; every later term equals tail=3, candidate=3. This is an explicitly specified infinite sequence, not an inference from a finite pattern. Give (prefix,tail,candidate,prefix length,converges?,witness). For equality use for any epsilon>0 choose N=prefix length; otherwise use tail-value-differs-from-candidate.","answer":{"kind":"expression","value":"([1, 2], 3, 3, 2, Yes, for any epsilon>0 choose N=prefix length)","explanation":"When the candidate equals the tail, every later distance is zero for every positive epsilon. Otherwise metric separation gives a positive tail distance, so the candidate is not a limit. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([1, 2], 3, 3, 2, Yes, for any epsilon>0 choose N=prefix length)."},"skills":["An Explicit Infinite Constant Tail Supplies a Quantified Witness"],"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-20261013-advance-20261023-r04-metric-convergence-0057","batch_id":"batch-20261013-advance-20261023-r04-metric-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f94b8f7fdf563b92bb05079a9c0ce5230e9267629c2fd12698961795e38be3e5","family_id":"qf-mod23-metricconv-ultimately-constant-limit","seed":202610234057,"semantic_template":"qf-mod23-metricconv-ultimately-constant-limit/{\"candidate\":1,\"distances\":[[0,7,15,24],[7,0,8,17],[15,8,0,9],[24,17,9,0]],\"n\":4,\"prefix\":[2,3],\"tail_value\":0}"},"fields":{"title":"An Explicit Infinite Constant Tail Supplies a Quantified Witness","stem":"Use the complete qualified finite metric table=[[0, 7, 15, 24], [7, 0, 8, 17], [15, 8, 0, 9], [24, 17, 9, 0]] on Fin(4)={0,…,3}. Preserve the convergence quantifiers: for every positive epsilon there exists N such that every n≥N satisfies d(x_n,candidate)<epsilon. A finite recorded tail alone does not certify this infinite statement. prefix=[2, 3]; every later term equals tail=0, candidate=1. This is an explicitly specified infinite sequence, not an inference from a finite pattern. Give (prefix,tail,candidate,prefix length,converges?,witness). For equality use for any epsilon>0 choose N=prefix length; otherwise use tail-value-differs-from-candidate.","answer":{"kind":"expression","value":"([2, 3], 0, 1, 2, No, tail-value-differs-from-candidate)","explanation":"When the candidate equals the tail, every later distance is zero for every positive epsilon. Otherwise metric separation gives a positive tail distance, so the candidate is not a limit. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([2, 3], 0, 1, 2, No, tail-value-differs-from-candidate)."},"skills":["An Explicit Infinite Constant Tail Supplies a Quantified Witness"],"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-20261013-advance-20261023-r04-metric-convergence-0058","batch_id":"batch-20261013-advance-20261023-r04-metric-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6f44513b12fa2605c26f6f5ef0f87dedd1f6b7aea991e3a00b3802edb07a4999","family_id":"qf-mod23-metricconv-ultimately-constant-limit","seed":202610234058,"semantic_template":"qf-mod23-metricconv-ultimately-constant-limit/{\"candidate\":1,\"distances\":[[0,8,17,27],[8,0,9,19],[17,9,0,10],[27,19,10,0]],\"n\":4,\"prefix\":[3,0],\"tail_value\":1}"},"fields":{"title":"An Explicit Infinite Constant Tail Supplies a Quantified Witness","stem":"Use the complete qualified finite metric table=[[0, 8, 17, 27], [8, 0, 9, 19], [17, 9, 0, 10], [27, 19, 10, 0]] on Fin(4)={0,…,3}. Preserve the convergence quantifiers: for every positive epsilon there exists N such that every n≥N satisfies d(x_n,candidate)<epsilon. A finite recorded tail alone does not certify this infinite statement. prefix=[3, 0]; every later term equals tail=1, candidate=1. This is an explicitly specified infinite sequence, not an inference from a finite pattern. Give (prefix,tail,candidate,prefix length,converges?,witness). For equality use for any epsilon>0 choose N=prefix length; otherwise use tail-value-differs-from-candidate.","answer":{"kind":"expression","value":"([3, 0], 1, 1, 2, Yes, for any epsilon>0 choose N=prefix length)","explanation":"When the candidate equals the tail, every later distance is zero for every positive epsilon. Otherwise metric separation gives a positive tail distance, so the candidate is not a limit. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([3, 0], 1, 1, 2, Yes, for any epsilon>0 choose N=prefix length)."},"skills":["An Explicit Infinite Constant Tail Supplies a Quantified Witness"],"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-20261013-advance-20261023-r04-metric-convergence-0059","batch_id":"batch-20261013-advance-20261023-r04-metric-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ab88beaf4d882ee299c2cc3c09cb032e2481266c635bd59baf97513b2e6cf2db","family_id":"qf-mod23-metricconv-ultimately-constant-limit","seed":202610234059,"semantic_template":"qf-mod23-metricconv-ultimately-constant-limit/{\"candidate\":2,\"distances\":[[0,9,19,30],[9,0,10,21],[19,10,0,11],[30,21,11,0]],\"n\":4,\"prefix\":[0,1],\"tail_value\":2}"},"fields":{"title":"An Explicit Infinite Constant Tail Supplies a Quantified Witness","stem":"Use the complete qualified finite metric table=[[0, 9, 19, 30], [9, 0, 10, 21], [19, 10, 0, 11], [30, 21, 11, 0]] on Fin(4)={0,…,3}. Preserve the convergence quantifiers: for every positive epsilon there exists N such that every n≥N satisfies d(x_n,candidate)<epsilon. A finite recorded tail alone does not certify this infinite statement. prefix=[0, 1]; every later term equals tail=2, candidate=2. This is an explicitly specified infinite sequence, not an inference from a finite pattern. Give (prefix,tail,candidate,prefix length,converges?,witness). For equality use for any epsilon>0 choose N=prefix length; otherwise use tail-value-differs-from-candidate.","answer":{"kind":"expression","value":"([0, 1], 2, 2, 2, Yes, for any epsilon>0 choose N=prefix length)","explanation":"When the candidate equals the tail, every later distance is zero for every positive epsilon. Otherwise metric separation gives a positive tail distance, so the candidate is not a limit. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([0, 1], 2, 2, 2, Yes, for any epsilon>0 choose N=prefix length)."},"skills":["An Explicit Infinite Constant Tail Supplies a Quantified Witness"],"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-20261013-advance-20261023-r04-metric-convergence-0060","batch_id":"batch-20261013-advance-20261023-r04-metric-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c85e5b27e345f6f3607cbac2a1a43b329f450143b9d317b835a09039de0d1d23","family_id":"qf-mod23-metricconv-ultimately-constant-limit","seed":202610234060,"semantic_template":"qf-mod23-metricconv-ultimately-constant-limit/{\"candidate\":0,\"distances\":[[0,10,21,33],[10,0,11,23],[21,11,0,12],[33,23,12,0]],\"n\":4,\"prefix\":[1,2],\"tail_value\":3}"},"fields":{"title":"An Explicit Infinite Constant Tail Supplies a Quantified Witness","stem":"Use the complete qualified finite metric table=[[0, 10, 21, 33], [10, 0, 11, 23], [21, 11, 0, 12], [33, 23, 12, 0]] on Fin(4)={0,…,3}. Preserve the convergence quantifiers: for every positive epsilon there exists N such that every n≥N satisfies d(x_n,candidate)<epsilon. A finite recorded tail alone does not certify this infinite statement. prefix=[1, 2]; every later term equals tail=3, candidate=0. This is an explicitly specified infinite sequence, not an inference from a finite pattern. Give (prefix,tail,candidate,prefix length,converges?,witness). For equality use for any epsilon>0 choose N=prefix length; otherwise use tail-value-differs-from-candidate.","answer":{"kind":"expression","value":"([1, 2], 3, 0, 2, No, tail-value-differs-from-candidate)","explanation":"When the candidate equals the tail, every later distance is zero for every positive epsilon. Otherwise metric separation gives a positive tail distance, so the candidate is not a limit. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([1, 2], 3, 0, 2, No, tail-value-differs-from-candidate)."},"skills":["An Explicit Infinite Constant Tail Supplies a Quantified Witness"],"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-20261013-advance-20261023-r04-metric-convergence-0061","batch_id":"batch-20261013-advance-20261023-r04-metric-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ee94dcf1d5653cb2a18d7146dc312a4c65ce714831f9daa1fb06a125b5065727","family_id":"qf-mod23-metricconv-ultimately-constant-limit","seed":202610234061,"semantic_template":"qf-mod23-metricconv-ultimately-constant-limit/{\"candidate\":0,\"distances\":[[0,11,23,36],[11,0,12,25],[23,12,0,13],[36,25,13,0]],\"n\":4,\"prefix\":[2,3],\"tail_value\":0}"},"fields":{"title":"An Explicit Infinite Constant Tail Supplies a Quantified Witness","stem":"Use the complete qualified finite metric table=[[0, 11, 23, 36], [11, 0, 12, 25], [23, 12, 0, 13], [36, 25, 13, 0]] on Fin(4)={0,…,3}. Preserve the convergence quantifiers: for every positive epsilon there exists N such that every n≥N satisfies d(x_n,candidate)<epsilon. A finite recorded tail alone does not certify this infinite statement. prefix=[2, 3]; every later term equals tail=0, candidate=0. This is an explicitly specified infinite sequence, not an inference from a finite pattern. Give (prefix,tail,candidate,prefix length,converges?,witness). For equality use for any epsilon>0 choose N=prefix length; otherwise use tail-value-differs-from-candidate.","answer":{"kind":"expression","value":"([2, 3], 0, 0, 2, Yes, for any epsilon>0 choose N=prefix length)","explanation":"When the candidate equals the tail, every later distance is zero for every positive epsilon. Otherwise metric separation gives a positive tail distance, so the candidate is not a limit. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([2, 3], 0, 0, 2, Yes, for any epsilon>0 choose N=prefix length)."},"skills":["An Explicit Infinite Constant Tail Supplies a Quantified Witness"],"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-20261013-advance-20261023-r04-metric-convergence-0062","batch_id":"batch-20261013-advance-20261023-r04-metric-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d30960d77d52f95260a0e34506ada6292f7e234901e04d9fd4364cfe5f4157f2","family_id":"qf-mod23-metricconv-ultimately-constant-limit","seed":202610234062,"semantic_template":"qf-mod23-metricconv-ultimately-constant-limit/{\"candidate\":1,\"distances\":[[0,12,25,39],[12,0,13,27],[25,13,0,14],[39,27,14,0]],\"n\":4,\"prefix\":[3,0],\"tail_value\":1}"},"fields":{"title":"An Explicit Infinite Constant Tail Supplies a Quantified Witness","stem":"Use the complete qualified finite metric table=[[0, 12, 25, 39], [12, 0, 13, 27], [25, 13, 0, 14], [39, 27, 14, 0]] on Fin(4)={0,…,3}. Preserve the convergence quantifiers: for every positive epsilon there exists N such that every n≥N satisfies d(x_n,candidate)<epsilon. A finite recorded tail alone does not certify this infinite statement. prefix=[3, 0]; every later term equals tail=1, candidate=1. This is an explicitly specified infinite sequence, not an inference from a finite pattern. Give (prefix,tail,candidate,prefix length,converges?,witness). For equality use for any epsilon>0 choose N=prefix length; otherwise use tail-value-differs-from-candidate.","answer":{"kind":"expression","value":"([3, 0], 1, 1, 2, Yes, for any epsilon>0 choose N=prefix length)","explanation":"When the candidate equals the tail, every later distance is zero for every positive epsilon. Otherwise metric separation gives a positive tail distance, so the candidate is not a limit. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([3, 0], 1, 1, 2, Yes, for any epsilon>0 choose N=prefix length)."},"skills":["An Explicit Infinite Constant Tail Supplies a Quantified Witness"],"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-20261013-advance-20261023-r04-metric-convergence-0063","batch_id":"batch-20261013-advance-20261023-r04-metric-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d4a5de0d628e08ea1235559a77ca7f324a41ffc9b58fa6d53ce37ae807e1398e","family_id":"qf-mod23-metricconv-ultimately-constant-limit","seed":202610234063,"semantic_template":"qf-mod23-metricconv-ultimately-constant-limit/{\"candidate\":3,\"distances\":[[0,13,27,42],[13,0,14,29],[27,14,0,15],[42,29,15,0]],\"n\":4,\"prefix\":[0,1],\"tail_value\":2}"},"fields":{"title":"An Explicit Infinite Constant Tail Supplies a Quantified Witness","stem":"Use the complete qualified finite metric table=[[0, 13, 27, 42], [13, 0, 14, 29], [27, 14, 0, 15], [42, 29, 15, 0]] on Fin(4)={0,…,3}. Preserve the convergence quantifiers: for every positive epsilon there exists N such that every n≥N satisfies d(x_n,candidate)<epsilon. A finite recorded tail alone does not certify this infinite statement. prefix=[0, 1]; every later term equals tail=2, candidate=3. This is an explicitly specified infinite sequence, not an inference from a finite pattern. Give (prefix,tail,candidate,prefix length,converges?,witness). For equality use for any epsilon>0 choose N=prefix length; otherwise use tail-value-differs-from-candidate.","answer":{"kind":"expression","value":"([0, 1], 2, 3, 2, No, tail-value-differs-from-candidate)","explanation":"When the candidate equals the tail, every later distance is zero for every positive epsilon. Otherwise metric separation gives a positive tail distance, so the candidate is not a limit. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([0, 1], 2, 3, 2, No, tail-value-differs-from-candidate)."},"skills":["An Explicit Infinite Constant Tail Supplies a Quantified Witness"],"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-20261013-advance-20261023-r04-metric-convergence-0064","batch_id":"batch-20261013-advance-20261023-r04-metric-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"88c474b10cf2188cd48fb18ba6c7facc5213e7e89e754b925a3ce6522192e2a0","family_id":"qf-mod23-metricconv-ultimately-constant-limit","seed":202610234064,"semantic_template":"qf-mod23-metricconv-ultimately-constant-limit/{\"candidate\":3,\"distances\":[[0,14,29,45],[14,0,15,31],[29,15,0,16],[45,31,16,0]],\"n\":4,\"prefix\":[1,2],\"tail_value\":3}"},"fields":{"title":"An Explicit Infinite Constant Tail Supplies a Quantified Witness","stem":"Use the complete qualified finite metric table=[[0, 14, 29, 45], [14, 0, 15, 31], [29, 15, 0, 16], [45, 31, 16, 0]] on Fin(4)={0,…,3}. Preserve the convergence quantifiers: for every positive epsilon there exists N such that every n≥N satisfies d(x_n,candidate)<epsilon. A finite recorded tail alone does not certify this infinite statement. prefix=[1, 2]; every later term equals tail=3, candidate=3. This is an explicitly specified infinite sequence, not an inference from a finite pattern. Give (prefix,tail,candidate,prefix length,converges?,witness). For equality use for any epsilon>0 choose N=prefix length; otherwise use tail-value-differs-from-candidate.","answer":{"kind":"expression","value":"([1, 2], 3, 3, 2, Yes, for any epsilon>0 choose N=prefix length)","explanation":"When the candidate equals the tail, every later distance is zero for every positive epsilon. Otherwise metric separation gives a positive tail distance, so the candidate is not a limit. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([1, 2], 3, 3, 2, Yes, for any epsilon>0 choose N=prefix length)."},"skills":["An Explicit Infinite Constant Tail Supplies a Quantified Witness"],"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-20261013-advance-20261023-r04-metric-convergence-0065","batch_id":"batch-20261013-advance-20261023-r04-metric-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3c7d911ce157feeffa720cc91148687b7b681654606ad4f1172b69d16f47b9fa","family_id":"qf-mod23-metricconv-ultimately-constant-limit","seed":202610234065,"semantic_template":"qf-mod23-metricconv-ultimately-constant-limit/{\"candidate\":0,\"distances\":[[0,15,31,48],[15,0,16,33],[31,16,0,17],[48,33,17,0]],\"n\":4,\"prefix\":[2,3],\"tail_value\":0}"},"fields":{"title":"An Explicit Infinite Constant Tail Supplies a Quantified Witness","stem":"Use the complete qualified finite metric table=[[0, 15, 31, 48], [15, 0, 16, 33], [31, 16, 0, 17], [48, 33, 17, 0]] on Fin(4)={0,…,3}. Preserve the convergence quantifiers: for every positive epsilon there exists N such that every n≥N satisfies d(x_n,candidate)<epsilon. A finite recorded tail alone does not certify this infinite statement. prefix=[2, 3]; every later term equals tail=0, candidate=0. This is an explicitly specified infinite sequence, not an inference from a finite pattern. Give (prefix,tail,candidate,prefix length,converges?,witness). For equality use for any epsilon>0 choose N=prefix length; otherwise use tail-value-differs-from-candidate.","answer":{"kind":"expression","value":"([2, 3], 0, 0, 2, Yes, for any epsilon>0 choose N=prefix length)","explanation":"When the candidate equals the tail, every later distance is zero for every positive epsilon. Otherwise metric separation gives a positive tail distance, so the candidate is not a limit. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([2, 3], 0, 0, 2, Yes, for any epsilon>0 choose N=prefix length)."},"skills":["An Explicit Infinite Constant Tail Supplies a Quantified Witness"],"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-20261013-advance-20261023-r04-metric-convergence-0066","batch_id":"batch-20261013-advance-20261023-r04-metric-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f1d86f86d76b60c71bbbc0297c3ef79553391b351d7f6d1544002ba8dd331150","family_id":"qf-mod23-metricconv-ultimately-constant-limit","seed":202610234066,"semantic_template":"qf-mod23-metricconv-ultimately-constant-limit/{\"candidate\":2,\"distances\":[[0,16,33,51],[16,0,17,35],[33,17,0,18],[51,35,18,0]],\"n\":4,\"prefix\":[3,0],\"tail_value\":1}"},"fields":{"title":"An Explicit Infinite Constant Tail Supplies a Quantified Witness","stem":"Use the complete qualified finite metric table=[[0, 16, 33, 51], [16, 0, 17, 35], [33, 17, 0, 18], [51, 35, 18, 0]] on Fin(4)={0,…,3}. Preserve the convergence quantifiers: for every positive epsilon there exists N such that every n≥N satisfies d(x_n,candidate)<epsilon. A finite recorded tail alone does not certify this infinite statement. prefix=[3, 0]; every later term equals tail=1, candidate=2. This is an explicitly specified infinite sequence, not an inference from a finite pattern. Give (prefix,tail,candidate,prefix length,converges?,witness). For equality use for any epsilon>0 choose N=prefix length; otherwise use tail-value-differs-from-candidate.","answer":{"kind":"expression","value":"([3, 0], 1, 2, 2, No, tail-value-differs-from-candidate)","explanation":"When the candidate equals the tail, every later distance is zero for every positive epsilon. Otherwise metric separation gives a positive tail distance, so the candidate is not a limit. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([3, 0], 1, 2, 2, No, tail-value-differs-from-candidate)."},"skills":["An Explicit Infinite Constant Tail Supplies a Quantified Witness"],"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-20261013-advance-20261023-r04-metric-convergence-0067","batch_id":"batch-20261013-advance-20261023-r04-metric-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1ecd900fc0a167afb82528f3d7fff230524c58a5d619b59bff3d53e3966ecbce","family_id":"qf-mod23-metricconv-ultimately-constant-limit","seed":202610234067,"semantic_template":"qf-mod23-metricconv-ultimately-constant-limit/{\"candidate\":2,\"distances\":[[0,17,35,54],[17,0,18,37],[35,18,0,19],[54,37,19,0]],\"n\":4,\"prefix\":[0,1],\"tail_value\":2}"},"fields":{"title":"An Explicit Infinite Constant Tail Supplies a Quantified Witness","stem":"Use the complete qualified finite metric table=[[0, 17, 35, 54], [17, 0, 18, 37], [35, 18, 0, 19], [54, 37, 19, 0]] on Fin(4)={0,…,3}. Preserve the convergence quantifiers: for every positive epsilon there exists N such that every n≥N satisfies d(x_n,candidate)<epsilon. A finite recorded tail alone does not certify this infinite statement. prefix=[0, 1]; every later term equals tail=2, candidate=2. This is an explicitly specified infinite sequence, not an inference from a finite pattern. Give (prefix,tail,candidate,prefix length,converges?,witness). For equality use for any epsilon>0 choose N=prefix length; otherwise use tail-value-differs-from-candidate.","answer":{"kind":"expression","value":"([0, 1], 2, 2, 2, Yes, for any epsilon>0 choose N=prefix length)","explanation":"When the candidate equals the tail, every later distance is zero for every positive epsilon. Otherwise metric separation gives a positive tail distance, so the candidate is not a limit. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([0, 1], 2, 2, 2, Yes, for any epsilon>0 choose N=prefix length)."},"skills":["An Explicit Infinite Constant Tail Supplies a Quantified Witness"],"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-20261013-advance-20261023-r04-metric-convergence-0068","batch_id":"batch-20261013-advance-20261023-r04-metric-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a4d4acb017784f6532585b29f8e795fdcc7bcd4bed2ad8abb9b53eb2087463e4","family_id":"qf-mod23-metricconv-ultimately-constant-limit","seed":202610234068,"semantic_template":"qf-mod23-metricconv-ultimately-constant-limit/{\"candidate\":3,\"distances\":[[0,18,37,57],[18,0,19,39],[37,19,0,20],[57,39,20,0]],\"n\":4,\"prefix\":[1,2],\"tail_value\":3}"},"fields":{"title":"An Explicit Infinite Constant Tail Supplies a Quantified Witness","stem":"Use the complete qualified finite metric table=[[0, 18, 37, 57], [18, 0, 19, 39], [37, 19, 0, 20], [57, 39, 20, 0]] on Fin(4)={0,…,3}. Preserve the convergence quantifiers: for every positive epsilon there exists N such that every n≥N satisfies d(x_n,candidate)<epsilon. A finite recorded tail alone does not certify this infinite statement. prefix=[1, 2]; every later term equals tail=3, candidate=3. This is an explicitly specified infinite sequence, not an inference from a finite pattern. Give (prefix,tail,candidate,prefix length,converges?,witness). For equality use for any epsilon>0 choose N=prefix length; otherwise use tail-value-differs-from-candidate.","answer":{"kind":"expression","value":"([1, 2], 3, 3, 2, Yes, for any epsilon>0 choose N=prefix length)","explanation":"When the candidate equals the tail, every later distance is zero for every positive epsilon. Otherwise metric separation gives a positive tail distance, so the candidate is not a limit. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([1, 2], 3, 3, 2, Yes, for any epsilon>0 choose N=prefix length)."},"skills":["An Explicit Infinite Constant Tail Supplies a Quantified Witness"],"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-20261013-advance-20261023-r04-metric-convergence-0069","batch_id":"batch-20261013-advance-20261023-r04-metric-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5c199dd96e553d8f23513a7031ee8a543117557f95674bf982c679d055f9f45a","family_id":"qf-mod23-metricconv-ultimately-constant-limit","seed":202610234069,"semantic_template":"qf-mod23-metricconv-ultimately-constant-limit/{\"candidate\":1,\"distances\":[[0,19,39,60],[19,0,20,41],[39,20,0,21],[60,41,21,0]],\"n\":4,\"prefix\":[2,3],\"tail_value\":0}"},"fields":{"title":"An Explicit Infinite Constant Tail Supplies a Quantified Witness","stem":"Use the complete qualified finite metric table=[[0, 19, 39, 60], [19, 0, 20, 41], [39, 20, 0, 21], [60, 41, 21, 0]] on Fin(4)={0,…,3}. Preserve the convergence quantifiers: for every positive epsilon there exists N such that every n≥N satisfies d(x_n,candidate)<epsilon. A finite recorded tail alone does not certify this infinite statement. prefix=[2, 3]; every later term equals tail=0, candidate=1. This is an explicitly specified infinite sequence, not an inference from a finite pattern. Give (prefix,tail,candidate,prefix length,converges?,witness). For equality use for any epsilon>0 choose N=prefix length; otherwise use tail-value-differs-from-candidate.","answer":{"kind":"expression","value":"([2, 3], 0, 1, 2, No, tail-value-differs-from-candidate)","explanation":"When the candidate equals the tail, every later distance is zero for every positive epsilon. Otherwise metric separation gives a positive tail distance, so the candidate is not a limit. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([2, 3], 0, 1, 2, No, tail-value-differs-from-candidate)."},"skills":["An Explicit Infinite Constant Tail Supplies a Quantified Witness"],"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-20261013-advance-20261023-r04-metric-convergence-0070","batch_id":"batch-20261013-advance-20261023-r04-metric-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b22214ab6fccc5af25f639bd1b7b308f905a37f565b8b46cd6e3a82a416e3a37","family_id":"qf-mod23-metricconv-ultimately-constant-limit","seed":202610234070,"semantic_template":"qf-mod23-metricconv-ultimately-constant-limit/{\"candidate\":1,\"distances\":[[0,20,41,63],[20,0,21,43],[41,21,0,22],[63,43,22,0]],\"n\":4,\"prefix\":[3,0],\"tail_value\":1}"},"fields":{"title":"An Explicit Infinite Constant Tail Supplies a Quantified Witness","stem":"Use the complete qualified finite metric table=[[0, 20, 41, 63], [20, 0, 21, 43], [41, 21, 0, 22], [63, 43, 22, 0]] on Fin(4)={0,…,3}. Preserve the convergence quantifiers: for every positive epsilon there exists N such that every n≥N satisfies d(x_n,candidate)<epsilon. A finite recorded tail alone does not certify this infinite statement. prefix=[3, 0]; every later term equals tail=1, candidate=1. This is an explicitly specified infinite sequence, not an inference from a finite pattern. Give (prefix,tail,candidate,prefix length,converges?,witness). For equality use for any epsilon>0 choose N=prefix length; otherwise use tail-value-differs-from-candidate.","answer":{"kind":"expression","value":"([3, 0], 1, 1, 2, Yes, for any epsilon>0 choose N=prefix length)","explanation":"When the candidate equals the tail, every later distance is zero for every positive epsilon. Otherwise metric separation gives a positive tail distance, so the candidate is not a limit. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([3, 0], 1, 1, 2, Yes, for any epsilon>0 choose N=prefix length)."},"skills":["An Explicit Infinite Constant Tail Supplies a Quantified Witness"],"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-20261013-advance-20261023-r04-metric-convergence-0071","batch_id":"batch-20261013-advance-20261023-r04-metric-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bbe1dcf32ce0a23977536f63f00c1e40e00b6f55496624c7cfec6b312dc68d48","family_id":"qf-mod23-metricconv-ultimately-constant-limit","seed":202610234071,"semantic_template":"qf-mod23-metricconv-ultimately-constant-limit/{\"candidate\":2,\"distances\":[[0,21,43,66],[21,0,22,45],[43,22,0,23],[66,45,23,0]],\"n\":4,\"prefix\":[0,1],\"tail_value\":2}"},"fields":{"title":"An Explicit Infinite Constant Tail Supplies a Quantified Witness","stem":"Use the complete qualified finite metric table=[[0, 21, 43, 66], [21, 0, 22, 45], [43, 22, 0, 23], [66, 45, 23, 0]] on Fin(4)={0,…,3}. Preserve the convergence quantifiers: for every positive epsilon there exists N such that every n≥N satisfies d(x_n,candidate)<epsilon. A finite recorded tail alone does not certify this infinite statement. prefix=[0, 1]; every later term equals tail=2, candidate=2. This is an explicitly specified infinite sequence, not an inference from a finite pattern. Give (prefix,tail,candidate,prefix length,converges?,witness). For equality use for any epsilon>0 choose N=prefix length; otherwise use tail-value-differs-from-candidate.","answer":{"kind":"expression","value":"([0, 1], 2, 2, 2, Yes, for any epsilon>0 choose N=prefix length)","explanation":"When the candidate equals the tail, every later distance is zero for every positive epsilon. Otherwise metric separation gives a positive tail distance, so the candidate is not a limit. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([0, 1], 2, 2, 2, Yes, for any epsilon>0 choose N=prefix length)."},"skills":["An Explicit Infinite Constant Tail Supplies a Quantified Witness"],"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-20261013-advance-20261023-r04-metric-convergence-0072","batch_id":"batch-20261013-advance-20261023-r04-metric-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7716abeb7cd9b45f3d3b9b454efd036fcde997d9861fb403d94e7cbbe52ee29c","family_id":"qf-mod23-metricconv-ultimately-constant-limit","seed":202610234072,"semantic_template":"qf-mod23-metricconv-ultimately-constant-limit/{\"candidate\":0,\"distances\":[[0,22,45,69],[22,0,23,47],[45,23,0,24],[69,47,24,0]],\"n\":4,\"prefix\":[1,2],\"tail_value\":3}"},"fields":{"title":"An Explicit Infinite Constant Tail Supplies a Quantified Witness","stem":"Use the complete qualified finite metric table=[[0, 22, 45, 69], [22, 0, 23, 47], [45, 23, 0, 24], [69, 47, 24, 0]] on Fin(4)={0,…,3}. Preserve the convergence quantifiers: for every positive epsilon there exists N such that every n≥N satisfies d(x_n,candidate)<epsilon. A finite recorded tail alone does not certify this infinite statement. prefix=[1, 2]; every later term equals tail=3, candidate=0. This is an explicitly specified infinite sequence, not an inference from a finite pattern. Give (prefix,tail,candidate,prefix length,converges?,witness). For equality use for any epsilon>0 choose N=prefix length; otherwise use tail-value-differs-from-candidate.","answer":{"kind":"expression","value":"([1, 2], 3, 0, 2, No, tail-value-differs-from-candidate)","explanation":"When the candidate equals the tail, every later distance is zero for every positive epsilon. Otherwise metric separation gives a positive tail distance, so the candidate is not a limit. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([1, 2], 3, 0, 2, No, tail-value-differs-from-candidate)."},"skills":["An Explicit Infinite Constant Tail Supplies a Quantified Witness"],"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-20261013-advance-20261023-r04-metric-convergence-0073","batch_id":"batch-20261013-advance-20261023-r04-metric-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1e1794580d86da8ea542b757475c05566b7217e4c0d278c343f570a94fcb36d6","family_id":"qf-mod23-metricconv-ultimately-constant-limit","seed":202610234073,"semantic_template":"qf-mod23-metricconv-ultimately-constant-limit/{\"candidate\":0,\"distances\":[[0,23,47,72],[23,0,24,49],[47,24,0,25],[72,49,25,0]],\"n\":4,\"prefix\":[2,3],\"tail_value\":0}"},"fields":{"title":"An Explicit Infinite Constant Tail Supplies a Quantified Witness","stem":"Use the complete qualified finite metric table=[[0, 23, 47, 72], [23, 0, 24, 49], [47, 24, 0, 25], [72, 49, 25, 0]] on Fin(4)={0,…,3}. Preserve the convergence quantifiers: for every positive epsilon there exists N such that every n≥N satisfies d(x_n,candidate)<epsilon. A finite recorded tail alone does not certify this infinite statement. prefix=[2, 3]; every later term equals tail=0, candidate=0. This is an explicitly specified infinite sequence, not an inference from a finite pattern. Give (prefix,tail,candidate,prefix length,converges?,witness). For equality use for any epsilon>0 choose N=prefix length; otherwise use tail-value-differs-from-candidate.","answer":{"kind":"expression","value":"([2, 3], 0, 0, 2, Yes, for any epsilon>0 choose N=prefix length)","explanation":"When the candidate equals the tail, every later distance is zero for every positive epsilon. Otherwise metric separation gives a positive tail distance, so the candidate is not a limit. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([2, 3], 0, 0, 2, Yes, for any epsilon>0 choose N=prefix length)."},"skills":["An Explicit Infinite Constant Tail Supplies a Quantified Witness"],"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-20261013-advance-20261023-r04-metric-convergence-0074","batch_id":"batch-20261013-advance-20261023-r04-metric-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8818e02a500bc5beff25ad965865e99d7f3583ac2a2b8b304fde6f3b25af8795","family_id":"qf-mod23-metricconv-ultimately-constant-limit","seed":202610234074,"semantic_template":"qf-mod23-metricconv-ultimately-constant-limit/{\"candidate\":1,\"distances\":[[0,24,49,75],[24,0,25,51],[49,25,0,26],[75,51,26,0]],\"n\":4,\"prefix\":[3,0],\"tail_value\":1}"},"fields":{"title":"An Explicit Infinite Constant Tail Supplies a Quantified Witness","stem":"Use the complete qualified finite metric table=[[0, 24, 49, 75], [24, 0, 25, 51], [49, 25, 0, 26], [75, 51, 26, 0]] on Fin(4)={0,…,3}. Preserve the convergence quantifiers: for every positive epsilon there exists N such that every n≥N satisfies d(x_n,candidate)<epsilon. A finite recorded tail alone does not certify this infinite statement. prefix=[3, 0]; every later term equals tail=1, candidate=1. This is an explicitly specified infinite sequence, not an inference from a finite pattern. Give (prefix,tail,candidate,prefix length,converges?,witness). For equality use for any epsilon>0 choose N=prefix length; otherwise use tail-value-differs-from-candidate.","answer":{"kind":"expression","value":"([3, 0], 1, 1, 2, Yes, for any epsilon>0 choose N=prefix length)","explanation":"When the candidate equals the tail, every later distance is zero for every positive epsilon. Otherwise metric separation gives a positive tail distance, so the candidate is not a limit. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([3, 0], 1, 1, 2, Yes, for any epsilon>0 choose N=prefix length)."},"skills":["An Explicit Infinite Constant Tail Supplies a Quantified Witness"],"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-20261013-advance-20261023-r04-metric-convergence-0075","batch_id":"batch-20261013-advance-20261023-r04-metric-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b42c93ecfd7420ce4e4b19f117885a20025e634a35631711a88186c806ed9be8","family_id":"qf-mod23-metricconv-ultimately-constant-limit","seed":202610234075,"semantic_template":"qf-mod23-metricconv-ultimately-constant-limit/{\"candidate\":3,\"distances\":[[0,25,51,78],[25,0,26,53],[51,26,0,27],[78,53,27,0]],\"n\":4,\"prefix\":[0,1],\"tail_value\":2}"},"fields":{"title":"An Explicit Infinite Constant Tail Supplies a Quantified Witness","stem":"Use the complete qualified finite metric table=[[0, 25, 51, 78], [25, 0, 26, 53], [51, 26, 0, 27], [78, 53, 27, 0]] on Fin(4)={0,…,3}. Preserve the convergence quantifiers: for every positive epsilon there exists N such that every n≥N satisfies d(x_n,candidate)<epsilon. A finite recorded tail alone does not certify this infinite statement. prefix=[0, 1]; every later term equals tail=2, candidate=3. This is an explicitly specified infinite sequence, not an inference from a finite pattern. Give (prefix,tail,candidate,prefix length,converges?,witness). For equality use for any epsilon>0 choose N=prefix length; otherwise use tail-value-differs-from-candidate.","answer":{"kind":"expression","value":"([0, 1], 2, 3, 2, No, tail-value-differs-from-candidate)","explanation":"When the candidate equals the tail, every later distance is zero for every positive epsilon. Otherwise metric separation gives a positive tail distance, so the candidate is not a limit. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([0, 1], 2, 3, 2, No, tail-value-differs-from-candidate)."},"skills":["An Explicit Infinite Constant Tail Supplies a Quantified Witness"],"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-20261013-advance-20261023-r04-metric-convergence-0076","batch_id":"batch-20261013-advance-20261023-r04-metric-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4159bb91269901a016326c13e74d977b323490678e6d8ced3107bbbd5c91c5b4","family_id":"qf-mod23-metricconv-limit-uniqueness-audit","seed":202610234076,"semantic_template":"qf-mod23-metricconv-limit-uniqueness-audit/{\"candidate_a\":2,\"candidate_b\":2,\"distances\":[[0,1,3,6],[1,0,2,5],[3,2,0,3],[6,5,3,0]],\"n\":4,\"prefix\":[0,1],\"tail_value\":2}"},"fields":{"title":"A Conditional Uniqueness Check Is Not Two Convergence Claims","stem":"Use the complete qualified finite metric table=[[0, 1, 3, 6], [1, 0, 2, 5], [3, 2, 0, 3], [6, 5, 3, 0]] on Fin(4)={0,…,3}. Preserve the convergence quantifiers: for every positive epsilon there exists N such that every n≥N satisfies d(x_n,candidate)<epsilon. A finite recorded tail alone does not certify this infinite statement. infinite sequence has prefix=[0, 1] and constant tail=2; candidates a=2, b=2. Give (prefix,tail,a,b,d(a,b),a is a limit?,b is a limit?,if both are limits then a=b?).","answer":{"kind":"expression","value":"([0, 1], 2, 2, 2, 0, Yes, Yes, Yes)","explanation":"Check each candidate against the declared constant tail. The final implication may be true when its antecedent is false; it does not assert both candidates converge. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([0, 1], 2, 2, 2, 0, Yes, Yes, Yes)."},"skills":["A Conditional Uniqueness Check Is Not Two Convergence Claims"],"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-20261013-advance-20261023-r04-metric-convergence-0077","batch_id":"batch-20261013-advance-20261023-r04-metric-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4fa3b7bba0d1d283f20bb2f4b5de703da108de256da633b1c6f66a5651f357fe","family_id":"qf-mod23-metricconv-limit-uniqueness-audit","seed":202610234077,"semantic_template":"qf-mod23-metricconv-limit-uniqueness-audit/{\"candidate_a\":3,\"candidate_b\":0,\"distances\":[[0,2,5,9],[2,0,3,7],[5,3,0,4],[9,7,4,0]],\"n\":4,\"prefix\":[1,2],\"tail_value\":3}"},"fields":{"title":"A Conditional Uniqueness Check Is Not Two Convergence Claims","stem":"Use the complete qualified finite metric table=[[0, 2, 5, 9], [2, 0, 3, 7], [5, 3, 0, 4], [9, 7, 4, 0]] on Fin(4)={0,…,3}. Preserve the convergence quantifiers: for every positive epsilon there exists N such that every n≥N satisfies d(x_n,candidate)<epsilon. A finite recorded tail alone does not certify this infinite statement. infinite sequence has prefix=[1, 2] and constant tail=3; candidates a=3, b=0. Give (prefix,tail,a,b,d(a,b),a is a limit?,b is a limit?,if both are limits then a=b?).","answer":{"kind":"expression","value":"([1, 2], 3, 3, 0, 9, Yes, No, Yes)","explanation":"Check each candidate against the declared constant tail. The final implication may be true when its antecedent is false; it does not assert both candidates converge. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([1, 2], 3, 3, 0, 9, Yes, No, Yes)."},"skills":["A Conditional Uniqueness Check Is Not Two Convergence Claims"],"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-20261013-advance-20261023-r04-metric-convergence-0078","batch_id":"batch-20261013-advance-20261023-r04-metric-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"78068c26f55312c85a26a6eec640764982c8526cf7b827fb97ad660078aa2243","family_id":"qf-mod23-metricconv-limit-uniqueness-audit","seed":202610234078,"semantic_template":"qf-mod23-metricconv-limit-uniqueness-audit/{\"candidate_a\":0,\"candidate_b\":1,\"distances\":[[0,3,7,12],[3,0,4,9],[7,4,0,5],[12,9,5,0]],\"n\":4,\"prefix\":[2,3],\"tail_value\":0}"},"fields":{"title":"A Conditional Uniqueness Check Is Not Two Convergence Claims","stem":"Use the complete qualified finite metric table=[[0, 3, 7, 12], [3, 0, 4, 9], [7, 4, 0, 5], [12, 9, 5, 0]] on Fin(4)={0,…,3}. Preserve the convergence quantifiers: for every positive epsilon there exists N such that every n≥N satisfies d(x_n,candidate)<epsilon. A finite recorded tail alone does not certify this infinite statement. infinite sequence has prefix=[2, 3] and constant tail=0; candidates a=0, b=1. Give (prefix,tail,a,b,d(a,b),a is a limit?,b is a limit?,if both are limits then a=b?).","answer":{"kind":"expression","value":"([2, 3], 0, 0, 1, 3, Yes, No, Yes)","explanation":"Check each candidate against the declared constant tail. The final implication may be true when its antecedent is false; it does not assert both candidates converge. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([2, 3], 0, 0, 1, 3, Yes, No, Yes)."},"skills":["A Conditional Uniqueness Check Is Not Two Convergence Claims"],"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-20261013-advance-20261023-r04-metric-convergence-0079","batch_id":"batch-20261013-advance-20261023-r04-metric-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"90c6c5ca3908100118f011c5c2f4d21a4656d5d568d45be09d9148ecfb568501","family_id":"qf-mod23-metricconv-limit-uniqueness-audit","seed":202610234079,"semantic_template":"qf-mod23-metricconv-limit-uniqueness-audit/{\"candidate_a\":1,\"candidate_b\":1,\"distances\":[[0,4,9,15],[4,0,5,11],[9,5,0,6],[15,11,6,0]],\"n\":4,\"prefix\":[3,0],\"tail_value\":1}"},"fields":{"title":"A Conditional Uniqueness Check Is Not Two Convergence Claims","stem":"Use the complete qualified finite metric table=[[0, 4, 9, 15], [4, 0, 5, 11], [9, 5, 0, 6], [15, 11, 6, 0]] on Fin(4)={0,…,3}. Preserve the convergence quantifiers: for every positive epsilon there exists N such that every n≥N satisfies d(x_n,candidate)<epsilon. A finite recorded tail alone does not certify this infinite statement. infinite sequence has prefix=[3, 0] and constant tail=1; candidates a=1, b=1. Give (prefix,tail,a,b,d(a,b),a is a limit?,b is a limit?,if both are limits then a=b?).","answer":{"kind":"expression","value":"([3, 0], 1, 1, 1, 0, Yes, Yes, Yes)","explanation":"Check each candidate against the declared constant tail. The final implication may be true when its antecedent is false; it does not assert both candidates converge. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([3, 0], 1, 1, 1, 0, Yes, Yes, Yes)."},"skills":["A Conditional Uniqueness Check Is Not Two Convergence Claims"],"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-20261013-advance-20261023-r04-metric-convergence-0080","batch_id":"batch-20261013-advance-20261023-r04-metric-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0f7af8fbb5ad118b1f954cd644ab93a9852c7b62233f9e9684b45a764b8ed33b","family_id":"qf-mod23-metricconv-limit-uniqueness-audit","seed":202610234080,"semantic_template":"qf-mod23-metricconv-limit-uniqueness-audit/{\"candidate_a\":2,\"candidate_b\":3,\"distances\":[[0,5,11,18],[5,0,6,13],[11,6,0,7],[18,13,7,0]],\"n\":4,\"prefix\":[0,1],\"tail_value\":2}"},"fields":{"title":"A Conditional Uniqueness Check Is Not Two Convergence Claims","stem":"Use the complete qualified finite metric table=[[0, 5, 11, 18], [5, 0, 6, 13], [11, 6, 0, 7], [18, 13, 7, 0]] on Fin(4)={0,…,3}. Preserve the convergence quantifiers: for every positive epsilon there exists N such that every n≥N satisfies d(x_n,candidate)<epsilon. A finite recorded tail alone does not certify this infinite statement. infinite sequence has prefix=[0, 1] and constant tail=2; candidates a=2, b=3. Give (prefix,tail,a,b,d(a,b),a is a limit?,b is a limit?,if both are limits then a=b?).","answer":{"kind":"expression","value":"([0, 1], 2, 2, 3, 7, Yes, No, Yes)","explanation":"Check each candidate against the declared constant tail. The final implication may be true when its antecedent is false; it does not assert both candidates converge. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([0, 1], 2, 2, 3, 7, Yes, No, Yes)."},"skills":["A Conditional Uniqueness Check Is Not Two Convergence Claims"],"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-20261013-advance-20261023-r04-metric-convergence-0081","batch_id":"batch-20261013-advance-20261023-r04-metric-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"95da3686492c6de7e07cdc5be83ef1633047d6b84580f498b9071755623cccbb","family_id":"qf-mod23-metricconv-limit-uniqueness-audit","seed":202610234081,"semantic_template":"qf-mod23-metricconv-limit-uniqueness-audit/{\"candidate_a\":3,\"candidate_b\":0,\"distances\":[[0,6,13,21],[6,0,7,15],[13,7,0,8],[21,15,8,0]],\"n\":4,\"prefix\":[1,2],\"tail_value\":3}"},"fields":{"title":"A Conditional Uniqueness Check Is Not Two Convergence Claims","stem":"Use the complete qualified finite metric table=[[0, 6, 13, 21], [6, 0, 7, 15], [13, 7, 0, 8], [21, 15, 8, 0]] on Fin(4)={0,…,3}. Preserve the convergence quantifiers: for every positive epsilon there exists N such that every n≥N satisfies d(x_n,candidate)<epsilon. A finite recorded tail alone does not certify this infinite statement. infinite sequence has prefix=[1, 2] and constant tail=3; candidates a=3, b=0. Give (prefix,tail,a,b,d(a,b),a is a limit?,b is a limit?,if both are limits then a=b?).","answer":{"kind":"expression","value":"([1, 2], 3, 3, 0, 21, Yes, No, Yes)","explanation":"Check each candidate against the declared constant tail. The final implication may be true when its antecedent is false; it does not assert both candidates converge. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([1, 2], 3, 3, 0, 21, Yes, No, Yes)."},"skills":["A Conditional Uniqueness Check Is Not Two Convergence Claims"],"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-20261013-advance-20261023-r04-metric-convergence-0082","batch_id":"batch-20261013-advance-20261023-r04-metric-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6cbff90af17c71962566913b7752af870eb817ebfd10057373ea57a89c542224","family_id":"qf-mod23-metricconv-limit-uniqueness-audit","seed":202610234082,"semantic_template":"qf-mod23-metricconv-limit-uniqueness-audit/{\"candidate_a\":0,\"candidate_b\":0,\"distances\":[[0,7,15,24],[7,0,8,17],[15,8,0,9],[24,17,9,0]],\"n\":4,\"prefix\":[2,3],\"tail_value\":0}"},"fields":{"title":"A Conditional Uniqueness Check Is Not Two Convergence Claims","stem":"Use the complete qualified finite metric table=[[0, 7, 15, 24], [7, 0, 8, 17], [15, 8, 0, 9], [24, 17, 9, 0]] on Fin(4)={0,…,3}. Preserve the convergence quantifiers: for every positive epsilon there exists N such that every n≥N satisfies d(x_n,candidate)<epsilon. A finite recorded tail alone does not certify this infinite statement. infinite sequence has prefix=[2, 3] and constant tail=0; candidates a=0, b=0. Give (prefix,tail,a,b,d(a,b),a is a limit?,b is a limit?,if both are limits then a=b?).","answer":{"kind":"expression","value":"([2, 3], 0, 0, 0, 0, Yes, Yes, Yes)","explanation":"Check each candidate against the declared constant tail. The final implication may be true when its antecedent is false; it does not assert both candidates converge. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([2, 3], 0, 0, 0, 0, Yes, Yes, Yes)."},"skills":["A Conditional Uniqueness Check Is Not Two Convergence Claims"],"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-20261013-advance-20261023-r04-metric-convergence-0083","batch_id":"batch-20261013-advance-20261023-r04-metric-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5bdbef80f2d627006f55b559af305e7b75996b7b94727234b4a28db0dde78f78","family_id":"qf-mod23-metricconv-limit-uniqueness-audit","seed":202610234083,"semantic_template":"qf-mod23-metricconv-limit-uniqueness-audit/{\"candidate_a\":1,\"candidate_b\":2,\"distances\":[[0,8,17,27],[8,0,9,19],[17,9,0,10],[27,19,10,0]],\"n\":4,\"prefix\":[3,0],\"tail_value\":1}"},"fields":{"title":"A Conditional Uniqueness Check Is Not Two Convergence Claims","stem":"Use the complete qualified finite metric table=[[0, 8, 17, 27], [8, 0, 9, 19], [17, 9, 0, 10], [27, 19, 10, 0]] on Fin(4)={0,…,3}. Preserve the convergence quantifiers: for every positive epsilon there exists N such that every n≥N satisfies d(x_n,candidate)<epsilon. A finite recorded tail alone does not certify this infinite statement. infinite sequence has prefix=[3, 0] and constant tail=1; candidates a=1, b=2. Give (prefix,tail,a,b,d(a,b),a is a limit?,b is a limit?,if both are limits then a=b?).","answer":{"kind":"expression","value":"([3, 0], 1, 1, 2, 9, Yes, No, Yes)","explanation":"Check each candidate against the declared constant tail. The final implication may be true when its antecedent is false; it does not assert both candidates converge. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([3, 0], 1, 1, 2, 9, Yes, No, Yes)."},"skills":["A Conditional Uniqueness Check Is Not Two Convergence Claims"],"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-20261013-advance-20261023-r04-metric-convergence-0084","batch_id":"batch-20261013-advance-20261023-r04-metric-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f6e2eec251bad3596ca668997938153cdab788b3ef9aa6397ff1a9ca2de098f2","family_id":"qf-mod23-metricconv-limit-uniqueness-audit","seed":202610234084,"semantic_template":"qf-mod23-metricconv-limit-uniqueness-audit/{\"candidate_a\":2,\"candidate_b\":3,\"distances\":[[0,9,19,30],[9,0,10,21],[19,10,0,11],[30,21,11,0]],\"n\":4,\"prefix\":[0,1],\"tail_value\":2}"},"fields":{"title":"A Conditional Uniqueness Check Is Not Two Convergence Claims","stem":"Use the complete qualified finite metric table=[[0, 9, 19, 30], [9, 0, 10, 21], [19, 10, 0, 11], [30, 21, 11, 0]] on Fin(4)={0,…,3}. Preserve the convergence quantifiers: for every positive epsilon there exists N such that every n≥N satisfies d(x_n,candidate)<epsilon. A finite recorded tail alone does not certify this infinite statement. infinite sequence has prefix=[0, 1] and constant tail=2; candidates a=2, b=3. Give (prefix,tail,a,b,d(a,b),a is a limit?,b is a limit?,if both are limits then a=b?).","answer":{"kind":"expression","value":"([0, 1], 2, 2, 3, 11, Yes, No, Yes)","explanation":"Check each candidate against the declared constant tail. The final implication may be true when its antecedent is false; it does not assert both candidates converge. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([0, 1], 2, 2, 3, 11, Yes, No, Yes)."},"skills":["A Conditional Uniqueness Check Is Not Two Convergence Claims"],"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-20261013-advance-20261023-r04-metric-convergence-0085","batch_id":"batch-20261013-advance-20261023-r04-metric-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"aa98f6fa19fcf383f45c09a37297c60051a54536b792b2686f0f41b092364395","family_id":"qf-mod23-metricconv-limit-uniqueness-audit","seed":202610234085,"semantic_template":"qf-mod23-metricconv-limit-uniqueness-audit/{\"candidate_a\":3,\"candidate_b\":3,\"distances\":[[0,10,21,33],[10,0,11,23],[21,11,0,12],[33,23,12,0]],\"n\":4,\"prefix\":[1,2],\"tail_value\":3}"},"fields":{"title":"A Conditional Uniqueness Check Is Not Two Convergence Claims","stem":"Use the complete qualified finite metric table=[[0, 10, 21, 33], [10, 0, 11, 23], [21, 11, 0, 12], [33, 23, 12, 0]] on Fin(4)={0,…,3}. Preserve the convergence quantifiers: for every positive epsilon there exists N such that every n≥N satisfies d(x_n,candidate)<epsilon. A finite recorded tail alone does not certify this infinite statement. infinite sequence has prefix=[1, 2] and constant tail=3; candidates a=3, b=3. Give (prefix,tail,a,b,d(a,b),a is a limit?,b is a limit?,if both are limits then a=b?).","answer":{"kind":"expression","value":"([1, 2], 3, 3, 3, 0, Yes, Yes, Yes)","explanation":"Check each candidate against the declared constant tail. The final implication may be true when its antecedent is false; it does not assert both candidates converge. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([1, 2], 3, 3, 3, 0, Yes, Yes, Yes)."},"skills":["A Conditional Uniqueness Check Is Not Two Convergence Claims"],"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-20261013-advance-20261023-r04-metric-convergence-0086","batch_id":"batch-20261013-advance-20261023-r04-metric-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4a54dbc952b991291223a5f89f8ab289e13e9728bb960e3e938a122fa4386251","family_id":"qf-mod23-metricconv-limit-uniqueness-audit","seed":202610234086,"semantic_template":"qf-mod23-metricconv-limit-uniqueness-audit/{\"candidate_a\":0,\"candidate_b\":1,\"distances\":[[0,11,23,36],[11,0,12,25],[23,12,0,13],[36,25,13,0]],\"n\":4,\"prefix\":[2,3],\"tail_value\":0}"},"fields":{"title":"A Conditional Uniqueness Check Is Not Two Convergence Claims","stem":"Use the complete qualified finite metric table=[[0, 11, 23, 36], [11, 0, 12, 25], [23, 12, 0, 13], [36, 25, 13, 0]] on Fin(4)={0,…,3}. Preserve the convergence quantifiers: for every positive epsilon there exists N such that every n≥N satisfies d(x_n,candidate)<epsilon. A finite recorded tail alone does not certify this infinite statement. infinite sequence has prefix=[2, 3] and constant tail=0; candidates a=0, b=1. Give (prefix,tail,a,b,d(a,b),a is a limit?,b is a limit?,if both are limits then a=b?).","answer":{"kind":"expression","value":"([2, 3], 0, 0, 1, 11, Yes, No, Yes)","explanation":"Check each candidate against the declared constant tail. The final implication may be true when its antecedent is false; it does not assert both candidates converge. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([2, 3], 0, 0, 1, 11, Yes, No, Yes)."},"skills":["A Conditional Uniqueness Check Is Not Two Convergence Claims"],"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-20261013-advance-20261023-r04-metric-convergence-0087","batch_id":"batch-20261013-advance-20261023-r04-metric-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4fda311ad649bba2e7d3f1775b38839a8a0d98f98f6ae7fd26ff4e70b2041460","family_id":"qf-mod23-metricconv-limit-uniqueness-audit","seed":202610234087,"semantic_template":"qf-mod23-metricconv-limit-uniqueness-audit/{\"candidate_a\":1,\"candidate_b\":2,\"distances\":[[0,12,25,39],[12,0,13,27],[25,13,0,14],[39,27,14,0]],\"n\":4,\"prefix\":[3,0],\"tail_value\":1}"},"fields":{"title":"A Conditional Uniqueness Check Is Not Two Convergence Claims","stem":"Use the complete qualified finite metric table=[[0, 12, 25, 39], [12, 0, 13, 27], [25, 13, 0, 14], [39, 27, 14, 0]] on Fin(4)={0,…,3}. Preserve the convergence quantifiers: for every positive epsilon there exists N such that every n≥N satisfies d(x_n,candidate)<epsilon. A finite recorded tail alone does not certify this infinite statement. infinite sequence has prefix=[3, 0] and constant tail=1; candidates a=1, b=2. Give (prefix,tail,a,b,d(a,b),a is a limit?,b is a limit?,if both are limits then a=b?).","answer":{"kind":"expression","value":"([3, 0], 1, 1, 2, 13, Yes, No, Yes)","explanation":"Check each candidate against the declared constant tail. The final implication may be true when its antecedent is false; it does not assert both candidates converge. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([3, 0], 1, 1, 2, 13, Yes, No, Yes)."},"skills":["A Conditional Uniqueness Check Is Not Two Convergence Claims"],"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-20261013-advance-20261023-r04-metric-convergence-0088","batch_id":"batch-20261013-advance-20261023-r04-metric-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2fb3c0bd869ead219df8a4d52b130f047ed11f61aa87dec87803ce8edbf81f4e","family_id":"qf-mod23-metricconv-limit-uniqueness-audit","seed":202610234088,"semantic_template":"qf-mod23-metricconv-limit-uniqueness-audit/{\"candidate_a\":2,\"candidate_b\":2,\"distances\":[[0,13,27,42],[13,0,14,29],[27,14,0,15],[42,29,15,0]],\"n\":4,\"prefix\":[0,1],\"tail_value\":2}"},"fields":{"title":"A Conditional Uniqueness Check Is Not Two Convergence Claims","stem":"Use the complete qualified finite metric table=[[0, 13, 27, 42], [13, 0, 14, 29], [27, 14, 0, 15], [42, 29, 15, 0]] on Fin(4)={0,…,3}. Preserve the convergence quantifiers: for every positive epsilon there exists N such that every n≥N satisfies d(x_n,candidate)<epsilon. A finite recorded tail alone does not certify this infinite statement. infinite sequence has prefix=[0, 1] and constant tail=2; candidates a=2, b=2. Give (prefix,tail,a,b,d(a,b),a is a limit?,b is a limit?,if both are limits then a=b?).","answer":{"kind":"expression","value":"([0, 1], 2, 2, 2, 0, Yes, Yes, Yes)","explanation":"Check each candidate against the declared constant tail. The final implication may be true when its antecedent is false; it does not assert both candidates converge. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([0, 1], 2, 2, 2, 0, Yes, Yes, Yes)."},"skills":["A Conditional Uniqueness Check Is Not Two Convergence Claims"],"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-20261013-advance-20261023-r04-metric-convergence-0089","batch_id":"batch-20261013-advance-20261023-r04-metric-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4302f7bc99c3aa436af4af1acdc7561818cfa8fd2a532cea097b545424f27ad3","family_id":"qf-mod23-metricconv-limit-uniqueness-audit","seed":202610234089,"semantic_template":"qf-mod23-metricconv-limit-uniqueness-audit/{\"candidate_a\":3,\"candidate_b\":0,\"distances\":[[0,14,29,45],[14,0,15,31],[29,15,0,16],[45,31,16,0]],\"n\":4,\"prefix\":[1,2],\"tail_value\":3}"},"fields":{"title":"A Conditional Uniqueness Check Is Not Two Convergence Claims","stem":"Use the complete qualified finite metric table=[[0, 14, 29, 45], [14, 0, 15, 31], [29, 15, 0, 16], [45, 31, 16, 0]] on Fin(4)={0,…,3}. Preserve the convergence quantifiers: for every positive epsilon there exists N such that every n≥N satisfies d(x_n,candidate)<epsilon. A finite recorded tail alone does not certify this infinite statement. infinite sequence has prefix=[1, 2] and constant tail=3; candidates a=3, b=0. Give (prefix,tail,a,b,d(a,b),a is a limit?,b is a limit?,if both are limits then a=b?).","answer":{"kind":"expression","value":"([1, 2], 3, 3, 0, 45, Yes, No, Yes)","explanation":"Check each candidate against the declared constant tail. The final implication may be true when its antecedent is false; it does not assert both candidates converge. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([1, 2], 3, 3, 0, 45, Yes, No, Yes)."},"skills":["A Conditional Uniqueness Check Is Not Two Convergence Claims"],"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-20261013-advance-20261023-r04-metric-convergence-0090","batch_id":"batch-20261013-advance-20261023-r04-metric-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f80c2b15afbe0cef6d8fd84de853a03180c064cba5afcbd36839f19d0ce59c41","family_id":"qf-mod23-metricconv-limit-uniqueness-audit","seed":202610234090,"semantic_template":"qf-mod23-metricconv-limit-uniqueness-audit/{\"candidate_a\":0,\"candidate_b\":1,\"distances\":[[0,15,31,48],[15,0,16,33],[31,16,0,17],[48,33,17,0]],\"n\":4,\"prefix\":[2,3],\"tail_value\":0}"},"fields":{"title":"A Conditional Uniqueness Check Is Not Two Convergence Claims","stem":"Use the complete qualified finite metric table=[[0, 15, 31, 48], [15, 0, 16, 33], [31, 16, 0, 17], [48, 33, 17, 0]] on Fin(4)={0,…,3}. Preserve the convergence quantifiers: for every positive epsilon there exists N such that every n≥N satisfies d(x_n,candidate)<epsilon. A finite recorded tail alone does not certify this infinite statement. infinite sequence has prefix=[2, 3] and constant tail=0; candidates a=0, b=1. Give (prefix,tail,a,b,d(a,b),a is a limit?,b is a limit?,if both are limits then a=b?).","answer":{"kind":"expression","value":"([2, 3], 0, 0, 1, 15, Yes, No, Yes)","explanation":"Check each candidate against the declared constant tail. The final implication may be true when its antecedent is false; it does not assert both candidates converge. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([2, 3], 0, 0, 1, 15, Yes, No, Yes)."},"skills":["A Conditional Uniqueness Check Is Not Two Convergence Claims"],"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-20261013-advance-20261023-r04-metric-convergence-0091","batch_id":"batch-20261013-advance-20261023-r04-metric-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a10cd9de3080c91d487e9497cf85ab3d6dc7e26b19523e0d10bbe1a19d778e5b","family_id":"qf-mod23-metricconv-limit-uniqueness-audit","seed":202610234091,"semantic_template":"qf-mod23-metricconv-limit-uniqueness-audit/{\"candidate_a\":1,\"candidate_b\":1,\"distances\":[[0,16,33,51],[16,0,17,35],[33,17,0,18],[51,35,18,0]],\"n\":4,\"prefix\":[3,0],\"tail_value\":1}"},"fields":{"title":"A Conditional Uniqueness Check Is Not Two Convergence Claims","stem":"Use the complete qualified finite metric table=[[0, 16, 33, 51], [16, 0, 17, 35], [33, 17, 0, 18], [51, 35, 18, 0]] on Fin(4)={0,…,3}. Preserve the convergence quantifiers: for every positive epsilon there exists N such that every n≥N satisfies d(x_n,candidate)<epsilon. A finite recorded tail alone does not certify this infinite statement. infinite sequence has prefix=[3, 0] and constant tail=1; candidates a=1, b=1. Give (prefix,tail,a,b,d(a,b),a is a limit?,b is a limit?,if both are limits then a=b?).","answer":{"kind":"expression","value":"([3, 0], 1, 1, 1, 0, Yes, Yes, Yes)","explanation":"Check each candidate against the declared constant tail. The final implication may be true when its antecedent is false; it does not assert both candidates converge. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([3, 0], 1, 1, 1, 0, Yes, Yes, Yes)."},"skills":["A Conditional Uniqueness Check Is Not Two Convergence Claims"],"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-20261013-advance-20261023-r04-metric-convergence-0092","batch_id":"batch-20261013-advance-20261023-r04-metric-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4b255bdd61deb6b1370074acedfbe26a7636d28f581fd67a43e46441866d004c","family_id":"qf-mod23-metricconv-limit-uniqueness-audit","seed":202610234092,"semantic_template":"qf-mod23-metricconv-limit-uniqueness-audit/{\"candidate_a\":2,\"candidate_b\":3,\"distances\":[[0,17,35,54],[17,0,18,37],[35,18,0,19],[54,37,19,0]],\"n\":4,\"prefix\":[0,1],\"tail_value\":2}"},"fields":{"title":"A Conditional Uniqueness Check Is Not Two Convergence Claims","stem":"Use the complete qualified finite metric table=[[0, 17, 35, 54], [17, 0, 18, 37], [35, 18, 0, 19], [54, 37, 19, 0]] on Fin(4)={0,…,3}. Preserve the convergence quantifiers: for every positive epsilon there exists N such that every n≥N satisfies d(x_n,candidate)<epsilon. A finite recorded tail alone does not certify this infinite statement. infinite sequence has prefix=[0, 1] and constant tail=2; candidates a=2, b=3. Give (prefix,tail,a,b,d(a,b),a is a limit?,b is a limit?,if both are limits then a=b?).","answer":{"kind":"expression","value":"([0, 1], 2, 2, 3, 19, Yes, No, Yes)","explanation":"Check each candidate against the declared constant tail. The final implication may be true when its antecedent is false; it does not assert both candidates converge. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([0, 1], 2, 2, 3, 19, Yes, No, Yes)."},"skills":["A Conditional Uniqueness Check Is Not Two Convergence Claims"],"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-20261013-advance-20261023-r04-metric-convergence-0093","batch_id":"batch-20261013-advance-20261023-r04-metric-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d532b51ad01cead01c0ab585bec2e0f668c7388e774f8bb209c0f44fe75df12f","family_id":"qf-mod23-metricconv-limit-uniqueness-audit","seed":202610234093,"semantic_template":"qf-mod23-metricconv-limit-uniqueness-audit/{\"candidate_a\":3,\"candidate_b\":0,\"distances\":[[0,18,37,57],[18,0,19,39],[37,19,0,20],[57,39,20,0]],\"n\":4,\"prefix\":[1,2],\"tail_value\":3}"},"fields":{"title":"A Conditional Uniqueness Check Is Not Two Convergence Claims","stem":"Use the complete qualified finite metric table=[[0, 18, 37, 57], [18, 0, 19, 39], [37, 19, 0, 20], [57, 39, 20, 0]] on Fin(4)={0,…,3}. Preserve the convergence quantifiers: for every positive epsilon there exists N such that every n≥N satisfies d(x_n,candidate)<epsilon. A finite recorded tail alone does not certify this infinite statement. infinite sequence has prefix=[1, 2] and constant tail=3; candidates a=3, b=0. Give (prefix,tail,a,b,d(a,b),a is a limit?,b is a limit?,if both are limits then a=b?).","answer":{"kind":"expression","value":"([1, 2], 3, 3, 0, 57, Yes, No, Yes)","explanation":"Check each candidate against the declared constant tail. The final implication may be true when its antecedent is false; it does not assert both candidates converge. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([1, 2], 3, 3, 0, 57, Yes, No, Yes)."},"skills":["A Conditional Uniqueness Check Is Not Two Convergence Claims"],"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-20261013-advance-20261023-r04-metric-convergence-0094","batch_id":"batch-20261013-advance-20261023-r04-metric-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2e7e3f47035cfbd9e6baa2898d783a2d46c221db70fcf665940e086737efca4f","family_id":"qf-mod23-metricconv-limit-uniqueness-audit","seed":202610234094,"semantic_template":"qf-mod23-metricconv-limit-uniqueness-audit/{\"candidate_a\":0,\"candidate_b\":0,\"distances\":[[0,19,39,60],[19,0,20,41],[39,20,0,21],[60,41,21,0]],\"n\":4,\"prefix\":[2,3],\"tail_value\":0}"},"fields":{"title":"A Conditional Uniqueness Check Is Not Two Convergence Claims","stem":"Use the complete qualified finite metric table=[[0, 19, 39, 60], [19, 0, 20, 41], [39, 20, 0, 21], [60, 41, 21, 0]] on Fin(4)={0,…,3}. Preserve the convergence quantifiers: for every positive epsilon there exists N such that every n≥N satisfies d(x_n,candidate)<epsilon. A finite recorded tail alone does not certify this infinite statement. infinite sequence has prefix=[2, 3] and constant tail=0; candidates a=0, b=0. Give (prefix,tail,a,b,d(a,b),a is a limit?,b is a limit?,if both are limits then a=b?).","answer":{"kind":"expression","value":"([2, 3], 0, 0, 0, 0, Yes, Yes, Yes)","explanation":"Check each candidate against the declared constant tail. The final implication may be true when its antecedent is false; it does not assert both candidates converge. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([2, 3], 0, 0, 0, 0, Yes, Yes, Yes)."},"skills":["A Conditional Uniqueness Check Is Not Two Convergence Claims"],"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-20261013-advance-20261023-r04-metric-convergence-0095","batch_id":"batch-20261013-advance-20261023-r04-metric-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2fc85fea837498029d076cb7f306dc8175323f7a681ad0bf5d29dcfc3725b5e2","family_id":"qf-mod23-metricconv-limit-uniqueness-audit","seed":202610234095,"semantic_template":"qf-mod23-metricconv-limit-uniqueness-audit/{\"candidate_a\":1,\"candidate_b\":2,\"distances\":[[0,20,41,63],[20,0,21,43],[41,21,0,22],[63,43,22,0]],\"n\":4,\"prefix\":[3,0],\"tail_value\":1}"},"fields":{"title":"A Conditional Uniqueness Check Is Not Two Convergence Claims","stem":"Use the complete qualified finite metric table=[[0, 20, 41, 63], [20, 0, 21, 43], [41, 21, 0, 22], [63, 43, 22, 0]] on Fin(4)={0,…,3}. Preserve the convergence quantifiers: for every positive epsilon there exists N such that every n≥N satisfies d(x_n,candidate)<epsilon. A finite recorded tail alone does not certify this infinite statement. infinite sequence has prefix=[3, 0] and constant tail=1; candidates a=1, b=2. Give (prefix,tail,a,b,d(a,b),a is a limit?,b is a limit?,if both are limits then a=b?).","answer":{"kind":"expression","value":"([3, 0], 1, 1, 2, 21, Yes, No, Yes)","explanation":"Check each candidate against the declared constant tail. The final implication may be true when its antecedent is false; it does not assert both candidates converge. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([3, 0], 1, 1, 2, 21, Yes, No, Yes)."},"skills":["A Conditional Uniqueness Check Is Not Two Convergence Claims"],"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-20261013-advance-20261023-r04-metric-convergence-0096","batch_id":"batch-20261013-advance-20261023-r04-metric-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"38eaaa5f2653a23d3950bb14de63bc9dd0d237b78af6ab07a8cce4ef6baf101b","family_id":"qf-mod23-metricconv-limit-uniqueness-audit","seed":202610234096,"semantic_template":"qf-mod23-metricconv-limit-uniqueness-audit/{\"candidate_a\":2,\"candidate_b\":3,\"distances\":[[0,21,43,66],[21,0,22,45],[43,22,0,23],[66,45,23,0]],\"n\":4,\"prefix\":[0,1],\"tail_value\":2}"},"fields":{"title":"A Conditional Uniqueness Check Is Not Two Convergence Claims","stem":"Use the complete qualified finite metric table=[[0, 21, 43, 66], [21, 0, 22, 45], [43, 22, 0, 23], [66, 45, 23, 0]] on Fin(4)={0,…,3}. Preserve the convergence quantifiers: for every positive epsilon there exists N such that every n≥N satisfies d(x_n,candidate)<epsilon. A finite recorded tail alone does not certify this infinite statement. infinite sequence has prefix=[0, 1] and constant tail=2; candidates a=2, b=3. Give (prefix,tail,a,b,d(a,b),a is a limit?,b is a limit?,if both are limits then a=b?).","answer":{"kind":"expression","value":"([0, 1], 2, 2, 3, 23, Yes, No, Yes)","explanation":"Check each candidate against the declared constant tail. The final implication may be true when its antecedent is false; it does not assert both candidates converge. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([0, 1], 2, 2, 3, 23, Yes, No, Yes)."},"skills":["A Conditional Uniqueness Check Is Not Two Convergence Claims"],"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-20261013-advance-20261023-r04-metric-convergence-0097","batch_id":"batch-20261013-advance-20261023-r04-metric-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a3d4179eb6f405f9173f24936a0b6dd48c6e5b0a824fd444a75507d17d6c0869","family_id":"qf-mod23-metricconv-limit-uniqueness-audit","seed":202610234097,"semantic_template":"qf-mod23-metricconv-limit-uniqueness-audit/{\"candidate_a\":3,\"candidate_b\":3,\"distances\":[[0,22,45,69],[22,0,23,47],[45,23,0,24],[69,47,24,0]],\"n\":4,\"prefix\":[1,2],\"tail_value\":3}"},"fields":{"title":"A Conditional Uniqueness Check Is Not Two Convergence Claims","stem":"Use the complete qualified finite metric table=[[0, 22, 45, 69], [22, 0, 23, 47], [45, 23, 0, 24], [69, 47, 24, 0]] on Fin(4)={0,…,3}. Preserve the convergence quantifiers: for every positive epsilon there exists N such that every n≥N satisfies d(x_n,candidate)<epsilon. A finite recorded tail alone does not certify this infinite statement. infinite sequence has prefix=[1, 2] and constant tail=3; candidates a=3, b=3. Give (prefix,tail,a,b,d(a,b),a is a limit?,b is a limit?,if both are limits then a=b?).","answer":{"kind":"expression","value":"([1, 2], 3, 3, 3, 0, Yes, Yes, Yes)","explanation":"Check each candidate against the declared constant tail. The final implication may be true when its antecedent is false; it does not assert both candidates converge. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([1, 2], 3, 3, 3, 0, Yes, Yes, Yes)."},"skills":["A Conditional Uniqueness Check Is Not Two Convergence Claims"],"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-20261013-advance-20261023-r04-metric-convergence-0098","batch_id":"batch-20261013-advance-20261023-r04-metric-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3b601177c7172d681bfd9596e919100e3b1fd7e88bf97576b432d7402115b5e9","family_id":"qf-mod23-metricconv-limit-uniqueness-audit","seed":202610234098,"semantic_template":"qf-mod23-metricconv-limit-uniqueness-audit/{\"candidate_a\":0,\"candidate_b\":1,\"distances\":[[0,23,47,72],[23,0,24,49],[47,24,0,25],[72,49,25,0]],\"n\":4,\"prefix\":[2,3],\"tail_value\":0}"},"fields":{"title":"A Conditional Uniqueness Check Is Not Two Convergence Claims","stem":"Use the complete qualified finite metric table=[[0, 23, 47, 72], [23, 0, 24, 49], [47, 24, 0, 25], [72, 49, 25, 0]] on Fin(4)={0,…,3}. Preserve the convergence quantifiers: for every positive epsilon there exists N such that every n≥N satisfies d(x_n,candidate)<epsilon. A finite recorded tail alone does not certify this infinite statement. infinite sequence has prefix=[2, 3] and constant tail=0; candidates a=0, b=1. Give (prefix,tail,a,b,d(a,b),a is a limit?,b is a limit?,if both are limits then a=b?).","answer":{"kind":"expression","value":"([2, 3], 0, 0, 1, 23, Yes, No, Yes)","explanation":"Check each candidate against the declared constant tail. The final implication may be true when its antecedent is false; it does not assert both candidates converge. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([2, 3], 0, 0, 1, 23, Yes, No, Yes)."},"skills":["A Conditional Uniqueness Check Is Not Two Convergence Claims"],"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-20261013-advance-20261023-r04-metric-convergence-0099","batch_id":"batch-20261013-advance-20261023-r04-metric-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"973a1b9670b19987f34fb27ee8998fd42281fc3c4aea29f0b8837dd45b1232da","family_id":"qf-mod23-metricconv-limit-uniqueness-audit","seed":202610234099,"semantic_template":"qf-mod23-metricconv-limit-uniqueness-audit/{\"candidate_a\":1,\"candidate_b\":2,\"distances\":[[0,24,49,75],[24,0,25,51],[49,25,0,26],[75,51,26,0]],\"n\":4,\"prefix\":[3,0],\"tail_value\":1}"},"fields":{"title":"A Conditional Uniqueness Check Is Not Two Convergence Claims","stem":"Use the complete qualified finite metric table=[[0, 24, 49, 75], [24, 0, 25, 51], [49, 25, 0, 26], [75, 51, 26, 0]] on Fin(4)={0,…,3}. Preserve the convergence quantifiers: for every positive epsilon there exists N such that every n≥N satisfies d(x_n,candidate)<epsilon. A finite recorded tail alone does not certify this infinite statement. infinite sequence has prefix=[3, 0] and constant tail=1; candidates a=1, b=2. Give (prefix,tail,a,b,d(a,b),a is a limit?,b is a limit?,if both are limits then a=b?).","answer":{"kind":"expression","value":"([3, 0], 1, 1, 2, 25, Yes, No, Yes)","explanation":"Check each candidate against the declared constant tail. The final implication may be true when its antecedent is false; it does not assert both candidates converge. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([3, 0], 1, 1, 2, 25, Yes, No, Yes)."},"skills":["A Conditional Uniqueness Check Is Not Two Convergence Claims"],"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-20261013-advance-20261023-r04-metric-convergence-0100","batch_id":"batch-20261013-advance-20261023-r04-metric-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"66641b59aa2b8efed821730931456847bf850af93a00f294f22e220a42f01515","family_id":"qf-mod23-metricconv-limit-uniqueness-audit","seed":202610234100,"semantic_template":"qf-mod23-metricconv-limit-uniqueness-audit/{\"candidate_a\":2,\"candidate_b\":2,\"distances\":[[0,25,51,78],[25,0,26,53],[51,26,0,27],[78,53,27,0]],\"n\":4,\"prefix\":[0,1],\"tail_value\":2}"},"fields":{"title":"A Conditional Uniqueness Check Is Not Two Convergence Claims","stem":"Use the complete qualified finite metric table=[[0, 25, 51, 78], [25, 0, 26, 53], [51, 26, 0, 27], [78, 53, 27, 0]] on Fin(4)={0,…,3}. Preserve the convergence quantifiers: for every positive epsilon there exists N such that every n≥N satisfies d(x_n,candidate)<epsilon. A finite recorded tail alone does not certify this infinite statement. infinite sequence has prefix=[0, 1] and constant tail=2; candidates a=2, b=2. Give (prefix,tail,a,b,d(a,b),a is a limit?,b is a limit?,if both are limits then a=b?).","answer":{"kind":"expression","value":"([0, 1], 2, 2, 2, 0, Yes, Yes, Yes)","explanation":"Check each candidate against the declared constant tail. The final implication may be true when its antecedent is false; it does not assert both candidates converge. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([0, 1], 2, 2, 2, 0, Yes, Yes, Yes)."},"skills":["A Conditional Uniqueness Check Is Not Two Convergence Claims"],"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."}}