{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0001","batch_id":"batch-20261014-advance-20261104-r02-uniform-convergence-of-functions-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"efefc04e60c319ef4adac962c95c87f73f23ec0e27395e8d3311ecb894a1b5ec","family_id":"qf-mod35-uniform-finite-domain-witness","seed":202611042001,"semantic_template":"qf-mod35-uniform-finite-domain-witness/{\"coefficients\":[1,-2],\"epsilon\":[1,10],\"limit\":[0,1],\"shift\":2}"},"fields":{"title":"One Strict Witness Controls All Points of the Finite Domain","stem":"Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[0, 1], coefficients=[1, -2], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. Positive epsilon=1/10. Give (epsilon, pointwise_witnesses, N, same_N_for_all_points, certificate).","answer":{"kind":"expression","value":"(1/10, [[0, 9], [1, 19]], 19, Yes, same_N_for_all_finite_domain_points)","explanation":"For each point find the least nonnegative N_x with |coefficient[x]|/(N_x+shift)<epsilon, then take their finite maximum. Increasing denominators preserve the strict bound for every later n. The exact result, retaining the requested order, is (1/10, [[0, 9], [1, 19]], 19, Yes, same_N_for_all_finite_domain_points)."},"skills":["One Strict Witness Controls All Points of the Finite Domain"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0002","batch_id":"batch-20261014-advance-20261104-r02-uniform-convergence-of-functions-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"72765a525bb24e07c248f7889669642549e503f9fd0f534aad391ea07f5aaa14","family_id":"qf-mod35-uniform-finite-domain-witness","seed":202611042002,"semantic_template":"qf-mod35-uniform-finite-domain-witness/{\"coefficients\":[2,-3],\"epsilon\":[1,11],\"limit\":[1,2],\"shift\":2}"},"fields":{"title":"One Strict Witness Controls All Points of the Finite Domain","stem":"Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[1, 2], coefficients=[2, -3], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. Positive epsilon=1/11. Give (epsilon, pointwise_witnesses, N, same_N_for_all_points, certificate).","answer":{"kind":"expression","value":"(1/11, [[0, 21], [1, 32]], 32, Yes, same_N_for_all_finite_domain_points)","explanation":"For each point find the least nonnegative N_x with |coefficient[x]|/(N_x+shift)<epsilon, then take their finite maximum. Increasing denominators preserve the strict bound for every later n. The exact result, retaining the requested order, is (1/11, [[0, 21], [1, 32]], 32, Yes, same_N_for_all_finite_domain_points)."},"skills":["One Strict Witness Controls All Points of the Finite Domain"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0003","batch_id":"batch-20261014-advance-20261104-r02-uniform-convergence-of-functions-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"65a711fe24759c8e4337da7cf37058b32e695bd9a3816c529657f4d6d578ead5","family_id":"qf-mod35-uniform-finite-domain-witness","seed":202611042003,"semantic_template":"qf-mod35-uniform-finite-domain-witness/{\"coefficients\":[3,-4],\"epsilon\":[1,12],\"limit\":[2,3],\"shift\":2}"},"fields":{"title":"One Strict Witness Controls All Points of the Finite Domain","stem":"Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[2, 3], coefficients=[3, -4], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. Positive epsilon=1/12. Give (epsilon, pointwise_witnesses, N, same_N_for_all_points, certificate).","answer":{"kind":"expression","value":"(1/12, [[0, 35], [1, 47]], 47, Yes, same_N_for_all_finite_domain_points)","explanation":"For each point find the least nonnegative N_x with |coefficient[x]|/(N_x+shift)<epsilon, then take their finite maximum. Increasing denominators preserve the strict bound for every later n. The exact result, retaining the requested order, is (1/12, [[0, 35], [1, 47]], 47, Yes, same_N_for_all_finite_domain_points)."},"skills":["One Strict Witness Controls All Points of the Finite Domain"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0004","batch_id":"batch-20261014-advance-20261104-r02-uniform-convergence-of-functions-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d18817e33db8f32e8a9b431441f70fab67e61db0159103c31c701b6cfb000223","family_id":"qf-mod35-uniform-finite-domain-witness","seed":202611042004,"semantic_template":"qf-mod35-uniform-finite-domain-witness/{\"coefficients\":[4,-5],\"epsilon\":[1,13],\"limit\":[3,4],\"shift\":2}"},"fields":{"title":"One Strict Witness Controls All Points of the Finite Domain","stem":"Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[3, 4], coefficients=[4, -5], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. Positive epsilon=1/13. Give (epsilon, pointwise_witnesses, N, same_N_for_all_points, certificate).","answer":{"kind":"expression","value":"(1/13, [[0, 51], [1, 64]], 64, Yes, same_N_for_all_finite_domain_points)","explanation":"For each point find the least nonnegative N_x with |coefficient[x]|/(N_x+shift)<epsilon, then take their finite maximum. Increasing denominators preserve the strict bound for every later n. The exact result, retaining the requested order, is (1/13, [[0, 51], [1, 64]], 64, Yes, same_N_for_all_finite_domain_points)."},"skills":["One Strict Witness Controls All Points of the Finite Domain"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0005","batch_id":"batch-20261014-advance-20261104-r02-uniform-convergence-of-functions-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9b50f5c51c2d055fec5a24698d0ecd948ae193f6680e345ec6d5d4c2d5cb04b0","family_id":"qf-mod35-uniform-finite-domain-witness","seed":202611042005,"semantic_template":"qf-mod35-uniform-finite-domain-witness/{\"coefficients\":[5,-6],\"epsilon\":[1,14],\"limit\":[4,5],\"shift\":2}"},"fields":{"title":"One Strict Witness Controls All Points of the Finite Domain","stem":"Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[4, 5], coefficients=[5, -6], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. Positive epsilon=1/14. Give (epsilon, pointwise_witnesses, N, same_N_for_all_points, certificate).","answer":{"kind":"expression","value":"(1/14, [[0, 69], [1, 83]], 83, Yes, same_N_for_all_finite_domain_points)","explanation":"For each point find the least nonnegative N_x with |coefficient[x]|/(N_x+shift)<epsilon, then take their finite maximum. Increasing denominators preserve the strict bound for every later n. The exact result, retaining the requested order, is (1/14, [[0, 69], [1, 83]], 83, Yes, same_N_for_all_finite_domain_points)."},"skills":["One Strict Witness Controls All Points of the Finite Domain"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0006","batch_id":"batch-20261014-advance-20261104-r02-uniform-convergence-of-functions-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1342a602d003860be9e7a23eaa992e900f5283511427ff57cfe6aae60bea4115","family_id":"qf-mod35-uniform-finite-domain-witness","seed":202611042006,"semantic_template":"qf-mod35-uniform-finite-domain-witness/{\"coefficients\":[6,-7],\"epsilon\":[1,15],\"limit\":[5,6],\"shift\":2}"},"fields":{"title":"One Strict Witness Controls All Points of the Finite Domain","stem":"Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[5, 6], coefficients=[6, -7], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. Positive epsilon=1/15. Give (epsilon, pointwise_witnesses, N, same_N_for_all_points, certificate).","answer":{"kind":"expression","value":"(1/15, [[0, 89], [1, 104]], 104, Yes, same_N_for_all_finite_domain_points)","explanation":"For each point find the least nonnegative N_x with |coefficient[x]|/(N_x+shift)<epsilon, then take their finite maximum. Increasing denominators preserve the strict bound for every later n. The exact result, retaining the requested order, is (1/15, [[0, 89], [1, 104]], 104, Yes, same_N_for_all_finite_domain_points)."},"skills":["One Strict Witness Controls All Points of the Finite Domain"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0007","batch_id":"batch-20261014-advance-20261104-r02-uniform-convergence-of-functions-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b8e478b04212199448b070279537defe64e07460d37055fee411859e234c004d","family_id":"qf-mod35-uniform-finite-domain-witness","seed":202611042007,"semantic_template":"qf-mod35-uniform-finite-domain-witness/{\"coefficients\":[7,-8],\"epsilon\":[1,16],\"limit\":[6,7],\"shift\":2}"},"fields":{"title":"One Strict Witness Controls All Points of the Finite Domain","stem":"Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[6, 7], coefficients=[7, -8], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. Positive epsilon=1/16. Give (epsilon, pointwise_witnesses, N, same_N_for_all_points, certificate).","answer":{"kind":"expression","value":"(1/16, [[0, 111], [1, 127]], 127, Yes, same_N_for_all_finite_domain_points)","explanation":"For each point find the least nonnegative N_x with |coefficient[x]|/(N_x+shift)<epsilon, then take their finite maximum. Increasing denominators preserve the strict bound for every later n. The exact result, retaining the requested order, is (1/16, [[0, 111], [1, 127]], 127, Yes, same_N_for_all_finite_domain_points)."},"skills":["One Strict Witness Controls All Points of the Finite Domain"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0008","batch_id":"batch-20261014-advance-20261104-r02-uniform-convergence-of-functions-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2e0de9713daa3b4ffb0c85a1dfaed0a0164ac7a87f1a24ce0b1346bcb5ea0550","family_id":"qf-mod35-uniform-finite-domain-witness","seed":202611042008,"semantic_template":"qf-mod35-uniform-finite-domain-witness/{\"coefficients\":[8,-9],\"epsilon\":[1,17],\"limit\":[7,8],\"shift\":2}"},"fields":{"title":"One Strict Witness Controls All Points of the Finite Domain","stem":"Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[7, 8], coefficients=[8, -9], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. Positive epsilon=1/17. Give (epsilon, pointwise_witnesses, N, same_N_for_all_points, certificate).","answer":{"kind":"expression","value":"(1/17, [[0, 135], [1, 152]], 152, Yes, same_N_for_all_finite_domain_points)","explanation":"For each point find the least nonnegative N_x with |coefficient[x]|/(N_x+shift)<epsilon, then take their finite maximum. Increasing denominators preserve the strict bound for every later n. The exact result, retaining the requested order, is (1/17, [[0, 135], [1, 152]], 152, Yes, same_N_for_all_finite_domain_points)."},"skills":["One Strict Witness Controls All Points of the Finite Domain"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0009","batch_id":"batch-20261014-advance-20261104-r02-uniform-convergence-of-functions-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9b51d1c716133122531eb22d49f15d8079f74e5ecb824f6be4d26c4bdcc46f29","family_id":"qf-mod35-uniform-finite-domain-witness","seed":202611042009,"semantic_template":"qf-mod35-uniform-finite-domain-witness/{\"coefficients\":[9,-10],\"epsilon\":[1,18],\"limit\":[8,9],\"shift\":2}"},"fields":{"title":"One Strict Witness Controls All Points of the Finite Domain","stem":"Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[8, 9], coefficients=[9, -10], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. Positive epsilon=1/18. Give (epsilon, pointwise_witnesses, N, same_N_for_all_points, certificate).","answer":{"kind":"expression","value":"(1/18, [[0, 161], [1, 179]], 179, Yes, same_N_for_all_finite_domain_points)","explanation":"For each point find the least nonnegative N_x with |coefficient[x]|/(N_x+shift)<epsilon, then take their finite maximum. Increasing denominators preserve the strict bound for every later n. The exact result, retaining the requested order, is (1/18, [[0, 161], [1, 179]], 179, Yes, same_N_for_all_finite_domain_points)."},"skills":["One Strict Witness Controls All Points of the Finite Domain"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0010","batch_id":"batch-20261014-advance-20261104-r02-uniform-convergence-of-functions-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3ff4400017bb3336bbfc61ac29cc7773e3c3cdd65bff385b211dcd3466c0ea4b","family_id":"qf-mod35-uniform-finite-domain-witness","seed":202611042010,"semantic_template":"qf-mod35-uniform-finite-domain-witness/{\"coefficients\":[10,-11],\"epsilon\":[1,19],\"limit\":[9,10],\"shift\":2}"},"fields":{"title":"One Strict Witness Controls All Points of the Finite Domain","stem":"Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[9, 10], coefficients=[10, -11], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. Positive epsilon=1/19. Give (epsilon, pointwise_witnesses, N, same_N_for_all_points, certificate).","answer":{"kind":"expression","value":"(1/19, [[0, 189], [1, 208]], 208, Yes, same_N_for_all_finite_domain_points)","explanation":"For each point find the least nonnegative N_x with |coefficient[x]|/(N_x+shift)<epsilon, then take their finite maximum. Increasing denominators preserve the strict bound for every later n. The exact result, retaining the requested order, is (1/19, [[0, 189], [1, 208]], 208, Yes, same_N_for_all_finite_domain_points)."},"skills":["One Strict Witness Controls All Points of the Finite Domain"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0011","batch_id":"batch-20261014-advance-20261104-r02-uniform-convergence-of-functions-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ed255653fe30dc7cb678001b8f05989355a4dae6b4756498c3c148b7cd75f037","family_id":"qf-mod35-uniform-finite-domain-witness","seed":202611042011,"semantic_template":"qf-mod35-uniform-finite-domain-witness/{\"coefficients\":[11,-12],\"epsilon\":[1,20],\"limit\":[10,11],\"shift\":2}"},"fields":{"title":"One Strict Witness Controls All Points of the Finite Domain","stem":"Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[10, 11], coefficients=[11, -12], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. Positive epsilon=1/20. Give (epsilon, pointwise_witnesses, N, same_N_for_all_points, certificate).","answer":{"kind":"expression","value":"(1/20, [[0, 219], [1, 239]], 239, Yes, same_N_for_all_finite_domain_points)","explanation":"For each point find the least nonnegative N_x with |coefficient[x]|/(N_x+shift)<epsilon, then take their finite maximum. Increasing denominators preserve the strict bound for every later n. The exact result, retaining the requested order, is (1/20, [[0, 219], [1, 239]], 239, Yes, same_N_for_all_finite_domain_points)."},"skills":["One Strict Witness Controls All Points of the Finite Domain"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0012","batch_id":"batch-20261014-advance-20261104-r02-uniform-convergence-of-functions-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1b2398fc7cf5418278cb96361ca3bc09fcfa07f20b2cff213251a26bacbb361f","family_id":"qf-mod35-uniform-finite-domain-witness","seed":202611042012,"semantic_template":"qf-mod35-uniform-finite-domain-witness/{\"coefficients\":[12,-13],\"epsilon\":[1,21],\"limit\":[11,12],\"shift\":2}"},"fields":{"title":"One Strict Witness Controls All Points of the Finite Domain","stem":"Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[11, 12], coefficients=[12, -13], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. Positive epsilon=1/21. Give (epsilon, pointwise_witnesses, N, same_N_for_all_points, certificate).","answer":{"kind":"expression","value":"(1/21, [[0, 251], [1, 272]], 272, Yes, same_N_for_all_finite_domain_points)","explanation":"For each point find the least nonnegative N_x with |coefficient[x]|/(N_x+shift)<epsilon, then take their finite maximum. Increasing denominators preserve the strict bound for every later n. The exact result, retaining the requested order, is (1/21, [[0, 251], [1, 272]], 272, Yes, same_N_for_all_finite_domain_points)."},"skills":["One Strict Witness Controls All Points of the Finite Domain"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0013","batch_id":"batch-20261014-advance-20261104-r02-uniform-convergence-of-functions-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"aa26b8c406e3e1ead34abc61ee757ca9ef567d6743f3e4d942338812285923cb","family_id":"qf-mod35-uniform-finite-domain-witness","seed":202611042013,"semantic_template":"qf-mod35-uniform-finite-domain-witness/{\"coefficients\":[13,-14],\"epsilon\":[1,22],\"limit\":[12,13],\"shift\":2}"},"fields":{"title":"One Strict Witness Controls All Points of the Finite Domain","stem":"Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[12, 13], coefficients=[13, -14], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. Positive epsilon=1/22. Give (epsilon, pointwise_witnesses, N, same_N_for_all_points, certificate).","answer":{"kind":"expression","value":"(1/22, [[0, 285], [1, 307]], 307, Yes, same_N_for_all_finite_domain_points)","explanation":"For each point find the least nonnegative N_x with |coefficient[x]|/(N_x+shift)<epsilon, then take their finite maximum. Increasing denominators preserve the strict bound for every later n. The exact result, retaining the requested order, is (1/22, [[0, 285], [1, 307]], 307, Yes, same_N_for_all_finite_domain_points)."},"skills":["One Strict Witness Controls All Points of the Finite Domain"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0014","batch_id":"batch-20261014-advance-20261104-r02-uniform-convergence-of-functions-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fc9f6b3b31c3d59d18aa63c722ab7b769ac3cea19b7bb485c3c2983451c018d1","family_id":"qf-mod35-uniform-finite-domain-witness","seed":202611042014,"semantic_template":"qf-mod35-uniform-finite-domain-witness/{\"coefficients\":[14,-15],\"epsilon\":[1,23],\"limit\":[13,14],\"shift\":2}"},"fields":{"title":"One Strict Witness Controls All Points of the Finite Domain","stem":"Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[13, 14], coefficients=[14, -15], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. Positive epsilon=1/23. Give (epsilon, pointwise_witnesses, N, same_N_for_all_points, certificate).","answer":{"kind":"expression","value":"(1/23, [[0, 321], [1, 344]], 344, Yes, same_N_for_all_finite_domain_points)","explanation":"For each point find the least nonnegative N_x with |coefficient[x]|/(N_x+shift)<epsilon, then take their finite maximum. Increasing denominators preserve the strict bound for every later n. The exact result, retaining the requested order, is (1/23, [[0, 321], [1, 344]], 344, Yes, same_N_for_all_finite_domain_points)."},"skills":["One Strict Witness Controls All Points of the Finite Domain"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0015","batch_id":"batch-20261014-advance-20261104-r02-uniform-convergence-of-functions-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8cc78fe3d3bab2b755fb68e32c06b53e2a2e20e876c543aebba6bd8a8e347d2f","family_id":"qf-mod35-uniform-finite-domain-witness","seed":202611042015,"semantic_template":"qf-mod35-uniform-finite-domain-witness/{\"coefficients\":[15,-16],\"epsilon\":[1,24],\"limit\":[14,15],\"shift\":2}"},"fields":{"title":"One Strict Witness Controls All Points of the Finite Domain","stem":"Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[14, 15], coefficients=[15, -16], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. Positive epsilon=1/24. Give (epsilon, pointwise_witnesses, N, same_N_for_all_points, certificate).","answer":{"kind":"expression","value":"(1/24, [[0, 359], [1, 383]], 383, Yes, same_N_for_all_finite_domain_points)","explanation":"For each point find the least nonnegative N_x with |coefficient[x]|/(N_x+shift)<epsilon, then take their finite maximum. Increasing denominators preserve the strict bound for every later n. The exact result, retaining the requested order, is (1/24, [[0, 359], [1, 383]], 383, Yes, same_N_for_all_finite_domain_points)."},"skills":["One Strict Witness Controls All Points of the Finite Domain"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0016","batch_id":"batch-20261014-advance-20261104-r02-uniform-convergence-of-functions-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b3c377717b08da6ea102521f44a485e1331f496d6782fda60caed28110faa63f","family_id":"qf-mod35-uniform-finite-domain-witness","seed":202611042016,"semantic_template":"qf-mod35-uniform-finite-domain-witness/{\"coefficients\":[16,-17],\"epsilon\":[1,25],\"limit\":[15,16],\"shift\":2}"},"fields":{"title":"One Strict Witness Controls All Points of the Finite Domain","stem":"Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[15, 16], coefficients=[16, -17], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. Positive epsilon=1/25. Give (epsilon, pointwise_witnesses, N, same_N_for_all_points, certificate).","answer":{"kind":"expression","value":"(1/25, [[0, 399], [1, 424]], 424, Yes, same_N_for_all_finite_domain_points)","explanation":"For each point find the least nonnegative N_x with |coefficient[x]|/(N_x+shift)<epsilon, then take their finite maximum. Increasing denominators preserve the strict bound for every later n. The exact result, retaining the requested order, is (1/25, [[0, 399], [1, 424]], 424, Yes, same_N_for_all_finite_domain_points)."},"skills":["One Strict Witness Controls All Points of the Finite Domain"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0017","batch_id":"batch-20261014-advance-20261104-r02-uniform-convergence-of-functions-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f00c0f9998ca796a039816058ea68b9aa5d74445415cd2b8ade3ff27576477cd","family_id":"qf-mod35-uniform-finite-domain-witness","seed":202611042017,"semantic_template":"qf-mod35-uniform-finite-domain-witness/{\"coefficients\":[17,-18],\"epsilon\":[1,26],\"limit\":[16,17],\"shift\":2}"},"fields":{"title":"One Strict Witness Controls All Points of the Finite Domain","stem":"Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[16, 17], coefficients=[17, -18], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. Positive epsilon=1/26. Give (epsilon, pointwise_witnesses, N, same_N_for_all_points, certificate).","answer":{"kind":"expression","value":"(1/26, [[0, 441], [1, 467]], 467, Yes, same_N_for_all_finite_domain_points)","explanation":"For each point find the least nonnegative N_x with |coefficient[x]|/(N_x+shift)<epsilon, then take their finite maximum. Increasing denominators preserve the strict bound for every later n. The exact result, retaining the requested order, is (1/26, [[0, 441], [1, 467]], 467, Yes, same_N_for_all_finite_domain_points)."},"skills":["One Strict Witness Controls All Points of the Finite Domain"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0018","batch_id":"batch-20261014-advance-20261104-r02-uniform-convergence-of-functions-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dadfda705ffe10cb4f8a2d8b6a685d8824beca6897c37e87e61b80f852739249","family_id":"qf-mod35-uniform-finite-domain-witness","seed":202611042018,"semantic_template":"qf-mod35-uniform-finite-domain-witness/{\"coefficients\":[18,-19],\"epsilon\":[1,27],\"limit\":[17,18],\"shift\":2}"},"fields":{"title":"One Strict Witness Controls All Points of the Finite Domain","stem":"Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[17, 18], coefficients=[18, -19], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. Positive epsilon=1/27. Give (epsilon, pointwise_witnesses, N, same_N_for_all_points, certificate).","answer":{"kind":"expression","value":"(1/27, [[0, 485], [1, 512]], 512, Yes, same_N_for_all_finite_domain_points)","explanation":"For each point find the least nonnegative N_x with |coefficient[x]|/(N_x+shift)<epsilon, then take their finite maximum. Increasing denominators preserve the strict bound for every later n. The exact result, retaining the requested order, is (1/27, [[0, 485], [1, 512]], 512, Yes, same_N_for_all_finite_domain_points)."},"skills":["One Strict Witness Controls All Points of the Finite Domain"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0019","batch_id":"batch-20261014-advance-20261104-r02-uniform-convergence-of-functions-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f963c1d01d570cf934a683138d805a7849b356467fc7cf84ce89228a04651b80","family_id":"qf-mod35-uniform-finite-domain-witness","seed":202611042019,"semantic_template":"qf-mod35-uniform-finite-domain-witness/{\"coefficients\":[19,-20],\"epsilon\":[1,28],\"limit\":[18,19],\"shift\":2}"},"fields":{"title":"One Strict Witness Controls All Points of the Finite Domain","stem":"Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[18, 19], coefficients=[19, -20], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. Positive epsilon=1/28. Give (epsilon, pointwise_witnesses, N, same_N_for_all_points, certificate).","answer":{"kind":"expression","value":"(1/28, [[0, 531], [1, 559]], 559, Yes, same_N_for_all_finite_domain_points)","explanation":"For each point find the least nonnegative N_x with |coefficient[x]|/(N_x+shift)<epsilon, then take their finite maximum. Increasing denominators preserve the strict bound for every later n. The exact result, retaining the requested order, is (1/28, [[0, 531], [1, 559]], 559, Yes, same_N_for_all_finite_domain_points)."},"skills":["One Strict Witness Controls All Points of the Finite Domain"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0020","batch_id":"batch-20261014-advance-20261104-r02-uniform-convergence-of-functions-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8b29bb74cb5b881c24ca55894521e09bd13feeb03145dce4b6be3f2ea6c62e43","family_id":"qf-mod35-uniform-finite-domain-witness","seed":202611042020,"semantic_template":"qf-mod35-uniform-finite-domain-witness/{\"coefficients\":[20,-21],\"epsilon\":[1,29],\"limit\":[19,20],\"shift\":2}"},"fields":{"title":"One Strict Witness Controls All Points of the Finite Domain","stem":"Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[19, 20], coefficients=[20, -21], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. Positive epsilon=1/29. Give (epsilon, pointwise_witnesses, N, same_N_for_all_points, certificate).","answer":{"kind":"expression","value":"(1/29, [[0, 579], [1, 608]], 608, Yes, same_N_for_all_finite_domain_points)","explanation":"For each point find the least nonnegative N_x with |coefficient[x]|/(N_x+shift)<epsilon, then take their finite maximum. Increasing denominators preserve the strict bound for every later n. The exact result, retaining the requested order, is (1/29, [[0, 579], [1, 608]], 608, Yes, same_N_for_all_finite_domain_points)."},"skills":["One Strict Witness Controls All Points of the Finite Domain"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0021","batch_id":"batch-20261014-advance-20261104-r02-uniform-convergence-of-functions-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0832db89a7d6cc6a9887a3e39ce208e518f4fbe91cadbd2237c0b1873042f571","family_id":"qf-mod35-uniform-finite-domain-witness","seed":202611042021,"semantic_template":"qf-mod35-uniform-finite-domain-witness/{\"coefficients\":[21,-22],\"epsilon\":[1,30],\"limit\":[20,21],\"shift\":2}"},"fields":{"title":"One Strict Witness Controls All Points of the Finite Domain","stem":"Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[20, 21], coefficients=[21, -22], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. Positive epsilon=1/30. Give (epsilon, pointwise_witnesses, N, same_N_for_all_points, certificate).","answer":{"kind":"expression","value":"(1/30, [[0, 629], [1, 659]], 659, Yes, same_N_for_all_finite_domain_points)","explanation":"For each point find the least nonnegative N_x with |coefficient[x]|/(N_x+shift)<epsilon, then take their finite maximum. Increasing denominators preserve the strict bound for every later n. The exact result, retaining the requested order, is (1/30, [[0, 629], [1, 659]], 659, Yes, same_N_for_all_finite_domain_points)."},"skills":["One Strict Witness Controls All Points of the Finite Domain"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0022","batch_id":"batch-20261014-advance-20261104-r02-uniform-convergence-of-functions-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d11f213683c75382d165d0e640a927d46eefaca2439d293806a81448c7df7705","family_id":"qf-mod35-uniform-finite-domain-witness","seed":202611042022,"semantic_template":"qf-mod35-uniform-finite-domain-witness/{\"coefficients\":[22,-23],\"epsilon\":[1,31],\"limit\":[21,22],\"shift\":2}"},"fields":{"title":"One Strict Witness Controls All Points of the Finite Domain","stem":"Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[21, 22], coefficients=[22, -23], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. Positive epsilon=1/31. Give (epsilon, pointwise_witnesses, N, same_N_for_all_points, certificate).","answer":{"kind":"expression","value":"(1/31, [[0, 681], [1, 712]], 712, Yes, same_N_for_all_finite_domain_points)","explanation":"For each point find the least nonnegative N_x with |coefficient[x]|/(N_x+shift)<epsilon, then take their finite maximum. Increasing denominators preserve the strict bound for every later n. The exact result, retaining the requested order, is (1/31, [[0, 681], [1, 712]], 712, Yes, same_N_for_all_finite_domain_points)."},"skills":["One Strict Witness Controls All Points of the Finite Domain"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0023","batch_id":"batch-20261014-advance-20261104-r02-uniform-convergence-of-functions-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"95321cab7b11757ba1b116d58bec1a0851c41cb649100014fed419ac6ea91ec6","family_id":"qf-mod35-uniform-finite-domain-witness","seed":202611042023,"semantic_template":"qf-mod35-uniform-finite-domain-witness/{\"coefficients\":[23,-24],\"epsilon\":[1,32],\"limit\":[22,23],\"shift\":2}"},"fields":{"title":"One Strict Witness Controls All Points of the Finite Domain","stem":"Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[22, 23], coefficients=[23, -24], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. Positive epsilon=1/32. Give (epsilon, pointwise_witnesses, N, same_N_for_all_points, certificate).","answer":{"kind":"expression","value":"(1/32, [[0, 735], [1, 767]], 767, Yes, same_N_for_all_finite_domain_points)","explanation":"For each point find the least nonnegative N_x with |coefficient[x]|/(N_x+shift)<epsilon, then take their finite maximum. Increasing denominators preserve the strict bound for every later n. The exact result, retaining the requested order, is (1/32, [[0, 735], [1, 767]], 767, Yes, same_N_for_all_finite_domain_points)."},"skills":["One Strict Witness Controls All Points of the Finite Domain"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0024","batch_id":"batch-20261014-advance-20261104-r02-uniform-convergence-of-functions-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dc83d807f829b069e061b3e959bf17630d8d34ac7c4a37ebdd373d8196a14bc3","family_id":"qf-mod35-uniform-finite-domain-witness","seed":202611042024,"semantic_template":"qf-mod35-uniform-finite-domain-witness/{\"coefficients\":[24,-25],\"epsilon\":[1,33],\"limit\":[23,24],\"shift\":2}"},"fields":{"title":"One Strict Witness Controls All Points of the Finite Domain","stem":"Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[23, 24], coefficients=[24, -25], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. Positive epsilon=1/33. Give (epsilon, pointwise_witnesses, N, same_N_for_all_points, certificate).","answer":{"kind":"expression","value":"(1/33, [[0, 791], [1, 824]], 824, Yes, same_N_for_all_finite_domain_points)","explanation":"For each point find the least nonnegative N_x with |coefficient[x]|/(N_x+shift)<epsilon, then take their finite maximum. Increasing denominators preserve the strict bound for every later n. The exact result, retaining the requested order, is (1/33, [[0, 791], [1, 824]], 824, Yes, same_N_for_all_finite_domain_points)."},"skills":["One Strict Witness Controls All Points of the Finite Domain"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0025","batch_id":"batch-20261014-advance-20261104-r02-uniform-convergence-of-functions-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"57b35b8f5b0c9c134442efca63163acb2e7a318a20f284fcf870356a42ddb89e","family_id":"qf-mod35-uniform-finite-domain-witness","seed":202611042025,"semantic_template":"qf-mod35-uniform-finite-domain-witness/{\"coefficients\":[25,-26],\"epsilon\":[1,34],\"limit\":[24,25],\"shift\":2}"},"fields":{"title":"One Strict Witness Controls All Points of the Finite Domain","stem":"Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[24, 25], coefficients=[25, -26], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. Positive epsilon=1/34. Give (epsilon, pointwise_witnesses, N, same_N_for_all_points, certificate).","answer":{"kind":"expression","value":"(1/34, [[0, 849], [1, 883]], 883, Yes, same_N_for_all_finite_domain_points)","explanation":"For each point find the least nonnegative N_x with |coefficient[x]|/(N_x+shift)<epsilon, then take their finite maximum. Increasing denominators preserve the strict bound for every later n. The exact result, retaining the requested order, is (1/34, [[0, 849], [1, 883]], 883, Yes, same_N_for_all_finite_domain_points)."},"skills":["One Strict Witness Controls All Points of the Finite Domain"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0026","batch_id":"batch-20261014-advance-20261104-r02-uniform-convergence-of-functions-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fd98f01d7b8f43c48c4ab87ae2e74be2220518f7a718bbda4074b2fe2aad7639","family_id":"qf-mod35-uniform-sup-gap","seed":202611042026,"semantic_template":"qf-mod35-uniform-sup-gap/{\"coefficients\":[1,-2],\"limit\":[0,1],\"n\":1,\"shift\":2}"},"fields":{"title":"Compute the Exact Finite-Domain Supremum without Reformatting the Source Ratio","stem":"Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[0, 1], coefficients=[1, -2], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. n=1. Give (n, sup_gap, certificate).","answer":{"kind":"expression","value":"(1, 2/3, exact_finite_domain_supremum)","explanation":"The exact supremum is max_x |coefficient[x]|/(n+shift). Retain the source direct-ratio text, which may be unreduced, instead of changing its numerical bytes during translation. The exact result, retaining the requested order, is (1, 2/3, exact_finite_domain_supremum)."},"skills":["Compute the Exact Finite-Domain Supremum without Reformatting the Source Ratio"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0027","batch_id":"batch-20261014-advance-20261104-r02-uniform-convergence-of-functions-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"33e4efa544025856a473ac7a35a7d99ed6a484402cfcd4e59c14b8f29373b66b","family_id":"qf-mod35-uniform-sup-gap","seed":202611042027,"semantic_template":"qf-mod35-uniform-sup-gap/{\"coefficients\":[2,-3],\"limit\":[1,2],\"n\":2,\"shift\":2}"},"fields":{"title":"Compute the Exact Finite-Domain Supremum without Reformatting the Source Ratio","stem":"Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[1, 2], coefficients=[2, -3], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. n=2. Give (n, sup_gap, certificate).","answer":{"kind":"expression","value":"(2, 3/4, exact_finite_domain_supremum)","explanation":"The exact supremum is max_x |coefficient[x]|/(n+shift). Retain the source direct-ratio text, which may be unreduced, instead of changing its numerical bytes during translation. The exact result, retaining the requested order, is (2, 3/4, exact_finite_domain_supremum)."},"skills":["Compute the Exact Finite-Domain Supremum without Reformatting the Source Ratio"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0028","batch_id":"batch-20261014-advance-20261104-r02-uniform-convergence-of-functions-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4b6eaa2e7b85b72ee6ec7b88db333fe9c82153b0745eede9865369cb660c3bfe","family_id":"qf-mod35-uniform-sup-gap","seed":202611042028,"semantic_template":"qf-mod35-uniform-sup-gap/{\"coefficients\":[3,-4],\"limit\":[2,3],\"n\":3,\"shift\":2}"},"fields":{"title":"Compute the Exact Finite-Domain Supremum without Reformatting the Source Ratio","stem":"Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[2, 3], coefficients=[3, -4], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. n=3. Give (n, sup_gap, certificate).","answer":{"kind":"expression","value":"(3, 4/5, exact_finite_domain_supremum)","explanation":"The exact supremum is max_x |coefficient[x]|/(n+shift). Retain the source direct-ratio text, which may be unreduced, instead of changing its numerical bytes during translation. The exact result, retaining the requested order, is (3, 4/5, exact_finite_domain_supremum)."},"skills":["Compute the Exact Finite-Domain Supremum without Reformatting the Source Ratio"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0029","batch_id":"batch-20261014-advance-20261104-r02-uniform-convergence-of-functions-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"da6342c16b8da1131cbf10c8dc23f614914b2e35f56c4e3c55d263b77fb22437","family_id":"qf-mod35-uniform-sup-gap","seed":202611042029,"semantic_template":"qf-mod35-uniform-sup-gap/{\"coefficients\":[4,-5],\"limit\":[3,4],\"n\":4,\"shift\":2}"},"fields":{"title":"Compute the Exact Finite-Domain Supremum without Reformatting the Source Ratio","stem":"Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[3, 4], coefficients=[4, -5], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. n=4. Give (n, sup_gap, certificate).","answer":{"kind":"expression","value":"(4, 5/6, exact_finite_domain_supremum)","explanation":"The exact supremum is max_x |coefficient[x]|/(n+shift). Retain the source direct-ratio text, which may be unreduced, instead of changing its numerical bytes during translation. The exact result, retaining the requested order, is (4, 5/6, exact_finite_domain_supremum)."},"skills":["Compute the Exact Finite-Domain Supremum without Reformatting the Source Ratio"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0030","batch_id":"batch-20261014-advance-20261104-r02-uniform-convergence-of-functions-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"639ebddf0f0e1d6b83a2aa07b9d9a850a94c0206c17b03469d17e415d0cd8c5e","family_id":"qf-mod35-uniform-sup-gap","seed":202611042030,"semantic_template":"qf-mod35-uniform-sup-gap/{\"coefficients\":[5,-6],\"limit\":[4,5],\"n\":5,\"shift\":2}"},"fields":{"title":"Compute the Exact Finite-Domain Supremum without Reformatting the Source Ratio","stem":"Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[4, 5], coefficients=[5, -6], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. n=5. Give (n, sup_gap, certificate).","answer":{"kind":"expression","value":"(5, 6/7, exact_finite_domain_supremum)","explanation":"The exact supremum is max_x |coefficient[x]|/(n+shift). Retain the source direct-ratio text, which may be unreduced, instead of changing its numerical bytes during translation. The exact result, retaining the requested order, is (5, 6/7, exact_finite_domain_supremum)."},"skills":["Compute the Exact Finite-Domain Supremum without Reformatting the Source Ratio"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0031","batch_id":"batch-20261014-advance-20261104-r02-uniform-convergence-of-functions-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5f119f8797cc72f9d35ab132858382e63883d7ff9c6fed5684c9eedc58084f3a","family_id":"qf-mod35-uniform-sup-gap","seed":202611042031,"semantic_template":"qf-mod35-uniform-sup-gap/{\"coefficients\":[6,-7],\"limit\":[5,6],\"n\":6,\"shift\":2}"},"fields":{"title":"Compute the Exact Finite-Domain Supremum without Reformatting the Source Ratio","stem":"Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[5, 6], coefficients=[6, -7], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. n=6. Give (n, sup_gap, certificate).","answer":{"kind":"expression","value":"(6, 7/8, exact_finite_domain_supremum)","explanation":"The exact supremum is max_x |coefficient[x]|/(n+shift). Retain the source direct-ratio text, which may be unreduced, instead of changing its numerical bytes during translation. The exact result, retaining the requested order, is (6, 7/8, exact_finite_domain_supremum)."},"skills":["Compute the Exact Finite-Domain Supremum without Reformatting the Source Ratio"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0032","batch_id":"batch-20261014-advance-20261104-r02-uniform-convergence-of-functions-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5d7d5eca7041082b23e71d727c0829d72cb90aed2bd823c267fd8ece58d2848f","family_id":"qf-mod35-uniform-sup-gap","seed":202611042032,"semantic_template":"qf-mod35-uniform-sup-gap/{\"coefficients\":[7,-8],\"limit\":[6,7],\"n\":7,\"shift\":2}"},"fields":{"title":"Compute the Exact Finite-Domain Supremum without Reformatting the Source Ratio","stem":"Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[6, 7], coefficients=[7, -8], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. n=7. Give (n, sup_gap, certificate).","answer":{"kind":"expression","value":"(7, 8/9, exact_finite_domain_supremum)","explanation":"The exact supremum is max_x |coefficient[x]|/(n+shift). Retain the source direct-ratio text, which may be unreduced, instead of changing its numerical bytes during translation. The exact result, retaining the requested order, is (7, 8/9, exact_finite_domain_supremum)."},"skills":["Compute the Exact Finite-Domain Supremum without Reformatting the Source Ratio"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0033","batch_id":"batch-20261014-advance-20261104-r02-uniform-convergence-of-functions-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"22969517adc5535832da0d3d2e348cc9a045e4499dbe520cfbda5bcc217b7fb0","family_id":"qf-mod35-uniform-sup-gap","seed":202611042033,"semantic_template":"qf-mod35-uniform-sup-gap/{\"coefficients\":[8,-9],\"limit\":[7,8],\"n\":8,\"shift\":2}"},"fields":{"title":"Compute the Exact Finite-Domain Supremum without Reformatting the Source Ratio","stem":"Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[7, 8], coefficients=[8, -9], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. n=8. Give (n, sup_gap, certificate).","answer":{"kind":"expression","value":"(8, 9/10, exact_finite_domain_supremum)","explanation":"The exact supremum is max_x |coefficient[x]|/(n+shift). Retain the source direct-ratio text, which may be unreduced, instead of changing its numerical bytes during translation. The exact result, retaining the requested order, is (8, 9/10, exact_finite_domain_supremum)."},"skills":["Compute the Exact Finite-Domain Supremum without Reformatting the Source Ratio"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0034","batch_id":"batch-20261014-advance-20261104-r02-uniform-convergence-of-functions-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6dbc9c6bd9bf7774a8d92a9b8cdf49e3246ed4b76981697663b185b847edee56","family_id":"qf-mod35-uniform-sup-gap","seed":202611042034,"semantic_template":"qf-mod35-uniform-sup-gap/{\"coefficients\":[9,-10],\"limit\":[8,9],\"n\":9,\"shift\":2}"},"fields":{"title":"Compute the Exact Finite-Domain Supremum without Reformatting the Source Ratio","stem":"Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[8, 9], coefficients=[9, -10], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. n=9. Give (n, sup_gap, certificate).","answer":{"kind":"expression","value":"(9, 10/11, exact_finite_domain_supremum)","explanation":"The exact supremum is max_x |coefficient[x]|/(n+shift). Retain the source direct-ratio text, which may be unreduced, instead of changing its numerical bytes during translation. The exact result, retaining the requested order, is (9, 10/11, exact_finite_domain_supremum)."},"skills":["Compute the Exact Finite-Domain Supremum without Reformatting the Source Ratio"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0035","batch_id":"batch-20261014-advance-20261104-r02-uniform-convergence-of-functions-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"38182227513742bf0633c9c833e4f0b409ec19c85d26e461ceba4701ac68aea6","family_id":"qf-mod35-uniform-sup-gap","seed":202611042035,"semantic_template":"qf-mod35-uniform-sup-gap/{\"coefficients\":[10,-11],\"limit\":[9,10],\"n\":10,\"shift\":2}"},"fields":{"title":"Compute the Exact Finite-Domain Supremum without Reformatting the Source Ratio","stem":"Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[9, 10], coefficients=[10, -11], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. n=10. Give (n, sup_gap, certificate).","answer":{"kind":"expression","value":"(10, 11/12, exact_finite_domain_supremum)","explanation":"The exact supremum is max_x |coefficient[x]|/(n+shift). Retain the source direct-ratio text, which may be unreduced, instead of changing its numerical bytes during translation. The exact result, retaining the requested order, is (10, 11/12, exact_finite_domain_supremum)."},"skills":["Compute the Exact Finite-Domain Supremum without Reformatting the Source Ratio"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0036","batch_id":"batch-20261014-advance-20261104-r02-uniform-convergence-of-functions-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a09aae1948fc82eece4d1a88f89603f8a3beb16e6418f5efa2e1555d570a69a4","family_id":"qf-mod35-uniform-sup-gap","seed":202611042036,"semantic_template":"qf-mod35-uniform-sup-gap/{\"coefficients\":[11,-12],\"limit\":[10,11],\"n\":11,\"shift\":2}"},"fields":{"title":"Compute the Exact Finite-Domain Supremum without Reformatting the Source Ratio","stem":"Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[10, 11], coefficients=[11, -12], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. n=11. Give (n, sup_gap, certificate).","answer":{"kind":"expression","value":"(11, 12/13, exact_finite_domain_supremum)","explanation":"The exact supremum is max_x |coefficient[x]|/(n+shift). Retain the source direct-ratio text, which may be unreduced, instead of changing its numerical bytes during translation. The exact result, retaining the requested order, is (11, 12/13, exact_finite_domain_supremum)."},"skills":["Compute the Exact Finite-Domain Supremum without Reformatting the Source Ratio"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0037","batch_id":"batch-20261014-advance-20261104-r02-uniform-convergence-of-functions-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4a1ec131d3e366a7054df984dd51a0b1a28e2c8ddad377d5f8d133c01d0f0ae1","family_id":"qf-mod35-uniform-sup-gap","seed":202611042037,"semantic_template":"qf-mod35-uniform-sup-gap/{\"coefficients\":[12,-13],\"limit\":[11,12],\"n\":12,\"shift\":2}"},"fields":{"title":"Compute the Exact Finite-Domain Supremum without Reformatting the Source Ratio","stem":"Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[11, 12], coefficients=[12, -13], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. n=12. Give (n, sup_gap, certificate).","answer":{"kind":"expression","value":"(12, 13/14, exact_finite_domain_supremum)","explanation":"The exact supremum is max_x |coefficient[x]|/(n+shift). Retain the source direct-ratio text, which may be unreduced, instead of changing its numerical bytes during translation. The exact result, retaining the requested order, is (12, 13/14, exact_finite_domain_supremum)."},"skills":["Compute the Exact Finite-Domain Supremum without Reformatting the Source Ratio"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0038","batch_id":"batch-20261014-advance-20261104-r02-uniform-convergence-of-functions-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f2d5fffeac0c2c34ec49bedb8c46dfa749de7a677bfa6ea8af5d071173c76788","family_id":"qf-mod35-uniform-sup-gap","seed":202611042038,"semantic_template":"qf-mod35-uniform-sup-gap/{\"coefficients\":[13,-14],\"limit\":[12,13],\"n\":13,\"shift\":2}"},"fields":{"title":"Compute the Exact Finite-Domain Supremum without Reformatting the Source Ratio","stem":"Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[12, 13], coefficients=[13, -14], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. n=13. Give (n, sup_gap, certificate).","answer":{"kind":"expression","value":"(13, 14/15, exact_finite_domain_supremum)","explanation":"The exact supremum is max_x |coefficient[x]|/(n+shift). Retain the source direct-ratio text, which may be unreduced, instead of changing its numerical bytes during translation. The exact result, retaining the requested order, is (13, 14/15, exact_finite_domain_supremum)."},"skills":["Compute the Exact Finite-Domain Supremum without Reformatting the Source Ratio"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0039","batch_id":"batch-20261014-advance-20261104-r02-uniform-convergence-of-functions-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d58196f40ed874a1de084bcf6d84f525412e373dd2719b64b9ef86e3b9d253a6","family_id":"qf-mod35-uniform-sup-gap","seed":202611042039,"semantic_template":"qf-mod35-uniform-sup-gap/{\"coefficients\":[14,-15],\"limit\":[13,14],\"n\":14,\"shift\":2}"},"fields":{"title":"Compute the Exact Finite-Domain Supremum without Reformatting the Source Ratio","stem":"Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[13, 14], coefficients=[14, -15], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. n=14. Give (n, sup_gap, certificate).","answer":{"kind":"expression","value":"(14, 15/16, exact_finite_domain_supremum)","explanation":"The exact supremum is max_x |coefficient[x]|/(n+shift). Retain the source direct-ratio text, which may be unreduced, instead of changing its numerical bytes during translation. The exact result, retaining the requested order, is (14, 15/16, exact_finite_domain_supremum)."},"skills":["Compute the Exact Finite-Domain Supremum without Reformatting the Source Ratio"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0040","batch_id":"batch-20261014-advance-20261104-r02-uniform-convergence-of-functions-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"daf6b9746d377663c721a13c3b0d9f23cfa2fd41a3abec9bffa4a648314340a4","family_id":"qf-mod35-uniform-sup-gap","seed":202611042040,"semantic_template":"qf-mod35-uniform-sup-gap/{\"coefficients\":[15,-16],\"limit\":[14,15],\"n\":15,\"shift\":2}"},"fields":{"title":"Compute the Exact Finite-Domain Supremum without Reformatting the Source Ratio","stem":"Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[14, 15], coefficients=[15, -16], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. n=15. Give (n, sup_gap, certificate).","answer":{"kind":"expression","value":"(15, 16/17, exact_finite_domain_supremum)","explanation":"The exact supremum is max_x |coefficient[x]|/(n+shift). Retain the source direct-ratio text, which may be unreduced, instead of changing its numerical bytes during translation. The exact result, retaining the requested order, is (15, 16/17, exact_finite_domain_supremum)."},"skills":["Compute the Exact Finite-Domain Supremum without Reformatting the Source Ratio"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0041","batch_id":"batch-20261014-advance-20261104-r02-uniform-convergence-of-functions-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"baaee9d55f573e32e646e46215490fbd1a74310cff0635c721862e173192ba6e","family_id":"qf-mod35-uniform-sup-gap","seed":202611042041,"semantic_template":"qf-mod35-uniform-sup-gap/{\"coefficients\":[16,-17],\"limit\":[15,16],\"n\":16,\"shift\":2}"},"fields":{"title":"Compute the Exact Finite-Domain Supremum without Reformatting the Source Ratio","stem":"Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[15, 16], coefficients=[16, -17], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. n=16. Give (n, sup_gap, certificate).","answer":{"kind":"expression","value":"(16, 17/18, exact_finite_domain_supremum)","explanation":"The exact supremum is max_x |coefficient[x]|/(n+shift). Retain the source direct-ratio text, which may be unreduced, instead of changing its numerical bytes during translation. The exact result, retaining the requested order, is (16, 17/18, exact_finite_domain_supremum)."},"skills":["Compute the Exact Finite-Domain Supremum without Reformatting the Source Ratio"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0042","batch_id":"batch-20261014-advance-20261104-r02-uniform-convergence-of-functions-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"80cb3e5bbd8ac8784f3f3652fc181270e6578acf47ca4bf7b5a770e724688059","family_id":"qf-mod35-uniform-sup-gap","seed":202611042042,"semantic_template":"qf-mod35-uniform-sup-gap/{\"coefficients\":[17,-18],\"limit\":[16,17],\"n\":17,\"shift\":2}"},"fields":{"title":"Compute the Exact Finite-Domain Supremum without Reformatting the Source Ratio","stem":"Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[16, 17], coefficients=[17, -18], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. n=17. Give (n, sup_gap, certificate).","answer":{"kind":"expression","value":"(17, 18/19, exact_finite_domain_supremum)","explanation":"The exact supremum is max_x |coefficient[x]|/(n+shift). Retain the source direct-ratio text, which may be unreduced, instead of changing its numerical bytes during translation. The exact result, retaining the requested order, is (17, 18/19, exact_finite_domain_supremum)."},"skills":["Compute the Exact Finite-Domain Supremum without Reformatting the Source Ratio"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0043","batch_id":"batch-20261014-advance-20261104-r02-uniform-convergence-of-functions-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"27b40205c6893038642e6c7c0028798065406227af757e7c2a5189d67efaf918","family_id":"qf-mod35-uniform-sup-gap","seed":202611042043,"semantic_template":"qf-mod35-uniform-sup-gap/{\"coefficients\":[18,-19],\"limit\":[17,18],\"n\":18,\"shift\":2}"},"fields":{"title":"Compute the Exact Finite-Domain Supremum without Reformatting the Source Ratio","stem":"Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[17, 18], coefficients=[18, -19], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. n=18. Give (n, sup_gap, certificate).","answer":{"kind":"expression","value":"(18, 19/20, exact_finite_domain_supremum)","explanation":"The exact supremum is max_x |coefficient[x]|/(n+shift). Retain the source direct-ratio text, which may be unreduced, instead of changing its numerical bytes during translation. The exact result, retaining the requested order, is (18, 19/20, exact_finite_domain_supremum)."},"skills":["Compute the Exact Finite-Domain Supremum without Reformatting the Source Ratio"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0044","batch_id":"batch-20261014-advance-20261104-r02-uniform-convergence-of-functions-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d4590cf598462597f264c6e75bca7112fe0c70ac8435a5672927818d3017d40b","family_id":"qf-mod35-uniform-sup-gap","seed":202611042044,"semantic_template":"qf-mod35-uniform-sup-gap/{\"coefficients\":[19,-20],\"limit\":[18,19],\"n\":19,\"shift\":2}"},"fields":{"title":"Compute the Exact Finite-Domain Supremum without Reformatting the Source Ratio","stem":"Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[18, 19], coefficients=[19, -20], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. n=19. Give (n, sup_gap, certificate).","answer":{"kind":"expression","value":"(19, 20/21, exact_finite_domain_supremum)","explanation":"The exact supremum is max_x |coefficient[x]|/(n+shift). Retain the source direct-ratio text, which may be unreduced, instead of changing its numerical bytes during translation. The exact result, retaining the requested order, is (19, 20/21, exact_finite_domain_supremum)."},"skills":["Compute the Exact Finite-Domain Supremum without Reformatting the Source Ratio"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0045","batch_id":"batch-20261014-advance-20261104-r02-uniform-convergence-of-functions-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"45507692fb314359a043bc80add1a07fb94835588fcca1891d73a85b582239b5","family_id":"qf-mod35-uniform-sup-gap","seed":202611042045,"semantic_template":"qf-mod35-uniform-sup-gap/{\"coefficients\":[20,-21],\"limit\":[19,20],\"n\":20,\"shift\":2}"},"fields":{"title":"Compute the Exact Finite-Domain Supremum without Reformatting the Source Ratio","stem":"Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[19, 20], coefficients=[20, -21], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. n=20. Give (n, sup_gap, certificate).","answer":{"kind":"expression","value":"(20, 21/22, exact_finite_domain_supremum)","explanation":"The exact supremum is max_x |coefficient[x]|/(n+shift). Retain the source direct-ratio text, which may be unreduced, instead of changing its numerical bytes during translation. The exact result, retaining the requested order, is (20, 21/22, exact_finite_domain_supremum)."},"skills":["Compute the Exact Finite-Domain Supremum without Reformatting the Source Ratio"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0046","batch_id":"batch-20261014-advance-20261104-r02-uniform-convergence-of-functions-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a66ede51c6708e4169196f4eaded207d8724f516e0f06db6638b7ae121a515fc","family_id":"qf-mod35-uniform-sup-gap","seed":202611042046,"semantic_template":"qf-mod35-uniform-sup-gap/{\"coefficients\":[21,-22],\"limit\":[20,21],\"n\":21,\"shift\":2}"},"fields":{"title":"Compute the Exact Finite-Domain Supremum without Reformatting the Source Ratio","stem":"Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[20, 21], coefficients=[21, -22], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. n=21. Give (n, sup_gap, certificate).","answer":{"kind":"expression","value":"(21, 22/23, exact_finite_domain_supremum)","explanation":"The exact supremum is max_x |coefficient[x]|/(n+shift). Retain the source direct-ratio text, which may be unreduced, instead of changing its numerical bytes during translation. The exact result, retaining the requested order, is (21, 22/23, exact_finite_domain_supremum)."},"skills":["Compute the Exact Finite-Domain Supremum without Reformatting the Source Ratio"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0047","batch_id":"batch-20261014-advance-20261104-r02-uniform-convergence-of-functions-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2c08f9124610ed4f30f3f892c6c327cbea6ff1dc7e76ba39a3798adc5fe9ba50","family_id":"qf-mod35-uniform-sup-gap","seed":202611042047,"semantic_template":"qf-mod35-uniform-sup-gap/{\"coefficients\":[22,-23],\"limit\":[21,22],\"n\":22,\"shift\":2}"},"fields":{"title":"Compute the Exact Finite-Domain Supremum without Reformatting the Source Ratio","stem":"Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[21, 22], coefficients=[22, -23], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. n=22. Give (n, sup_gap, certificate).","answer":{"kind":"expression","value":"(22, 23/24, exact_finite_domain_supremum)","explanation":"The exact supremum is max_x |coefficient[x]|/(n+shift). Retain the source direct-ratio text, which may be unreduced, instead of changing its numerical bytes during translation. The exact result, retaining the requested order, is (22, 23/24, exact_finite_domain_supremum)."},"skills":["Compute the Exact Finite-Domain Supremum without Reformatting the Source Ratio"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0048","batch_id":"batch-20261014-advance-20261104-r02-uniform-convergence-of-functions-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"34b19a42d320c419498709d1f076b452541f17caf2bd8f2f7981644c235121f6","family_id":"qf-mod35-uniform-sup-gap","seed":202611042048,"semantic_template":"qf-mod35-uniform-sup-gap/{\"coefficients\":[23,-24],\"limit\":[22,23],\"n\":23,\"shift\":2}"},"fields":{"title":"Compute the Exact Finite-Domain Supremum without Reformatting the Source Ratio","stem":"Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[22, 23], coefficients=[23, -24], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. n=23. Give (n, sup_gap, certificate).","answer":{"kind":"expression","value":"(23, 24/25, exact_finite_domain_supremum)","explanation":"The exact supremum is max_x |coefficient[x]|/(n+shift). Retain the source direct-ratio text, which may be unreduced, instead of changing its numerical bytes during translation. The exact result, retaining the requested order, is (23, 24/25, exact_finite_domain_supremum)."},"skills":["Compute the Exact Finite-Domain Supremum without Reformatting the Source Ratio"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0049","batch_id":"batch-20261014-advance-20261104-r02-uniform-convergence-of-functions-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"897a56893ec916feeeae636d47173c40bd70c61c1ba399d9a07f17e9b06b7677","family_id":"qf-mod35-uniform-sup-gap","seed":202611042049,"semantic_template":"qf-mod35-uniform-sup-gap/{\"coefficients\":[24,-25],\"limit\":[23,24],\"n\":24,\"shift\":2}"},"fields":{"title":"Compute the Exact Finite-Domain Supremum without Reformatting the Source Ratio","stem":"Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[23, 24], coefficients=[24, -25], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. n=24. Give (n, sup_gap, certificate).","answer":{"kind":"expression","value":"(24, 25/26, exact_finite_domain_supremum)","explanation":"The exact supremum is max_x |coefficient[x]|/(n+shift). Retain the source direct-ratio text, which may be unreduced, instead of changing its numerical bytes during translation. The exact result, retaining the requested order, is (24, 25/26, exact_finite_domain_supremum)."},"skills":["Compute the Exact Finite-Domain Supremum without Reformatting the Source Ratio"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0050","batch_id":"batch-20261014-advance-20261104-r02-uniform-convergence-of-functions-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f49c468a5713e15e90520ec1a76b7402898da1676bf9339cdd96f7c9c18921b8","family_id":"qf-mod35-uniform-sup-gap","seed":202611042050,"semantic_template":"qf-mod35-uniform-sup-gap/{\"coefficients\":[25,-26],\"limit\":[24,25],\"n\":25,\"shift\":2}"},"fields":{"title":"Compute the Exact Finite-Domain Supremum without Reformatting the Source Ratio","stem":"Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[24, 25], coefficients=[25, -26], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. n=25. Give (n, sup_gap, certificate).","answer":{"kind":"expression","value":"(25, 26/27, exact_finite_domain_supremum)","explanation":"The exact supremum is max_x |coefficient[x]|/(n+shift). Retain the source direct-ratio text, which may be unreduced, instead of changing its numerical bytes during translation. The exact result, retaining the requested order, is (25, 26/27, exact_finite_domain_supremum)."},"skills":["Compute the Exact Finite-Domain Supremum without Reformatting the Source Ratio"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0051","batch_id":"batch-20261014-advance-20261104-r02-uniform-convergence-of-functions-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4b50b569815bd62c16df2609d44944567b9857455dec29de17b7b1d3fc81d2f5","family_id":"qf-mod35-uniform-pointwise-max","seed":202611042051,"semantic_template":"qf-mod35-uniform-pointwise-max/{\"coefficients\":[1,-2],\"epsilon\":[1,10],\"limit\":[0,1],\"shift\":2}"},"fields":{"title":"A Finite Maximum Is Not an Infinite-Domain Quantifier Exchange","stem":"Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[0, 1], coefficients=[1, -2], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. Positive epsilon=1/10. Give (pointwise_witnesses, N, uniform_on_this_finite_domain, boundary).","answer":{"kind":"expression","value":"([[0, 9], [1, 19]], 19, Yes, finite_maximum_only_not_infinite_domain_implication)","explanation":"List pointwise strict witnesses in domain order and take their maximum. That supplies a common N on this complete finite domain only, not a general implication from pointwise to uniform convergence. The exact result, retaining the requested order, is ([[0, 9], [1, 19]], 19, Yes, finite_maximum_only_not_infinite_domain_implication)."},"skills":["A Finite Maximum Is Not an Infinite-Domain Quantifier Exchange"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0052","batch_id":"batch-20261014-advance-20261104-r02-uniform-convergence-of-functions-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e1bc71a65f9086654671261153cc9e581b4c50bdf76a55b63e7ca328a102c5ce","family_id":"qf-mod35-uniform-pointwise-max","seed":202611042052,"semantic_template":"qf-mod35-uniform-pointwise-max/{\"coefficients\":[2,-3],\"epsilon\":[1,11],\"limit\":[1,2],\"shift\":2}"},"fields":{"title":"A Finite Maximum Is Not an Infinite-Domain Quantifier Exchange","stem":"Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[1, 2], coefficients=[2, -3], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. Positive epsilon=1/11. Give (pointwise_witnesses, N, uniform_on_this_finite_domain, boundary).","answer":{"kind":"expression","value":"([[0, 21], [1, 32]], 32, Yes, finite_maximum_only_not_infinite_domain_implication)","explanation":"List pointwise strict witnesses in domain order and take their maximum. That supplies a common N on this complete finite domain only, not a general implication from pointwise to uniform convergence. The exact result, retaining the requested order, is ([[0, 21], [1, 32]], 32, Yes, finite_maximum_only_not_infinite_domain_implication)."},"skills":["A Finite Maximum Is Not an Infinite-Domain Quantifier Exchange"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0053","batch_id":"batch-20261014-advance-20261104-r02-uniform-convergence-of-functions-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"26e1f168008ff667dd8bd9a6a38c797ec719177e3c6fe1b6113f0fe0c83b451b","family_id":"qf-mod35-uniform-pointwise-max","seed":202611042053,"semantic_template":"qf-mod35-uniform-pointwise-max/{\"coefficients\":[3,-4],\"epsilon\":[1,12],\"limit\":[2,3],\"shift\":2}"},"fields":{"title":"A Finite Maximum Is Not an Infinite-Domain Quantifier Exchange","stem":"Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[2, 3], coefficients=[3, -4], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. Positive epsilon=1/12. Give (pointwise_witnesses, N, uniform_on_this_finite_domain, boundary).","answer":{"kind":"expression","value":"([[0, 35], [1, 47]], 47, Yes, finite_maximum_only_not_infinite_domain_implication)","explanation":"List pointwise strict witnesses in domain order and take their maximum. That supplies a common N on this complete finite domain only, not a general implication from pointwise to uniform convergence. The exact result, retaining the requested order, is ([[0, 35], [1, 47]], 47, Yes, finite_maximum_only_not_infinite_domain_implication)."},"skills":["A Finite Maximum Is Not an Infinite-Domain Quantifier Exchange"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0054","batch_id":"batch-20261014-advance-20261104-r02-uniform-convergence-of-functions-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d20747b5de55bf7a0999de950980ca7ccbfb3a210543eac9fdbbd17da15049fb","family_id":"qf-mod35-uniform-pointwise-max","seed":202611042054,"semantic_template":"qf-mod35-uniform-pointwise-max/{\"coefficients\":[4,-5],\"epsilon\":[1,13],\"limit\":[3,4],\"shift\":2}"},"fields":{"title":"A Finite Maximum Is Not an Infinite-Domain Quantifier Exchange","stem":"Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[3, 4], coefficients=[4, -5], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. Positive epsilon=1/13. Give (pointwise_witnesses, N, uniform_on_this_finite_domain, boundary).","answer":{"kind":"expression","value":"([[0, 51], [1, 64]], 64, Yes, finite_maximum_only_not_infinite_domain_implication)","explanation":"List pointwise strict witnesses in domain order and take their maximum. That supplies a common N on this complete finite domain only, not a general implication from pointwise to uniform convergence. The exact result, retaining the requested order, is ([[0, 51], [1, 64]], 64, Yes, finite_maximum_only_not_infinite_domain_implication)."},"skills":["A Finite Maximum Is Not an Infinite-Domain Quantifier Exchange"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0055","batch_id":"batch-20261014-advance-20261104-r02-uniform-convergence-of-functions-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"46a6854611f0d02a067106039f7965ddbfe70aff5e6695d7905604a1fcbbe8b6","family_id":"qf-mod35-uniform-pointwise-max","seed":202611042055,"semantic_template":"qf-mod35-uniform-pointwise-max/{\"coefficients\":[5,-6],\"epsilon\":[1,14],\"limit\":[4,5],\"shift\":2}"},"fields":{"title":"A Finite Maximum Is Not an Infinite-Domain Quantifier Exchange","stem":"Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[4, 5], coefficients=[5, -6], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. Positive epsilon=1/14. Give (pointwise_witnesses, N, uniform_on_this_finite_domain, boundary).","answer":{"kind":"expression","value":"([[0, 69], [1, 83]], 83, Yes, finite_maximum_only_not_infinite_domain_implication)","explanation":"List pointwise strict witnesses in domain order and take their maximum. That supplies a common N on this complete finite domain only, not a general implication from pointwise to uniform convergence. The exact result, retaining the requested order, is ([[0, 69], [1, 83]], 83, Yes, finite_maximum_only_not_infinite_domain_implication)."},"skills":["A Finite Maximum Is Not an Infinite-Domain Quantifier Exchange"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0056","batch_id":"batch-20261014-advance-20261104-r02-uniform-convergence-of-functions-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"41e802a1e1879b18bf409e9f0a1e6df23b7f8f8d9fa9b355fdc2cc3c4bbd3c4e","family_id":"qf-mod35-uniform-pointwise-max","seed":202611042056,"semantic_template":"qf-mod35-uniform-pointwise-max/{\"coefficients\":[6,-7],\"epsilon\":[1,15],\"limit\":[5,6],\"shift\":2}"},"fields":{"title":"A Finite Maximum Is Not an Infinite-Domain Quantifier Exchange","stem":"Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[5, 6], coefficients=[6, -7], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. Positive epsilon=1/15. Give (pointwise_witnesses, N, uniform_on_this_finite_domain, boundary).","answer":{"kind":"expression","value":"([[0, 89], [1, 104]], 104, Yes, finite_maximum_only_not_infinite_domain_implication)","explanation":"List pointwise strict witnesses in domain order and take their maximum. That supplies a common N on this complete finite domain only, not a general implication from pointwise to uniform convergence. The exact result, retaining the requested order, is ([[0, 89], [1, 104]], 104, Yes, finite_maximum_only_not_infinite_domain_implication)."},"skills":["A Finite Maximum Is Not an Infinite-Domain Quantifier Exchange"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0057","batch_id":"batch-20261014-advance-20261104-r02-uniform-convergence-of-functions-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"62c3d063299a200035fd4f90f04f3469f7e88662ab7ef46e394188e902d5a537","family_id":"qf-mod35-uniform-pointwise-max","seed":202611042057,"semantic_template":"qf-mod35-uniform-pointwise-max/{\"coefficients\":[7,-8],\"epsilon\":[1,16],\"limit\":[6,7],\"shift\":2}"},"fields":{"title":"A Finite Maximum Is Not an Infinite-Domain Quantifier Exchange","stem":"Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[6, 7], coefficients=[7, -8], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. Positive epsilon=1/16. Give (pointwise_witnesses, N, uniform_on_this_finite_domain, boundary).","answer":{"kind":"expression","value":"([[0, 111], [1, 127]], 127, Yes, finite_maximum_only_not_infinite_domain_implication)","explanation":"List pointwise strict witnesses in domain order and take their maximum. That supplies a common N on this complete finite domain only, not a general implication from pointwise to uniform convergence. The exact result, retaining the requested order, is ([[0, 111], [1, 127]], 127, Yes, finite_maximum_only_not_infinite_domain_implication)."},"skills":["A Finite Maximum Is Not an Infinite-Domain Quantifier Exchange"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0058","batch_id":"batch-20261014-advance-20261104-r02-uniform-convergence-of-functions-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3a257a8d99114846549879228bddb3988a6a918761fdd1e463b02d591d19a74a","family_id":"qf-mod35-uniform-pointwise-max","seed":202611042058,"semantic_template":"qf-mod35-uniform-pointwise-max/{\"coefficients\":[8,-9],\"epsilon\":[1,17],\"limit\":[7,8],\"shift\":2}"},"fields":{"title":"A Finite Maximum Is Not an Infinite-Domain Quantifier Exchange","stem":"Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[7, 8], coefficients=[8, -9], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. Positive epsilon=1/17. Give (pointwise_witnesses, N, uniform_on_this_finite_domain, boundary).","answer":{"kind":"expression","value":"([[0, 135], [1, 152]], 152, Yes, finite_maximum_only_not_infinite_domain_implication)","explanation":"List pointwise strict witnesses in domain order and take their maximum. That supplies a common N on this complete finite domain only, not a general implication from pointwise to uniform convergence. The exact result, retaining the requested order, is ([[0, 135], [1, 152]], 152, Yes, finite_maximum_only_not_infinite_domain_implication)."},"skills":["A Finite Maximum Is Not an Infinite-Domain Quantifier Exchange"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0059","batch_id":"batch-20261014-advance-20261104-r02-uniform-convergence-of-functions-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c3298f70fe7c731e179a4e4a98748a191cb11d425394b7a068966e482734cea5","family_id":"qf-mod35-uniform-pointwise-max","seed":202611042059,"semantic_template":"qf-mod35-uniform-pointwise-max/{\"coefficients\":[9,-10],\"epsilon\":[1,18],\"limit\":[8,9],\"shift\":2}"},"fields":{"title":"A Finite Maximum Is Not an Infinite-Domain Quantifier Exchange","stem":"Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[8, 9], coefficients=[9, -10], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. Positive epsilon=1/18. Give (pointwise_witnesses, N, uniform_on_this_finite_domain, boundary).","answer":{"kind":"expression","value":"([[0, 161], [1, 179]], 179, Yes, finite_maximum_only_not_infinite_domain_implication)","explanation":"List pointwise strict witnesses in domain order and take their maximum. That supplies a common N on this complete finite domain only, not a general implication from pointwise to uniform convergence. The exact result, retaining the requested order, is ([[0, 161], [1, 179]], 179, Yes, finite_maximum_only_not_infinite_domain_implication)."},"skills":["A Finite Maximum Is Not an Infinite-Domain Quantifier Exchange"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0060","batch_id":"batch-20261014-advance-20261104-r02-uniform-convergence-of-functions-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4b82b8725fecdb4f357574f8ebf97a3ba7c595f7fec15c9c8de320ba976a6faa","family_id":"qf-mod35-uniform-pointwise-max","seed":202611042060,"semantic_template":"qf-mod35-uniform-pointwise-max/{\"coefficients\":[10,-11],\"epsilon\":[1,19],\"limit\":[9,10],\"shift\":2}"},"fields":{"title":"A Finite Maximum Is Not an Infinite-Domain Quantifier Exchange","stem":"Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[9, 10], coefficients=[10, -11], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. Positive epsilon=1/19. Give (pointwise_witnesses, N, uniform_on_this_finite_domain, boundary).","answer":{"kind":"expression","value":"([[0, 189], [1, 208]], 208, Yes, finite_maximum_only_not_infinite_domain_implication)","explanation":"List pointwise strict witnesses in domain order and take their maximum. That supplies a common N on this complete finite domain only, not a general implication from pointwise to uniform convergence. The exact result, retaining the requested order, is ([[0, 189], [1, 208]], 208, Yes, finite_maximum_only_not_infinite_domain_implication)."},"skills":["A Finite Maximum Is Not an Infinite-Domain Quantifier Exchange"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0061","batch_id":"batch-20261014-advance-20261104-r02-uniform-convergence-of-functions-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e2b9efb110d9c09c8f824182aea0c51d51fe3fa463729f4883c32531633d0e91","family_id":"qf-mod35-uniform-pointwise-max","seed":202611042061,"semantic_template":"qf-mod35-uniform-pointwise-max/{\"coefficients\":[11,-12],\"epsilon\":[1,20],\"limit\":[10,11],\"shift\":2}"},"fields":{"title":"A Finite Maximum Is Not an Infinite-Domain Quantifier Exchange","stem":"Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[10, 11], coefficients=[11, -12], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. Positive epsilon=1/20. Give (pointwise_witnesses, N, uniform_on_this_finite_domain, boundary).","answer":{"kind":"expression","value":"([[0, 219], [1, 239]], 239, Yes, finite_maximum_only_not_infinite_domain_implication)","explanation":"List pointwise strict witnesses in domain order and take their maximum. That supplies a common N on this complete finite domain only, not a general implication from pointwise to uniform convergence. The exact result, retaining the requested order, is ([[0, 219], [1, 239]], 239, Yes, finite_maximum_only_not_infinite_domain_implication)."},"skills":["A Finite Maximum Is Not an Infinite-Domain Quantifier Exchange"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0062","batch_id":"batch-20261014-advance-20261104-r02-uniform-convergence-of-functions-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"19f0b7081844b67e79285f93649aab1bd9a1f86b29bec149c2814978c86fc553","family_id":"qf-mod35-uniform-pointwise-max","seed":202611042062,"semantic_template":"qf-mod35-uniform-pointwise-max/{\"coefficients\":[12,-13],\"epsilon\":[1,21],\"limit\":[11,12],\"shift\":2}"},"fields":{"title":"A Finite Maximum Is Not an Infinite-Domain Quantifier Exchange","stem":"Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[11, 12], coefficients=[12, -13], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. Positive epsilon=1/21. Give (pointwise_witnesses, N, uniform_on_this_finite_domain, boundary).","answer":{"kind":"expression","value":"([[0, 251], [1, 272]], 272, Yes, finite_maximum_only_not_infinite_domain_implication)","explanation":"List pointwise strict witnesses in domain order and take their maximum. That supplies a common N on this complete finite domain only, not a general implication from pointwise to uniform convergence. The exact result, retaining the requested order, is ([[0, 251], [1, 272]], 272, Yes, finite_maximum_only_not_infinite_domain_implication)."},"skills":["A Finite Maximum Is Not an Infinite-Domain Quantifier Exchange"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0063","batch_id":"batch-20261014-advance-20261104-r02-uniform-convergence-of-functions-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d1773b7eb7c1a9bc43066a5cde962ecbaa48e653f59a634684a8c1792ff6ad72","family_id":"qf-mod35-uniform-pointwise-max","seed":202611042063,"semantic_template":"qf-mod35-uniform-pointwise-max/{\"coefficients\":[13,-14],\"epsilon\":[1,22],\"limit\":[12,13],\"shift\":2}"},"fields":{"title":"A Finite Maximum Is Not an Infinite-Domain Quantifier Exchange","stem":"Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[12, 13], coefficients=[13, -14], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. Positive epsilon=1/22. Give (pointwise_witnesses, N, uniform_on_this_finite_domain, boundary).","answer":{"kind":"expression","value":"([[0, 285], [1, 307]], 307, Yes, finite_maximum_only_not_infinite_domain_implication)","explanation":"List pointwise strict witnesses in domain order and take their maximum. That supplies a common N on this complete finite domain only, not a general implication from pointwise to uniform convergence. The exact result, retaining the requested order, is ([[0, 285], [1, 307]], 307, Yes, finite_maximum_only_not_infinite_domain_implication)."},"skills":["A Finite Maximum Is Not an Infinite-Domain Quantifier Exchange"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0064","batch_id":"batch-20261014-advance-20261104-r02-uniform-convergence-of-functions-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"49f18a0ab432b15cb670fe2722cd3d2e0914ccf6f8347a9036c119bc68aed49f","family_id":"qf-mod35-uniform-pointwise-max","seed":202611042064,"semantic_template":"qf-mod35-uniform-pointwise-max/{\"coefficients\":[14,-15],\"epsilon\":[1,23],\"limit\":[13,14],\"shift\":2}"},"fields":{"title":"A Finite Maximum Is Not an Infinite-Domain Quantifier Exchange","stem":"Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[13, 14], coefficients=[14, -15], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. Positive epsilon=1/23. Give (pointwise_witnesses, N, uniform_on_this_finite_domain, boundary).","answer":{"kind":"expression","value":"([[0, 321], [1, 344]], 344, Yes, finite_maximum_only_not_infinite_domain_implication)","explanation":"List pointwise strict witnesses in domain order and take their maximum. That supplies a common N on this complete finite domain only, not a general implication from pointwise to uniform convergence. The exact result, retaining the requested order, is ([[0, 321], [1, 344]], 344, Yes, finite_maximum_only_not_infinite_domain_implication)."},"skills":["A Finite Maximum Is Not an Infinite-Domain Quantifier Exchange"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0065","batch_id":"batch-20261014-advance-20261104-r02-uniform-convergence-of-functions-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"095864e117ce1b4bbb64cd958c4eac164540bb6fd286165d325ebe2b8b237801","family_id":"qf-mod35-uniform-pointwise-max","seed":202611042065,"semantic_template":"qf-mod35-uniform-pointwise-max/{\"coefficients\":[15,-16],\"epsilon\":[1,24],\"limit\":[14,15],\"shift\":2}"},"fields":{"title":"A Finite Maximum Is Not an Infinite-Domain Quantifier Exchange","stem":"Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[14, 15], coefficients=[15, -16], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. Positive epsilon=1/24. Give (pointwise_witnesses, N, uniform_on_this_finite_domain, boundary).","answer":{"kind":"expression","value":"([[0, 359], [1, 383]], 383, Yes, finite_maximum_only_not_infinite_domain_implication)","explanation":"List pointwise strict witnesses in domain order and take their maximum. That supplies a common N on this complete finite domain only, not a general implication from pointwise to uniform convergence. The exact result, retaining the requested order, is ([[0, 359], [1, 383]], 383, Yes, finite_maximum_only_not_infinite_domain_implication)."},"skills":["A Finite Maximum Is Not an Infinite-Domain Quantifier Exchange"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0066","batch_id":"batch-20261014-advance-20261104-r02-uniform-convergence-of-functions-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4bf22e3916c333930212f8b9588eeb2567f9bd037098529311f0dbb401603e1d","family_id":"qf-mod35-uniform-pointwise-max","seed":202611042066,"semantic_template":"qf-mod35-uniform-pointwise-max/{\"coefficients\":[16,-17],\"epsilon\":[1,25],\"limit\":[15,16],\"shift\":2}"},"fields":{"title":"A Finite Maximum Is Not an Infinite-Domain Quantifier Exchange","stem":"Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[15, 16], coefficients=[16, -17], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. Positive epsilon=1/25. Give (pointwise_witnesses, N, uniform_on_this_finite_domain, boundary).","answer":{"kind":"expression","value":"([[0, 399], [1, 424]], 424, Yes, finite_maximum_only_not_infinite_domain_implication)","explanation":"List pointwise strict witnesses in domain order and take their maximum. That supplies a common N on this complete finite domain only, not a general implication from pointwise to uniform convergence. The exact result, retaining the requested order, is ([[0, 399], [1, 424]], 424, Yes, finite_maximum_only_not_infinite_domain_implication)."},"skills":["A Finite Maximum Is Not an Infinite-Domain Quantifier Exchange"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0067","batch_id":"batch-20261014-advance-20261104-r02-uniform-convergence-of-functions-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"90d11c065b92dc2ff1796dfa2be8e08b589b7228ff37cbd50159f134fac65655","family_id":"qf-mod35-uniform-pointwise-max","seed":202611042067,"semantic_template":"qf-mod35-uniform-pointwise-max/{\"coefficients\":[17,-18],\"epsilon\":[1,26],\"limit\":[16,17],\"shift\":2}"},"fields":{"title":"A Finite Maximum Is Not an Infinite-Domain Quantifier Exchange","stem":"Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[16, 17], coefficients=[17, -18], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. Positive epsilon=1/26. Give (pointwise_witnesses, N, uniform_on_this_finite_domain, boundary).","answer":{"kind":"expression","value":"([[0, 441], [1, 467]], 467, Yes, finite_maximum_only_not_infinite_domain_implication)","explanation":"List pointwise strict witnesses in domain order and take their maximum. That supplies a common N on this complete finite domain only, not a general implication from pointwise to uniform convergence. The exact result, retaining the requested order, is ([[0, 441], [1, 467]], 467, Yes, finite_maximum_only_not_infinite_domain_implication)."},"skills":["A Finite Maximum Is Not an Infinite-Domain Quantifier Exchange"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0068","batch_id":"batch-20261014-advance-20261104-r02-uniform-convergence-of-functions-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4484668949d3e1c5cef56b6ae6f9db335b672400f69c213763f718f823123d25","family_id":"qf-mod35-uniform-pointwise-max","seed":202611042068,"semantic_template":"qf-mod35-uniform-pointwise-max/{\"coefficients\":[18,-19],\"epsilon\":[1,27],\"limit\":[17,18],\"shift\":2}"},"fields":{"title":"A Finite Maximum Is Not an Infinite-Domain Quantifier Exchange","stem":"Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[17, 18], coefficients=[18, -19], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. Positive epsilon=1/27. Give (pointwise_witnesses, N, uniform_on_this_finite_domain, boundary).","answer":{"kind":"expression","value":"([[0, 485], [1, 512]], 512, Yes, finite_maximum_only_not_infinite_domain_implication)","explanation":"List pointwise strict witnesses in domain order and take their maximum. That supplies a common N on this complete finite domain only, not a general implication from pointwise to uniform convergence. The exact result, retaining the requested order, is ([[0, 485], [1, 512]], 512, Yes, finite_maximum_only_not_infinite_domain_implication)."},"skills":["A Finite Maximum Is Not an Infinite-Domain Quantifier Exchange"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0069","batch_id":"batch-20261014-advance-20261104-r02-uniform-convergence-of-functions-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e20b0635d2dfc787de4e764bd705c2cc821ee177af7da70b8490aa24bb387897","family_id":"qf-mod35-uniform-pointwise-max","seed":202611042069,"semantic_template":"qf-mod35-uniform-pointwise-max/{\"coefficients\":[19,-20],\"epsilon\":[1,28],\"limit\":[18,19],\"shift\":2}"},"fields":{"title":"A Finite Maximum Is Not an Infinite-Domain Quantifier Exchange","stem":"Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[18, 19], coefficients=[19, -20], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. Positive epsilon=1/28. Give (pointwise_witnesses, N, uniform_on_this_finite_domain, boundary).","answer":{"kind":"expression","value":"([[0, 531], [1, 559]], 559, Yes, finite_maximum_only_not_infinite_domain_implication)","explanation":"List pointwise strict witnesses in domain order and take their maximum. That supplies a common N on this complete finite domain only, not a general implication from pointwise to uniform convergence. The exact result, retaining the requested order, is ([[0, 531], [1, 559]], 559, Yes, finite_maximum_only_not_infinite_domain_implication)."},"skills":["A Finite Maximum Is Not an Infinite-Domain Quantifier Exchange"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0070","batch_id":"batch-20261014-advance-20261104-r02-uniform-convergence-of-functions-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fd932959a2ea0d26ac98e595bdaad1f18069e794c062baaa00f0d0e2cc5f4c91","family_id":"qf-mod35-uniform-pointwise-max","seed":202611042070,"semantic_template":"qf-mod35-uniform-pointwise-max/{\"coefficients\":[20,-21],\"epsilon\":[1,29],\"limit\":[19,20],\"shift\":2}"},"fields":{"title":"A Finite Maximum Is Not an Infinite-Domain Quantifier Exchange","stem":"Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[19, 20], coefficients=[20, -21], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. Positive epsilon=1/29. Give (pointwise_witnesses, N, uniform_on_this_finite_domain, boundary).","answer":{"kind":"expression","value":"([[0, 579], [1, 608]], 608, Yes, finite_maximum_only_not_infinite_domain_implication)","explanation":"List pointwise strict witnesses in domain order and take their maximum. That supplies a common N on this complete finite domain only, not a general implication from pointwise to uniform convergence. The exact result, retaining the requested order, is ([[0, 579], [1, 608]], 608, Yes, finite_maximum_only_not_infinite_domain_implication)."},"skills":["A Finite Maximum Is Not an Infinite-Domain Quantifier Exchange"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0071","batch_id":"batch-20261014-advance-20261104-r02-uniform-convergence-of-functions-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"11f09286332280d4117fe4558caf16705b41ac108e0939ff13674deca908364e","family_id":"qf-mod35-uniform-pointwise-max","seed":202611042071,"semantic_template":"qf-mod35-uniform-pointwise-max/{\"coefficients\":[21,-22],\"epsilon\":[1,30],\"limit\":[20,21],\"shift\":2}"},"fields":{"title":"A Finite Maximum Is Not an Infinite-Domain Quantifier Exchange","stem":"Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[20, 21], coefficients=[21, -22], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. Positive epsilon=1/30. Give (pointwise_witnesses, N, uniform_on_this_finite_domain, boundary).","answer":{"kind":"expression","value":"([[0, 629], [1, 659]], 659, Yes, finite_maximum_only_not_infinite_domain_implication)","explanation":"List pointwise strict witnesses in domain order and take their maximum. That supplies a common N on this complete finite domain only, not a general implication from pointwise to uniform convergence. The exact result, retaining the requested order, is ([[0, 629], [1, 659]], 659, Yes, finite_maximum_only_not_infinite_domain_implication)."},"skills":["A Finite Maximum Is Not an Infinite-Domain Quantifier Exchange"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0072","batch_id":"batch-20261014-advance-20261104-r02-uniform-convergence-of-functions-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"570b492b3f30f850f20a506d5a50bd44e241cea1c06a4e6bec15845703933250","family_id":"qf-mod35-uniform-pointwise-max","seed":202611042072,"semantic_template":"qf-mod35-uniform-pointwise-max/{\"coefficients\":[22,-23],\"epsilon\":[1,31],\"limit\":[21,22],\"shift\":2}"},"fields":{"title":"A Finite Maximum Is Not an Infinite-Domain Quantifier Exchange","stem":"Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[21, 22], coefficients=[22, -23], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. Positive epsilon=1/31. Give (pointwise_witnesses, N, uniform_on_this_finite_domain, boundary).","answer":{"kind":"expression","value":"([[0, 681], [1, 712]], 712, Yes, finite_maximum_only_not_infinite_domain_implication)","explanation":"List pointwise strict witnesses in domain order and take their maximum. That supplies a common N on this complete finite domain only, not a general implication from pointwise to uniform convergence. The exact result, retaining the requested order, is ([[0, 681], [1, 712]], 712, Yes, finite_maximum_only_not_infinite_domain_implication)."},"skills":["A Finite Maximum Is Not an Infinite-Domain Quantifier Exchange"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0073","batch_id":"batch-20261014-advance-20261104-r02-uniform-convergence-of-functions-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9a9b96e2ba7d725de422c873019b8bfe97130175c5ee3cbe462729d83baf84b0","family_id":"qf-mod35-uniform-pointwise-max","seed":202611042073,"semantic_template":"qf-mod35-uniform-pointwise-max/{\"coefficients\":[23,-24],\"epsilon\":[1,32],\"limit\":[22,23],\"shift\":2}"},"fields":{"title":"A Finite Maximum Is Not an Infinite-Domain Quantifier Exchange","stem":"Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[22, 23], coefficients=[23, -24], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. Positive epsilon=1/32. Give (pointwise_witnesses, N, uniform_on_this_finite_domain, boundary).","answer":{"kind":"expression","value":"([[0, 735], [1, 767]], 767, Yes, finite_maximum_only_not_infinite_domain_implication)","explanation":"List pointwise strict witnesses in domain order and take their maximum. That supplies a common N on this complete finite domain only, not a general implication from pointwise to uniform convergence. The exact result, retaining the requested order, is ([[0, 735], [1, 767]], 767, Yes, finite_maximum_only_not_infinite_domain_implication)."},"skills":["A Finite Maximum Is Not an Infinite-Domain Quantifier Exchange"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0074","batch_id":"batch-20261014-advance-20261104-r02-uniform-convergence-of-functions-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ace5a161407651e1c3f3231c04cf976057792dfaeb8de71001ceb56d18d634e1","family_id":"qf-mod35-uniform-pointwise-max","seed":202611042074,"semantic_template":"qf-mod35-uniform-pointwise-max/{\"coefficients\":[24,-25],\"epsilon\":[1,33],\"limit\":[23,24],\"shift\":2}"},"fields":{"title":"A Finite Maximum Is Not an Infinite-Domain Quantifier Exchange","stem":"Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[23, 24], coefficients=[24, -25], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. Positive epsilon=1/33. Give (pointwise_witnesses, N, uniform_on_this_finite_domain, boundary).","answer":{"kind":"expression","value":"([[0, 791], [1, 824]], 824, Yes, finite_maximum_only_not_infinite_domain_implication)","explanation":"List pointwise strict witnesses in domain order and take their maximum. That supplies a common N on this complete finite domain only, not a general implication from pointwise to uniform convergence. The exact result, retaining the requested order, is ([[0, 791], [1, 824]], 824, Yes, finite_maximum_only_not_infinite_domain_implication)."},"skills":["A Finite Maximum Is Not an Infinite-Domain Quantifier Exchange"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0075","batch_id":"batch-20261014-advance-20261104-r02-uniform-convergence-of-functions-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2b922c048d4d19f490c546f0560d36227fdd47019251ad55ee8199a32c5eef99","family_id":"qf-mod35-uniform-pointwise-max","seed":202611042075,"semantic_template":"qf-mod35-uniform-pointwise-max/{\"coefficients\":[25,-26],\"epsilon\":[1,34],\"limit\":[24,25],\"shift\":2}"},"fields":{"title":"A Finite Maximum Is Not an Infinite-Domain Quantifier Exchange","stem":"Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[24, 25], coefficients=[25, -26], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. Positive epsilon=1/34. Give (pointwise_witnesses, N, uniform_on_this_finite_domain, boundary).","answer":{"kind":"expression","value":"([[0, 849], [1, 883]], 883, Yes, finite_maximum_only_not_infinite_domain_implication)","explanation":"List pointwise strict witnesses in domain order and take their maximum. That supplies a common N on this complete finite domain only, not a general implication from pointwise to uniform convergence. The exact result, retaining the requested order, is ([[0, 849], [1, 883]], 883, Yes, finite_maximum_only_not_infinite_domain_implication)."},"skills":["A Finite Maximum Is Not an Infinite-Domain Quantifier Exchange"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0076","batch_id":"batch-20261014-advance-20261104-r02-uniform-convergence-of-functions-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a0d646fd488ed68aead2aeed02d09220ef983bc00a337b50fae0086ad7d08e22","family_id":"qf-mod35-uniform-finite-prefix-boundary","seed":202611042076,"semantic_template":"qf-mod35-uniform-finite-prefix-boundary/{\"coefficients\":[1,-2],\"limit\":[0,1],\"prefix_length\":1,\"shift\":2}"},"fields":{"title":"A Finite Prefix Does Not Supply the Uniform All-n Obligation","stem":"Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[0, 1], coefficients=[1, -2], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. Observed prefix_length=1. Give (prefix_length, sufficient, boundary).","answer":{"kind":"expression","value":"(1, No, finite_prefix_not_uniform_convergence_certificate)","explanation":"The No concerns sufficiency of finite-prefix evidence, not failure of the separately supplied symbolic family. Uniform convergence needs one witness controlling all points and every later n. The exact result, retaining the requested order, is (1, No, finite_prefix_not_uniform_convergence_certificate)."},"skills":["A Finite Prefix Does Not Supply the Uniform All-n Obligation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0077","batch_id":"batch-20261014-advance-20261104-r02-uniform-convergence-of-functions-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cdf8f1cdd4cc90014a8fa32f100b0cd46954d3b217a30fc1fc92b2172c79a4ca","family_id":"qf-mod35-uniform-finite-prefix-boundary","seed":202611042077,"semantic_template":"qf-mod35-uniform-finite-prefix-boundary/{\"coefficients\":[2,-3],\"limit\":[1,2],\"prefix_length\":2,\"shift\":2}"},"fields":{"title":"A Finite Prefix Does Not Supply the Uniform All-n Obligation","stem":"Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[1, 2], coefficients=[2, -3], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. Observed prefix_length=2. Give (prefix_length, sufficient, boundary).","answer":{"kind":"expression","value":"(2, No, finite_prefix_not_uniform_convergence_certificate)","explanation":"The No concerns sufficiency of finite-prefix evidence, not failure of the separately supplied symbolic family. Uniform convergence needs one witness controlling all points and every later n. The exact result, retaining the requested order, is (2, No, finite_prefix_not_uniform_convergence_certificate)."},"skills":["A Finite Prefix Does Not Supply the Uniform All-n Obligation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0078","batch_id":"batch-20261014-advance-20261104-r02-uniform-convergence-of-functions-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cfe693d52cf80c92b4fe5f69bd09e11d6d6d2b1d7b3a47e3e22728591b428ac4","family_id":"qf-mod35-uniform-finite-prefix-boundary","seed":202611042078,"semantic_template":"qf-mod35-uniform-finite-prefix-boundary/{\"coefficients\":[3,-4],\"limit\":[2,3],\"prefix_length\":3,\"shift\":2}"},"fields":{"title":"A Finite Prefix Does Not Supply the Uniform All-n Obligation","stem":"Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[2, 3], coefficients=[3, -4], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. Observed prefix_length=3. Give (prefix_length, sufficient, boundary).","answer":{"kind":"expression","value":"(3, No, finite_prefix_not_uniform_convergence_certificate)","explanation":"The No concerns sufficiency of finite-prefix evidence, not failure of the separately supplied symbolic family. Uniform convergence needs one witness controlling all points and every later n. The exact result, retaining the requested order, is (3, No, finite_prefix_not_uniform_convergence_certificate)."},"skills":["A Finite Prefix Does Not Supply the Uniform All-n Obligation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0079","batch_id":"batch-20261014-advance-20261104-r02-uniform-convergence-of-functions-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"796c77f6f4aa1f1f9a72d2ec3a705b502accd0440b33e86a7d1d9c252caed98a","family_id":"qf-mod35-uniform-finite-prefix-boundary","seed":202611042079,"semantic_template":"qf-mod35-uniform-finite-prefix-boundary/{\"coefficients\":[4,-5],\"limit\":[3,4],\"prefix_length\":4,\"shift\":2}"},"fields":{"title":"A Finite Prefix Does Not Supply the Uniform All-n Obligation","stem":"Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[3, 4], coefficients=[4, -5], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. Observed prefix_length=4. Give (prefix_length, sufficient, boundary).","answer":{"kind":"expression","value":"(4, No, finite_prefix_not_uniform_convergence_certificate)","explanation":"The No concerns sufficiency of finite-prefix evidence, not failure of the separately supplied symbolic family. Uniform convergence needs one witness controlling all points and every later n. The exact result, retaining the requested order, is (4, No, finite_prefix_not_uniform_convergence_certificate)."},"skills":["A Finite Prefix Does Not Supply the Uniform All-n Obligation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0080","batch_id":"batch-20261014-advance-20261104-r02-uniform-convergence-of-functions-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5d66025658fb2370877d5a2528dfdf1d43e95eca3d86691c30b11441cf2670ef","family_id":"qf-mod35-uniform-finite-prefix-boundary","seed":202611042080,"semantic_template":"qf-mod35-uniform-finite-prefix-boundary/{\"coefficients\":[5,-6],\"limit\":[4,5],\"prefix_length\":5,\"shift\":2}"},"fields":{"title":"A Finite Prefix Does Not Supply the Uniform All-n Obligation","stem":"Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[4, 5], coefficients=[5, -6], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. Observed prefix_length=5. Give (prefix_length, sufficient, boundary).","answer":{"kind":"expression","value":"(5, No, finite_prefix_not_uniform_convergence_certificate)","explanation":"The No concerns sufficiency of finite-prefix evidence, not failure of the separately supplied symbolic family. Uniform convergence needs one witness controlling all points and every later n. The exact result, retaining the requested order, is (5, No, finite_prefix_not_uniform_convergence_certificate)."},"skills":["A Finite Prefix Does Not Supply the Uniform All-n Obligation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0081","batch_id":"batch-20261014-advance-20261104-r02-uniform-convergence-of-functions-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a6552baa98fc6a67eff880b099e5f59838360a0ff27730af699289dacb19f167","family_id":"qf-mod35-uniform-finite-prefix-boundary","seed":202611042081,"semantic_template":"qf-mod35-uniform-finite-prefix-boundary/{\"coefficients\":[6,-7],\"limit\":[5,6],\"prefix_length\":6,\"shift\":2}"},"fields":{"title":"A Finite Prefix Does Not Supply the Uniform All-n Obligation","stem":"Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[5, 6], coefficients=[6, -7], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. Observed prefix_length=6. Give (prefix_length, sufficient, boundary).","answer":{"kind":"expression","value":"(6, No, finite_prefix_not_uniform_convergence_certificate)","explanation":"The No concerns sufficiency of finite-prefix evidence, not failure of the separately supplied symbolic family. Uniform convergence needs one witness controlling all points and every later n. The exact result, retaining the requested order, is (6, No, finite_prefix_not_uniform_convergence_certificate)."},"skills":["A Finite Prefix Does Not Supply the Uniform All-n Obligation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0082","batch_id":"batch-20261014-advance-20261104-r02-uniform-convergence-of-functions-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"afe0291d6d7317190b8162dd0c5130cbed00d3386c6667eff77ecd83754b402d","family_id":"qf-mod35-uniform-finite-prefix-boundary","seed":202611042082,"semantic_template":"qf-mod35-uniform-finite-prefix-boundary/{\"coefficients\":[7,-8],\"limit\":[6,7],\"prefix_length\":7,\"shift\":2}"},"fields":{"title":"A Finite Prefix Does Not Supply the Uniform All-n Obligation","stem":"Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[6, 7], coefficients=[7, -8], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. Observed prefix_length=7. Give (prefix_length, sufficient, boundary).","answer":{"kind":"expression","value":"(7, No, finite_prefix_not_uniform_convergence_certificate)","explanation":"The No concerns sufficiency of finite-prefix evidence, not failure of the separately supplied symbolic family. Uniform convergence needs one witness controlling all points and every later n. The exact result, retaining the requested order, is (7, No, finite_prefix_not_uniform_convergence_certificate)."},"skills":["A Finite Prefix Does Not Supply the Uniform All-n Obligation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0083","batch_id":"batch-20261014-advance-20261104-r02-uniform-convergence-of-functions-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"42a68f49e57ebb404ca8dd4ba4e2b634bb8db6caf56775ea0a617a281ec35700","family_id":"qf-mod35-uniform-finite-prefix-boundary","seed":202611042083,"semantic_template":"qf-mod35-uniform-finite-prefix-boundary/{\"coefficients\":[8,-9],\"limit\":[7,8],\"prefix_length\":8,\"shift\":2}"},"fields":{"title":"A Finite Prefix Does Not Supply the Uniform All-n Obligation","stem":"Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[7, 8], coefficients=[8, -9], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. Observed prefix_length=8. Give (prefix_length, sufficient, boundary).","answer":{"kind":"expression","value":"(8, No, finite_prefix_not_uniform_convergence_certificate)","explanation":"The No concerns sufficiency of finite-prefix evidence, not failure of the separately supplied symbolic family. Uniform convergence needs one witness controlling all points and every later n. The exact result, retaining the requested order, is (8, No, finite_prefix_not_uniform_convergence_certificate)."},"skills":["A Finite Prefix Does Not Supply the Uniform All-n Obligation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0084","batch_id":"batch-20261014-advance-20261104-r02-uniform-convergence-of-functions-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"16b803624075d6aa3b2937bc4621cffee9292063c929c5505dbaa69c3fe6542c","family_id":"qf-mod35-uniform-finite-prefix-boundary","seed":202611042084,"semantic_template":"qf-mod35-uniform-finite-prefix-boundary/{\"coefficients\":[9,-10],\"limit\":[8,9],\"prefix_length\":9,\"shift\":2}"},"fields":{"title":"A Finite Prefix Does Not Supply the Uniform All-n Obligation","stem":"Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[8, 9], coefficients=[9, -10], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. Observed prefix_length=9. Give (prefix_length, sufficient, boundary).","answer":{"kind":"expression","value":"(9, No, finite_prefix_not_uniform_convergence_certificate)","explanation":"The No concerns sufficiency of finite-prefix evidence, not failure of the separately supplied symbolic family. Uniform convergence needs one witness controlling all points and every later n. The exact result, retaining the requested order, is (9, No, finite_prefix_not_uniform_convergence_certificate)."},"skills":["A Finite Prefix Does Not Supply the Uniform All-n Obligation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0085","batch_id":"batch-20261014-advance-20261104-r02-uniform-convergence-of-functions-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"49e2b4a0df225f22d5d1b8cf6b1bec436465da38cda4b8260bb369cef5a63998","family_id":"qf-mod35-uniform-finite-prefix-boundary","seed":202611042085,"semantic_template":"qf-mod35-uniform-finite-prefix-boundary/{\"coefficients\":[10,-11],\"limit\":[9,10],\"prefix_length\":10,\"shift\":2}"},"fields":{"title":"A Finite Prefix Does Not Supply the Uniform All-n Obligation","stem":"Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[9, 10], coefficients=[10, -11], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. Observed prefix_length=10. Give (prefix_length, sufficient, boundary).","answer":{"kind":"expression","value":"(10, No, finite_prefix_not_uniform_convergence_certificate)","explanation":"The No concerns sufficiency of finite-prefix evidence, not failure of the separately supplied symbolic family. Uniform convergence needs one witness controlling all points and every later n. The exact result, retaining the requested order, is (10, No, finite_prefix_not_uniform_convergence_certificate)."},"skills":["A Finite Prefix Does Not Supply the Uniform All-n Obligation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0086","batch_id":"batch-20261014-advance-20261104-r02-uniform-convergence-of-functions-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a657ef015c50e0500c4b24d3fe5d88d92a4ac6cf8662395becd1c9517db4b84d","family_id":"qf-mod35-uniform-finite-prefix-boundary","seed":202611042086,"semantic_template":"qf-mod35-uniform-finite-prefix-boundary/{\"coefficients\":[11,-12],\"limit\":[10,11],\"prefix_length\":11,\"shift\":2}"},"fields":{"title":"A Finite Prefix Does Not Supply the Uniform All-n Obligation","stem":"Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[10, 11], coefficients=[11, -12], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. Observed prefix_length=11. Give (prefix_length, sufficient, boundary).","answer":{"kind":"expression","value":"(11, No, finite_prefix_not_uniform_convergence_certificate)","explanation":"The No concerns sufficiency of finite-prefix evidence, not failure of the separately supplied symbolic family. Uniform convergence needs one witness controlling all points and every later n. The exact result, retaining the requested order, is (11, No, finite_prefix_not_uniform_convergence_certificate)."},"skills":["A Finite Prefix Does Not Supply the Uniform All-n Obligation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0087","batch_id":"batch-20261014-advance-20261104-r02-uniform-convergence-of-functions-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"df0d0ee8c5a82e5afa726cfaf9d138b3890c33d603330afde5c4923bb6eab803","family_id":"qf-mod35-uniform-finite-prefix-boundary","seed":202611042087,"semantic_template":"qf-mod35-uniform-finite-prefix-boundary/{\"coefficients\":[12,-13],\"limit\":[11,12],\"prefix_length\":12,\"shift\":2}"},"fields":{"title":"A Finite Prefix Does Not Supply the Uniform All-n Obligation","stem":"Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[11, 12], coefficients=[12, -13], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. Observed prefix_length=12. Give (prefix_length, sufficient, boundary).","answer":{"kind":"expression","value":"(12, No, finite_prefix_not_uniform_convergence_certificate)","explanation":"The No concerns sufficiency of finite-prefix evidence, not failure of the separately supplied symbolic family. Uniform convergence needs one witness controlling all points and every later n. The exact result, retaining the requested order, is (12, No, finite_prefix_not_uniform_convergence_certificate)."},"skills":["A Finite Prefix Does Not Supply the Uniform All-n Obligation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0088","batch_id":"batch-20261014-advance-20261104-r02-uniform-convergence-of-functions-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b9f9f44df37a516747dc1ef385cf9da000d62e5fad2de67e7bf78290b012b64b","family_id":"qf-mod35-uniform-finite-prefix-boundary","seed":202611042088,"semantic_template":"qf-mod35-uniform-finite-prefix-boundary/{\"coefficients\":[13,-14],\"limit\":[12,13],\"prefix_length\":13,\"shift\":2}"},"fields":{"title":"A Finite Prefix Does Not Supply the Uniform All-n Obligation","stem":"Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[12, 13], coefficients=[13, -14], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. Observed prefix_length=13. Give (prefix_length, sufficient, boundary).","answer":{"kind":"expression","value":"(13, No, finite_prefix_not_uniform_convergence_certificate)","explanation":"The No concerns sufficiency of finite-prefix evidence, not failure of the separately supplied symbolic family. Uniform convergence needs one witness controlling all points and every later n. The exact result, retaining the requested order, is (13, No, finite_prefix_not_uniform_convergence_certificate)."},"skills":["A Finite Prefix Does Not Supply the Uniform All-n Obligation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0089","batch_id":"batch-20261014-advance-20261104-r02-uniform-convergence-of-functions-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"00a46043759cccc57e07191605b82a9e5e006866fe99a5bc1b93fab27998dd7c","family_id":"qf-mod35-uniform-finite-prefix-boundary","seed":202611042089,"semantic_template":"qf-mod35-uniform-finite-prefix-boundary/{\"coefficients\":[14,-15],\"limit\":[13,14],\"prefix_length\":14,\"shift\":2}"},"fields":{"title":"A Finite Prefix Does Not Supply the Uniform All-n Obligation","stem":"Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[13, 14], coefficients=[14, -15], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. Observed prefix_length=14. Give (prefix_length, sufficient, boundary).","answer":{"kind":"expression","value":"(14, No, finite_prefix_not_uniform_convergence_certificate)","explanation":"The No concerns sufficiency of finite-prefix evidence, not failure of the separately supplied symbolic family. Uniform convergence needs one witness controlling all points and every later n. The exact result, retaining the requested order, is (14, No, finite_prefix_not_uniform_convergence_certificate)."},"skills":["A Finite Prefix Does Not Supply the Uniform All-n Obligation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0090","batch_id":"batch-20261014-advance-20261104-r02-uniform-convergence-of-functions-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1b3611d787a959f20a94801ec6ed95999a421314f9c9af2b6150a11d8c45939c","family_id":"qf-mod35-uniform-finite-prefix-boundary","seed":202611042090,"semantic_template":"qf-mod35-uniform-finite-prefix-boundary/{\"coefficients\":[15,-16],\"limit\":[14,15],\"prefix_length\":15,\"shift\":2}"},"fields":{"title":"A Finite Prefix Does Not Supply the Uniform All-n Obligation","stem":"Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[14, 15], coefficients=[15, -16], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. Observed prefix_length=15. Give (prefix_length, sufficient, boundary).","answer":{"kind":"expression","value":"(15, No, finite_prefix_not_uniform_convergence_certificate)","explanation":"The No concerns sufficiency of finite-prefix evidence, not failure of the separately supplied symbolic family. Uniform convergence needs one witness controlling all points and every later n. The exact result, retaining the requested order, is (15, No, finite_prefix_not_uniform_convergence_certificate)."},"skills":["A Finite Prefix Does Not Supply the Uniform All-n Obligation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0091","batch_id":"batch-20261014-advance-20261104-r02-uniform-convergence-of-functions-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a78fd6624272be725c32834afbe0aeaec2b80d4881b9a2096f90ac55291b3983","family_id":"qf-mod35-uniform-finite-prefix-boundary","seed":202611042091,"semantic_template":"qf-mod35-uniform-finite-prefix-boundary/{\"coefficients\":[16,-17],\"limit\":[15,16],\"prefix_length\":16,\"shift\":2}"},"fields":{"title":"A Finite Prefix Does Not Supply the Uniform All-n Obligation","stem":"Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[15, 16], coefficients=[16, -17], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. Observed prefix_length=16. Give (prefix_length, sufficient, boundary).","answer":{"kind":"expression","value":"(16, No, finite_prefix_not_uniform_convergence_certificate)","explanation":"The No concerns sufficiency of finite-prefix evidence, not failure of the separately supplied symbolic family. Uniform convergence needs one witness controlling all points and every later n. The exact result, retaining the requested order, is (16, No, finite_prefix_not_uniform_convergence_certificate)."},"skills":["A Finite Prefix Does Not Supply the Uniform All-n Obligation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0092","batch_id":"batch-20261014-advance-20261104-r02-uniform-convergence-of-functions-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4e234fd04be15832c40ac3c7ddeb3fad3a15ced7130ab4b2699658b5e15f3307","family_id":"qf-mod35-uniform-finite-prefix-boundary","seed":202611042092,"semantic_template":"qf-mod35-uniform-finite-prefix-boundary/{\"coefficients\":[17,-18],\"limit\":[16,17],\"prefix_length\":17,\"shift\":2}"},"fields":{"title":"A Finite Prefix Does Not Supply the Uniform All-n Obligation","stem":"Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[16, 17], coefficients=[17, -18], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. Observed prefix_length=17. Give (prefix_length, sufficient, boundary).","answer":{"kind":"expression","value":"(17, No, finite_prefix_not_uniform_convergence_certificate)","explanation":"The No concerns sufficiency of finite-prefix evidence, not failure of the separately supplied symbolic family. Uniform convergence needs one witness controlling all points and every later n. The exact result, retaining the requested order, is (17, No, finite_prefix_not_uniform_convergence_certificate)."},"skills":["A Finite Prefix Does Not Supply the Uniform All-n Obligation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0093","batch_id":"batch-20261014-advance-20261104-r02-uniform-convergence-of-functions-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0bddd109e39e0a69915614690889a778927652ec8fffa2c0e3a4a4353c0bad37","family_id":"qf-mod35-uniform-finite-prefix-boundary","seed":202611042093,"semantic_template":"qf-mod35-uniform-finite-prefix-boundary/{\"coefficients\":[18,-19],\"limit\":[17,18],\"prefix_length\":18,\"shift\":2}"},"fields":{"title":"A Finite Prefix Does Not Supply the Uniform All-n Obligation","stem":"Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[17, 18], coefficients=[18, -19], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. Observed prefix_length=18. Give (prefix_length, sufficient, boundary).","answer":{"kind":"expression","value":"(18, No, finite_prefix_not_uniform_convergence_certificate)","explanation":"The No concerns sufficiency of finite-prefix evidence, not failure of the separately supplied symbolic family. Uniform convergence needs one witness controlling all points and every later n. The exact result, retaining the requested order, is (18, No, finite_prefix_not_uniform_convergence_certificate)."},"skills":["A Finite Prefix Does Not Supply the Uniform All-n Obligation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0094","batch_id":"batch-20261014-advance-20261104-r02-uniform-convergence-of-functions-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"aca06dcd16692f751d17d5721a34e58c9467e4ead2f059423967752da45eb0b1","family_id":"qf-mod35-uniform-finite-prefix-boundary","seed":202611042094,"semantic_template":"qf-mod35-uniform-finite-prefix-boundary/{\"coefficients\":[19,-20],\"limit\":[18,19],\"prefix_length\":19,\"shift\":2}"},"fields":{"title":"A Finite Prefix Does Not Supply the Uniform All-n Obligation","stem":"Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[18, 19], coefficients=[19, -20], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. Observed prefix_length=19. Give (prefix_length, sufficient, boundary).","answer":{"kind":"expression","value":"(19, No, finite_prefix_not_uniform_convergence_certificate)","explanation":"The No concerns sufficiency of finite-prefix evidence, not failure of the separately supplied symbolic family. Uniform convergence needs one witness controlling all points and every later n. The exact result, retaining the requested order, is (19, No, finite_prefix_not_uniform_convergence_certificate)."},"skills":["A Finite Prefix Does Not Supply the Uniform All-n Obligation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0095","batch_id":"batch-20261014-advance-20261104-r02-uniform-convergence-of-functions-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2ec4bbdaab2b0144a4ac863dbf49aa45589c89a4c06f2519aa20db9e8607a62d","family_id":"qf-mod35-uniform-finite-prefix-boundary","seed":202611042095,"semantic_template":"qf-mod35-uniform-finite-prefix-boundary/{\"coefficients\":[20,-21],\"limit\":[19,20],\"prefix_length\":20,\"shift\":2}"},"fields":{"title":"A Finite Prefix Does Not Supply the Uniform All-n Obligation","stem":"Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[19, 20], coefficients=[20, -21], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. Observed prefix_length=20. Give (prefix_length, sufficient, boundary).","answer":{"kind":"expression","value":"(20, No, finite_prefix_not_uniform_convergence_certificate)","explanation":"The No concerns sufficiency of finite-prefix evidence, not failure of the separately supplied symbolic family. Uniform convergence needs one witness controlling all points and every later n. The exact result, retaining the requested order, is (20, No, finite_prefix_not_uniform_convergence_certificate)."},"skills":["A Finite Prefix Does Not Supply the Uniform All-n Obligation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0096","batch_id":"batch-20261014-advance-20261104-r02-uniform-convergence-of-functions-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b5bf7c39c7c99b1a2f418bacb6f1e2853ff52b227ea46ecd5e07363fbbbc211b","family_id":"qf-mod35-uniform-finite-prefix-boundary","seed":202611042096,"semantic_template":"qf-mod35-uniform-finite-prefix-boundary/{\"coefficients\":[21,-22],\"limit\":[20,21],\"prefix_length\":21,\"shift\":2}"},"fields":{"title":"A Finite Prefix Does Not Supply the Uniform All-n Obligation","stem":"Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[20, 21], coefficients=[21, -22], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. Observed prefix_length=21. Give (prefix_length, sufficient, boundary).","answer":{"kind":"expression","value":"(21, No, finite_prefix_not_uniform_convergence_certificate)","explanation":"The No concerns sufficiency of finite-prefix evidence, not failure of the separately supplied symbolic family. Uniform convergence needs one witness controlling all points and every later n. The exact result, retaining the requested order, is (21, No, finite_prefix_not_uniform_convergence_certificate)."},"skills":["A Finite Prefix Does Not Supply the Uniform All-n Obligation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0097","batch_id":"batch-20261014-advance-20261104-r02-uniform-convergence-of-functions-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4847401a6bb87c416ed949a53876d5535f4193f7e28b818cfe0257da591e7ff9","family_id":"qf-mod35-uniform-finite-prefix-boundary","seed":202611042097,"semantic_template":"qf-mod35-uniform-finite-prefix-boundary/{\"coefficients\":[22,-23],\"limit\":[21,22],\"prefix_length\":22,\"shift\":2}"},"fields":{"title":"A Finite Prefix Does Not Supply the Uniform All-n Obligation","stem":"Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[21, 22], coefficients=[22, -23], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. Observed prefix_length=22. Give (prefix_length, sufficient, boundary).","answer":{"kind":"expression","value":"(22, No, finite_prefix_not_uniform_convergence_certificate)","explanation":"The No concerns sufficiency of finite-prefix evidence, not failure of the separately supplied symbolic family. Uniform convergence needs one witness controlling all points and every later n. The exact result, retaining the requested order, is (22, No, finite_prefix_not_uniform_convergence_certificate)."},"skills":["A Finite Prefix Does Not Supply the Uniform All-n Obligation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0098","batch_id":"batch-20261014-advance-20261104-r02-uniform-convergence-of-functions-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e006a60e765fd653f21b381a94571ce7bab57dbd6e7149187518253d2cec936b","family_id":"qf-mod35-uniform-finite-prefix-boundary","seed":202611042098,"semantic_template":"qf-mod35-uniform-finite-prefix-boundary/{\"coefficients\":[23,-24],\"limit\":[22,23],\"prefix_length\":23,\"shift\":2}"},"fields":{"title":"A Finite Prefix Does Not Supply the Uniform All-n Obligation","stem":"Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[22, 23], coefficients=[23, -24], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. Observed prefix_length=23. Give (prefix_length, sufficient, boundary).","answer":{"kind":"expression","value":"(23, No, finite_prefix_not_uniform_convergence_certificate)","explanation":"The No concerns sufficiency of finite-prefix evidence, not failure of the separately supplied symbolic family. Uniform convergence needs one witness controlling all points and every later n. The exact result, retaining the requested order, is (23, No, finite_prefix_not_uniform_convergence_certificate)."},"skills":["A Finite Prefix Does Not Supply the Uniform All-n Obligation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0099","batch_id":"batch-20261014-advance-20261104-r02-uniform-convergence-of-functions-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4436f31776223bf7184d00b0ac818d55f63ccfc4ebbd603630856afcafc9efe1","family_id":"qf-mod35-uniform-finite-prefix-boundary","seed":202611042099,"semantic_template":"qf-mod35-uniform-finite-prefix-boundary/{\"coefficients\":[24,-25],\"limit\":[23,24],\"prefix_length\":24,\"shift\":2}"},"fields":{"title":"A Finite Prefix Does Not Supply the Uniform All-n Obligation","stem":"Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[23, 24], coefficients=[24, -25], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. Observed prefix_length=24. Give (prefix_length, sufficient, boundary).","answer":{"kind":"expression","value":"(24, No, finite_prefix_not_uniform_convergence_certificate)","explanation":"The No concerns sufficiency of finite-prefix evidence, not failure of the separately supplied symbolic family. Uniform convergence needs one witness controlling all points and every later n. The exact result, retaining the requested order, is (24, No, finite_prefix_not_uniform_convergence_certificate)."},"skills":["A Finite Prefix Does Not Supply the Uniform All-n Obligation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0100","batch_id":"batch-20261014-advance-20261104-r02-uniform-convergence-of-functions-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"933486f9caec59ec175c05b0ce67139217a35546f227cbde6b7430cb128cb493","family_id":"qf-mod35-uniform-finite-prefix-boundary","seed":202611042100,"semantic_template":"qf-mod35-uniform-finite-prefix-boundary/{\"coefficients\":[25,-26],\"limit\":[24,25],\"prefix_length\":25,\"shift\":2}"},"fields":{"title":"A Finite Prefix Does Not Supply the Uniform All-n Obligation","stem":"Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[24, 25], coefficients=[25, -26], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. Observed prefix_length=25. Give (prefix_length, sufficient, boundary).","answer":{"kind":"expression","value":"(25, No, finite_prefix_not_uniform_convergence_certificate)","explanation":"The No concerns sufficiency of finite-prefix evidence, not failure of the separately supplied symbolic family. Uniform convergence needs one witness controlling all points and every later n. The exact result, retaining the requested order, is (25, No, finite_prefix_not_uniform_convergence_certificate)."},"skills":["A Finite Prefix Does Not Supply the Uniform All-n Obligation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}