{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r01-series-comparison-test-0001","batch_id":"batch-20261015-advance-20261117-r01-series-comparison-test-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c8dcdf0c3ddabb6f880d1c1369df1da0977872f61e0c784c54370f49a9cddb42","family_id":"qf-mod48-series-upper-geometric","seed":202611171001,"semantic_template":"series-upper-geometric|{a_scale:[1,1],b_scale:[2,1],ratio:[1,7]}"},"fields":{"title":"A Convergent Nonnegative Geometric Upper Series Controls the Smaller Series","stem":"Use exact scales A=1,B=2, rational ratio r=1/7. For n≥0 use a_n=A*(r)^n,b_n=B*(r)^n, with0≤A≤B and0≤r<1. Give (a_scale, b_scale, ratio, sum_a, sum_b, sum_comparison, certificate).","answer":{"kind":"expression","value":"(1, 2, 1/7, 7/6, 7/3, Yes, upper_comparison_proves_convergence)","explanation":"Compute both exact sums and compare them. The first term has exponent0, including r=0. The full nonnegative comparison and convergent reference are premises; finite samples do not prove the general theorem. The exact result, retaining the requested order, is (1, 2, 1/7, 7/6, 7/3, Yes, upper_comparison_proves_convergence)."},"skills":["A Convergent Nonnegative Geometric Upper Series Controls the Smaller Series"],"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-20261015-advance-20261117-r01-series-comparison-test-0002","batch_id":"batch-20261015-advance-20261117-r01-series-comparison-test-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8ce7d0d060f22a4e1f1dd74783cbe8878d3e02e8abf0714e77cb0c1bec995637","family_id":"qf-mod48-series-upper-geometric","seed":202611171002,"semantic_template":"series-upper-geometric|{a_scale:[2,1],b_scale:[3,1],ratio:[1,4]}"},"fields":{"title":"A Convergent Nonnegative Geometric Upper Series Controls the Smaller Series","stem":"Use exact scales A=2,B=3, rational ratio r=2/8. For n≥0 use a_n=A*(r)^n,b_n=B*(r)^n, with0≤A≤B and0≤r<1. Give (a_scale, b_scale, ratio, sum_a, sum_b, sum_comparison, certificate).","answer":{"kind":"expression","value":"(2, 3, 1/4, 8/3, 4, Yes, upper_comparison_proves_convergence)","explanation":"Compute both exact sums and compare them. The first term has exponent0, including r=0. The full nonnegative comparison and convergent reference are premises; finite samples do not prove the general theorem. The exact result, retaining the requested order, is (2, 3, 1/4, 8/3, 4, Yes, upper_comparison_proves_convergence)."},"skills":["A Convergent Nonnegative Geometric Upper Series Controls the Smaller Series"],"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-20261015-advance-20261117-r01-series-comparison-test-0003","batch_id":"batch-20261015-advance-20261117-r01-series-comparison-test-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3377ce8a78b0ed212da509e7254163943b1a9ef67e0bd036cfdea3d9f1a4acb0","family_id":"qf-mod48-series-upper-geometric","seed":202611171003,"semantic_template":"series-upper-geometric|{a_scale:[3,1],b_scale:[4,1],ratio:[1,3]}"},"fields":{"title":"A Convergent Nonnegative Geometric Upper Series Controls the Smaller Series","stem":"Use exact scales A=3,B=4, rational ratio r=3/9. For n≥0 use a_n=A*(r)^n,b_n=B*(r)^n, with0≤A≤B and0≤r<1. Give (a_scale, b_scale, ratio, sum_a, sum_b, sum_comparison, certificate).","answer":{"kind":"expression","value":"(3, 4, 1/3, 9/2, 6, Yes, upper_comparison_proves_convergence)","explanation":"Compute both exact sums and compare them. The first term has exponent0, including r=0. The full nonnegative comparison and convergent reference are premises; finite samples do not prove the general theorem. The exact result, retaining the requested order, is (3, 4, 1/3, 9/2, 6, Yes, upper_comparison_proves_convergence)."},"skills":["A Convergent Nonnegative Geometric Upper Series Controls the Smaller Series"],"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-20261015-advance-20261117-r01-series-comparison-test-0004","batch_id":"batch-20261015-advance-20261117-r01-series-comparison-test-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5571f45a66bb12af7564636fd5949b7e447341ed9a0bf41cd79e4c2cd848f5ef","family_id":"qf-mod48-series-upper-geometric","seed":202611171004,"semantic_template":"series-upper-geometric|{a_scale:[4,1],b_scale:[5,1],ratio:[2,5]}"},"fields":{"title":"A Convergent Nonnegative Geometric Upper Series Controls the Smaller Series","stem":"Use exact scales A=4,B=5, rational ratio r=4/10. For n≥0 use a_n=A*(r)^n,b_n=B*(r)^n, with0≤A≤B and0≤r<1. Give (a_scale, b_scale, ratio, sum_a, sum_b, sum_comparison, certificate).","answer":{"kind":"expression","value":"(4, 5, 2/5, 20/3, 25/3, Yes, upper_comparison_proves_convergence)","explanation":"Compute both exact sums and compare them. The first term has exponent0, including r=0. The full nonnegative comparison and convergent reference are premises; finite samples do not prove the general theorem. The exact result, retaining the requested order, is (4, 5, 2/5, 20/3, 25/3, Yes, upper_comparison_proves_convergence)."},"skills":["A Convergent Nonnegative Geometric Upper Series Controls the Smaller Series"],"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-20261015-advance-20261117-r01-series-comparison-test-0005","batch_id":"batch-20261015-advance-20261117-r01-series-comparison-test-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f7fec17c9be4683c200787263b136b009bf2be49d19ccfc00aa84519c6c52cef","family_id":"qf-mod48-series-upper-geometric","seed":202611171005,"semantic_template":"series-upper-geometric|{a_scale:[5,1],b_scale:[6,1],ratio:[5,11]}"},"fields":{"title":"A Convergent Nonnegative Geometric Upper Series Controls the Smaller Series","stem":"Use exact scales A=5,B=6, rational ratio r=5/11. For n≥0 use a_n=A*(r)^n,b_n=B*(r)^n, with0≤A≤B and0≤r<1. Give (a_scale, b_scale, ratio, sum_a, sum_b, sum_comparison, certificate).","answer":{"kind":"expression","value":"(5, 6, 5/11, 55/6, 11, Yes, upper_comparison_proves_convergence)","explanation":"Compute both exact sums and compare them. The first term has exponent0, including r=0. The full nonnegative comparison and convergent reference are premises; finite samples do not prove the general theorem. The exact result, retaining the requested order, is (5, 6, 5/11, 55/6, 11, Yes, upper_comparison_proves_convergence)."},"skills":["A Convergent Nonnegative Geometric Upper Series Controls the Smaller Series"],"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-20261015-advance-20261117-r01-series-comparison-test-0006","batch_id":"batch-20261015-advance-20261117-r01-series-comparison-test-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9893cd411395199b7d2b795bdd7152f26e0cb83661028f6e5150149c45d3e9fd","family_id":"qf-mod48-series-upper-geometric","seed":202611171006,"semantic_template":"series-upper-geometric|{a_scale:[6,1],b_scale:[7,1],ratio:[1,7]}"},"fields":{"title":"A Convergent Nonnegative Geometric Upper Series Controls the Smaller Series","stem":"Use exact scales A=6,B=7, rational ratio r=1/7. For n≥0 use a_n=A*(r)^n,b_n=B*(r)^n, with0≤A≤B and0≤r<1. Give (a_scale, b_scale, ratio, sum_a, sum_b, sum_comparison, certificate).","answer":{"kind":"expression","value":"(6, 7, 1/7, 7, 49/6, Yes, upper_comparison_proves_convergence)","explanation":"Compute both exact sums and compare them. The first term has exponent0, including r=0. The full nonnegative comparison and convergent reference are premises; finite samples do not prove the general theorem. The exact result, retaining the requested order, is (6, 7, 1/7, 7, 49/6, Yes, upper_comparison_proves_convergence)."},"skills":["A Convergent Nonnegative Geometric Upper Series Controls the Smaller Series"],"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-20261015-advance-20261117-r01-series-comparison-test-0007","batch_id":"batch-20261015-advance-20261117-r01-series-comparison-test-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a43e6741e3aa4649b6ef08c5ad80f15111ef98690ebce579bf99b3afa4054ca0","family_id":"qf-mod48-series-upper-geometric","seed":202611171007,"semantic_template":"series-upper-geometric|{a_scale:[7,1],b_scale:[8,1],ratio:[1,4]}"},"fields":{"title":"A Convergent Nonnegative Geometric Upper Series Controls the Smaller Series","stem":"Use exact scales A=7,B=8, rational ratio r=2/8. For n≥0 use a_n=A*(r)^n,b_n=B*(r)^n, with0≤A≤B and0≤r<1. Give (a_scale, b_scale, ratio, sum_a, sum_b, sum_comparison, certificate).","answer":{"kind":"expression","value":"(7, 8, 1/4, 28/3, 32/3, Yes, upper_comparison_proves_convergence)","explanation":"Compute both exact sums and compare them. The first term has exponent0, including r=0. The full nonnegative comparison and convergent reference are premises; finite samples do not prove the general theorem. The exact result, retaining the requested order, is (7, 8, 1/4, 28/3, 32/3, Yes, upper_comparison_proves_convergence)."},"skills":["A Convergent Nonnegative Geometric Upper Series Controls the Smaller Series"],"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-20261015-advance-20261117-r01-series-comparison-test-0008","batch_id":"batch-20261015-advance-20261117-r01-series-comparison-test-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"06133d90a36891f69080e765425177e42a6e030687ad5145da03111f5a39f77a","family_id":"qf-mod48-series-upper-geometric","seed":202611171008,"semantic_template":"series-upper-geometric|{a_scale:[8,1],b_scale:[9,1],ratio:[1,3]}"},"fields":{"title":"A Convergent Nonnegative Geometric Upper Series Controls the Smaller Series","stem":"Use exact scales A=8,B=9, rational ratio r=3/9. For n≥0 use a_n=A*(r)^n,b_n=B*(r)^n, with0≤A≤B and0≤r<1. Give (a_scale, b_scale, ratio, sum_a, sum_b, sum_comparison, certificate).","answer":{"kind":"expression","value":"(8, 9, 1/3, 12, 27/2, Yes, upper_comparison_proves_convergence)","explanation":"Compute both exact sums and compare them. The first term has exponent0, including r=0. The full nonnegative comparison and convergent reference are premises; finite samples do not prove the general theorem. The exact result, retaining the requested order, is (8, 9, 1/3, 12, 27/2, Yes, upper_comparison_proves_convergence)."},"skills":["A Convergent Nonnegative Geometric Upper Series Controls the Smaller Series"],"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-20261015-advance-20261117-r01-series-comparison-test-0009","batch_id":"batch-20261015-advance-20261117-r01-series-comparison-test-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b755cd22f58b895a19543733115ae83b3b75d66aa2ece2777d439b157b15ad54","family_id":"qf-mod48-series-upper-geometric","seed":202611171009,"semantic_template":"series-upper-geometric|{a_scale:[9,1],b_scale:[10,1],ratio:[2,5]}"},"fields":{"title":"A Convergent Nonnegative Geometric Upper Series Controls the Smaller Series","stem":"Use exact scales A=9,B=10, rational ratio r=4/10. For n≥0 use a_n=A*(r)^n,b_n=B*(r)^n, with0≤A≤B and0≤r<1. Give (a_scale, b_scale, ratio, sum_a, sum_b, sum_comparison, certificate).","answer":{"kind":"expression","value":"(9, 10, 2/5, 15, 50/3, Yes, upper_comparison_proves_convergence)","explanation":"Compute both exact sums and compare them. The first term has exponent0, including r=0. The full nonnegative comparison and convergent reference are premises; finite samples do not prove the general theorem. The exact result, retaining the requested order, is (9, 10, 2/5, 15, 50/3, Yes, upper_comparison_proves_convergence)."},"skills":["A Convergent Nonnegative Geometric Upper Series Controls the Smaller Series"],"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-20261015-advance-20261117-r01-series-comparison-test-0010","batch_id":"batch-20261015-advance-20261117-r01-series-comparison-test-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7d4daa200fed225f396f9846ff33760622490d65e2f88cf1abe816cac8d7076a","family_id":"qf-mod48-series-upper-geometric","seed":202611171010,"semantic_template":"series-upper-geometric|{a_scale:[10,1],b_scale:[11,1],ratio:[5,11]}"},"fields":{"title":"A Convergent Nonnegative Geometric Upper Series Controls the Smaller Series","stem":"Use exact scales A=10,B=11, rational ratio r=5/11. For n≥0 use a_n=A*(r)^n,b_n=B*(r)^n, with0≤A≤B and0≤r<1. Give (a_scale, b_scale, ratio, sum_a, sum_b, sum_comparison, certificate).","answer":{"kind":"expression","value":"(10, 11, 5/11, 55/3, 121/6, Yes, upper_comparison_proves_convergence)","explanation":"Compute both exact sums and compare them. The first term has exponent0, including r=0. The full nonnegative comparison and convergent reference are premises; finite samples do not prove the general theorem. The exact result, retaining the requested order, is (10, 11, 5/11, 55/3, 121/6, Yes, upper_comparison_proves_convergence)."},"skills":["A Convergent Nonnegative Geometric Upper Series Controls the Smaller Series"],"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-20261015-advance-20261117-r01-series-comparison-test-0011","batch_id":"batch-20261015-advance-20261117-r01-series-comparison-test-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0159d76df0215cfc6f497c5b8b30ec42a25dbfb2d3c815596ecf83739842c5c2","family_id":"qf-mod48-series-upper-geometric","seed":202611171011,"semantic_template":"series-upper-geometric|{a_scale:[11,1],b_scale:[12,1],ratio:[1,7]}"},"fields":{"title":"A Convergent Nonnegative Geometric Upper Series Controls the Smaller Series","stem":"Use exact scales A=11,B=12, rational ratio r=1/7. For n≥0 use a_n=A*(r)^n,b_n=B*(r)^n, with0≤A≤B and0≤r<1. Give (a_scale, b_scale, ratio, sum_a, sum_b, sum_comparison, certificate).","answer":{"kind":"expression","value":"(11, 12, 1/7, 77/6, 14, Yes, upper_comparison_proves_convergence)","explanation":"Compute both exact sums and compare them. The first term has exponent0, including r=0. The full nonnegative comparison and convergent reference are premises; finite samples do not prove the general theorem. The exact result, retaining the requested order, is (11, 12, 1/7, 77/6, 14, Yes, upper_comparison_proves_convergence)."},"skills":["A Convergent Nonnegative Geometric Upper Series Controls the Smaller Series"],"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-20261015-advance-20261117-r01-series-comparison-test-0012","batch_id":"batch-20261015-advance-20261117-r01-series-comparison-test-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e6aafc4119bfafd7c199c184af53e28a0763a47659047acfe467b45f7f73e9ac","family_id":"qf-mod48-series-upper-geometric","seed":202611171012,"semantic_template":"series-upper-geometric|{a_scale:[12,1],b_scale:[13,1],ratio:[1,4]}"},"fields":{"title":"A Convergent Nonnegative Geometric Upper Series Controls the Smaller Series","stem":"Use exact scales A=12,B=13, rational ratio r=2/8. For n≥0 use a_n=A*(r)^n,b_n=B*(r)^n, with0≤A≤B and0≤r<1. Give (a_scale, b_scale, ratio, sum_a, sum_b, sum_comparison, certificate).","answer":{"kind":"expression","value":"(12, 13, 1/4, 16, 52/3, Yes, upper_comparison_proves_convergence)","explanation":"Compute both exact sums and compare them. The first term has exponent0, including r=0. The full nonnegative comparison and convergent reference are premises; finite samples do not prove the general theorem. The exact result, retaining the requested order, is (12, 13, 1/4, 16, 52/3, Yes, upper_comparison_proves_convergence)."},"skills":["A Convergent Nonnegative Geometric Upper Series Controls the Smaller Series"],"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-20261015-advance-20261117-r01-series-comparison-test-0013","batch_id":"batch-20261015-advance-20261117-r01-series-comparison-test-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"023d61a8802c6ab4ae6530c3fe6faad0c0ad05180c5b883cbd90e0c7ba15cb58","family_id":"qf-mod48-series-upper-geometric","seed":202611171013,"semantic_template":"series-upper-geometric|{a_scale:[13,1],b_scale:[14,1],ratio:[1,3]}"},"fields":{"title":"A Convergent Nonnegative Geometric Upper Series Controls the Smaller Series","stem":"Use exact scales A=13,B=14, rational ratio r=3/9. For n≥0 use a_n=A*(r)^n,b_n=B*(r)^n, with0≤A≤B and0≤r<1. Give (a_scale, b_scale, ratio, sum_a, sum_b, sum_comparison, certificate).","answer":{"kind":"expression","value":"(13, 14, 1/3, 39/2, 21, Yes, upper_comparison_proves_convergence)","explanation":"Compute both exact sums and compare them. The first term has exponent0, including r=0. The full nonnegative comparison and convergent reference are premises; finite samples do not prove the general theorem. The exact result, retaining the requested order, is (13, 14, 1/3, 39/2, 21, Yes, upper_comparison_proves_convergence)."},"skills":["A Convergent Nonnegative Geometric Upper Series Controls the Smaller Series"],"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-20261015-advance-20261117-r01-series-comparison-test-0014","batch_id":"batch-20261015-advance-20261117-r01-series-comparison-test-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b7a8b7e8b575bcaa15f16e6461df3747a8b1d78da3fb2dcfec9e53caa9750f5b","family_id":"qf-mod48-series-upper-geometric","seed":202611171014,"semantic_template":"series-upper-geometric|{a_scale:[14,1],b_scale:[15,1],ratio:[2,5]}"},"fields":{"title":"A Convergent Nonnegative Geometric Upper Series Controls the Smaller Series","stem":"Use exact scales A=14,B=15, rational ratio r=4/10. For n≥0 use a_n=A*(r)^n,b_n=B*(r)^n, with0≤A≤B and0≤r<1. Give (a_scale, b_scale, ratio, sum_a, sum_b, sum_comparison, certificate).","answer":{"kind":"expression","value":"(14, 15, 2/5, 70/3, 25, Yes, upper_comparison_proves_convergence)","explanation":"Compute both exact sums and compare them. The first term has exponent0, including r=0. The full nonnegative comparison and convergent reference are premises; finite samples do not prove the general theorem. The exact result, retaining the requested order, is (14, 15, 2/5, 70/3, 25, Yes, upper_comparison_proves_convergence)."},"skills":["A Convergent Nonnegative Geometric Upper Series Controls the Smaller Series"],"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-20261015-advance-20261117-r01-series-comparison-test-0015","batch_id":"batch-20261015-advance-20261117-r01-series-comparison-test-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"58282ae1dbfff3933231edf877d8a8c2c99543a0aff79d0470208f71e171a427","family_id":"qf-mod48-series-upper-geometric","seed":202611171015,"semantic_template":"series-upper-geometric|{a_scale:[15,1],b_scale:[16,1],ratio:[5,11]}"},"fields":{"title":"A Convergent Nonnegative Geometric Upper Series Controls the Smaller Series","stem":"Use exact scales A=15,B=16, rational ratio r=5/11. For n≥0 use a_n=A*(r)^n,b_n=B*(r)^n, with0≤A≤B and0≤r<1. Give (a_scale, b_scale, ratio, sum_a, sum_b, sum_comparison, certificate).","answer":{"kind":"expression","value":"(15, 16, 5/11, 55/2, 88/3, Yes, upper_comparison_proves_convergence)","explanation":"Compute both exact sums and compare them. The first term has exponent0, including r=0. The full nonnegative comparison and convergent reference are premises; finite samples do not prove the general theorem. The exact result, retaining the requested order, is (15, 16, 5/11, 55/2, 88/3, Yes, upper_comparison_proves_convergence)."},"skills":["A Convergent Nonnegative Geometric Upper Series Controls the Smaller Series"],"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-20261015-advance-20261117-r01-series-comparison-test-0016","batch_id":"batch-20261015-advance-20261117-r01-series-comparison-test-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5e0d0238221244a8e2208a58b09cdf93acfc732fdfdfadeca0e7ff0e34576eba","family_id":"qf-mod48-series-upper-geometric","seed":202611171016,"semantic_template":"series-upper-geometric|{a_scale:[16,1],b_scale:[17,1],ratio:[1,7]}"},"fields":{"title":"A Convergent Nonnegative Geometric Upper Series Controls the Smaller Series","stem":"Use exact scales A=16,B=17, rational ratio r=1/7. For n≥0 use a_n=A*(r)^n,b_n=B*(r)^n, with0≤A≤B and0≤r<1. Give (a_scale, b_scale, ratio, sum_a, sum_b, sum_comparison, certificate).","answer":{"kind":"expression","value":"(16, 17, 1/7, 56/3, 119/6, Yes, upper_comparison_proves_convergence)","explanation":"Compute both exact sums and compare them. The first term has exponent0, including r=0. The full nonnegative comparison and convergent reference are premises; finite samples do not prove the general theorem. The exact result, retaining the requested order, is (16, 17, 1/7, 56/3, 119/6, Yes, upper_comparison_proves_convergence)."},"skills":["A Convergent Nonnegative Geometric Upper Series Controls the Smaller Series"],"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-20261015-advance-20261117-r01-series-comparison-test-0017","batch_id":"batch-20261015-advance-20261117-r01-series-comparison-test-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d9cf5f96116a3f8506f1b23d5d2252f540541a161fe129a7b2917982284363a6","family_id":"qf-mod48-series-upper-geometric","seed":202611171017,"semantic_template":"series-upper-geometric|{a_scale:[17,1],b_scale:[18,1],ratio:[1,4]}"},"fields":{"title":"A Convergent Nonnegative Geometric Upper Series Controls the Smaller Series","stem":"Use exact scales A=17,B=18, rational ratio r=2/8. For n≥0 use a_n=A*(r)^n,b_n=B*(r)^n, with0≤A≤B and0≤r<1. Give (a_scale, b_scale, ratio, sum_a, sum_b, sum_comparison, certificate).","answer":{"kind":"expression","value":"(17, 18, 1/4, 68/3, 24, Yes, upper_comparison_proves_convergence)","explanation":"Compute both exact sums and compare them. The first term has exponent0, including r=0. The full nonnegative comparison and convergent reference are premises; finite samples do not prove the general theorem. The exact result, retaining the requested order, is (17, 18, 1/4, 68/3, 24, Yes, upper_comparison_proves_convergence)."},"skills":["A Convergent Nonnegative Geometric Upper Series Controls the Smaller Series"],"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-20261015-advance-20261117-r01-series-comparison-test-0018","batch_id":"batch-20261015-advance-20261117-r01-series-comparison-test-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a0bbba5d15d4fce8bb64ece54eec6556bd64aafcb714b039989f5906c206fbb6","family_id":"qf-mod48-series-upper-geometric","seed":202611171018,"semantic_template":"series-upper-geometric|{a_scale:[18,1],b_scale:[19,1],ratio:[1,3]}"},"fields":{"title":"A Convergent Nonnegative Geometric Upper Series Controls the Smaller Series","stem":"Use exact scales A=18,B=19, rational ratio r=3/9. For n≥0 use a_n=A*(r)^n,b_n=B*(r)^n, with0≤A≤B and0≤r<1. Give (a_scale, b_scale, ratio, sum_a, sum_b, sum_comparison, certificate).","answer":{"kind":"expression","value":"(18, 19, 1/3, 27, 57/2, Yes, upper_comparison_proves_convergence)","explanation":"Compute both exact sums and compare them. The first term has exponent0, including r=0. The full nonnegative comparison and convergent reference are premises; finite samples do not prove the general theorem. The exact result, retaining the requested order, is (18, 19, 1/3, 27, 57/2, Yes, upper_comparison_proves_convergence)."},"skills":["A Convergent Nonnegative Geometric Upper Series Controls the Smaller Series"],"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-20261015-advance-20261117-r01-series-comparison-test-0019","batch_id":"batch-20261015-advance-20261117-r01-series-comparison-test-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8f6482f2e41f91f72d58d8e65f6ac8810cd9941d488769ccb76519b35ab23d0e","family_id":"qf-mod48-series-upper-geometric","seed":202611171019,"semantic_template":"series-upper-geometric|{a_scale:[19,1],b_scale:[20,1],ratio:[2,5]}"},"fields":{"title":"A Convergent Nonnegative Geometric Upper Series Controls the Smaller Series","stem":"Use exact scales A=19,B=20, rational ratio r=4/10. For n≥0 use a_n=A*(r)^n,b_n=B*(r)^n, with0≤A≤B and0≤r<1. Give (a_scale, b_scale, ratio, sum_a, sum_b, sum_comparison, certificate).","answer":{"kind":"expression","value":"(19, 20, 2/5, 95/3, 100/3, Yes, upper_comparison_proves_convergence)","explanation":"Compute both exact sums and compare them. The first term has exponent0, including r=0. The full nonnegative comparison and convergent reference are premises; finite samples do not prove the general theorem. The exact result, retaining the requested order, is (19, 20, 2/5, 95/3, 100/3, Yes, upper_comparison_proves_convergence)."},"skills":["A Convergent Nonnegative Geometric Upper Series Controls the Smaller Series"],"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-20261015-advance-20261117-r01-series-comparison-test-0020","batch_id":"batch-20261015-advance-20261117-r01-series-comparison-test-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d051adae837e432bb2b828d1b6357205b2fcab973137cc9538bc53cc3c476040","family_id":"qf-mod48-series-upper-geometric","seed":202611171020,"semantic_template":"series-upper-geometric|{a_scale:[20,1],b_scale:[21,1],ratio:[5,11]}"},"fields":{"title":"A Convergent Nonnegative Geometric Upper Series Controls the Smaller Series","stem":"Use exact scales A=20,B=21, rational ratio r=5/11. For n≥0 use a_n=A*(r)^n,b_n=B*(r)^n, with0≤A≤B and0≤r<1. Give (a_scale, b_scale, ratio, sum_a, sum_b, sum_comparison, certificate).","answer":{"kind":"expression","value":"(20, 21, 5/11, 110/3, 77/2, Yes, upper_comparison_proves_convergence)","explanation":"Compute both exact sums and compare them. The first term has exponent0, including r=0. The full nonnegative comparison and convergent reference are premises; finite samples do not prove the general theorem. The exact result, retaining the requested order, is (20, 21, 5/11, 110/3, 77/2, Yes, upper_comparison_proves_convergence)."},"skills":["A Convergent Nonnegative Geometric Upper Series Controls the Smaller Series"],"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-20261015-advance-20261117-r01-series-comparison-test-0021","batch_id":"batch-20261015-advance-20261117-r01-series-comparison-test-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ebd2f363a1f740844f5fb74d79b5d8fe0c950f86655c159e13240fc9f9f361b6","family_id":"qf-mod48-series-upper-geometric","seed":202611171021,"semantic_template":"series-upper-geometric|{a_scale:[21,1],b_scale:[22,1],ratio:[1,7]}"},"fields":{"title":"A Convergent Nonnegative Geometric Upper Series Controls the Smaller Series","stem":"Use exact scales A=21,B=22, rational ratio r=1/7. For n≥0 use a_n=A*(r)^n,b_n=B*(r)^n, with0≤A≤B and0≤r<1. Give (a_scale, b_scale, ratio, sum_a, sum_b, sum_comparison, certificate).","answer":{"kind":"expression","value":"(21, 22, 1/7, 49/2, 77/3, Yes, upper_comparison_proves_convergence)","explanation":"Compute both exact sums and compare them. The first term has exponent0, including r=0. The full nonnegative comparison and convergent reference are premises; finite samples do not prove the general theorem. The exact result, retaining the requested order, is (21, 22, 1/7, 49/2, 77/3, Yes, upper_comparison_proves_convergence)."},"skills":["A Convergent Nonnegative Geometric Upper Series Controls the Smaller Series"],"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-20261015-advance-20261117-r01-series-comparison-test-0022","batch_id":"batch-20261015-advance-20261117-r01-series-comparison-test-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"121dedfc62203c0f65bce9a62a4c2a618f77ad0a36338d58afe50b7a7ba858e3","family_id":"qf-mod48-series-upper-geometric","seed":202611171022,"semantic_template":"series-upper-geometric|{a_scale:[22,1],b_scale:[23,1],ratio:[1,4]}"},"fields":{"title":"A Convergent Nonnegative Geometric Upper Series Controls the Smaller Series","stem":"Use exact scales A=22,B=23, rational ratio r=2/8. For n≥0 use a_n=A*(r)^n,b_n=B*(r)^n, with0≤A≤B and0≤r<1. Give (a_scale, b_scale, ratio, sum_a, sum_b, sum_comparison, certificate).","answer":{"kind":"expression","value":"(22, 23, 1/4, 88/3, 92/3, Yes, upper_comparison_proves_convergence)","explanation":"Compute both exact sums and compare them. The first term has exponent0, including r=0. The full nonnegative comparison and convergent reference are premises; finite samples do not prove the general theorem. The exact result, retaining the requested order, is (22, 23, 1/4, 88/3, 92/3, Yes, upper_comparison_proves_convergence)."},"skills":["A Convergent Nonnegative Geometric Upper Series Controls the Smaller Series"],"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-20261015-advance-20261117-r01-series-comparison-test-0023","batch_id":"batch-20261015-advance-20261117-r01-series-comparison-test-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cd7f7553b95209b07be8f6faf53f22065ae0c41839333eb51859eee4dcf43a70","family_id":"qf-mod48-series-upper-geometric","seed":202611171023,"semantic_template":"series-upper-geometric|{a_scale:[23,1],b_scale:[24,1],ratio:[1,3]}"},"fields":{"title":"A Convergent Nonnegative Geometric Upper Series Controls the Smaller Series","stem":"Use exact scales A=23,B=24, rational ratio r=3/9. For n≥0 use a_n=A*(r)^n,b_n=B*(r)^n, with0≤A≤B and0≤r<1. Give (a_scale, b_scale, ratio, sum_a, sum_b, sum_comparison, certificate).","answer":{"kind":"expression","value":"(23, 24, 1/3, 69/2, 36, Yes, upper_comparison_proves_convergence)","explanation":"Compute both exact sums and compare them. The first term has exponent0, including r=0. The full nonnegative comparison and convergent reference are premises; finite samples do not prove the general theorem. The exact result, retaining the requested order, is (23, 24, 1/3, 69/2, 36, Yes, upper_comparison_proves_convergence)."},"skills":["A Convergent Nonnegative Geometric Upper Series Controls the Smaller Series"],"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-20261015-advance-20261117-r01-series-comparison-test-0024","batch_id":"batch-20261015-advance-20261117-r01-series-comparison-test-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0a0a54b132645c000dd6a3274a017aaa30c9ad969f4f7359f51609955df070b0","family_id":"qf-mod48-series-upper-geometric","seed":202611171024,"semantic_template":"series-upper-geometric|{a_scale:[24,1],b_scale:[25,1],ratio:[2,5]}"},"fields":{"title":"A Convergent Nonnegative Geometric Upper Series Controls the Smaller Series","stem":"Use exact scales A=24,B=25, rational ratio r=4/10. For n≥0 use a_n=A*(r)^n,b_n=B*(r)^n, with0≤A≤B and0≤r<1. Give (a_scale, b_scale, ratio, sum_a, sum_b, sum_comparison, certificate).","answer":{"kind":"expression","value":"(24, 25, 2/5, 40, 125/3, Yes, upper_comparison_proves_convergence)","explanation":"Compute both exact sums and compare them. The first term has exponent0, including r=0. The full nonnegative comparison and convergent reference are premises; finite samples do not prove the general theorem. The exact result, retaining the requested order, is (24, 25, 2/5, 40, 125/3, Yes, upper_comparison_proves_convergence)."},"skills":["A Convergent Nonnegative Geometric Upper Series Controls the Smaller Series"],"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-20261015-advance-20261117-r01-series-comparison-test-0025","batch_id":"batch-20261015-advance-20261117-r01-series-comparison-test-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"40bf8050746ad3c9d8f8cff4c01e3467ca483a4e1eeaebf17dc4e86a19368abe","family_id":"qf-mod48-series-upper-geometric","seed":202611171025,"semantic_template":"series-upper-geometric|{a_scale:[25,1],b_scale:[26,1],ratio:[5,11]}"},"fields":{"title":"A Convergent Nonnegative Geometric Upper Series Controls the Smaller Series","stem":"Use exact scales A=25,B=26, rational ratio r=5/11. For n≥0 use a_n=A*(r)^n,b_n=B*(r)^n, with0≤A≤B and0≤r<1. Give (a_scale, b_scale, ratio, sum_a, sum_b, sum_comparison, certificate).","answer":{"kind":"expression","value":"(25, 26, 5/11, 275/6, 143/3, Yes, upper_comparison_proves_convergence)","explanation":"Compute both exact sums and compare them. The first term has exponent0, including r=0. The full nonnegative comparison and convergent reference are premises; finite samples do not prove the general theorem. The exact result, retaining the requested order, is (25, 26, 5/11, 275/6, 143/3, Yes, upper_comparison_proves_convergence)."},"skills":["A Convergent Nonnegative Geometric Upper Series Controls the Smaller Series"],"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-20261015-advance-20261117-r01-series-comparison-test-0026","batch_id":"batch-20261015-advance-20261117-r01-series-comparison-test-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3513c59b0e77cfaee050dc9ae4bfe0bdac54aa52b3af3764f9ed6d7e0f6f0b56","family_id":"qf-mod48-series-lower-divergence","seed":202611171026,"semantic_template":"series-lower-divergence|{a_scale:[2,1],b_scale:[1,1]}"},"fields":{"title":"A Positive Constant Lower Bound Gives Unbounded Partial Sums","stem":"Use exact scales A=2,B=1. This family uses a_n=A≥b_n=B>0 for every n, rather than reversing the conclusion of an upper comparison. Give (a_scale, b_scale, term_lower_bound, unbounded_partial_sum_statement, divergence_certificate, certificate).","answer":{"kind":"expression","value":"(2, 1, a_n>=b_n>0_for_all_n, partial_sum_b=N*b_unbounded, Yes, lower_comparison_proves_divergence)","explanation":"The reference partial sums with N terms equal N*B and are unbounded. The larger series therefore diverges. Preserve the lower-bound direction and do not infer divergence from a divergent upper bound. The exact result, retaining the requested order, is (2, 1, a_n>=b_n>0_for_all_n, partial_sum_b=N*b_unbounded, Yes, lower_comparison_proves_divergence)."},"skills":["A Positive Constant Lower Bound Gives Unbounded Partial Sums"],"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-20261015-advance-20261117-r01-series-comparison-test-0027","batch_id":"batch-20261015-advance-20261117-r01-series-comparison-test-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5e4a3ca5da90d9aed931609b1a4ef118e5212c99637a49bea35ecabc2e12e557","family_id":"qf-mod48-series-lower-divergence","seed":202611171027,"semantic_template":"series-lower-divergence|{a_scale:[4,1],b_scale:[2,1]}"},"fields":{"title":"A Positive Constant Lower Bound Gives Unbounded Partial Sums","stem":"Use exact scales A=4,B=2. This family uses a_n=A≥b_n=B>0 for every n, rather than reversing the conclusion of an upper comparison. Give (a_scale, b_scale, term_lower_bound, unbounded_partial_sum_statement, divergence_certificate, certificate).","answer":{"kind":"expression","value":"(4, 2, a_n>=b_n>0_for_all_n, partial_sum_b=N*b_unbounded, Yes, lower_comparison_proves_divergence)","explanation":"The reference partial sums with N terms equal N*B and are unbounded. The larger series therefore diverges. Preserve the lower-bound direction and do not infer divergence from a divergent upper bound. The exact result, retaining the requested order, is (4, 2, a_n>=b_n>0_for_all_n, partial_sum_b=N*b_unbounded, Yes, lower_comparison_proves_divergence)."},"skills":["A Positive Constant Lower Bound Gives Unbounded Partial Sums"],"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-20261015-advance-20261117-r01-series-comparison-test-0028","batch_id":"batch-20261015-advance-20261117-r01-series-comparison-test-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6b418a8575b79dea386136edbf3c065b50675c56f9b4da38d4d547d983b48123","family_id":"qf-mod48-series-lower-divergence","seed":202611171028,"semantic_template":"series-lower-divergence|{a_scale:[6,1],b_scale:[3,1]}"},"fields":{"title":"A Positive Constant Lower Bound Gives Unbounded Partial Sums","stem":"Use exact scales A=6,B=3. This family uses a_n=A≥b_n=B>0 for every n, rather than reversing the conclusion of an upper comparison. Give (a_scale, b_scale, term_lower_bound, unbounded_partial_sum_statement, divergence_certificate, certificate).","answer":{"kind":"expression","value":"(6, 3, a_n>=b_n>0_for_all_n, partial_sum_b=N*b_unbounded, Yes, lower_comparison_proves_divergence)","explanation":"The reference partial sums with N terms equal N*B and are unbounded. The larger series therefore diverges. Preserve the lower-bound direction and do not infer divergence from a divergent upper bound. The exact result, retaining the requested order, is (6, 3, a_n>=b_n>0_for_all_n, partial_sum_b=N*b_unbounded, Yes, lower_comparison_proves_divergence)."},"skills":["A Positive Constant Lower Bound Gives Unbounded Partial Sums"],"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-20261015-advance-20261117-r01-series-comparison-test-0029","batch_id":"batch-20261015-advance-20261117-r01-series-comparison-test-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f43f99e8f27ba1f8aa651cbff189d2f4a517fccdd57ec364f80963f8cbabe0a8","family_id":"qf-mod48-series-lower-divergence","seed":202611171029,"semantic_template":"series-lower-divergence|{a_scale:[8,1],b_scale:[4,1]}"},"fields":{"title":"A Positive Constant Lower Bound Gives Unbounded Partial Sums","stem":"Use exact scales A=8,B=4. This family uses a_n=A≥b_n=B>0 for every n, rather than reversing the conclusion of an upper comparison. Give (a_scale, b_scale, term_lower_bound, unbounded_partial_sum_statement, divergence_certificate, certificate).","answer":{"kind":"expression","value":"(8, 4, a_n>=b_n>0_for_all_n, partial_sum_b=N*b_unbounded, Yes, lower_comparison_proves_divergence)","explanation":"The reference partial sums with N terms equal N*B and are unbounded. The larger series therefore diverges. Preserve the lower-bound direction and do not infer divergence from a divergent upper bound. The exact result, retaining the requested order, is (8, 4, a_n>=b_n>0_for_all_n, partial_sum_b=N*b_unbounded, Yes, lower_comparison_proves_divergence)."},"skills":["A Positive Constant Lower Bound Gives Unbounded Partial Sums"],"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-20261015-advance-20261117-r01-series-comparison-test-0030","batch_id":"batch-20261015-advance-20261117-r01-series-comparison-test-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cf39aef48c0090191bb6b2ed9dee1582b70d394348a774499efec1e696ea8424","family_id":"qf-mod48-series-lower-divergence","seed":202611171030,"semantic_template":"series-lower-divergence|{a_scale:[10,1],b_scale:[5,1]}"},"fields":{"title":"A Positive Constant Lower Bound Gives Unbounded Partial Sums","stem":"Use exact scales A=10,B=5. This family uses a_n=A≥b_n=B>0 for every n, rather than reversing the conclusion of an upper comparison. Give (a_scale, b_scale, term_lower_bound, unbounded_partial_sum_statement, divergence_certificate, certificate).","answer":{"kind":"expression","value":"(10, 5, a_n>=b_n>0_for_all_n, partial_sum_b=N*b_unbounded, Yes, lower_comparison_proves_divergence)","explanation":"The reference partial sums with N terms equal N*B and are unbounded. The larger series therefore diverges. Preserve the lower-bound direction and do not infer divergence from a divergent upper bound. The exact result, retaining the requested order, is (10, 5, a_n>=b_n>0_for_all_n, partial_sum_b=N*b_unbounded, Yes, lower_comparison_proves_divergence)."},"skills":["A Positive Constant Lower Bound Gives Unbounded Partial Sums"],"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-20261015-advance-20261117-r01-series-comparison-test-0031","batch_id":"batch-20261015-advance-20261117-r01-series-comparison-test-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2a405b0a3ceae7de788c239dec272880d908cedc6595061bf5775e4f31acc635","family_id":"qf-mod48-series-lower-divergence","seed":202611171031,"semantic_template":"series-lower-divergence|{a_scale:[12,1],b_scale:[6,1]}"},"fields":{"title":"A Positive Constant Lower Bound Gives Unbounded Partial Sums","stem":"Use exact scales A=12,B=6. This family uses a_n=A≥b_n=B>0 for every n, rather than reversing the conclusion of an upper comparison. Give (a_scale, b_scale, term_lower_bound, unbounded_partial_sum_statement, divergence_certificate, certificate).","answer":{"kind":"expression","value":"(12, 6, a_n>=b_n>0_for_all_n, partial_sum_b=N*b_unbounded, Yes, lower_comparison_proves_divergence)","explanation":"The reference partial sums with N terms equal N*B and are unbounded. The larger series therefore diverges. Preserve the lower-bound direction and do not infer divergence from a divergent upper bound. The exact result, retaining the requested order, is (12, 6, a_n>=b_n>0_for_all_n, partial_sum_b=N*b_unbounded, Yes, lower_comparison_proves_divergence)."},"skills":["A Positive Constant Lower Bound Gives Unbounded Partial Sums"],"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-20261015-advance-20261117-r01-series-comparison-test-0032","batch_id":"batch-20261015-advance-20261117-r01-series-comparison-test-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"304b72ab84fa48d3408f89ce928f0e5f62861ca09b2d0bcc689da98e8765be62","family_id":"qf-mod48-series-lower-divergence","seed":202611171032,"semantic_template":"series-lower-divergence|{a_scale:[14,1],b_scale:[7,1]}"},"fields":{"title":"A Positive Constant Lower Bound Gives Unbounded Partial Sums","stem":"Use exact scales A=14,B=7. This family uses a_n=A≥b_n=B>0 for every n, rather than reversing the conclusion of an upper comparison. Give (a_scale, b_scale, term_lower_bound, unbounded_partial_sum_statement, divergence_certificate, certificate).","answer":{"kind":"expression","value":"(14, 7, a_n>=b_n>0_for_all_n, partial_sum_b=N*b_unbounded, Yes, lower_comparison_proves_divergence)","explanation":"The reference partial sums with N terms equal N*B and are unbounded. The larger series therefore diverges. Preserve the lower-bound direction and do not infer divergence from a divergent upper bound. The exact result, retaining the requested order, is (14, 7, a_n>=b_n>0_for_all_n, partial_sum_b=N*b_unbounded, Yes, lower_comparison_proves_divergence)."},"skills":["A Positive Constant Lower Bound Gives Unbounded Partial Sums"],"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-20261015-advance-20261117-r01-series-comparison-test-0033","batch_id":"batch-20261015-advance-20261117-r01-series-comparison-test-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7e42aeef91350cb26343f19b52a2b8c1a172126a2b038c5d1ea7497c9a4f1e53","family_id":"qf-mod48-series-lower-divergence","seed":202611171033,"semantic_template":"series-lower-divergence|{a_scale:[16,1],b_scale:[8,1]}"},"fields":{"title":"A Positive Constant Lower Bound Gives Unbounded Partial Sums","stem":"Use exact scales A=16,B=8. This family uses a_n=A≥b_n=B>0 for every n, rather than reversing the conclusion of an upper comparison. Give (a_scale, b_scale, term_lower_bound, unbounded_partial_sum_statement, divergence_certificate, certificate).","answer":{"kind":"expression","value":"(16, 8, a_n>=b_n>0_for_all_n, partial_sum_b=N*b_unbounded, Yes, lower_comparison_proves_divergence)","explanation":"The reference partial sums with N terms equal N*B and are unbounded. The larger series therefore diverges. Preserve the lower-bound direction and do not infer divergence from a divergent upper bound. The exact result, retaining the requested order, is (16, 8, a_n>=b_n>0_for_all_n, partial_sum_b=N*b_unbounded, Yes, lower_comparison_proves_divergence)."},"skills":["A Positive Constant Lower Bound Gives Unbounded Partial Sums"],"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-20261015-advance-20261117-r01-series-comparison-test-0034","batch_id":"batch-20261015-advance-20261117-r01-series-comparison-test-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c0e09493de0d2e0ce5fc389b54375266029205dcdcceb9420949c8e10fa93e05","family_id":"qf-mod48-series-lower-divergence","seed":202611171034,"semantic_template":"series-lower-divergence|{a_scale:[18,1],b_scale:[9,1]}"},"fields":{"title":"A Positive Constant Lower Bound Gives Unbounded Partial Sums","stem":"Use exact scales A=18,B=9. This family uses a_n=A≥b_n=B>0 for every n, rather than reversing the conclusion of an upper comparison. Give (a_scale, b_scale, term_lower_bound, unbounded_partial_sum_statement, divergence_certificate, certificate).","answer":{"kind":"expression","value":"(18, 9, a_n>=b_n>0_for_all_n, partial_sum_b=N*b_unbounded, Yes, lower_comparison_proves_divergence)","explanation":"The reference partial sums with N terms equal N*B and are unbounded. The larger series therefore diverges. Preserve the lower-bound direction and do not infer divergence from a divergent upper bound. The exact result, retaining the requested order, is (18, 9, a_n>=b_n>0_for_all_n, partial_sum_b=N*b_unbounded, Yes, lower_comparison_proves_divergence)."},"skills":["A Positive Constant Lower Bound Gives Unbounded Partial Sums"],"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-20261015-advance-20261117-r01-series-comparison-test-0035","batch_id":"batch-20261015-advance-20261117-r01-series-comparison-test-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"78ad6b2a47ca12cdfbcce8facd5708fb36ab2a5faf98f73811719f26372d0ad1","family_id":"qf-mod48-series-lower-divergence","seed":202611171035,"semantic_template":"series-lower-divergence|{a_scale:[20,1],b_scale:[10,1]}"},"fields":{"title":"A Positive Constant Lower Bound Gives Unbounded Partial Sums","stem":"Use exact scales A=20,B=10. This family uses a_n=A≥b_n=B>0 for every n, rather than reversing the conclusion of an upper comparison. Give (a_scale, b_scale, term_lower_bound, unbounded_partial_sum_statement, divergence_certificate, certificate).","answer":{"kind":"expression","value":"(20, 10, a_n>=b_n>0_for_all_n, partial_sum_b=N*b_unbounded, Yes, lower_comparison_proves_divergence)","explanation":"The reference partial sums with N terms equal N*B and are unbounded. The larger series therefore diverges. Preserve the lower-bound direction and do not infer divergence from a divergent upper bound. The exact result, retaining the requested order, is (20, 10, a_n>=b_n>0_for_all_n, partial_sum_b=N*b_unbounded, Yes, lower_comparison_proves_divergence)."},"skills":["A Positive Constant Lower Bound Gives Unbounded Partial Sums"],"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-20261015-advance-20261117-r01-series-comparison-test-0036","batch_id":"batch-20261015-advance-20261117-r01-series-comparison-test-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f759bb20dfd8203eb823109cfb64c167eb0cd085e6c4c3e0adbae0e8e77df8a5","family_id":"qf-mod48-series-lower-divergence","seed":202611171036,"semantic_template":"series-lower-divergence|{a_scale:[22,1],b_scale:[11,1]}"},"fields":{"title":"A Positive Constant Lower Bound Gives Unbounded Partial Sums","stem":"Use exact scales A=22,B=11. This family uses a_n=A≥b_n=B>0 for every n, rather than reversing the conclusion of an upper comparison. Give (a_scale, b_scale, term_lower_bound, unbounded_partial_sum_statement, divergence_certificate, certificate).","answer":{"kind":"expression","value":"(22, 11, a_n>=b_n>0_for_all_n, partial_sum_b=N*b_unbounded, Yes, lower_comparison_proves_divergence)","explanation":"The reference partial sums with N terms equal N*B and are unbounded. The larger series therefore diverges. Preserve the lower-bound direction and do not infer divergence from a divergent upper bound. The exact result, retaining the requested order, is (22, 11, a_n>=b_n>0_for_all_n, partial_sum_b=N*b_unbounded, Yes, lower_comparison_proves_divergence)."},"skills":["A Positive Constant Lower Bound Gives Unbounded Partial Sums"],"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-20261015-advance-20261117-r01-series-comparison-test-0037","batch_id":"batch-20261015-advance-20261117-r01-series-comparison-test-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"174050782839516a9532f101a7bdb94445b8e9b33d51b9256283fdbfe6feaf21","family_id":"qf-mod48-series-lower-divergence","seed":202611171037,"semantic_template":"series-lower-divergence|{a_scale:[24,1],b_scale:[12,1]}"},"fields":{"title":"A Positive Constant Lower Bound Gives Unbounded Partial Sums","stem":"Use exact scales A=24,B=12. This family uses a_n=A≥b_n=B>0 for every n, rather than reversing the conclusion of an upper comparison. Give (a_scale, b_scale, term_lower_bound, unbounded_partial_sum_statement, divergence_certificate, certificate).","answer":{"kind":"expression","value":"(24, 12, a_n>=b_n>0_for_all_n, partial_sum_b=N*b_unbounded, Yes, lower_comparison_proves_divergence)","explanation":"The reference partial sums with N terms equal N*B and are unbounded. The larger series therefore diverges. Preserve the lower-bound direction and do not infer divergence from a divergent upper bound. The exact result, retaining the requested order, is (24, 12, a_n>=b_n>0_for_all_n, partial_sum_b=N*b_unbounded, Yes, lower_comparison_proves_divergence)."},"skills":["A Positive Constant Lower Bound Gives Unbounded Partial Sums"],"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-20261015-advance-20261117-r01-series-comparison-test-0038","batch_id":"batch-20261015-advance-20261117-r01-series-comparison-test-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8d64da15f2755690ecc0748c6c3878e76da74e77f252e2e40c9b16b38aa2ccf7","family_id":"qf-mod48-series-lower-divergence","seed":202611171038,"semantic_template":"series-lower-divergence|{a_scale:[26,1],b_scale:[13,1]}"},"fields":{"title":"A Positive Constant Lower Bound Gives Unbounded Partial Sums","stem":"Use exact scales A=26,B=13. This family uses a_n=A≥b_n=B>0 for every n, rather than reversing the conclusion of an upper comparison. Give (a_scale, b_scale, term_lower_bound, unbounded_partial_sum_statement, divergence_certificate, certificate).","answer":{"kind":"expression","value":"(26, 13, a_n>=b_n>0_for_all_n, partial_sum_b=N*b_unbounded, Yes, lower_comparison_proves_divergence)","explanation":"The reference partial sums with N terms equal N*B and are unbounded. The larger series therefore diverges. Preserve the lower-bound direction and do not infer divergence from a divergent upper bound. The exact result, retaining the requested order, is (26, 13, a_n>=b_n>0_for_all_n, partial_sum_b=N*b_unbounded, Yes, lower_comparison_proves_divergence)."},"skills":["A Positive Constant Lower Bound Gives Unbounded Partial Sums"],"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-20261015-advance-20261117-r01-series-comparison-test-0039","batch_id":"batch-20261015-advance-20261117-r01-series-comparison-test-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"acc072ba28b75b9b3ec260fa5715a7caf8cd8399dd329bd719f771dad4a26100","family_id":"qf-mod48-series-lower-divergence","seed":202611171039,"semantic_template":"series-lower-divergence|{a_scale:[28,1],b_scale:[14,1]}"},"fields":{"title":"A Positive Constant Lower Bound Gives Unbounded Partial Sums","stem":"Use exact scales A=28,B=14. This family uses a_n=A≥b_n=B>0 for every n, rather than reversing the conclusion of an upper comparison. Give (a_scale, b_scale, term_lower_bound, unbounded_partial_sum_statement, divergence_certificate, certificate).","answer":{"kind":"expression","value":"(28, 14, a_n>=b_n>0_for_all_n, partial_sum_b=N*b_unbounded, Yes, lower_comparison_proves_divergence)","explanation":"The reference partial sums with N terms equal N*B and are unbounded. The larger series therefore diverges. Preserve the lower-bound direction and do not infer divergence from a divergent upper bound. The exact result, retaining the requested order, is (28, 14, a_n>=b_n>0_for_all_n, partial_sum_b=N*b_unbounded, Yes, lower_comparison_proves_divergence)."},"skills":["A Positive Constant Lower Bound Gives Unbounded Partial Sums"],"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-20261015-advance-20261117-r01-series-comparison-test-0040","batch_id":"batch-20261015-advance-20261117-r01-series-comparison-test-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1c14661a7c9aa97ed302c730a5c108904c73650b10b1e5d5c5fed958241779e7","family_id":"qf-mod48-series-lower-divergence","seed":202611171040,"semantic_template":"series-lower-divergence|{a_scale:[30,1],b_scale:[15,1]}"},"fields":{"title":"A Positive Constant Lower Bound Gives Unbounded Partial Sums","stem":"Use exact scales A=30,B=15. This family uses a_n=A≥b_n=B>0 for every n, rather than reversing the conclusion of an upper comparison. Give (a_scale, b_scale, term_lower_bound, unbounded_partial_sum_statement, divergence_certificate, certificate).","answer":{"kind":"expression","value":"(30, 15, a_n>=b_n>0_for_all_n, partial_sum_b=N*b_unbounded, Yes, lower_comparison_proves_divergence)","explanation":"The reference partial sums with N terms equal N*B and are unbounded. The larger series therefore diverges. Preserve the lower-bound direction and do not infer divergence from a divergent upper bound. The exact result, retaining the requested order, is (30, 15, a_n>=b_n>0_for_all_n, partial_sum_b=N*b_unbounded, Yes, lower_comparison_proves_divergence)."},"skills":["A Positive Constant Lower Bound Gives Unbounded Partial Sums"],"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-20261015-advance-20261117-r01-series-comparison-test-0041","batch_id":"batch-20261015-advance-20261117-r01-series-comparison-test-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"57bbde1b1e1d0bb143811a431b16f5d35813ad4b6e8eec80fa4b32894eaf803d","family_id":"qf-mod48-series-lower-divergence","seed":202611171041,"semantic_template":"series-lower-divergence|{a_scale:[32,1],b_scale:[16,1]}"},"fields":{"title":"A Positive Constant Lower Bound Gives Unbounded Partial Sums","stem":"Use exact scales A=32,B=16. This family uses a_n=A≥b_n=B>0 for every n, rather than reversing the conclusion of an upper comparison. Give (a_scale, b_scale, term_lower_bound, unbounded_partial_sum_statement, divergence_certificate, certificate).","answer":{"kind":"expression","value":"(32, 16, a_n>=b_n>0_for_all_n, partial_sum_b=N*b_unbounded, Yes, lower_comparison_proves_divergence)","explanation":"The reference partial sums with N terms equal N*B and are unbounded. The larger series therefore diverges. Preserve the lower-bound direction and do not infer divergence from a divergent upper bound. The exact result, retaining the requested order, is (32, 16, a_n>=b_n>0_for_all_n, partial_sum_b=N*b_unbounded, Yes, lower_comparison_proves_divergence)."},"skills":["A Positive Constant Lower Bound Gives Unbounded Partial Sums"],"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-20261015-advance-20261117-r01-series-comparison-test-0042","batch_id":"batch-20261015-advance-20261117-r01-series-comparison-test-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"825b3ae04e3a21599c6eb8111b141d5994a703e4bc8987c7e817f21f78555ce7","family_id":"qf-mod48-series-lower-divergence","seed":202611171042,"semantic_template":"series-lower-divergence|{a_scale:[34,1],b_scale:[17,1]}"},"fields":{"title":"A Positive Constant Lower Bound Gives Unbounded Partial Sums","stem":"Use exact scales A=34,B=17. This family uses a_n=A≥b_n=B>0 for every n, rather than reversing the conclusion of an upper comparison. Give (a_scale, b_scale, term_lower_bound, unbounded_partial_sum_statement, divergence_certificate, certificate).","answer":{"kind":"expression","value":"(34, 17, a_n>=b_n>0_for_all_n, partial_sum_b=N*b_unbounded, Yes, lower_comparison_proves_divergence)","explanation":"The reference partial sums with N terms equal N*B and are unbounded. The larger series therefore diverges. Preserve the lower-bound direction and do not infer divergence from a divergent upper bound. The exact result, retaining the requested order, is (34, 17, a_n>=b_n>0_for_all_n, partial_sum_b=N*b_unbounded, Yes, lower_comparison_proves_divergence)."},"skills":["A Positive Constant Lower Bound Gives Unbounded Partial Sums"],"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-20261015-advance-20261117-r01-series-comparison-test-0043","batch_id":"batch-20261015-advance-20261117-r01-series-comparison-test-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e6063696ce8ce2e344250e5fd53f2bd9a0321dd0b9c881f78bdcd6a2a72aa931","family_id":"qf-mod48-series-lower-divergence","seed":202611171043,"semantic_template":"series-lower-divergence|{a_scale:[36,1],b_scale:[18,1]}"},"fields":{"title":"A Positive Constant Lower Bound Gives Unbounded Partial Sums","stem":"Use exact scales A=36,B=18. This family uses a_n=A≥b_n=B>0 for every n, rather than reversing the conclusion of an upper comparison. Give (a_scale, b_scale, term_lower_bound, unbounded_partial_sum_statement, divergence_certificate, certificate).","answer":{"kind":"expression","value":"(36, 18, a_n>=b_n>0_for_all_n, partial_sum_b=N*b_unbounded, Yes, lower_comparison_proves_divergence)","explanation":"The reference partial sums with N terms equal N*B and are unbounded. The larger series therefore diverges. Preserve the lower-bound direction and do not infer divergence from a divergent upper bound. The exact result, retaining the requested order, is (36, 18, a_n>=b_n>0_for_all_n, partial_sum_b=N*b_unbounded, Yes, lower_comparison_proves_divergence)."},"skills":["A Positive Constant Lower Bound Gives Unbounded Partial Sums"],"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-20261015-advance-20261117-r01-series-comparison-test-0044","batch_id":"batch-20261015-advance-20261117-r01-series-comparison-test-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"76fd743f95e5335bf1546cd473a55961c7593e10f2e90cf997a6f8e75d4df812","family_id":"qf-mod48-series-lower-divergence","seed":202611171044,"semantic_template":"series-lower-divergence|{a_scale:[38,1],b_scale:[19,1]}"},"fields":{"title":"A Positive Constant Lower Bound Gives Unbounded Partial Sums","stem":"Use exact scales A=38,B=19. This family uses a_n=A≥b_n=B>0 for every n, rather than reversing the conclusion of an upper comparison. Give (a_scale, b_scale, term_lower_bound, unbounded_partial_sum_statement, divergence_certificate, certificate).","answer":{"kind":"expression","value":"(38, 19, a_n>=b_n>0_for_all_n, partial_sum_b=N*b_unbounded, Yes, lower_comparison_proves_divergence)","explanation":"The reference partial sums with N terms equal N*B and are unbounded. The larger series therefore diverges. Preserve the lower-bound direction and do not infer divergence from a divergent upper bound. The exact result, retaining the requested order, is (38, 19, a_n>=b_n>0_for_all_n, partial_sum_b=N*b_unbounded, Yes, lower_comparison_proves_divergence)."},"skills":["A Positive Constant Lower Bound Gives Unbounded Partial Sums"],"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-20261015-advance-20261117-r01-series-comparison-test-0045","batch_id":"batch-20261015-advance-20261117-r01-series-comparison-test-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"daaaa06a4046c190076df90440ecadb6937f3a5084e4c1bedced65520eec2e0a","family_id":"qf-mod48-series-lower-divergence","seed":202611171045,"semantic_template":"series-lower-divergence|{a_scale:[40,1],b_scale:[20,1]}"},"fields":{"title":"A Positive Constant Lower Bound Gives Unbounded Partial Sums","stem":"Use exact scales A=40,B=20. This family uses a_n=A≥b_n=B>0 for every n, rather than reversing the conclusion of an upper comparison. Give (a_scale, b_scale, term_lower_bound, unbounded_partial_sum_statement, divergence_certificate, certificate).","answer":{"kind":"expression","value":"(40, 20, a_n>=b_n>0_for_all_n, partial_sum_b=N*b_unbounded, Yes, lower_comparison_proves_divergence)","explanation":"The reference partial sums with N terms equal N*B and are unbounded. The larger series therefore diverges. Preserve the lower-bound direction and do not infer divergence from a divergent upper bound. The exact result, retaining the requested order, is (40, 20, a_n>=b_n>0_for_all_n, partial_sum_b=N*b_unbounded, Yes, lower_comparison_proves_divergence)."},"skills":["A Positive Constant Lower Bound Gives Unbounded Partial Sums"],"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-20261015-advance-20261117-r01-series-comparison-test-0046","batch_id":"batch-20261015-advance-20261117-r01-series-comparison-test-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"174f3da0581724e17e535c5540d72649e723ea0542cbc46198edbf37d7ac4f44","family_id":"qf-mod48-series-lower-divergence","seed":202611171046,"semantic_template":"series-lower-divergence|{a_scale:[42,1],b_scale:[21,1]}"},"fields":{"title":"A Positive Constant Lower Bound Gives Unbounded Partial Sums","stem":"Use exact scales A=42,B=21. This family uses a_n=A≥b_n=B>0 for every n, rather than reversing the conclusion of an upper comparison. Give (a_scale, b_scale, term_lower_bound, unbounded_partial_sum_statement, divergence_certificate, certificate).","answer":{"kind":"expression","value":"(42, 21, a_n>=b_n>0_for_all_n, partial_sum_b=N*b_unbounded, Yes, lower_comparison_proves_divergence)","explanation":"The reference partial sums with N terms equal N*B and are unbounded. The larger series therefore diverges. Preserve the lower-bound direction and do not infer divergence from a divergent upper bound. The exact result, retaining the requested order, is (42, 21, a_n>=b_n>0_for_all_n, partial_sum_b=N*b_unbounded, Yes, lower_comparison_proves_divergence)."},"skills":["A Positive Constant Lower Bound Gives Unbounded Partial Sums"],"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-20261015-advance-20261117-r01-series-comparison-test-0047","batch_id":"batch-20261015-advance-20261117-r01-series-comparison-test-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"33c99e17b4cb94755f93dccef23552cc0382183b1863d30dc27dd4845e1562e3","family_id":"qf-mod48-series-lower-divergence","seed":202611171047,"semantic_template":"series-lower-divergence|{a_scale:[44,1],b_scale:[22,1]}"},"fields":{"title":"A Positive Constant Lower Bound Gives Unbounded Partial Sums","stem":"Use exact scales A=44,B=22. This family uses a_n=A≥b_n=B>0 for every n, rather than reversing the conclusion of an upper comparison. Give (a_scale, b_scale, term_lower_bound, unbounded_partial_sum_statement, divergence_certificate, certificate).","answer":{"kind":"expression","value":"(44, 22, a_n>=b_n>0_for_all_n, partial_sum_b=N*b_unbounded, Yes, lower_comparison_proves_divergence)","explanation":"The reference partial sums with N terms equal N*B and are unbounded. The larger series therefore diverges. Preserve the lower-bound direction and do not infer divergence from a divergent upper bound. The exact result, retaining the requested order, is (44, 22, a_n>=b_n>0_for_all_n, partial_sum_b=N*b_unbounded, Yes, lower_comparison_proves_divergence)."},"skills":["A Positive Constant Lower Bound Gives Unbounded Partial Sums"],"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-20261015-advance-20261117-r01-series-comparison-test-0048","batch_id":"batch-20261015-advance-20261117-r01-series-comparison-test-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"892896a68b2c6ab81f5e667e20747d9f4bc4ef59f9fb495d75663805db824b5e","family_id":"qf-mod48-series-lower-divergence","seed":202611171048,"semantic_template":"series-lower-divergence|{a_scale:[46,1],b_scale:[23,1]}"},"fields":{"title":"A Positive Constant Lower Bound Gives Unbounded Partial Sums","stem":"Use exact scales A=46,B=23. This family uses a_n=A≥b_n=B>0 for every n, rather than reversing the conclusion of an upper comparison. Give (a_scale, b_scale, term_lower_bound, unbounded_partial_sum_statement, divergence_certificate, certificate).","answer":{"kind":"expression","value":"(46, 23, a_n>=b_n>0_for_all_n, partial_sum_b=N*b_unbounded, Yes, lower_comparison_proves_divergence)","explanation":"The reference partial sums with N terms equal N*B and are unbounded. The larger series therefore diverges. Preserve the lower-bound direction and do not infer divergence from a divergent upper bound. The exact result, retaining the requested order, is (46, 23, a_n>=b_n>0_for_all_n, partial_sum_b=N*b_unbounded, Yes, lower_comparison_proves_divergence)."},"skills":["A Positive Constant Lower Bound Gives Unbounded Partial Sums"],"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-20261015-advance-20261117-r01-series-comparison-test-0049","batch_id":"batch-20261015-advance-20261117-r01-series-comparison-test-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fc3599a07d1de5e7159d8334861898e34b27364a80683ef0e0973fc87b5fe274","family_id":"qf-mod48-series-lower-divergence","seed":202611171049,"semantic_template":"series-lower-divergence|{a_scale:[48,1],b_scale:[24,1]}"},"fields":{"title":"A Positive Constant Lower Bound Gives Unbounded Partial Sums","stem":"Use exact scales A=48,B=24. This family uses a_n=A≥b_n=B>0 for every n, rather than reversing the conclusion of an upper comparison. Give (a_scale, b_scale, term_lower_bound, unbounded_partial_sum_statement, divergence_certificate, certificate).","answer":{"kind":"expression","value":"(48, 24, a_n>=b_n>0_for_all_n, partial_sum_b=N*b_unbounded, Yes, lower_comparison_proves_divergence)","explanation":"The reference partial sums with N terms equal N*B and are unbounded. The larger series therefore diverges. Preserve the lower-bound direction and do not infer divergence from a divergent upper bound. The exact result, retaining the requested order, is (48, 24, a_n>=b_n>0_for_all_n, partial_sum_b=N*b_unbounded, Yes, lower_comparison_proves_divergence)."},"skills":["A Positive Constant Lower Bound Gives Unbounded Partial Sums"],"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-20261015-advance-20261117-r01-series-comparison-test-0050","batch_id":"batch-20261015-advance-20261117-r01-series-comparison-test-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c10995d64c98f5d7c4e7da9999f82c641cbe15b3b24b32f17ee50b8f32b21e0c","family_id":"qf-mod48-series-lower-divergence","seed":202611171050,"semantic_template":"series-lower-divergence|{a_scale:[50,1],b_scale:[25,1]}"},"fields":{"title":"A Positive Constant Lower Bound Gives Unbounded Partial Sums","stem":"Use exact scales A=50,B=25. This family uses a_n=A≥b_n=B>0 for every n, rather than reversing the conclusion of an upper comparison. Give (a_scale, b_scale, term_lower_bound, unbounded_partial_sum_statement, divergence_certificate, certificate).","answer":{"kind":"expression","value":"(50, 25, a_n>=b_n>0_for_all_n, partial_sum_b=N*b_unbounded, Yes, lower_comparison_proves_divergence)","explanation":"The reference partial sums with N terms equal N*B and are unbounded. The larger series therefore diverges. Preserve the lower-bound direction and do not infer divergence from a divergent upper bound. The exact result, retaining the requested order, is (50, 25, a_n>=b_n>0_for_all_n, partial_sum_b=N*b_unbounded, Yes, lower_comparison_proves_divergence)."},"skills":["A Positive Constant Lower Bound Gives Unbounded Partial Sums"],"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-20261015-advance-20261117-r01-series-comparison-test-0051","batch_id":"batch-20261015-advance-20261117-r01-series-comparison-test-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"318e95a9e689e7b6ff68f24026ee767b116b395e9b22832b7066254dc7493197","family_id":"qf-mod48-series-eventual-tail","seed":202611171051,"semantic_template":"series-eventual-tail|{a_scale:[1,1],b_scale:[2,1],cutoff:0,prefix_a:[],prefix_b:[],ratio:[1,8]}"},"fields":{"title":"Arbitrary Finite Prefixes Do Not Change Convergence of Dominated Tails","stem":"Use exact scales A=1,B=2, rational ratio r=1/8. Prefix a=[], prefix b=[], each with cutoff=0 terms. Starting at that index use A*(r)^(n−cutoff) and B*(r)^(n−cutoff), so the first tail exponent is0. Give (cutoff, prefix_sum_a, prefix_sum_b, tail_sum_a, tail_sum_b, total_sum_a, tail_convergence, certificate).","answer":{"kind":"expression","value":"(0, 0, 0, 8/7, 16/7, 8/7, Yes, finite_prefix_does_not_change_tail_convergence)","explanation":"Add each finite prefix separately and compute both geometric tails. The tail comparison does not require the prefixes or total sums to have the same ordering. The convergence flag concerns the full valid tail, not finitely sampled terms. The exact result, retaining the requested order, is (0, 0, 0, 8/7, 16/7, 8/7, Yes, finite_prefix_does_not_change_tail_convergence)."},"skills":["Arbitrary Finite Prefixes Do Not Change Convergence of Dominated Tails"],"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-20261015-advance-20261117-r01-series-comparison-test-0052","batch_id":"batch-20261015-advance-20261117-r01-series-comparison-test-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6408a4a4454fe120105d637f38104d20b1508f09a9836d4ce9159b32aed3dc85","family_id":"qf-mod48-series-eventual-tail","seed":202611171052,"semantic_template":"series-eventual-tail|{a_scale:[2,1],b_scale:[3,1],cutoff:1,prefix_a:[[2,1]],prefix_b:[[1,1]],ratio:[2,9]}"},"fields":{"title":"Arbitrary Finite Prefixes Do Not Change Convergence of Dominated Tails","stem":"Use exact scales A=2,B=3, rational ratio r=2/9. Prefix a=[2], prefix b=[1], each with cutoff=1 terms. Starting at that index use A*(r)^(n−cutoff) and B*(r)^(n−cutoff), so the first tail exponent is0. Give (cutoff, prefix_sum_a, prefix_sum_b, tail_sum_a, tail_sum_b, total_sum_a, tail_convergence, certificate).","answer":{"kind":"expression","value":"(1, 2, 1, 18/7, 27/7, 32/7, Yes, finite_prefix_does_not_change_tail_convergence)","explanation":"Add each finite prefix separately and compute both geometric tails. The tail comparison does not require the prefixes or total sums to have the same ordering. The convergence flag concerns the full valid tail, not finitely sampled terms. The exact result, retaining the requested order, is (1, 2, 1, 18/7, 27/7, 32/7, Yes, finite_prefix_does_not_change_tail_convergence)."},"skills":["Arbitrary Finite Prefixes Do Not Change Convergence of Dominated Tails"],"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-20261015-advance-20261117-r01-series-comparison-test-0053","batch_id":"batch-20261015-advance-20261117-r01-series-comparison-test-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b4084a25655f3cb1337c3d5f64eddd2580ca670114763f0705749e5b603b211f","family_id":"qf-mod48-series-eventual-tail","seed":202611171053,"semantic_template":"series-eventual-tail|{a_scale:[3,1],b_scale:[4,1],cutoff:2,prefix_a:[[3,1],[4,1]],prefix_b:[[1,1],[2,1]],ratio:[3,10]}"},"fields":{"title":"Arbitrary Finite Prefixes Do Not Change Convergence of Dominated Tails","stem":"Use exact scales A=3,B=4, rational ratio r=3/10. Prefix a=[3, 4], prefix b=[1, 2], each with cutoff=2 terms. Starting at that index use A*(r)^(n−cutoff) and B*(r)^(n−cutoff), so the first tail exponent is0. Give (cutoff, prefix_sum_a, prefix_sum_b, tail_sum_a, tail_sum_b, total_sum_a, tail_convergence, certificate).","answer":{"kind":"expression","value":"(2, 7, 3, 30/7, 40/7, 79/7, Yes, finite_prefix_does_not_change_tail_convergence)","explanation":"Add each finite prefix separately and compute both geometric tails. The tail comparison does not require the prefixes or total sums to have the same ordering. The convergence flag concerns the full valid tail, not finitely sampled terms. The exact result, retaining the requested order, is (2, 7, 3, 30/7, 40/7, 79/7, Yes, finite_prefix_does_not_change_tail_convergence)."},"skills":["Arbitrary Finite Prefixes Do Not Change Convergence of Dominated Tails"],"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-20261015-advance-20261117-r01-series-comparison-test-0054","batch_id":"batch-20261015-advance-20261117-r01-series-comparison-test-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3c2911aadb4f0391f239ad3eb450eca88b9483ccf2e85f1843757700d39b9242","family_id":"qf-mod48-series-eventual-tail","seed":202611171054,"semantic_template":"series-eventual-tail|{a_scale:[4,1],b_scale:[5,1],cutoff:3,prefix_a:[[4,1],[5,1],[6,1]],prefix_b:[[1,1],[2,1],[3,1]],ratio:[4,11]}"},"fields":{"title":"Arbitrary Finite Prefixes Do Not Change Convergence of Dominated Tails","stem":"Use exact scales A=4,B=5, rational ratio r=4/11. Prefix a=[4, 5, 6], prefix b=[1, 2, 3], each with cutoff=3 terms. Starting at that index use A*(r)^(n−cutoff) and B*(r)^(n−cutoff), so the first tail exponent is0. Give (cutoff, prefix_sum_a, prefix_sum_b, tail_sum_a, tail_sum_b, total_sum_a, tail_convergence, certificate).","answer":{"kind":"expression","value":"(3, 15, 6, 44/7, 55/7, 149/7, Yes, finite_prefix_does_not_change_tail_convergence)","explanation":"Add each finite prefix separately and compute both geometric tails. The tail comparison does not require the prefixes or total sums to have the same ordering. The convergence flag concerns the full valid tail, not finitely sampled terms. The exact result, retaining the requested order, is (3, 15, 6, 44/7, 55/7, 149/7, Yes, finite_prefix_does_not_change_tail_convergence)."},"skills":["Arbitrary Finite Prefixes Do Not Change Convergence of Dominated Tails"],"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-20261015-advance-20261117-r01-series-comparison-test-0055","batch_id":"batch-20261015-advance-20261117-r01-series-comparison-test-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8373dae6f00d85aa439884b5526e8a4c3a6124354d01ee4e748a7dc99dbd3a5a","family_id":"qf-mod48-series-eventual-tail","seed":202611171055,"semantic_template":"series-eventual-tail|{a_scale:[5,1],b_scale:[6,1],cutoff:4,prefix_a:[[5,1],[6,1],[7,1],[8,1]],prefix_b:[[1,1],[2,1],[3,1],[4,1]],ratio:[5,12]}"},"fields":{"title":"Arbitrary Finite Prefixes Do Not Change Convergence of Dominated Tails","stem":"Use exact scales A=5,B=6, rational ratio r=5/12. Prefix a=[5, 6, 7, 8], prefix b=[1, 2, 3, 4], each with cutoff=4 terms. Starting at that index use A*(r)^(n−cutoff) and B*(r)^(n−cutoff), so the first tail exponent is0. Give (cutoff, prefix_sum_a, prefix_sum_b, tail_sum_a, tail_sum_b, total_sum_a, tail_convergence, certificate).","answer":{"kind":"expression","value":"(4, 26, 10, 60/7, 72/7, 242/7, Yes, finite_prefix_does_not_change_tail_convergence)","explanation":"Add each finite prefix separately and compute both geometric tails. The tail comparison does not require the prefixes or total sums to have the same ordering. The convergence flag concerns the full valid tail, not finitely sampled terms. The exact result, retaining the requested order, is (4, 26, 10, 60/7, 72/7, 242/7, Yes, finite_prefix_does_not_change_tail_convergence)."},"skills":["Arbitrary Finite Prefixes Do Not Change Convergence of Dominated Tails"],"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-20261015-advance-20261117-r01-series-comparison-test-0056","batch_id":"batch-20261015-advance-20261117-r01-series-comparison-test-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bf2c09fd0529972252a6add5dcffe08528909ee343bfffced08680f846e2f62a","family_id":"qf-mod48-series-eventual-tail","seed":202611171056,"semantic_template":"series-eventual-tail|{a_scale:[6,1],b_scale:[7,1],cutoff:0,prefix_a:[],prefix_b:[],ratio:[1,8]}"},"fields":{"title":"Arbitrary Finite Prefixes Do Not Change Convergence of Dominated Tails","stem":"Use exact scales A=6,B=7, rational ratio r=1/8. Prefix a=[], prefix b=[], each with cutoff=0 terms. Starting at that index use A*(r)^(n−cutoff) and B*(r)^(n−cutoff), so the first tail exponent is0. Give (cutoff, prefix_sum_a, prefix_sum_b, tail_sum_a, tail_sum_b, total_sum_a, tail_convergence, certificate).","answer":{"kind":"expression","value":"(0, 0, 0, 48/7, 8, 48/7, Yes, finite_prefix_does_not_change_tail_convergence)","explanation":"Add each finite prefix separately and compute both geometric tails. The tail comparison does not require the prefixes or total sums to have the same ordering. The convergence flag concerns the full valid tail, not finitely sampled terms. The exact result, retaining the requested order, is (0, 0, 0, 48/7, 8, 48/7, Yes, finite_prefix_does_not_change_tail_convergence)."},"skills":["Arbitrary Finite Prefixes Do Not Change Convergence of Dominated Tails"],"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-20261015-advance-20261117-r01-series-comparison-test-0057","batch_id":"batch-20261015-advance-20261117-r01-series-comparison-test-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0b03ffceb0400cedabb89f60b361f9a4ca7a3cc5d8d7eb2fac22bc2da49c24f5","family_id":"qf-mod48-series-eventual-tail","seed":202611171057,"semantic_template":"series-eventual-tail|{a_scale:[7,1],b_scale:[8,1],cutoff:1,prefix_a:[[7,1]],prefix_b:[[1,1]],ratio:[2,9]}"},"fields":{"title":"Arbitrary Finite Prefixes Do Not Change Convergence of Dominated Tails","stem":"Use exact scales A=7,B=8, rational ratio r=2/9. Prefix a=[7], prefix b=[1], each with cutoff=1 terms. Starting at that index use A*(r)^(n−cutoff) and B*(r)^(n−cutoff), so the first tail exponent is0. Give (cutoff, prefix_sum_a, prefix_sum_b, tail_sum_a, tail_sum_b, total_sum_a, tail_convergence, certificate).","answer":{"kind":"expression","value":"(1, 7, 1, 9, 72/7, 16, Yes, finite_prefix_does_not_change_tail_convergence)","explanation":"Add each finite prefix separately and compute both geometric tails. The tail comparison does not require the prefixes or total sums to have the same ordering. The convergence flag concerns the full valid tail, not finitely sampled terms. The exact result, retaining the requested order, is (1, 7, 1, 9, 72/7, 16, Yes, finite_prefix_does_not_change_tail_convergence)."},"skills":["Arbitrary Finite Prefixes Do Not Change Convergence of Dominated Tails"],"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-20261015-advance-20261117-r01-series-comparison-test-0058","batch_id":"batch-20261015-advance-20261117-r01-series-comparison-test-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ceb31c14bab170f53e2e0d190c165f1d0de729821e029717bbc681e3027e06b8","family_id":"qf-mod48-series-eventual-tail","seed":202611171058,"semantic_template":"series-eventual-tail|{a_scale:[8,1],b_scale:[9,1],cutoff:2,prefix_a:[[8,1],[9,1]],prefix_b:[[1,1],[2,1]],ratio:[3,10]}"},"fields":{"title":"Arbitrary Finite Prefixes Do Not Change Convergence of Dominated Tails","stem":"Use exact scales A=8,B=9, rational ratio r=3/10. Prefix a=[8, 9], prefix b=[1, 2], each with cutoff=2 terms. Starting at that index use A*(r)^(n−cutoff) and B*(r)^(n−cutoff), so the first tail exponent is0. Give (cutoff, prefix_sum_a, prefix_sum_b, tail_sum_a, tail_sum_b, total_sum_a, tail_convergence, certificate).","answer":{"kind":"expression","value":"(2, 17, 3, 80/7, 90/7, 199/7, Yes, finite_prefix_does_not_change_tail_convergence)","explanation":"Add each finite prefix separately and compute both geometric tails. The tail comparison does not require the prefixes or total sums to have the same ordering. The convergence flag concerns the full valid tail, not finitely sampled terms. The exact result, retaining the requested order, is (2, 17, 3, 80/7, 90/7, 199/7, Yes, finite_prefix_does_not_change_tail_convergence)."},"skills":["Arbitrary Finite Prefixes Do Not Change Convergence of Dominated Tails"],"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-20261015-advance-20261117-r01-series-comparison-test-0059","batch_id":"batch-20261015-advance-20261117-r01-series-comparison-test-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"446314c418999ea7f39048447473548f674dbe2ced96a841d8b95b384a49f4fb","family_id":"qf-mod48-series-eventual-tail","seed":202611171059,"semantic_template":"series-eventual-tail|{a_scale:[9,1],b_scale:[10,1],cutoff:3,prefix_a:[[9,1],[10,1],[11,1]],prefix_b:[[1,1],[2,1],[3,1]],ratio:[4,11]}"},"fields":{"title":"Arbitrary Finite Prefixes Do Not Change Convergence of Dominated Tails","stem":"Use exact scales A=9,B=10, rational ratio r=4/11. Prefix a=[9, 10, 11], prefix b=[1, 2, 3], each with cutoff=3 terms. Starting at that index use A*(r)^(n−cutoff) and B*(r)^(n−cutoff), so the first tail exponent is0. Give (cutoff, prefix_sum_a, prefix_sum_b, tail_sum_a, tail_sum_b, total_sum_a, tail_convergence, certificate).","answer":{"kind":"expression","value":"(3, 30, 6, 99/7, 110/7, 309/7, Yes, finite_prefix_does_not_change_tail_convergence)","explanation":"Add each finite prefix separately and compute both geometric tails. The tail comparison does not require the prefixes or total sums to have the same ordering. The convergence flag concerns the full valid tail, not finitely sampled terms. The exact result, retaining the requested order, is (3, 30, 6, 99/7, 110/7, 309/7, Yes, finite_prefix_does_not_change_tail_convergence)."},"skills":["Arbitrary Finite Prefixes Do Not Change Convergence of Dominated Tails"],"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-20261015-advance-20261117-r01-series-comparison-test-0060","batch_id":"batch-20261015-advance-20261117-r01-series-comparison-test-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4b619ff63b96d5ac594660546da6700b1dfa63fccaa43fb65d78130fe2745dfd","family_id":"qf-mod48-series-eventual-tail","seed":202611171060,"semantic_template":"series-eventual-tail|{a_scale:[10,1],b_scale:[11,1],cutoff:4,prefix_a:[[10,1],[11,1],[12,1],[13,1]],prefix_b:[[1,1],[2,1],[3,1],[4,1]],ratio:[5,12]}"},"fields":{"title":"Arbitrary Finite Prefixes Do Not Change Convergence of Dominated Tails","stem":"Use exact scales A=10,B=11, rational ratio r=5/12. Prefix a=[10, 11, 12, 13], prefix b=[1, 2, 3, 4], each with cutoff=4 terms. Starting at that index use A*(r)^(n−cutoff) and B*(r)^(n−cutoff), so the first tail exponent is0. Give (cutoff, prefix_sum_a, prefix_sum_b, tail_sum_a, tail_sum_b, total_sum_a, tail_convergence, certificate).","answer":{"kind":"expression","value":"(4, 46, 10, 120/7, 132/7, 442/7, Yes, finite_prefix_does_not_change_tail_convergence)","explanation":"Add each finite prefix separately and compute both geometric tails. The tail comparison does not require the prefixes or total sums to have the same ordering. The convergence flag concerns the full valid tail, not finitely sampled terms. The exact result, retaining the requested order, is (4, 46, 10, 120/7, 132/7, 442/7, Yes, finite_prefix_does_not_change_tail_convergence)."},"skills":["Arbitrary Finite Prefixes Do Not Change Convergence of Dominated Tails"],"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-20261015-advance-20261117-r01-series-comparison-test-0061","batch_id":"batch-20261015-advance-20261117-r01-series-comparison-test-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b6cacffdff2aff95cedee1dcdb6726629b82f8eb6b862aa9190ae599a4e3d09a","family_id":"qf-mod48-series-eventual-tail","seed":202611171061,"semantic_template":"series-eventual-tail|{a_scale:[11,1],b_scale:[12,1],cutoff:0,prefix_a:[],prefix_b:[],ratio:[1,8]}"},"fields":{"title":"Arbitrary Finite Prefixes Do Not Change Convergence of Dominated Tails","stem":"Use exact scales A=11,B=12, rational ratio r=1/8. Prefix a=[], prefix b=[], each with cutoff=0 terms. Starting at that index use A*(r)^(n−cutoff) and B*(r)^(n−cutoff), so the first tail exponent is0. Give (cutoff, prefix_sum_a, prefix_sum_b, tail_sum_a, tail_sum_b, total_sum_a, tail_convergence, certificate).","answer":{"kind":"expression","value":"(0, 0, 0, 88/7, 96/7, 88/7, Yes, finite_prefix_does_not_change_tail_convergence)","explanation":"Add each finite prefix separately and compute both geometric tails. The tail comparison does not require the prefixes or total sums to have the same ordering. The convergence flag concerns the full valid tail, not finitely sampled terms. The exact result, retaining the requested order, is (0, 0, 0, 88/7, 96/7, 88/7, Yes, finite_prefix_does_not_change_tail_convergence)."},"skills":["Arbitrary Finite Prefixes Do Not Change Convergence of Dominated Tails"],"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-20261015-advance-20261117-r01-series-comparison-test-0062","batch_id":"batch-20261015-advance-20261117-r01-series-comparison-test-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3263c1112ad601a91a7143887c35d91d45f0828b3a3ed50f8707cae52e6bc14b","family_id":"qf-mod48-series-eventual-tail","seed":202611171062,"semantic_template":"series-eventual-tail|{a_scale:[12,1],b_scale:[13,1],cutoff:1,prefix_a:[[12,1]],prefix_b:[[1,1]],ratio:[2,9]}"},"fields":{"title":"Arbitrary Finite Prefixes Do Not Change Convergence of Dominated Tails","stem":"Use exact scales A=12,B=13, rational ratio r=2/9. Prefix a=[12], prefix b=[1], each with cutoff=1 terms. Starting at that index use A*(r)^(n−cutoff) and B*(r)^(n−cutoff), so the first tail exponent is0. Give (cutoff, prefix_sum_a, prefix_sum_b, tail_sum_a, tail_sum_b, total_sum_a, tail_convergence, certificate).","answer":{"kind":"expression","value":"(1, 12, 1, 108/7, 117/7, 192/7, Yes, finite_prefix_does_not_change_tail_convergence)","explanation":"Add each finite prefix separately and compute both geometric tails. The tail comparison does not require the prefixes or total sums to have the same ordering. The convergence flag concerns the full valid tail, not finitely sampled terms. The exact result, retaining the requested order, is (1, 12, 1, 108/7, 117/7, 192/7, Yes, finite_prefix_does_not_change_tail_convergence)."},"skills":["Arbitrary Finite Prefixes Do Not Change Convergence of Dominated Tails"],"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-20261015-advance-20261117-r01-series-comparison-test-0063","batch_id":"batch-20261015-advance-20261117-r01-series-comparison-test-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d1c9f545204674d66284c49afb10763b92f3891e347e99807b5f8f6a48324564","family_id":"qf-mod48-series-eventual-tail","seed":202611171063,"semantic_template":"series-eventual-tail|{a_scale:[13,1],b_scale:[14,1],cutoff:2,prefix_a:[[13,1],[14,1]],prefix_b:[[1,1],[2,1]],ratio:[3,10]}"},"fields":{"title":"Arbitrary Finite Prefixes Do Not Change Convergence of Dominated Tails","stem":"Use exact scales A=13,B=14, rational ratio r=3/10. Prefix a=[13, 14], prefix b=[1, 2], each with cutoff=2 terms. Starting at that index use A*(r)^(n−cutoff) and B*(r)^(n−cutoff), so the first tail exponent is0. Give (cutoff, prefix_sum_a, prefix_sum_b, tail_sum_a, tail_sum_b, total_sum_a, tail_convergence, certificate).","answer":{"kind":"expression","value":"(2, 27, 3, 130/7, 20, 319/7, Yes, finite_prefix_does_not_change_tail_convergence)","explanation":"Add each finite prefix separately and compute both geometric tails. The tail comparison does not require the prefixes or total sums to have the same ordering. The convergence flag concerns the full valid tail, not finitely sampled terms. The exact result, retaining the requested order, is (2, 27, 3, 130/7, 20, 319/7, Yes, finite_prefix_does_not_change_tail_convergence)."},"skills":["Arbitrary Finite Prefixes Do Not Change Convergence of Dominated Tails"],"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-20261015-advance-20261117-r01-series-comparison-test-0064","batch_id":"batch-20261015-advance-20261117-r01-series-comparison-test-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6875d6df65d6e911299e1a8a1a26c021465fc9cc27f6ac4fd687bcba6cea4675","family_id":"qf-mod48-series-eventual-tail","seed":202611171064,"semantic_template":"series-eventual-tail|{a_scale:[14,1],b_scale:[15,1],cutoff:3,prefix_a:[[14,1],[15,1],[16,1]],prefix_b:[[1,1],[2,1],[3,1]],ratio:[4,11]}"},"fields":{"title":"Arbitrary Finite Prefixes Do Not Change Convergence of Dominated Tails","stem":"Use exact scales A=14,B=15, rational ratio r=4/11. Prefix a=[14, 15, 16], prefix b=[1, 2, 3], each with cutoff=3 terms. Starting at that index use A*(r)^(n−cutoff) and B*(r)^(n−cutoff), so the first tail exponent is0. Give (cutoff, prefix_sum_a, prefix_sum_b, tail_sum_a, tail_sum_b, total_sum_a, tail_convergence, certificate).","answer":{"kind":"expression","value":"(3, 45, 6, 22, 165/7, 67, Yes, finite_prefix_does_not_change_tail_convergence)","explanation":"Add each finite prefix separately and compute both geometric tails. The tail comparison does not require the prefixes or total sums to have the same ordering. The convergence flag concerns the full valid tail, not finitely sampled terms. The exact result, retaining the requested order, is (3, 45, 6, 22, 165/7, 67, Yes, finite_prefix_does_not_change_tail_convergence)."},"skills":["Arbitrary Finite Prefixes Do Not Change Convergence of Dominated Tails"],"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-20261015-advance-20261117-r01-series-comparison-test-0065","batch_id":"batch-20261015-advance-20261117-r01-series-comparison-test-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1c05e323406aa5f6c869b4cffd54ddc2e1255256d281c270891ead321e0e6457","family_id":"qf-mod48-series-eventual-tail","seed":202611171065,"semantic_template":"series-eventual-tail|{a_scale:[15,1],b_scale:[16,1],cutoff:4,prefix_a:[[15,1],[16,1],[17,1],[18,1]],prefix_b:[[1,1],[2,1],[3,1],[4,1]],ratio:[5,12]}"},"fields":{"title":"Arbitrary Finite Prefixes Do Not Change Convergence of Dominated Tails","stem":"Use exact scales A=15,B=16, rational ratio r=5/12. Prefix a=[15, 16, 17, 18], prefix b=[1, 2, 3, 4], each with cutoff=4 terms. Starting at that index use A*(r)^(n−cutoff) and B*(r)^(n−cutoff), so the first tail exponent is0. Give (cutoff, prefix_sum_a, prefix_sum_b, tail_sum_a, tail_sum_b, total_sum_a, tail_convergence, certificate).","answer":{"kind":"expression","value":"(4, 66, 10, 180/7, 192/7, 642/7, Yes, finite_prefix_does_not_change_tail_convergence)","explanation":"Add each finite prefix separately and compute both geometric tails. The tail comparison does not require the prefixes or total sums to have the same ordering. The convergence flag concerns the full valid tail, not finitely sampled terms. The exact result, retaining the requested order, is (4, 66, 10, 180/7, 192/7, 642/7, Yes, finite_prefix_does_not_change_tail_convergence)."},"skills":["Arbitrary Finite Prefixes Do Not Change Convergence of Dominated Tails"],"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-20261015-advance-20261117-r01-series-comparison-test-0066","batch_id":"batch-20261015-advance-20261117-r01-series-comparison-test-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2c41660cf0f5018ff3be035f7c29d234fe75b591550b8c4323c51f1a3c679144","family_id":"qf-mod48-series-eventual-tail","seed":202611171066,"semantic_template":"series-eventual-tail|{a_scale:[16,1],b_scale:[17,1],cutoff:0,prefix_a:[],prefix_b:[],ratio:[1,8]}"},"fields":{"title":"Arbitrary Finite Prefixes Do Not Change Convergence of Dominated Tails","stem":"Use exact scales A=16,B=17, rational ratio r=1/8. Prefix a=[], prefix b=[], each with cutoff=0 terms. Starting at that index use A*(r)^(n−cutoff) and B*(r)^(n−cutoff), so the first tail exponent is0. Give (cutoff, prefix_sum_a, prefix_sum_b, tail_sum_a, tail_sum_b, total_sum_a, tail_convergence, certificate).","answer":{"kind":"expression","value":"(0, 0, 0, 128/7, 136/7, 128/7, Yes, finite_prefix_does_not_change_tail_convergence)","explanation":"Add each finite prefix separately and compute both geometric tails. The tail comparison does not require the prefixes or total sums to have the same ordering. The convergence flag concerns the full valid tail, not finitely sampled terms. The exact result, retaining the requested order, is (0, 0, 0, 128/7, 136/7, 128/7, Yes, finite_prefix_does_not_change_tail_convergence)."},"skills":["Arbitrary Finite Prefixes Do Not Change Convergence of Dominated Tails"],"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-20261015-advance-20261117-r01-series-comparison-test-0067","batch_id":"batch-20261015-advance-20261117-r01-series-comparison-test-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d60d40b23da9ca8b00265d725cbbb871d3ccb4d82bebacf09375875115533400","family_id":"qf-mod48-series-eventual-tail","seed":202611171067,"semantic_template":"series-eventual-tail|{a_scale:[17,1],b_scale:[18,1],cutoff:1,prefix_a:[[17,1]],prefix_b:[[1,1]],ratio:[2,9]}"},"fields":{"title":"Arbitrary Finite Prefixes Do Not Change Convergence of Dominated Tails","stem":"Use exact scales A=17,B=18, rational ratio r=2/9. Prefix a=[17], prefix b=[1], each with cutoff=1 terms. Starting at that index use A*(r)^(n−cutoff) and B*(r)^(n−cutoff), so the first tail exponent is0. Give (cutoff, prefix_sum_a, prefix_sum_b, tail_sum_a, tail_sum_b, total_sum_a, tail_convergence, certificate).","answer":{"kind":"expression","value":"(1, 17, 1, 153/7, 162/7, 272/7, Yes, finite_prefix_does_not_change_tail_convergence)","explanation":"Add each finite prefix separately and compute both geometric tails. The tail comparison does not require the prefixes or total sums to have the same ordering. The convergence flag concerns the full valid tail, not finitely sampled terms. The exact result, retaining the requested order, is (1, 17, 1, 153/7, 162/7, 272/7, Yes, finite_prefix_does_not_change_tail_convergence)."},"skills":["Arbitrary Finite Prefixes Do Not Change Convergence of Dominated Tails"],"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-20261015-advance-20261117-r01-series-comparison-test-0068","batch_id":"batch-20261015-advance-20261117-r01-series-comparison-test-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"494742de3075215e16539be033571052c1d1f2bec443b224afc04b82391cf446","family_id":"qf-mod48-series-eventual-tail","seed":202611171068,"semantic_template":"series-eventual-tail|{a_scale:[18,1],b_scale:[19,1],cutoff:2,prefix_a:[[18,1],[19,1]],prefix_b:[[1,1],[2,1]],ratio:[3,10]}"},"fields":{"title":"Arbitrary Finite Prefixes Do Not Change Convergence of Dominated Tails","stem":"Use exact scales A=18,B=19, rational ratio r=3/10. Prefix a=[18, 19], prefix b=[1, 2], each with cutoff=2 terms. Starting at that index use A*(r)^(n−cutoff) and B*(r)^(n−cutoff), so the first tail exponent is0. Give (cutoff, prefix_sum_a, prefix_sum_b, tail_sum_a, tail_sum_b, total_sum_a, tail_convergence, certificate).","answer":{"kind":"expression","value":"(2, 37, 3, 180/7, 190/7, 439/7, Yes, finite_prefix_does_not_change_tail_convergence)","explanation":"Add each finite prefix separately and compute both geometric tails. The tail comparison does not require the prefixes or total sums to have the same ordering. The convergence flag concerns the full valid tail, not finitely sampled terms. The exact result, retaining the requested order, is (2, 37, 3, 180/7, 190/7, 439/7, Yes, finite_prefix_does_not_change_tail_convergence)."},"skills":["Arbitrary Finite Prefixes Do Not Change Convergence of Dominated Tails"],"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-20261015-advance-20261117-r01-series-comparison-test-0069","batch_id":"batch-20261015-advance-20261117-r01-series-comparison-test-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"39a4e99357047b2254f8a1ec0692f0832b1ffd48115bf6dd87215fe409c90198","family_id":"qf-mod48-series-eventual-tail","seed":202611171069,"semantic_template":"series-eventual-tail|{a_scale:[19,1],b_scale:[20,1],cutoff:3,prefix_a:[[19,1],[20,1],[21,1]],prefix_b:[[1,1],[2,1],[3,1]],ratio:[4,11]}"},"fields":{"title":"Arbitrary Finite Prefixes Do Not Change Convergence of Dominated Tails","stem":"Use exact scales A=19,B=20, rational ratio r=4/11. Prefix a=[19, 20, 21], prefix b=[1, 2, 3], each with cutoff=3 terms. Starting at that index use A*(r)^(n−cutoff) and B*(r)^(n−cutoff), so the first tail exponent is0. Give (cutoff, prefix_sum_a, prefix_sum_b, tail_sum_a, tail_sum_b, total_sum_a, tail_convergence, certificate).","answer":{"kind":"expression","value":"(3, 60, 6, 209/7, 220/7, 629/7, Yes, finite_prefix_does_not_change_tail_convergence)","explanation":"Add each finite prefix separately and compute both geometric tails. The tail comparison does not require the prefixes or total sums to have the same ordering. The convergence flag concerns the full valid tail, not finitely sampled terms. The exact result, retaining the requested order, is (3, 60, 6, 209/7, 220/7, 629/7, Yes, finite_prefix_does_not_change_tail_convergence)."},"skills":["Arbitrary Finite Prefixes Do Not Change Convergence of Dominated Tails"],"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-20261015-advance-20261117-r01-series-comparison-test-0070","batch_id":"batch-20261015-advance-20261117-r01-series-comparison-test-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"58e2d4da0ac8e09f87e43d14f68352c72b4c2f5c3028c52037d58b5020d56451","family_id":"qf-mod48-series-eventual-tail","seed":202611171070,"semantic_template":"series-eventual-tail|{a_scale:[20,1],b_scale:[21,1],cutoff:4,prefix_a:[[20,1],[21,1],[22,1],[23,1]],prefix_b:[[1,1],[2,1],[3,1],[4,1]],ratio:[5,12]}"},"fields":{"title":"Arbitrary Finite Prefixes Do Not Change Convergence of Dominated Tails","stem":"Use exact scales A=20,B=21, rational ratio r=5/12. Prefix a=[20, 21, 22, 23], prefix b=[1, 2, 3, 4], each with cutoff=4 terms. Starting at that index use A*(r)^(n−cutoff) and B*(r)^(n−cutoff), so the first tail exponent is0. Give (cutoff, prefix_sum_a, prefix_sum_b, tail_sum_a, tail_sum_b, total_sum_a, tail_convergence, certificate).","answer":{"kind":"expression","value":"(4, 86, 10, 240/7, 36, 842/7, Yes, finite_prefix_does_not_change_tail_convergence)","explanation":"Add each finite prefix separately and compute both geometric tails. The tail comparison does not require the prefixes or total sums to have the same ordering. The convergence flag concerns the full valid tail, not finitely sampled terms. The exact result, retaining the requested order, is (4, 86, 10, 240/7, 36, 842/7, Yes, finite_prefix_does_not_change_tail_convergence)."},"skills":["Arbitrary Finite Prefixes Do Not Change Convergence of Dominated Tails"],"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-20261015-advance-20261117-r01-series-comparison-test-0071","batch_id":"batch-20261015-advance-20261117-r01-series-comparison-test-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7b32cc6e50baa4775af5260ba05e871506d0fd130c58bfb39e772785c9f0d5eb","family_id":"qf-mod48-series-eventual-tail","seed":202611171071,"semantic_template":"series-eventual-tail|{a_scale:[21,1],b_scale:[22,1],cutoff:0,prefix_a:[],prefix_b:[],ratio:[1,8]}"},"fields":{"title":"Arbitrary Finite Prefixes Do Not Change Convergence of Dominated Tails","stem":"Use exact scales A=21,B=22, rational ratio r=1/8. Prefix a=[], prefix b=[], each with cutoff=0 terms. Starting at that index use A*(r)^(n−cutoff) and B*(r)^(n−cutoff), so the first tail exponent is0. Give (cutoff, prefix_sum_a, prefix_sum_b, tail_sum_a, tail_sum_b, total_sum_a, tail_convergence, certificate).","answer":{"kind":"expression","value":"(0, 0, 0, 24, 176/7, 24, Yes, finite_prefix_does_not_change_tail_convergence)","explanation":"Add each finite prefix separately and compute both geometric tails. The tail comparison does not require the prefixes or total sums to have the same ordering. The convergence flag concerns the full valid tail, not finitely sampled terms. The exact result, retaining the requested order, is (0, 0, 0, 24, 176/7, 24, Yes, finite_prefix_does_not_change_tail_convergence)."},"skills":["Arbitrary Finite Prefixes Do Not Change Convergence of Dominated Tails"],"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-20261015-advance-20261117-r01-series-comparison-test-0072","batch_id":"batch-20261015-advance-20261117-r01-series-comparison-test-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f26e549bc6866b450aa6f84d5f8690c323eeec9feeb156a412672a4f91af4837","family_id":"qf-mod48-series-eventual-tail","seed":202611171072,"semantic_template":"series-eventual-tail|{a_scale:[22,1],b_scale:[23,1],cutoff:1,prefix_a:[[22,1]],prefix_b:[[1,1]],ratio:[2,9]}"},"fields":{"title":"Arbitrary Finite Prefixes Do Not Change Convergence of Dominated Tails","stem":"Use exact scales A=22,B=23, rational ratio r=2/9. Prefix a=[22], prefix b=[1], each with cutoff=1 terms. Starting at that index use A*(r)^(n−cutoff) and B*(r)^(n−cutoff), so the first tail exponent is0. Give (cutoff, prefix_sum_a, prefix_sum_b, tail_sum_a, tail_sum_b, total_sum_a, tail_convergence, certificate).","answer":{"kind":"expression","value":"(1, 22, 1, 198/7, 207/7, 352/7, Yes, finite_prefix_does_not_change_tail_convergence)","explanation":"Add each finite prefix separately and compute both geometric tails. The tail comparison does not require the prefixes or total sums to have the same ordering. The convergence flag concerns the full valid tail, not finitely sampled terms. The exact result, retaining the requested order, is (1, 22, 1, 198/7, 207/7, 352/7, Yes, finite_prefix_does_not_change_tail_convergence)."},"skills":["Arbitrary Finite Prefixes Do Not Change Convergence of Dominated Tails"],"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-20261015-advance-20261117-r01-series-comparison-test-0073","batch_id":"batch-20261015-advance-20261117-r01-series-comparison-test-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"05bc7f9949b891d9b23db623f73ad9adfa291b79b06c7f1637ecc50aea3a4dd6","family_id":"qf-mod48-series-eventual-tail","seed":202611171073,"semantic_template":"series-eventual-tail|{a_scale:[23,1],b_scale:[24,1],cutoff:2,prefix_a:[[23,1],[24,1]],prefix_b:[[1,1],[2,1]],ratio:[3,10]}"},"fields":{"title":"Arbitrary Finite Prefixes Do Not Change Convergence of Dominated Tails","stem":"Use exact scales A=23,B=24, rational ratio r=3/10. Prefix a=[23, 24], prefix b=[1, 2], each with cutoff=2 terms. Starting at that index use A*(r)^(n−cutoff) and B*(r)^(n−cutoff), so the first tail exponent is0. Give (cutoff, prefix_sum_a, prefix_sum_b, tail_sum_a, tail_sum_b, total_sum_a, tail_convergence, certificate).","answer":{"kind":"expression","value":"(2, 47, 3, 230/7, 240/7, 559/7, Yes, finite_prefix_does_not_change_tail_convergence)","explanation":"Add each finite prefix separately and compute both geometric tails. The tail comparison does not require the prefixes or total sums to have the same ordering. The convergence flag concerns the full valid tail, not finitely sampled terms. The exact result, retaining the requested order, is (2, 47, 3, 230/7, 240/7, 559/7, Yes, finite_prefix_does_not_change_tail_convergence)."},"skills":["Arbitrary Finite Prefixes Do Not Change Convergence of Dominated Tails"],"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-20261015-advance-20261117-r01-series-comparison-test-0074","batch_id":"batch-20261015-advance-20261117-r01-series-comparison-test-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"41062f2f4237107ed2a4be007353b48f1f0186fece27de611ff7acfe7ed9c24c","family_id":"qf-mod48-series-eventual-tail","seed":202611171074,"semantic_template":"series-eventual-tail|{a_scale:[24,1],b_scale:[25,1],cutoff:3,prefix_a:[[24,1],[25,1],[26,1]],prefix_b:[[1,1],[2,1],[3,1]],ratio:[4,11]}"},"fields":{"title":"Arbitrary Finite Prefixes Do Not Change Convergence of Dominated Tails","stem":"Use exact scales A=24,B=25, rational ratio r=4/11. Prefix a=[24, 25, 26], prefix b=[1, 2, 3], each with cutoff=3 terms. Starting at that index use A*(r)^(n−cutoff) and B*(r)^(n−cutoff), so the first tail exponent is0. Give (cutoff, prefix_sum_a, prefix_sum_b, tail_sum_a, tail_sum_b, total_sum_a, tail_convergence, certificate).","answer":{"kind":"expression","value":"(3, 75, 6, 264/7, 275/7, 789/7, Yes, finite_prefix_does_not_change_tail_convergence)","explanation":"Add each finite prefix separately and compute both geometric tails. The tail comparison does not require the prefixes or total sums to have the same ordering. The convergence flag concerns the full valid tail, not finitely sampled terms. The exact result, retaining the requested order, is (3, 75, 6, 264/7, 275/7, 789/7, Yes, finite_prefix_does_not_change_tail_convergence)."},"skills":["Arbitrary Finite Prefixes Do Not Change Convergence of Dominated Tails"],"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-20261015-advance-20261117-r01-series-comparison-test-0075","batch_id":"batch-20261015-advance-20261117-r01-series-comparison-test-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9eadd0dcb60bb0c2d6359eafaa53bc32f5660414cfecc9cccbd3f5290c4d33e1","family_id":"qf-mod48-series-eventual-tail","seed":202611171075,"semantic_template":"series-eventual-tail|{a_scale:[25,1],b_scale:[26,1],cutoff:4,prefix_a:[[25,1],[26,1],[27,1],[28,1]],prefix_b:[[1,1],[2,1],[3,1],[4,1]],ratio:[5,12]}"},"fields":{"title":"Arbitrary Finite Prefixes Do Not Change Convergence of Dominated Tails","stem":"Use exact scales A=25,B=26, rational ratio r=5/12. Prefix a=[25, 26, 27, 28], prefix b=[1, 2, 3, 4], each with cutoff=4 terms. Starting at that index use A*(r)^(n−cutoff) and B*(r)^(n−cutoff), so the first tail exponent is0. Give (cutoff, prefix_sum_a, prefix_sum_b, tail_sum_a, tail_sum_b, total_sum_a, tail_convergence, certificate).","answer":{"kind":"expression","value":"(4, 106, 10, 300/7, 312/7, 1042/7, Yes, finite_prefix_does_not_change_tail_convergence)","explanation":"Add each finite prefix separately and compute both geometric tails. The tail comparison does not require the prefixes or total sums to have the same ordering. The convergence flag concerns the full valid tail, not finitely sampled terms. The exact result, retaining the requested order, is (4, 106, 10, 300/7, 312/7, 1042/7, Yes, finite_prefix_does_not_change_tail_convergence)."},"skills":["Arbitrary Finite Prefixes Do Not Change Convergence of Dominated Tails"],"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-20261015-advance-20261117-r01-series-comparison-test-0076","batch_id":"batch-20261015-advance-20261117-r01-series-comparison-test-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"05a96c8a4c78e00b1d597e640fdf5d2a0493d8e47996493426665b8af2ae145a","family_id":"qf-mod48-series-comparison-boundary","seed":202611171076,"semantic_template":"series-comparison-boundary|{mode:\"finite_sample\",scale:[1,1]}"},"fields":{"title":"Comparison Direction, Nonnegativity and Reference Status Cannot Be Omitted","stem":"Boundary mode=finite_sample, positive scale=1. Give (mode, scale, sample_scope, eventual_comparison_claim, boundary).","answer":{"kind":"expression","value":"(finite_sample, 1, first_N_terms_compare_next_term_fails, No, finite_samples_do_not_establish_eventual_forall)","explanation":"finite_sample records insufficient eventual-forall evidence. A scaled1/n^2 series can converge below divergent scaled1/n; signed terms violate the nonnegative premise; an unproved reference status supplies no convergence certificate. Keep the distinct branch field roles, not one padded tuple. The exact result, retaining the requested order, is (finite_sample, 1, first_N_terms_compare_next_term_fails, No, finite_samples_do_not_establish_eventual_forall)."},"skills":["Comparison Direction, Nonnegativity and Reference Status Cannot Be Omitted"],"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-20261015-advance-20261117-r01-series-comparison-test-0077","batch_id":"batch-20261015-advance-20261117-r01-series-comparison-test-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9d5749da6f8ee8492629366a1378317db2904fc488f699f269cdca6cb35a52a3","family_id":"qf-mod48-series-comparison-boundary","seed":202611171077,"semantic_template":"series-comparison-boundary|{mode:\"divergent_upper\",scale:[2,1]}"},"fields":{"title":"Comparison Direction, Nonnegativity and Reference Status Cannot Be Omitted","stem":"Boundary mode=divergent_upper, positive scale=2. Give (mode, series_a_description, series_b_description, term_comparison, actual_series_status, divergence_inference_claim).","answer":{"kind":"expression","value":"(divergent_upper, a_n=2/n^2, b_n=2/n, a_n<=b_n, b_diverges_but_a_converges, No)","explanation":"finite_sample records insufficient eventual-forall evidence. A scaled1/n^2 series can converge below divergent scaled1/n; signed terms violate the nonnegative premise; an unproved reference status supplies no convergence certificate. Keep the distinct branch field roles, not one padded tuple. The exact result, retaining the requested order, is (divergent_upper, a_n=2/n^2, b_n=2/n, a_n<=b_n, b_diverges_but_a_converges, No)."},"skills":["Comparison Direction, Nonnegativity and Reference Status Cannot Be Omitted"],"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-20261015-advance-20261117-r01-series-comparison-test-0078","batch_id":"batch-20261015-advance-20261117-r01-series-comparison-test-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ffe3b329a20f7e25b04ae33641d284eb42f56135f095e1419f0e0dbed403d497","family_id":"qf-mod48-series-comparison-boundary","seed":202611171078,"semantic_template":"series-comparison-boundary|{mode:\"signed_terms\",scale:[3,1]}"},"fields":{"title":"Comparison Direction, Nonnegativity and Reference Status Cannot Be Omitted","stem":"Boundary mode=signed_terms, positive scale=3. Give (mode, series_description, failed_hypothesis, comparison_test_claim).","answer":{"kind":"expression","value":"(signed_terms, a_n=(-1)^n*3/n, nonnegative_hypothesis_fails, No)","explanation":"finite_sample records insufficient eventual-forall evidence. A scaled1/n^2 series can converge below divergent scaled1/n; signed terms violate the nonnegative premise; an unproved reference status supplies no convergence certificate. Keep the distinct branch field roles, not one padded tuple. The exact result, retaining the requested order, is (signed_terms, a_n=(-1)^n*3/n, nonnegative_hypothesis_fails, No)."},"skills":["Comparison Direction, Nonnegativity and Reference Status Cannot Be Omitted"],"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-20261015-advance-20261117-r01-series-comparison-test-0079","batch_id":"batch-20261015-advance-20261117-r01-series-comparison-test-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"360b9e3407e9ad7e8d1004cd3d4de07c21e2d82aa8588950201203fec2e8c9da","family_id":"qf-mod48-series-comparison-boundary","seed":202611171079,"semantic_template":"series-comparison-boundary|{mode:\"unknown_reference\",scale:[4,1]}"},"fields":{"title":"Comparison Direction, Nonnegativity and Reference Status Cannot Be Omitted","stem":"Boundary mode=unknown_reference, positive scale=4. Give (mode, scale, reference_status_scope, comparison_test_claim).","answer":{"kind":"expression","value":"(unknown_reference, 4, comparison_holds_reference_status_not_proved, No)","explanation":"finite_sample records insufficient eventual-forall evidence. A scaled1/n^2 series can converge below divergent scaled1/n; signed terms violate the nonnegative premise; an unproved reference status supplies no convergence certificate. Keep the distinct branch field roles, not one padded tuple. The exact result, retaining the requested order, is (unknown_reference, 4, comparison_holds_reference_status_not_proved, No)."},"skills":["Comparison Direction, Nonnegativity and Reference Status Cannot Be Omitted"],"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-20261015-advance-20261117-r01-series-comparison-test-0080","batch_id":"batch-20261015-advance-20261117-r01-series-comparison-test-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"739dc051f0cdbc32511cb0620f1133e5fb63082fb13ffe5ea4cd0e084a82ca69","family_id":"qf-mod48-series-comparison-boundary","seed":202611171080,"semantic_template":"series-comparison-boundary|{mode:\"finite_sample\",scale:[5,1]}"},"fields":{"title":"Comparison Direction, Nonnegativity and Reference Status Cannot Be Omitted","stem":"Boundary mode=finite_sample, positive scale=5. Give (mode, scale, sample_scope, eventual_comparison_claim, boundary).","answer":{"kind":"expression","value":"(finite_sample, 5, first_N_terms_compare_next_term_fails, No, finite_samples_do_not_establish_eventual_forall)","explanation":"finite_sample records insufficient eventual-forall evidence. A scaled1/n^2 series can converge below divergent scaled1/n; signed terms violate the nonnegative premise; an unproved reference status supplies no convergence certificate. Keep the distinct branch field roles, not one padded tuple. The exact result, retaining the requested order, is (finite_sample, 5, first_N_terms_compare_next_term_fails, No, finite_samples_do_not_establish_eventual_forall)."},"skills":["Comparison Direction, Nonnegativity and Reference Status Cannot Be Omitted"],"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-20261015-advance-20261117-r01-series-comparison-test-0081","batch_id":"batch-20261015-advance-20261117-r01-series-comparison-test-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"49fd0a8800d5f471eb23f614bbb230ede2a14e8ced4d00079e01224830f62fee","family_id":"qf-mod48-series-comparison-boundary","seed":202611171081,"semantic_template":"series-comparison-boundary|{mode:\"divergent_upper\",scale:[6,1]}"},"fields":{"title":"Comparison Direction, Nonnegativity and Reference Status Cannot Be Omitted","stem":"Boundary mode=divergent_upper, positive scale=6. Give (mode, series_a_description, series_b_description, term_comparison, actual_series_status, divergence_inference_claim).","answer":{"kind":"expression","value":"(divergent_upper, a_n=6/n^2, b_n=6/n, a_n<=b_n, b_diverges_but_a_converges, No)","explanation":"finite_sample records insufficient eventual-forall evidence. A scaled1/n^2 series can converge below divergent scaled1/n; signed terms violate the nonnegative premise; an unproved reference status supplies no convergence certificate. Keep the distinct branch field roles, not one padded tuple. The exact result, retaining the requested order, is (divergent_upper, a_n=6/n^2, b_n=6/n, a_n<=b_n, b_diverges_but_a_converges, No)."},"skills":["Comparison Direction, Nonnegativity and Reference Status Cannot Be Omitted"],"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-20261015-advance-20261117-r01-series-comparison-test-0082","batch_id":"batch-20261015-advance-20261117-r01-series-comparison-test-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"aaf3d40ff883fb83d575fd17f1d11852e83eae7f24ad59d6eade4370bff237ab","family_id":"qf-mod48-series-comparison-boundary","seed":202611171082,"semantic_template":"series-comparison-boundary|{mode:\"signed_terms\",scale:[7,1]}"},"fields":{"title":"Comparison Direction, Nonnegativity and Reference Status Cannot Be Omitted","stem":"Boundary mode=signed_terms, positive scale=7. Give (mode, series_description, failed_hypothesis, comparison_test_claim).","answer":{"kind":"expression","value":"(signed_terms, a_n=(-1)^n*7/n, nonnegative_hypothesis_fails, No)","explanation":"finite_sample records insufficient eventual-forall evidence. A scaled1/n^2 series can converge below divergent scaled1/n; signed terms violate the nonnegative premise; an unproved reference status supplies no convergence certificate. Keep the distinct branch field roles, not one padded tuple. The exact result, retaining the requested order, is (signed_terms, a_n=(-1)^n*7/n, nonnegative_hypothesis_fails, No)."},"skills":["Comparison Direction, Nonnegativity and Reference Status Cannot Be Omitted"],"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-20261015-advance-20261117-r01-series-comparison-test-0083","batch_id":"batch-20261015-advance-20261117-r01-series-comparison-test-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b1893ffa62bec72ffc0b0503f8c72e48c828d9eb3d0015b8d1e85d7f6f9cf431","family_id":"qf-mod48-series-comparison-boundary","seed":202611171083,"semantic_template":"series-comparison-boundary|{mode:\"unknown_reference\",scale:[8,1]}"},"fields":{"title":"Comparison Direction, Nonnegativity and Reference Status Cannot Be Omitted","stem":"Boundary mode=unknown_reference, positive scale=8. Give (mode, scale, reference_status_scope, comparison_test_claim).","answer":{"kind":"expression","value":"(unknown_reference, 8, comparison_holds_reference_status_not_proved, No)","explanation":"finite_sample records insufficient eventual-forall evidence. A scaled1/n^2 series can converge below divergent scaled1/n; signed terms violate the nonnegative premise; an unproved reference status supplies no convergence certificate. Keep the distinct branch field roles, not one padded tuple. The exact result, retaining the requested order, is (unknown_reference, 8, comparison_holds_reference_status_not_proved, No)."},"skills":["Comparison Direction, Nonnegativity and Reference Status Cannot Be Omitted"],"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-20261015-advance-20261117-r01-series-comparison-test-0084","batch_id":"batch-20261015-advance-20261117-r01-series-comparison-test-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b6491feea48bed964f6efe97ef7e4b6912db8589216f0383feb4de06460ee3ff","family_id":"qf-mod48-series-comparison-boundary","seed":202611171084,"semantic_template":"series-comparison-boundary|{mode:\"finite_sample\",scale:[9,1]}"},"fields":{"title":"Comparison Direction, Nonnegativity and Reference Status Cannot Be Omitted","stem":"Boundary mode=finite_sample, positive scale=9. Give (mode, scale, sample_scope, eventual_comparison_claim, boundary).","answer":{"kind":"expression","value":"(finite_sample, 9, first_N_terms_compare_next_term_fails, No, finite_samples_do_not_establish_eventual_forall)","explanation":"finite_sample records insufficient eventual-forall evidence. A scaled1/n^2 series can converge below divergent scaled1/n; signed terms violate the nonnegative premise; an unproved reference status supplies no convergence certificate. Keep the distinct branch field roles, not one padded tuple. The exact result, retaining the requested order, is (finite_sample, 9, first_N_terms_compare_next_term_fails, No, finite_samples_do_not_establish_eventual_forall)."},"skills":["Comparison Direction, Nonnegativity and Reference Status Cannot Be Omitted"],"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-20261015-advance-20261117-r01-series-comparison-test-0085","batch_id":"batch-20261015-advance-20261117-r01-series-comparison-test-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e41834bd5db751b334efb316053f2295b3b93e6cfec749ba19a2cf5663e3ba2d","family_id":"qf-mod48-series-comparison-boundary","seed":202611171085,"semantic_template":"series-comparison-boundary|{mode:\"divergent_upper\",scale:[10,1]}"},"fields":{"title":"Comparison Direction, Nonnegativity and Reference Status Cannot Be Omitted","stem":"Boundary mode=divergent_upper, positive scale=10. Give (mode, series_a_description, series_b_description, term_comparison, actual_series_status, divergence_inference_claim).","answer":{"kind":"expression","value":"(divergent_upper, a_n=10/n^2, b_n=10/n, a_n<=b_n, b_diverges_but_a_converges, No)","explanation":"finite_sample records insufficient eventual-forall evidence. A scaled1/n^2 series can converge below divergent scaled1/n; signed terms violate the nonnegative premise; an unproved reference status supplies no convergence certificate. Keep the distinct branch field roles, not one padded tuple. The exact result, retaining the requested order, is (divergent_upper, a_n=10/n^2, b_n=10/n, a_n<=b_n, b_diverges_but_a_converges, No)."},"skills":["Comparison Direction, Nonnegativity and Reference Status Cannot Be Omitted"],"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-20261015-advance-20261117-r01-series-comparison-test-0086","batch_id":"batch-20261015-advance-20261117-r01-series-comparison-test-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2d28caf44affb99512e1d23bf67762b80a6698020e5a7a74a30c29d90739c474","family_id":"qf-mod48-series-comparison-boundary","seed":202611171086,"semantic_template":"series-comparison-boundary|{mode:\"signed_terms\",scale:[11,1]}"},"fields":{"title":"Comparison Direction, Nonnegativity and Reference Status Cannot Be Omitted","stem":"Boundary mode=signed_terms, positive scale=11. Give (mode, series_description, failed_hypothesis, comparison_test_claim).","answer":{"kind":"expression","value":"(signed_terms, a_n=(-1)^n*11/n, nonnegative_hypothesis_fails, No)","explanation":"finite_sample records insufficient eventual-forall evidence. A scaled1/n^2 series can converge below divergent scaled1/n; signed terms violate the nonnegative premise; an unproved reference status supplies no convergence certificate. Keep the distinct branch field roles, not one padded tuple. The exact result, retaining the requested order, is (signed_terms, a_n=(-1)^n*11/n, nonnegative_hypothesis_fails, No)."},"skills":["Comparison Direction, Nonnegativity and Reference Status Cannot Be Omitted"],"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-20261015-advance-20261117-r01-series-comparison-test-0087","batch_id":"batch-20261015-advance-20261117-r01-series-comparison-test-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c9317321d1202bebe1f06b80ec2feea48b73c7c977158d08101c9dd3ba64e363","family_id":"qf-mod48-series-comparison-boundary","seed":202611171087,"semantic_template":"series-comparison-boundary|{mode:\"unknown_reference\",scale:[12,1]}"},"fields":{"title":"Comparison Direction, Nonnegativity and Reference Status Cannot Be Omitted","stem":"Boundary mode=unknown_reference, positive scale=12. Give (mode, scale, reference_status_scope, comparison_test_claim).","answer":{"kind":"expression","value":"(unknown_reference, 12, comparison_holds_reference_status_not_proved, No)","explanation":"finite_sample records insufficient eventual-forall evidence. A scaled1/n^2 series can converge below divergent scaled1/n; signed terms violate the nonnegative premise; an unproved reference status supplies no convergence certificate. Keep the distinct branch field roles, not one padded tuple. The exact result, retaining the requested order, is (unknown_reference, 12, comparison_holds_reference_status_not_proved, No)."},"skills":["Comparison Direction, Nonnegativity and Reference Status Cannot Be Omitted"],"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-20261015-advance-20261117-r01-series-comparison-test-0088","batch_id":"batch-20261015-advance-20261117-r01-series-comparison-test-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ca3d79937e03572342a715fefb8b2616b3626bea444c6a8951f73a1e61353351","family_id":"qf-mod48-series-comparison-boundary","seed":202611171088,"semantic_template":"series-comparison-boundary|{mode:\"finite_sample\",scale:[13,1]}"},"fields":{"title":"Comparison Direction, Nonnegativity and Reference Status Cannot Be Omitted","stem":"Boundary mode=finite_sample, positive scale=13. Give (mode, scale, sample_scope, eventual_comparison_claim, boundary).","answer":{"kind":"expression","value":"(finite_sample, 13, first_N_terms_compare_next_term_fails, No, finite_samples_do_not_establish_eventual_forall)","explanation":"finite_sample records insufficient eventual-forall evidence. A scaled1/n^2 series can converge below divergent scaled1/n; signed terms violate the nonnegative premise; an unproved reference status supplies no convergence certificate. Keep the distinct branch field roles, not one padded tuple. The exact result, retaining the requested order, is (finite_sample, 13, first_N_terms_compare_next_term_fails, No, finite_samples_do_not_establish_eventual_forall)."},"skills":["Comparison Direction, Nonnegativity and Reference Status Cannot Be Omitted"],"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-20261015-advance-20261117-r01-series-comparison-test-0089","batch_id":"batch-20261015-advance-20261117-r01-series-comparison-test-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e13fc6ebcde4073483e49e6eceffc489b03bebd1175b76f8f0ff672f4222a95b","family_id":"qf-mod48-series-comparison-boundary","seed":202611171089,"semantic_template":"series-comparison-boundary|{mode:\"divergent_upper\",scale:[14,1]}"},"fields":{"title":"Comparison Direction, Nonnegativity and Reference Status Cannot Be Omitted","stem":"Boundary mode=divergent_upper, positive scale=14. Give (mode, series_a_description, series_b_description, term_comparison, actual_series_status, divergence_inference_claim).","answer":{"kind":"expression","value":"(divergent_upper, a_n=14/n^2, b_n=14/n, a_n<=b_n, b_diverges_but_a_converges, No)","explanation":"finite_sample records insufficient eventual-forall evidence. A scaled1/n^2 series can converge below divergent scaled1/n; signed terms violate the nonnegative premise; an unproved reference status supplies no convergence certificate. Keep the distinct branch field roles, not one padded tuple. The exact result, retaining the requested order, is (divergent_upper, a_n=14/n^2, b_n=14/n, a_n<=b_n, b_diverges_but_a_converges, No)."},"skills":["Comparison Direction, Nonnegativity and Reference Status Cannot Be Omitted"],"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-20261015-advance-20261117-r01-series-comparison-test-0090","batch_id":"batch-20261015-advance-20261117-r01-series-comparison-test-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2f4626c275cb0313d52deb8da25bdc7f76498c136a67c85267366e7bd0d7c812","family_id":"qf-mod48-series-comparison-boundary","seed":202611171090,"semantic_template":"series-comparison-boundary|{mode:\"signed_terms\",scale:[15,1]}"},"fields":{"title":"Comparison Direction, Nonnegativity and Reference Status Cannot Be Omitted","stem":"Boundary mode=signed_terms, positive scale=15. Give (mode, series_description, failed_hypothesis, comparison_test_claim).","answer":{"kind":"expression","value":"(signed_terms, a_n=(-1)^n*15/n, nonnegative_hypothesis_fails, No)","explanation":"finite_sample records insufficient eventual-forall evidence. A scaled1/n^2 series can converge below divergent scaled1/n; signed terms violate the nonnegative premise; an unproved reference status supplies no convergence certificate. Keep the distinct branch field roles, not one padded tuple. The exact result, retaining the requested order, is (signed_terms, a_n=(-1)^n*15/n, nonnegative_hypothesis_fails, No)."},"skills":["Comparison Direction, Nonnegativity and Reference Status Cannot Be Omitted"],"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-20261015-advance-20261117-r01-series-comparison-test-0091","batch_id":"batch-20261015-advance-20261117-r01-series-comparison-test-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6bbe43ef72b71d29d201b2c2582c1df1af88dc1948c90d76fae2b1fdabd40d23","family_id":"qf-mod48-series-comparison-boundary","seed":202611171091,"semantic_template":"series-comparison-boundary|{mode:\"unknown_reference\",scale:[16,1]}"},"fields":{"title":"Comparison Direction, Nonnegativity and Reference Status Cannot Be Omitted","stem":"Boundary mode=unknown_reference, positive scale=16. Give (mode, scale, reference_status_scope, comparison_test_claim).","answer":{"kind":"expression","value":"(unknown_reference, 16, comparison_holds_reference_status_not_proved, No)","explanation":"finite_sample records insufficient eventual-forall evidence. A scaled1/n^2 series can converge below divergent scaled1/n; signed terms violate the nonnegative premise; an unproved reference status supplies no convergence certificate. Keep the distinct branch field roles, not one padded tuple. The exact result, retaining the requested order, is (unknown_reference, 16, comparison_holds_reference_status_not_proved, No)."},"skills":["Comparison Direction, Nonnegativity and Reference Status Cannot Be Omitted"],"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-20261015-advance-20261117-r01-series-comparison-test-0092","batch_id":"batch-20261015-advance-20261117-r01-series-comparison-test-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f5bd1d9a42526517ae2170e1d47466577e9c2ba9df682ee60d361fd71f9fc4ee","family_id":"qf-mod48-series-comparison-boundary","seed":202611171092,"semantic_template":"series-comparison-boundary|{mode:\"finite_sample\",scale:[17,1]}"},"fields":{"title":"Comparison Direction, Nonnegativity and Reference Status Cannot Be Omitted","stem":"Boundary mode=finite_sample, positive scale=17. Give (mode, scale, sample_scope, eventual_comparison_claim, boundary).","answer":{"kind":"expression","value":"(finite_sample, 17, first_N_terms_compare_next_term_fails, No, finite_samples_do_not_establish_eventual_forall)","explanation":"finite_sample records insufficient eventual-forall evidence. A scaled1/n^2 series can converge below divergent scaled1/n; signed terms violate the nonnegative premise; an unproved reference status supplies no convergence certificate. Keep the distinct branch field roles, not one padded tuple. The exact result, retaining the requested order, is (finite_sample, 17, first_N_terms_compare_next_term_fails, No, finite_samples_do_not_establish_eventual_forall)."},"skills":["Comparison Direction, Nonnegativity and Reference Status Cannot Be Omitted"],"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-20261015-advance-20261117-r01-series-comparison-test-0093","batch_id":"batch-20261015-advance-20261117-r01-series-comparison-test-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a8ba16eec22d46a78273fdb38e554e955f9441147e43b63a3f4e48a11dd2176c","family_id":"qf-mod48-series-comparison-boundary","seed":202611171093,"semantic_template":"series-comparison-boundary|{mode:\"divergent_upper\",scale:[18,1]}"},"fields":{"title":"Comparison Direction, Nonnegativity and Reference Status Cannot Be Omitted","stem":"Boundary mode=divergent_upper, positive scale=18. Give (mode, series_a_description, series_b_description, term_comparison, actual_series_status, divergence_inference_claim).","answer":{"kind":"expression","value":"(divergent_upper, a_n=18/n^2, b_n=18/n, a_n<=b_n, b_diverges_but_a_converges, No)","explanation":"finite_sample records insufficient eventual-forall evidence. A scaled1/n^2 series can converge below divergent scaled1/n; signed terms violate the nonnegative premise; an unproved reference status supplies no convergence certificate. Keep the distinct branch field roles, not one padded tuple. The exact result, retaining the requested order, is (divergent_upper, a_n=18/n^2, b_n=18/n, a_n<=b_n, b_diverges_but_a_converges, No)."},"skills":["Comparison Direction, Nonnegativity and Reference Status Cannot Be Omitted"],"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-20261015-advance-20261117-r01-series-comparison-test-0094","batch_id":"batch-20261015-advance-20261117-r01-series-comparison-test-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f854336cfb8a0ae40c8544119194a117967aed0e6123d816a33d11b1c761c7bc","family_id":"qf-mod48-series-comparison-boundary","seed":202611171094,"semantic_template":"series-comparison-boundary|{mode:\"signed_terms\",scale:[19,1]}"},"fields":{"title":"Comparison Direction, Nonnegativity and Reference Status Cannot Be Omitted","stem":"Boundary mode=signed_terms, positive scale=19. Give (mode, series_description, failed_hypothesis, comparison_test_claim).","answer":{"kind":"expression","value":"(signed_terms, a_n=(-1)^n*19/n, nonnegative_hypothesis_fails, No)","explanation":"finite_sample records insufficient eventual-forall evidence. A scaled1/n^2 series can converge below divergent scaled1/n; signed terms violate the nonnegative premise; an unproved reference status supplies no convergence certificate. Keep the distinct branch field roles, not one padded tuple. The exact result, retaining the requested order, is (signed_terms, a_n=(-1)^n*19/n, nonnegative_hypothesis_fails, No)."},"skills":["Comparison Direction, Nonnegativity and Reference Status Cannot Be Omitted"],"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-20261015-advance-20261117-r01-series-comparison-test-0095","batch_id":"batch-20261015-advance-20261117-r01-series-comparison-test-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6a0928eee2e90014a5660e92765a661529ba50b477f18b4a929db55349703098","family_id":"qf-mod48-series-comparison-boundary","seed":202611171095,"semantic_template":"series-comparison-boundary|{mode:\"unknown_reference\",scale:[20,1]}"},"fields":{"title":"Comparison Direction, Nonnegativity and Reference Status Cannot Be Omitted","stem":"Boundary mode=unknown_reference, positive scale=20. Give (mode, scale, reference_status_scope, comparison_test_claim).","answer":{"kind":"expression","value":"(unknown_reference, 20, comparison_holds_reference_status_not_proved, No)","explanation":"finite_sample records insufficient eventual-forall evidence. A scaled1/n^2 series can converge below divergent scaled1/n; signed terms violate the nonnegative premise; an unproved reference status supplies no convergence certificate. Keep the distinct branch field roles, not one padded tuple. The exact result, retaining the requested order, is (unknown_reference, 20, comparison_holds_reference_status_not_proved, No)."},"skills":["Comparison Direction, Nonnegativity and Reference Status Cannot Be Omitted"],"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-20261015-advance-20261117-r01-series-comparison-test-0096","batch_id":"batch-20261015-advance-20261117-r01-series-comparison-test-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"274e8ad1f959e204b04ae175ac8daa0934713b629ecbf66a9e19d4fbc5e31d41","family_id":"qf-mod48-series-comparison-boundary","seed":202611171096,"semantic_template":"series-comparison-boundary|{mode:\"finite_sample\",scale:[21,1]}"},"fields":{"title":"Comparison Direction, Nonnegativity and Reference Status Cannot Be Omitted","stem":"Boundary mode=finite_sample, positive scale=21. Give (mode, scale, sample_scope, eventual_comparison_claim, boundary).","answer":{"kind":"expression","value":"(finite_sample, 21, first_N_terms_compare_next_term_fails, No, finite_samples_do_not_establish_eventual_forall)","explanation":"finite_sample records insufficient eventual-forall evidence. A scaled1/n^2 series can converge below divergent scaled1/n; signed terms violate the nonnegative premise; an unproved reference status supplies no convergence certificate. Keep the distinct branch field roles, not one padded tuple. The exact result, retaining the requested order, is (finite_sample, 21, first_N_terms_compare_next_term_fails, No, finite_samples_do_not_establish_eventual_forall)."},"skills":["Comparison Direction, Nonnegativity and Reference Status Cannot Be Omitted"],"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-20261015-advance-20261117-r01-series-comparison-test-0097","batch_id":"batch-20261015-advance-20261117-r01-series-comparison-test-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7207a1eff4bd24878163d446e17927930bf2996293bb204b3a557b0437a78cc6","family_id":"qf-mod48-series-comparison-boundary","seed":202611171097,"semantic_template":"series-comparison-boundary|{mode:\"divergent_upper\",scale:[22,1]}"},"fields":{"title":"Comparison Direction, Nonnegativity and Reference Status Cannot Be Omitted","stem":"Boundary mode=divergent_upper, positive scale=22. Give (mode, series_a_description, series_b_description, term_comparison, actual_series_status, divergence_inference_claim).","answer":{"kind":"expression","value":"(divergent_upper, a_n=22/n^2, b_n=22/n, a_n<=b_n, b_diverges_but_a_converges, No)","explanation":"finite_sample records insufficient eventual-forall evidence. A scaled1/n^2 series can converge below divergent scaled1/n; signed terms violate the nonnegative premise; an unproved reference status supplies no convergence certificate. Keep the distinct branch field roles, not one padded tuple. The exact result, retaining the requested order, is (divergent_upper, a_n=22/n^2, b_n=22/n, a_n<=b_n, b_diverges_but_a_converges, No)."},"skills":["Comparison Direction, Nonnegativity and Reference Status Cannot Be Omitted"],"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-20261015-advance-20261117-r01-series-comparison-test-0098","batch_id":"batch-20261015-advance-20261117-r01-series-comparison-test-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a5d24be7307aa4e20dbd3d3f39904222a2b8f16dadbc190c2ed435554211e227","family_id":"qf-mod48-series-comparison-boundary","seed":202611171098,"semantic_template":"series-comparison-boundary|{mode:\"signed_terms\",scale:[23,1]}"},"fields":{"title":"Comparison Direction, Nonnegativity and Reference Status Cannot Be Omitted","stem":"Boundary mode=signed_terms, positive scale=23. Give (mode, series_description, failed_hypothesis, comparison_test_claim).","answer":{"kind":"expression","value":"(signed_terms, a_n=(-1)^n*23/n, nonnegative_hypothesis_fails, No)","explanation":"finite_sample records insufficient eventual-forall evidence. A scaled1/n^2 series can converge below divergent scaled1/n; signed terms violate the nonnegative premise; an unproved reference status supplies no convergence certificate. Keep the distinct branch field roles, not one padded tuple. The exact result, retaining the requested order, is (signed_terms, a_n=(-1)^n*23/n, nonnegative_hypothesis_fails, No)."},"skills":["Comparison Direction, Nonnegativity and Reference Status Cannot Be Omitted"],"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-20261015-advance-20261117-r01-series-comparison-test-0099","batch_id":"batch-20261015-advance-20261117-r01-series-comparison-test-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9e1ab07529cb1458a6d51336ea5b0f26e9157f69d26ad610bcbe3b904ce9bf45","family_id":"qf-mod48-series-comparison-boundary","seed":202611171099,"semantic_template":"series-comparison-boundary|{mode:\"unknown_reference\",scale:[24,1]}"},"fields":{"title":"Comparison Direction, Nonnegativity and Reference Status Cannot Be Omitted","stem":"Boundary mode=unknown_reference, positive scale=24. Give (mode, scale, reference_status_scope, comparison_test_claim).","answer":{"kind":"expression","value":"(unknown_reference, 24, comparison_holds_reference_status_not_proved, No)","explanation":"finite_sample records insufficient eventual-forall evidence. A scaled1/n^2 series can converge below divergent scaled1/n; signed terms violate the nonnegative premise; an unproved reference status supplies no convergence certificate. Keep the distinct branch field roles, not one padded tuple. The exact result, retaining the requested order, is (unknown_reference, 24, comparison_holds_reference_status_not_proved, No)."},"skills":["Comparison Direction, Nonnegativity and Reference Status Cannot Be Omitted"],"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-20261015-advance-20261117-r01-series-comparison-test-0100","batch_id":"batch-20261015-advance-20261117-r01-series-comparison-test-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1e283e9e1b3cf61d9d625cd72722e37e8aa0f78d78d68989d20dc068d7bf8ab9","family_id":"qf-mod48-series-comparison-boundary","seed":202611171100,"semantic_template":"series-comparison-boundary|{mode:\"finite_sample\",scale:[25,1]}"},"fields":{"title":"Comparison Direction, Nonnegativity and Reference Status Cannot Be Omitted","stem":"Boundary mode=finite_sample, positive scale=25. Give (mode, scale, sample_scope, eventual_comparison_claim, boundary).","answer":{"kind":"expression","value":"(finite_sample, 25, first_N_terms_compare_next_term_fails, No, finite_samples_do_not_establish_eventual_forall)","explanation":"finite_sample records insufficient eventual-forall evidence. A scaled1/n^2 series can converge below divergent scaled1/n; signed terms violate the nonnegative premise; an unproved reference status supplies no convergence certificate. Keep the distinct branch field roles, not one padded tuple. The exact result, retaining the requested order, is (finite_sample, 25, first_N_terms_compare_next_term_fails, No, finite_samples_do_not_establish_eventual_forall)."},"skills":["Comparison Direction, Nonnegativity and Reference Status Cannot Be Omitted"],"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."}}