{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r07-weak-law-large-numbers-0001","batch_id":"batch-20261014-advance-20261105-r07-weak-law-large-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"12eafb8a866e4e6b77c1243bebf14a8329232e13991a75510fc409b4a80f2ced","family_id":"qf-mod36-wlln-variance-average","seed":202611057001,"semantic_template":"qf-mod36-wlln-variance-average/{\"iid_certificate\":\"symbolic_all_finite_subfamilies\",\"mean\":[0,1],\"n\":5,\"variance\":[1,2]}"},"fields":{"title":"Independence Gives Variance of the Average, Not a Finite-Sample Limit","stem":"Use declared mean=0/1, positive finite variance=1/2, and iid_certificate=symbolic_all_finite_subfamilies. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive sample size n=5. Give (mean, variance, n, average_variance, certificate).","answer":{"kind":"expression","value":"(0, 1/2, 5, 1/10, independence_gives_variance_sigma_squared_over_n)","explanation":"Use the stated IID premise to divide the common variance by n. This is an exact conditional variance identity, not evidence that a particular observed sample mean equals its expectation. The exact result, retaining the requested order, is (0, 1/2, 5, 1/10, independence_gives_variance_sigma_squared_over_n)."},"skills":["Independence Gives Variance of the Average, Not a Finite-Sample Limit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261105-r07-weak-law-large-numbers-0002","batch_id":"batch-20261014-advance-20261105-r07-weak-law-large-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b9eb3712618140cd6302828d308df12072e736cf608ef73179c0a852977381ae","family_id":"qf-mod36-wlln-variance-average","seed":202611057002,"semantic_template":"qf-mod36-wlln-variance-average/{\"iid_certificate\":\"symbolic_all_finite_subfamilies\",\"mean\":[1,2],\"n\":6,\"variance\":[2,3]}"},"fields":{"title":"Independence Gives Variance of the Average, Not a Finite-Sample Limit","stem":"Use declared mean=1/2, positive finite variance=2/3, and iid_certificate=symbolic_all_finite_subfamilies. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive sample size n=6. Give (mean, variance, n, average_variance, certificate).","answer":{"kind":"expression","value":"(1/2, 2/3, 6, 1/9, independence_gives_variance_sigma_squared_over_n)","explanation":"Use the stated IID premise to divide the common variance by n. This is an exact conditional variance identity, not evidence that a particular observed sample mean equals its expectation. The exact result, retaining the requested order, is (1/2, 2/3, 6, 1/9, independence_gives_variance_sigma_squared_over_n)."},"skills":["Independence Gives Variance of the Average, Not a Finite-Sample Limit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261105-r07-weak-law-large-numbers-0003","batch_id":"batch-20261014-advance-20261105-r07-weak-law-large-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6cbf6dba0af6a15efcf5432f3e09fe58ba700191c76cdf500e7da7d90fe96ffa","family_id":"qf-mod36-wlln-variance-average","seed":202611057003,"semantic_template":"qf-mod36-wlln-variance-average/{\"iid_certificate\":\"symbolic_all_finite_subfamilies\",\"mean\":[2,3],\"n\":7,\"variance\":[3,4]}"},"fields":{"title":"Independence Gives Variance of the Average, Not a Finite-Sample Limit","stem":"Use declared mean=2/3, positive finite variance=3/4, and iid_certificate=symbolic_all_finite_subfamilies. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive sample size n=7. Give (mean, variance, n, average_variance, certificate).","answer":{"kind":"expression","value":"(2/3, 3/4, 7, 3/28, independence_gives_variance_sigma_squared_over_n)","explanation":"Use the stated IID premise to divide the common variance by n. This is an exact conditional variance identity, not evidence that a particular observed sample mean equals its expectation. The exact result, retaining the requested order, is (2/3, 3/4, 7, 3/28, independence_gives_variance_sigma_squared_over_n)."},"skills":["Independence Gives Variance of the Average, Not a Finite-Sample Limit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261105-r07-weak-law-large-numbers-0004","batch_id":"batch-20261014-advance-20261105-r07-weak-law-large-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a62d21919ef55d5f7ed18325458faaf600206031f106742a4455d00261b3e6bc","family_id":"qf-mod36-wlln-variance-average","seed":202611057004,"semantic_template":"qf-mod36-wlln-variance-average/{\"iid_certificate\":\"symbolic_all_finite_subfamilies\",\"mean\":[3,4],\"n\":8,\"variance\":[4,5]}"},"fields":{"title":"Independence Gives Variance of the Average, Not a Finite-Sample Limit","stem":"Use declared mean=3/4, positive finite variance=4/5, and iid_certificate=symbolic_all_finite_subfamilies. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive sample size n=8. Give (mean, variance, n, average_variance, certificate).","answer":{"kind":"expression","value":"(3/4, 4/5, 8, 1/10, independence_gives_variance_sigma_squared_over_n)","explanation":"Use the stated IID premise to divide the common variance by n. This is an exact conditional variance identity, not evidence that a particular observed sample mean equals its expectation. The exact result, retaining the requested order, is (3/4, 4/5, 8, 1/10, independence_gives_variance_sigma_squared_over_n)."},"skills":["Independence Gives Variance of the Average, Not a Finite-Sample Limit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261105-r07-weak-law-large-numbers-0005","batch_id":"batch-20261014-advance-20261105-r07-weak-law-large-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b764bd23e7484d61eb553041cc175b4324267df75092a535a636ecb221b2c47b","family_id":"qf-mod36-wlln-variance-average","seed":202611057005,"semantic_template":"qf-mod36-wlln-variance-average/{\"iid_certificate\":\"symbolic_all_finite_subfamilies\",\"mean\":[4,5],\"n\":9,\"variance\":[5,6]}"},"fields":{"title":"Independence Gives Variance of the Average, Not a Finite-Sample Limit","stem":"Use declared mean=4/5, positive finite variance=5/6, and iid_certificate=symbolic_all_finite_subfamilies. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive sample size n=9. Give (mean, variance, n, average_variance, certificate).","answer":{"kind":"expression","value":"(4/5, 5/6, 9, 5/54, independence_gives_variance_sigma_squared_over_n)","explanation":"Use the stated IID premise to divide the common variance by n. This is an exact conditional variance identity, not evidence that a particular observed sample mean equals its expectation. The exact result, retaining the requested order, is (4/5, 5/6, 9, 5/54, independence_gives_variance_sigma_squared_over_n)."},"skills":["Independence Gives Variance of the Average, Not a Finite-Sample Limit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261105-r07-weak-law-large-numbers-0006","batch_id":"batch-20261014-advance-20261105-r07-weak-law-large-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"10b023a4f62c0c12f4dc62e66323131ff75c36314d52c9cac481298f41e9f9b0","family_id":"qf-mod36-wlln-variance-average","seed":202611057006,"semantic_template":"qf-mod36-wlln-variance-average/{\"iid_certificate\":\"symbolic_all_finite_subfamilies\",\"mean\":[5,6],\"n\":10,\"variance\":[6,7]}"},"fields":{"title":"Independence Gives Variance of the Average, Not a Finite-Sample Limit","stem":"Use declared mean=5/6, positive finite variance=6/7, and iid_certificate=symbolic_all_finite_subfamilies. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive sample size n=10. Give (mean, variance, n, average_variance, certificate).","answer":{"kind":"expression","value":"(5/6, 6/7, 10, 3/35, independence_gives_variance_sigma_squared_over_n)","explanation":"Use the stated IID premise to divide the common variance by n. This is an exact conditional variance identity, not evidence that a particular observed sample mean equals its expectation. The exact result, retaining the requested order, is (5/6, 6/7, 10, 3/35, independence_gives_variance_sigma_squared_over_n)."},"skills":["Independence Gives Variance of the Average, Not a Finite-Sample Limit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261105-r07-weak-law-large-numbers-0007","batch_id":"batch-20261014-advance-20261105-r07-weak-law-large-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"984187c38dedd065275f2613761781732af4c521e01677040d7cc01966b8121f","family_id":"qf-mod36-wlln-variance-average","seed":202611057007,"semantic_template":"qf-mod36-wlln-variance-average/{\"iid_certificate\":\"symbolic_all_finite_subfamilies\",\"mean\":[6,7],\"n\":11,\"variance\":[7,8]}"},"fields":{"title":"Independence Gives Variance of the Average, Not a Finite-Sample Limit","stem":"Use declared mean=6/7, positive finite variance=7/8, and iid_certificate=symbolic_all_finite_subfamilies. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive sample size n=11. Give (mean, variance, n, average_variance, certificate).","answer":{"kind":"expression","value":"(6/7, 7/8, 11, 7/88, independence_gives_variance_sigma_squared_over_n)","explanation":"Use the stated IID premise to divide the common variance by n. This is an exact conditional variance identity, not evidence that a particular observed sample mean equals its expectation. The exact result, retaining the requested order, is (6/7, 7/8, 11, 7/88, independence_gives_variance_sigma_squared_over_n)."},"skills":["Independence Gives Variance of the Average, Not a Finite-Sample Limit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261105-r07-weak-law-large-numbers-0008","batch_id":"batch-20261014-advance-20261105-r07-weak-law-large-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f68160487db285e8b652d10c49d4b87202b34967296a0e6d5595a7a8e91158ab","family_id":"qf-mod36-wlln-variance-average","seed":202611057008,"semantic_template":"qf-mod36-wlln-variance-average/{\"iid_certificate\":\"symbolic_all_finite_subfamilies\",\"mean\":[7,8],\"n\":12,\"variance\":[8,9]}"},"fields":{"title":"Independence Gives Variance of the Average, Not a Finite-Sample Limit","stem":"Use declared mean=7/8, positive finite variance=8/9, and iid_certificate=symbolic_all_finite_subfamilies. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive sample size n=12. Give (mean, variance, n, average_variance, certificate).","answer":{"kind":"expression","value":"(7/8, 8/9, 12, 2/27, independence_gives_variance_sigma_squared_over_n)","explanation":"Use the stated IID premise to divide the common variance by n. This is an exact conditional variance identity, not evidence that a particular observed sample mean equals its expectation. The exact result, retaining the requested order, is (7/8, 8/9, 12, 2/27, independence_gives_variance_sigma_squared_over_n)."},"skills":["Independence Gives Variance of the Average, Not a Finite-Sample Limit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261105-r07-weak-law-large-numbers-0009","batch_id":"batch-20261014-advance-20261105-r07-weak-law-large-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b7e41de1d02c222344f2f0484c97a21d9328d2b19e8fc78310721a482edbdf00","family_id":"qf-mod36-wlln-variance-average","seed":202611057009,"semantic_template":"qf-mod36-wlln-variance-average/{\"iid_certificate\":\"symbolic_all_finite_subfamilies\",\"mean\":[8,9],\"n\":13,\"variance\":[9,10]}"},"fields":{"title":"Independence Gives Variance of the Average, Not a Finite-Sample Limit","stem":"Use declared mean=8/9, positive finite variance=9/10, and iid_certificate=symbolic_all_finite_subfamilies. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive sample size n=13. Give (mean, variance, n, average_variance, certificate).","answer":{"kind":"expression","value":"(8/9, 9/10, 13, 9/130, independence_gives_variance_sigma_squared_over_n)","explanation":"Use the stated IID premise to divide the common variance by n. This is an exact conditional variance identity, not evidence that a particular observed sample mean equals its expectation. The exact result, retaining the requested order, is (8/9, 9/10, 13, 9/130, independence_gives_variance_sigma_squared_over_n)."},"skills":["Independence Gives Variance of the Average, Not a Finite-Sample Limit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261105-r07-weak-law-large-numbers-0010","batch_id":"batch-20261014-advance-20261105-r07-weak-law-large-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6ea2f1d7aae1deb395f6cc4662df8c75d26c0608aae7b188f8ab722a031c9ad6","family_id":"qf-mod36-wlln-variance-average","seed":202611057010,"semantic_template":"qf-mod36-wlln-variance-average/{\"iid_certificate\":\"symbolic_all_finite_subfamilies\",\"mean\":[9,10],\"n\":14,\"variance\":[10,11]}"},"fields":{"title":"Independence Gives Variance of the Average, Not a Finite-Sample Limit","stem":"Use declared mean=9/10, positive finite variance=10/11, and iid_certificate=symbolic_all_finite_subfamilies. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive sample size n=14. Give (mean, variance, n, average_variance, certificate).","answer":{"kind":"expression","value":"(9/10, 10/11, 14, 5/77, independence_gives_variance_sigma_squared_over_n)","explanation":"Use the stated IID premise to divide the common variance by n. This is an exact conditional variance identity, not evidence that a particular observed sample mean equals its expectation. The exact result, retaining the requested order, is (9/10, 10/11, 14, 5/77, independence_gives_variance_sigma_squared_over_n)."},"skills":["Independence Gives Variance of the Average, Not a Finite-Sample Limit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261105-r07-weak-law-large-numbers-0011","batch_id":"batch-20261014-advance-20261105-r07-weak-law-large-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6eaeed1fc02595b3824db1505f50a7b85e64477021d2dbf4e2d304fef60f9f6e","family_id":"qf-mod36-wlln-variance-average","seed":202611057011,"semantic_template":"qf-mod36-wlln-variance-average/{\"iid_certificate\":\"symbolic_all_finite_subfamilies\",\"mean\":[10,11],\"n\":15,\"variance\":[11,12]}"},"fields":{"title":"Independence Gives Variance of the Average, Not a Finite-Sample Limit","stem":"Use declared mean=10/11, positive finite variance=11/12, and iid_certificate=symbolic_all_finite_subfamilies. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive sample size n=15. Give (mean, variance, n, average_variance, certificate).","answer":{"kind":"expression","value":"(10/11, 11/12, 15, 11/180, independence_gives_variance_sigma_squared_over_n)","explanation":"Use the stated IID premise to divide the common variance by n. This is an exact conditional variance identity, not evidence that a particular observed sample mean equals its expectation. The exact result, retaining the requested order, is (10/11, 11/12, 15, 11/180, independence_gives_variance_sigma_squared_over_n)."},"skills":["Independence Gives Variance of the Average, Not a Finite-Sample Limit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261105-r07-weak-law-large-numbers-0012","batch_id":"batch-20261014-advance-20261105-r07-weak-law-large-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9be548cd17c135a8ae99bc53b1c15d8089ca094fd3c3b633a79abc21796eedca","family_id":"qf-mod36-wlln-variance-average","seed":202611057012,"semantic_template":"qf-mod36-wlln-variance-average/{\"iid_certificate\":\"symbolic_all_finite_subfamilies\",\"mean\":[11,12],\"n\":16,\"variance\":[12,13]}"},"fields":{"title":"Independence Gives Variance of the Average, Not a Finite-Sample Limit","stem":"Use declared mean=11/12, positive finite variance=12/13, and iid_certificate=symbolic_all_finite_subfamilies. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive sample size n=16. Give (mean, variance, n, average_variance, certificate).","answer":{"kind":"expression","value":"(11/12, 12/13, 16, 3/52, independence_gives_variance_sigma_squared_over_n)","explanation":"Use the stated IID premise to divide the common variance by n. This is an exact conditional variance identity, not evidence that a particular observed sample mean equals its expectation. The exact result, retaining the requested order, is (11/12, 12/13, 16, 3/52, independence_gives_variance_sigma_squared_over_n)."},"skills":["Independence Gives Variance of the Average, Not a Finite-Sample Limit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261105-r07-weak-law-large-numbers-0013","batch_id":"batch-20261014-advance-20261105-r07-weak-law-large-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e5170cdaf35e0c74be3133e3f0c1f8f4e81eb37919b0964baea1bc0054a6c17e","family_id":"qf-mod36-wlln-variance-average","seed":202611057013,"semantic_template":"qf-mod36-wlln-variance-average/{\"iid_certificate\":\"symbolic_all_finite_subfamilies\",\"mean\":[12,13],\"n\":17,\"variance\":[13,14]}"},"fields":{"title":"Independence Gives Variance of the Average, Not a Finite-Sample Limit","stem":"Use declared mean=12/13, positive finite variance=13/14, and iid_certificate=symbolic_all_finite_subfamilies. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive sample size n=17. Give (mean, variance, n, average_variance, certificate).","answer":{"kind":"expression","value":"(12/13, 13/14, 17, 13/238, independence_gives_variance_sigma_squared_over_n)","explanation":"Use the stated IID premise to divide the common variance by n. This is an exact conditional variance identity, not evidence that a particular observed sample mean equals its expectation. The exact result, retaining the requested order, is (12/13, 13/14, 17, 13/238, independence_gives_variance_sigma_squared_over_n)."},"skills":["Independence Gives Variance of the Average, Not a Finite-Sample Limit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261105-r07-weak-law-large-numbers-0014","batch_id":"batch-20261014-advance-20261105-r07-weak-law-large-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3e666ad279a438ea0a1212d39e09bc6ee8412cb3a2b92b7c62259b2b8697ff7d","family_id":"qf-mod36-wlln-variance-average","seed":202611057014,"semantic_template":"qf-mod36-wlln-variance-average/{\"iid_certificate\":\"symbolic_all_finite_subfamilies\",\"mean\":[13,14],\"n\":18,\"variance\":[14,15]}"},"fields":{"title":"Independence Gives Variance of the Average, Not a Finite-Sample Limit","stem":"Use declared mean=13/14, positive finite variance=14/15, and iid_certificate=symbolic_all_finite_subfamilies. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive sample size n=18. Give (mean, variance, n, average_variance, certificate).","answer":{"kind":"expression","value":"(13/14, 14/15, 18, 7/135, independence_gives_variance_sigma_squared_over_n)","explanation":"Use the stated IID premise to divide the common variance by n. This is an exact conditional variance identity, not evidence that a particular observed sample mean equals its expectation. The exact result, retaining the requested order, is (13/14, 14/15, 18, 7/135, independence_gives_variance_sigma_squared_over_n)."},"skills":["Independence Gives Variance of the Average, Not a Finite-Sample Limit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261105-r07-weak-law-large-numbers-0015","batch_id":"batch-20261014-advance-20261105-r07-weak-law-large-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6880c9d4e2661a08fa06ce4be84da0afc68ad4ced8d9c7c04cb362d89e285252","family_id":"qf-mod36-wlln-variance-average","seed":202611057015,"semantic_template":"qf-mod36-wlln-variance-average/{\"iid_certificate\":\"symbolic_all_finite_subfamilies\",\"mean\":[14,15],\"n\":19,\"variance\":[15,16]}"},"fields":{"title":"Independence Gives Variance of the Average, Not a Finite-Sample Limit","stem":"Use declared mean=14/15, positive finite variance=15/16, and iid_certificate=symbolic_all_finite_subfamilies. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive sample size n=19. Give (mean, variance, n, average_variance, certificate).","answer":{"kind":"expression","value":"(14/15, 15/16, 19, 15/304, independence_gives_variance_sigma_squared_over_n)","explanation":"Use the stated IID premise to divide the common variance by n. This is an exact conditional variance identity, not evidence that a particular observed sample mean equals its expectation. The exact result, retaining the requested order, is (14/15, 15/16, 19, 15/304, independence_gives_variance_sigma_squared_over_n)."},"skills":["Independence Gives Variance of the Average, Not a Finite-Sample Limit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261105-r07-weak-law-large-numbers-0016","batch_id":"batch-20261014-advance-20261105-r07-weak-law-large-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ac935a480cd27200b0e3bd93c6969ad09eeeae182c3937c1a47e970319c58100","family_id":"qf-mod36-wlln-variance-average","seed":202611057016,"semantic_template":"qf-mod36-wlln-variance-average/{\"iid_certificate\":\"symbolic_all_finite_subfamilies\",\"mean\":[15,16],\"n\":20,\"variance\":[16,17]}"},"fields":{"title":"Independence Gives Variance of the Average, Not a Finite-Sample Limit","stem":"Use declared mean=15/16, positive finite variance=16/17, and iid_certificate=symbolic_all_finite_subfamilies. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive sample size n=20. Give (mean, variance, n, average_variance, certificate).","answer":{"kind":"expression","value":"(15/16, 16/17, 20, 4/85, independence_gives_variance_sigma_squared_over_n)","explanation":"Use the stated IID premise to divide the common variance by n. This is an exact conditional variance identity, not evidence that a particular observed sample mean equals its expectation. The exact result, retaining the requested order, is (15/16, 16/17, 20, 4/85, independence_gives_variance_sigma_squared_over_n)."},"skills":["Independence Gives Variance of the Average, Not a Finite-Sample Limit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261105-r07-weak-law-large-numbers-0017","batch_id":"batch-20261014-advance-20261105-r07-weak-law-large-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"eb8c8cc97b084941946db2f67b616e970804811d444d2f57be2259796041d52a","family_id":"qf-mod36-wlln-variance-average","seed":202611057017,"semantic_template":"qf-mod36-wlln-variance-average/{\"iid_certificate\":\"symbolic_all_finite_subfamilies\",\"mean\":[16,17],\"n\":21,\"variance\":[17,18]}"},"fields":{"title":"Independence Gives Variance of the Average, Not a Finite-Sample Limit","stem":"Use declared mean=16/17, positive finite variance=17/18, and iid_certificate=symbolic_all_finite_subfamilies. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive sample size n=21. Give (mean, variance, n, average_variance, certificate).","answer":{"kind":"expression","value":"(16/17, 17/18, 21, 17/378, independence_gives_variance_sigma_squared_over_n)","explanation":"Use the stated IID premise to divide the common variance by n. This is an exact conditional variance identity, not evidence that a particular observed sample mean equals its expectation. The exact result, retaining the requested order, is (16/17, 17/18, 21, 17/378, independence_gives_variance_sigma_squared_over_n)."},"skills":["Independence Gives Variance of the Average, Not a Finite-Sample Limit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261105-r07-weak-law-large-numbers-0018","batch_id":"batch-20261014-advance-20261105-r07-weak-law-large-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6dd94a3b1f933b59285c21bfe970caaee960583100ee6af8e8af748c1de0952e","family_id":"qf-mod36-wlln-variance-average","seed":202611057018,"semantic_template":"qf-mod36-wlln-variance-average/{\"iid_certificate\":\"symbolic_all_finite_subfamilies\",\"mean\":[17,18],\"n\":22,\"variance\":[18,19]}"},"fields":{"title":"Independence Gives Variance of the Average, Not a Finite-Sample Limit","stem":"Use declared mean=17/18, positive finite variance=18/19, and iid_certificate=symbolic_all_finite_subfamilies. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive sample size n=22. Give (mean, variance, n, average_variance, certificate).","answer":{"kind":"expression","value":"(17/18, 18/19, 22, 9/209, independence_gives_variance_sigma_squared_over_n)","explanation":"Use the stated IID premise to divide the common variance by n. This is an exact conditional variance identity, not evidence that a particular observed sample mean equals its expectation. The exact result, retaining the requested order, is (17/18, 18/19, 22, 9/209, independence_gives_variance_sigma_squared_over_n)."},"skills":["Independence Gives Variance of the Average, Not a Finite-Sample Limit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261105-r07-weak-law-large-numbers-0019","batch_id":"batch-20261014-advance-20261105-r07-weak-law-large-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"acb7a51f9a635c0a9eef0ba609159f7b1c03514379f1703127abe92a424467ba","family_id":"qf-mod36-wlln-variance-average","seed":202611057019,"semantic_template":"qf-mod36-wlln-variance-average/{\"iid_certificate\":\"symbolic_all_finite_subfamilies\",\"mean\":[18,19],\"n\":23,\"variance\":[19,20]}"},"fields":{"title":"Independence Gives Variance of the Average, Not a Finite-Sample Limit","stem":"Use declared mean=18/19, positive finite variance=19/20, and iid_certificate=symbolic_all_finite_subfamilies. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive sample size n=23. Give (mean, variance, n, average_variance, certificate).","answer":{"kind":"expression","value":"(18/19, 19/20, 23, 19/460, independence_gives_variance_sigma_squared_over_n)","explanation":"Use the stated IID premise to divide the common variance by n. This is an exact conditional variance identity, not evidence that a particular observed sample mean equals its expectation. The exact result, retaining the requested order, is (18/19, 19/20, 23, 19/460, independence_gives_variance_sigma_squared_over_n)."},"skills":["Independence Gives Variance of the Average, Not a Finite-Sample Limit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261105-r07-weak-law-large-numbers-0020","batch_id":"batch-20261014-advance-20261105-r07-weak-law-large-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"217a147b4b36498af9b58184b9eea171dbf3f309cb3fad984f4db5d4eb02166a","family_id":"qf-mod36-wlln-variance-average","seed":202611057020,"semantic_template":"qf-mod36-wlln-variance-average/{\"iid_certificate\":\"symbolic_all_finite_subfamilies\",\"mean\":[19,20],\"n\":24,\"variance\":[20,21]}"},"fields":{"title":"Independence Gives Variance of the Average, Not a Finite-Sample Limit","stem":"Use declared mean=19/20, positive finite variance=20/21, and iid_certificate=symbolic_all_finite_subfamilies. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive sample size n=24. Give (mean, variance, n, average_variance, certificate).","answer":{"kind":"expression","value":"(19/20, 20/21, 24, 5/126, independence_gives_variance_sigma_squared_over_n)","explanation":"Use the stated IID premise to divide the common variance by n. This is an exact conditional variance identity, not evidence that a particular observed sample mean equals its expectation. The exact result, retaining the requested order, is (19/20, 20/21, 24, 5/126, independence_gives_variance_sigma_squared_over_n)."},"skills":["Independence Gives Variance of the Average, Not a Finite-Sample Limit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261105-r07-weak-law-large-numbers-0021","batch_id":"batch-20261014-advance-20261105-r07-weak-law-large-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2dd79cc3c683a4010b7565ea3e0132b7b38f2eeba4c3e8c45c4629e80ae46d02","family_id":"qf-mod36-wlln-variance-average","seed":202611057021,"semantic_template":"qf-mod36-wlln-variance-average/{\"iid_certificate\":\"symbolic_all_finite_subfamilies\",\"mean\":[20,21],\"n\":25,\"variance\":[21,22]}"},"fields":{"title":"Independence Gives Variance of the Average, Not a Finite-Sample Limit","stem":"Use declared mean=20/21, positive finite variance=21/22, and iid_certificate=symbolic_all_finite_subfamilies. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive sample size n=25. Give (mean, variance, n, average_variance, certificate).","answer":{"kind":"expression","value":"(20/21, 21/22, 25, 21/550, independence_gives_variance_sigma_squared_over_n)","explanation":"Use the stated IID premise to divide the common variance by n. This is an exact conditional variance identity, not evidence that a particular observed sample mean equals its expectation. The exact result, retaining the requested order, is (20/21, 21/22, 25, 21/550, independence_gives_variance_sigma_squared_over_n)."},"skills":["Independence Gives Variance of the Average, Not a Finite-Sample Limit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261105-r07-weak-law-large-numbers-0022","batch_id":"batch-20261014-advance-20261105-r07-weak-law-large-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"761111798adbf2e3dd70f9c1556cb5e059dc3426e674d86569ba785f94f5ae08","family_id":"qf-mod36-wlln-variance-average","seed":202611057022,"semantic_template":"qf-mod36-wlln-variance-average/{\"iid_certificate\":\"symbolic_all_finite_subfamilies\",\"mean\":[21,22],\"n\":26,\"variance\":[22,23]}"},"fields":{"title":"Independence Gives Variance of the Average, Not a Finite-Sample Limit","stem":"Use declared mean=21/22, positive finite variance=22/23, and iid_certificate=symbolic_all_finite_subfamilies. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive sample size n=26. Give (mean, variance, n, average_variance, certificate).","answer":{"kind":"expression","value":"(21/22, 22/23, 26, 11/299, independence_gives_variance_sigma_squared_over_n)","explanation":"Use the stated IID premise to divide the common variance by n. This is an exact conditional variance identity, not evidence that a particular observed sample mean equals its expectation. The exact result, retaining the requested order, is (21/22, 22/23, 26, 11/299, independence_gives_variance_sigma_squared_over_n)."},"skills":["Independence Gives Variance of the Average, Not a Finite-Sample Limit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261105-r07-weak-law-large-numbers-0023","batch_id":"batch-20261014-advance-20261105-r07-weak-law-large-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"445bf3e263a086798153ce983fc6b18c0e4caaaf20a773eb44ca9382d05f9d58","family_id":"qf-mod36-wlln-variance-average","seed":202611057023,"semantic_template":"qf-mod36-wlln-variance-average/{\"iid_certificate\":\"symbolic_all_finite_subfamilies\",\"mean\":[22,23],\"n\":27,\"variance\":[23,24]}"},"fields":{"title":"Independence Gives Variance of the Average, Not a Finite-Sample Limit","stem":"Use declared mean=22/23, positive finite variance=23/24, and iid_certificate=symbolic_all_finite_subfamilies. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive sample size n=27. Give (mean, variance, n, average_variance, certificate).","answer":{"kind":"expression","value":"(22/23, 23/24, 27, 23/648, independence_gives_variance_sigma_squared_over_n)","explanation":"Use the stated IID premise to divide the common variance by n. This is an exact conditional variance identity, not evidence that a particular observed sample mean equals its expectation. The exact result, retaining the requested order, is (22/23, 23/24, 27, 23/648, independence_gives_variance_sigma_squared_over_n)."},"skills":["Independence Gives Variance of the Average, Not a Finite-Sample Limit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261105-r07-weak-law-large-numbers-0024","batch_id":"batch-20261014-advance-20261105-r07-weak-law-large-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8f7dfbdb3e678cdf7aebac7bdc19489c3afacb880c00da26e6ad1dad429375e6","family_id":"qf-mod36-wlln-variance-average","seed":202611057024,"semantic_template":"qf-mod36-wlln-variance-average/{\"iid_certificate\":\"symbolic_all_finite_subfamilies\",\"mean\":[23,24],\"n\":28,\"variance\":[24,25]}"},"fields":{"title":"Independence Gives Variance of the Average, Not a Finite-Sample Limit","stem":"Use declared mean=23/24, positive finite variance=24/25, and iid_certificate=symbolic_all_finite_subfamilies. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive sample size n=28. Give (mean, variance, n, average_variance, certificate).","answer":{"kind":"expression","value":"(23/24, 24/25, 28, 6/175, independence_gives_variance_sigma_squared_over_n)","explanation":"Use the stated IID premise to divide the common variance by n. This is an exact conditional variance identity, not evidence that a particular observed sample mean equals its expectation. The exact result, retaining the requested order, is (23/24, 24/25, 28, 6/175, independence_gives_variance_sigma_squared_over_n)."},"skills":["Independence Gives Variance of the Average, Not a Finite-Sample Limit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261105-r07-weak-law-large-numbers-0025","batch_id":"batch-20261014-advance-20261105-r07-weak-law-large-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1d43e5b795a8b2e0ad94531a97ca65e676389ba0f520e8cf54f3ee8765e81e19","family_id":"qf-mod36-wlln-variance-average","seed":202611057025,"semantic_template":"qf-mod36-wlln-variance-average/{\"iid_certificate\":\"symbolic_all_finite_subfamilies\",\"mean\":[24,25],\"n\":29,\"variance\":[25,26]}"},"fields":{"title":"Independence Gives Variance of the Average, Not a Finite-Sample Limit","stem":"Use declared mean=24/25, positive finite variance=25/26, and iid_certificate=symbolic_all_finite_subfamilies. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive sample size n=29. Give (mean, variance, n, average_variance, certificate).","answer":{"kind":"expression","value":"(24/25, 25/26, 29, 25/754, independence_gives_variance_sigma_squared_over_n)","explanation":"Use the stated IID premise to divide the common variance by n. This is an exact conditional variance identity, not evidence that a particular observed sample mean equals its expectation. The exact result, retaining the requested order, is (24/25, 25/26, 29, 25/754, independence_gives_variance_sigma_squared_over_n)."},"skills":["Independence Gives Variance of the Average, Not a Finite-Sample Limit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261105-r07-weak-law-large-numbers-0026","batch_id":"batch-20261014-advance-20261105-r07-weak-law-large-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b3a7f638b0833e7c02c13388f5bdc06cf963eacf738fba43ff2a2ad6ff19e9f0","family_id":"qf-mod36-wlln-chebyshev-bound","seed":202611057026,"semantic_template":"qf-mod36-wlln-chebyshev-bound/{\"epsilon\":[1,2],\"iid_certificate\":\"symbolic_all_finite_subfamilies\",\"mean\":[0,1],\"n\":20,\"variance\":[1,2]}"},"fields":{"title":"A Chebyshev Upper Bound May Exceed One without Being a Probability","stem":"Use declared mean=0/1, positive finite variance=1/2, and iid_certificate=symbolic_all_finite_subfamilies. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive epsilon=1/2 and sample size n=20. Report the raw upper bound without clipping it to1. Give (mean, epsilon, n, probability_upper_bound, conditional_upper_bound_valid, certificate).","answer":{"kind":"expression","value":"(0, 1/2, 20, 1/10, Yes, probability_upper_bound_not_finite_n_equality)","explanation":"Compute variance/(n*epsilon^2). It is an upper bound, not the event probability or a finite-n equality claim; a bound above1 is loose but not an invalid probability value. The exact result, retaining the requested order, is (0, 1/2, 20, 1/10, Yes, probability_upper_bound_not_finite_n_equality)."},"skills":["A Chebyshev Upper Bound May Exceed One without Being a Probability"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261105-r07-weak-law-large-numbers-0027","batch_id":"batch-20261014-advance-20261105-r07-weak-law-large-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1d582ec05c54aeb134e1b9466008dc282f18de77ab45dcb9e7be65a9ee8c5c1b","family_id":"qf-mod36-wlln-chebyshev-bound","seed":202611057027,"semantic_template":"qf-mod36-wlln-chebyshev-bound/{\"epsilon\":[1,3],\"iid_certificate\":\"symbolic_all_finite_subfamilies\",\"mean\":[1,2],\"n\":21,\"variance\":[2,3]}"},"fields":{"title":"A Chebyshev Upper Bound May Exceed One without Being a Probability","stem":"Use declared mean=1/2, positive finite variance=2/3, and iid_certificate=symbolic_all_finite_subfamilies. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive epsilon=1/3 and sample size n=21. Report the raw upper bound without clipping it to1. Give (mean, epsilon, n, probability_upper_bound, conditional_upper_bound_valid, certificate).","answer":{"kind":"expression","value":"(1/2, 1/3, 21, 2/7, Yes, probability_upper_bound_not_finite_n_equality)","explanation":"Compute variance/(n*epsilon^2). It is an upper bound, not the event probability or a finite-n equality claim; a bound above1 is loose but not an invalid probability value. The exact result, retaining the requested order, is (1/2, 1/3, 21, 2/7, Yes, probability_upper_bound_not_finite_n_equality)."},"skills":["A Chebyshev Upper Bound May Exceed One without Being a Probability"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261105-r07-weak-law-large-numbers-0028","batch_id":"batch-20261014-advance-20261105-r07-weak-law-large-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ecceae7bb3cf49b914bc3ddf693cd7986a8f5d160c934607342bdf09e9d2ddb5","family_id":"qf-mod36-wlln-chebyshev-bound","seed":202611057028,"semantic_template":"qf-mod36-wlln-chebyshev-bound/{\"epsilon\":[1,4],\"iid_certificate\":\"symbolic_all_finite_subfamilies\",\"mean\":[2,3],\"n\":22,\"variance\":[3,4]}"},"fields":{"title":"A Chebyshev Upper Bound May Exceed One without Being a Probability","stem":"Use declared mean=2/3, positive finite variance=3/4, and iid_certificate=symbolic_all_finite_subfamilies. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive epsilon=1/4 and sample size n=22. Report the raw upper bound without clipping it to1. Give (mean, epsilon, n, probability_upper_bound, conditional_upper_bound_valid, certificate).","answer":{"kind":"expression","value":"(2/3, 1/4, 22, 6/11, Yes, probability_upper_bound_not_finite_n_equality)","explanation":"Compute variance/(n*epsilon^2). It is an upper bound, not the event probability or a finite-n equality claim; a bound above1 is loose but not an invalid probability value. The exact result, retaining the requested order, is (2/3, 1/4, 22, 6/11, Yes, probability_upper_bound_not_finite_n_equality)."},"skills":["A Chebyshev Upper Bound May Exceed One without Being a Probability"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261105-r07-weak-law-large-numbers-0029","batch_id":"batch-20261014-advance-20261105-r07-weak-law-large-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d51fff5c9f38ae2e24e7d7950267ec42f22f3c799dc6493d4d42ee48c3d4d72a","family_id":"qf-mod36-wlln-chebyshev-bound","seed":202611057029,"semantic_template":"qf-mod36-wlln-chebyshev-bound/{\"epsilon\":[1,5],\"iid_certificate\":\"symbolic_all_finite_subfamilies\",\"mean\":[3,4],\"n\":23,\"variance\":[4,5]}"},"fields":{"title":"A Chebyshev Upper Bound May Exceed One without Being a Probability","stem":"Use declared mean=3/4, positive finite variance=4/5, and iid_certificate=symbolic_all_finite_subfamilies. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive epsilon=1/5 and sample size n=23. Report the raw upper bound without clipping it to1. Give (mean, epsilon, n, probability_upper_bound, conditional_upper_bound_valid, certificate).","answer":{"kind":"expression","value":"(3/4, 1/5, 23, 20/23, Yes, probability_upper_bound_not_finite_n_equality)","explanation":"Compute variance/(n*epsilon^2). It is an upper bound, not the event probability or a finite-n equality claim; a bound above1 is loose but not an invalid probability value. The exact result, retaining the requested order, is (3/4, 1/5, 23, 20/23, Yes, probability_upper_bound_not_finite_n_equality)."},"skills":["A Chebyshev Upper Bound May Exceed One without Being a Probability"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261105-r07-weak-law-large-numbers-0030","batch_id":"batch-20261014-advance-20261105-r07-weak-law-large-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4a4d9eb9fd21e8cfb3d4bb96992ad2450fb681c4be48030c2b2f09a2205501fe","family_id":"qf-mod36-wlln-chebyshev-bound","seed":202611057030,"semantic_template":"qf-mod36-wlln-chebyshev-bound/{\"epsilon\":[1,6],\"iid_certificate\":\"symbolic_all_finite_subfamilies\",\"mean\":[4,5],\"n\":24,\"variance\":[5,6]}"},"fields":{"title":"A Chebyshev Upper Bound May Exceed One without Being a Probability","stem":"Use declared mean=4/5, positive finite variance=5/6, and iid_certificate=symbolic_all_finite_subfamilies. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive epsilon=1/6 and sample size n=24. Report the raw upper bound without clipping it to1. Give (mean, epsilon, n, probability_upper_bound, conditional_upper_bound_valid, certificate).","answer":{"kind":"expression","value":"(4/5, 1/6, 24, 5/4, Yes, probability_upper_bound_not_finite_n_equality)","explanation":"Compute variance/(n*epsilon^2). It is an upper bound, not the event probability or a finite-n equality claim; a bound above1 is loose but not an invalid probability value. The exact result, retaining the requested order, is (4/5, 1/6, 24, 5/4, Yes, probability_upper_bound_not_finite_n_equality)."},"skills":["A Chebyshev Upper Bound May Exceed One without Being a Probability"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261105-r07-weak-law-large-numbers-0031","batch_id":"batch-20261014-advance-20261105-r07-weak-law-large-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"49c4278aa1ec3edab3a5e6f38bb1fe8c8d9cb3b0579a43f1b8663a356ebea9ef","family_id":"qf-mod36-wlln-chebyshev-bound","seed":202611057031,"semantic_template":"qf-mod36-wlln-chebyshev-bound/{\"epsilon\":[1,7],\"iid_certificate\":\"symbolic_all_finite_subfamilies\",\"mean\":[5,6],\"n\":25,\"variance\":[6,7]}"},"fields":{"title":"A Chebyshev Upper Bound May Exceed One without Being a Probability","stem":"Use declared mean=5/6, positive finite variance=6/7, and iid_certificate=symbolic_all_finite_subfamilies. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive epsilon=1/7 and sample size n=25. Report the raw upper bound without clipping it to1. Give (mean, epsilon, n, probability_upper_bound, conditional_upper_bound_valid, certificate).","answer":{"kind":"expression","value":"(5/6, 1/7, 25, 42/25, Yes, probability_upper_bound_not_finite_n_equality)","explanation":"Compute variance/(n*epsilon^2). It is an upper bound, not the event probability or a finite-n equality claim; a bound above1 is loose but not an invalid probability value. The exact result, retaining the requested order, is (5/6, 1/7, 25, 42/25, Yes, probability_upper_bound_not_finite_n_equality)."},"skills":["A Chebyshev Upper Bound May Exceed One without Being a Probability"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261105-r07-weak-law-large-numbers-0032","batch_id":"batch-20261014-advance-20261105-r07-weak-law-large-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"02629cf932e917fb735c0cfaed3570beb3e4c21659883a72d2aa854b518549ae","family_id":"qf-mod36-wlln-chebyshev-bound","seed":202611057032,"semantic_template":"qf-mod36-wlln-chebyshev-bound/{\"epsilon\":[1,8],\"iid_certificate\":\"symbolic_all_finite_subfamilies\",\"mean\":[6,7],\"n\":26,\"variance\":[7,8]}"},"fields":{"title":"A Chebyshev Upper Bound May Exceed One without Being a Probability","stem":"Use declared mean=6/7, positive finite variance=7/8, and iid_certificate=symbolic_all_finite_subfamilies. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive epsilon=1/8 and sample size n=26. Report the raw upper bound without clipping it to1. Give (mean, epsilon, n, probability_upper_bound, conditional_upper_bound_valid, certificate).","answer":{"kind":"expression","value":"(6/7, 1/8, 26, 28/13, Yes, probability_upper_bound_not_finite_n_equality)","explanation":"Compute variance/(n*epsilon^2). It is an upper bound, not the event probability or a finite-n equality claim; a bound above1 is loose but not an invalid probability value. The exact result, retaining the requested order, is (6/7, 1/8, 26, 28/13, Yes, probability_upper_bound_not_finite_n_equality)."},"skills":["A Chebyshev Upper Bound May Exceed One without Being a Probability"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261105-r07-weak-law-large-numbers-0033","batch_id":"batch-20261014-advance-20261105-r07-weak-law-large-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ff7175c6344393ffe4a7a0d9950faa548cca14bd48fe2b2f2f997e27d70be9a3","family_id":"qf-mod36-wlln-chebyshev-bound","seed":202611057033,"semantic_template":"qf-mod36-wlln-chebyshev-bound/{\"epsilon\":[1,9],\"iid_certificate\":\"symbolic_all_finite_subfamilies\",\"mean\":[7,8],\"n\":27,\"variance\":[8,9]}"},"fields":{"title":"A Chebyshev Upper Bound May Exceed One without Being a Probability","stem":"Use declared mean=7/8, positive finite variance=8/9, and iid_certificate=symbolic_all_finite_subfamilies. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive epsilon=1/9 and sample size n=27. Report the raw upper bound without clipping it to1. Give (mean, epsilon, n, probability_upper_bound, conditional_upper_bound_valid, certificate).","answer":{"kind":"expression","value":"(7/8, 1/9, 27, 8/3, Yes, probability_upper_bound_not_finite_n_equality)","explanation":"Compute variance/(n*epsilon^2). It is an upper bound, not the event probability or a finite-n equality claim; a bound above1 is loose but not an invalid probability value. The exact result, retaining the requested order, is (7/8, 1/9, 27, 8/3, Yes, probability_upper_bound_not_finite_n_equality)."},"skills":["A Chebyshev Upper Bound May Exceed One without Being a Probability"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261105-r07-weak-law-large-numbers-0034","batch_id":"batch-20261014-advance-20261105-r07-weak-law-large-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7abddef422ef95cf2e12f63b3bebded297b96d9c86a3f3c92a162b7ed85d9655","family_id":"qf-mod36-wlln-chebyshev-bound","seed":202611057034,"semantic_template":"qf-mod36-wlln-chebyshev-bound/{\"epsilon\":[1,10],\"iid_certificate\":\"symbolic_all_finite_subfamilies\",\"mean\":[8,9],\"n\":28,\"variance\":[9,10]}"},"fields":{"title":"A Chebyshev Upper Bound May Exceed One without Being a Probability","stem":"Use declared mean=8/9, positive finite variance=9/10, and iid_certificate=symbolic_all_finite_subfamilies. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive epsilon=1/10 and sample size n=28. Report the raw upper bound without clipping it to1. Give (mean, epsilon, n, probability_upper_bound, conditional_upper_bound_valid, certificate).","answer":{"kind":"expression","value":"(8/9, 1/10, 28, 45/14, Yes, probability_upper_bound_not_finite_n_equality)","explanation":"Compute variance/(n*epsilon^2). It is an upper bound, not the event probability or a finite-n equality claim; a bound above1 is loose but not an invalid probability value. The exact result, retaining the requested order, is (8/9, 1/10, 28, 45/14, Yes, probability_upper_bound_not_finite_n_equality)."},"skills":["A Chebyshev Upper Bound May Exceed One without Being a Probability"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261105-r07-weak-law-large-numbers-0035","batch_id":"batch-20261014-advance-20261105-r07-weak-law-large-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1419e824490e3f4e916dfe5dd7797547ea7da52467afc0a069863c612025dab1","family_id":"qf-mod36-wlln-chebyshev-bound","seed":202611057035,"semantic_template":"qf-mod36-wlln-chebyshev-bound/{\"epsilon\":[1,11],\"iid_certificate\":\"symbolic_all_finite_subfamilies\",\"mean\":[9,10],\"n\":29,\"variance\":[10,11]}"},"fields":{"title":"A Chebyshev Upper Bound May Exceed One without Being a Probability","stem":"Use declared mean=9/10, positive finite variance=10/11, and iid_certificate=symbolic_all_finite_subfamilies. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive epsilon=1/11 and sample size n=29. Report the raw upper bound without clipping it to1. Give (mean, epsilon, n, probability_upper_bound, conditional_upper_bound_valid, certificate).","answer":{"kind":"expression","value":"(9/10, 1/11, 29, 110/29, Yes, probability_upper_bound_not_finite_n_equality)","explanation":"Compute variance/(n*epsilon^2). It is an upper bound, not the event probability or a finite-n equality claim; a bound above1 is loose but not an invalid probability value. The exact result, retaining the requested order, is (9/10, 1/11, 29, 110/29, Yes, probability_upper_bound_not_finite_n_equality)."},"skills":["A Chebyshev Upper Bound May Exceed One without Being a Probability"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261105-r07-weak-law-large-numbers-0036","batch_id":"batch-20261014-advance-20261105-r07-weak-law-large-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"91e59b1fbdb632e8fdaa3bf2f1b675936f0a05bac856f91d785eba646907c4a7","family_id":"qf-mod36-wlln-chebyshev-bound","seed":202611057036,"semantic_template":"qf-mod36-wlln-chebyshev-bound/{\"epsilon\":[1,12],\"iid_certificate\":\"symbolic_all_finite_subfamilies\",\"mean\":[10,11],\"n\":30,\"variance\":[11,12]}"},"fields":{"title":"A Chebyshev Upper Bound May Exceed One without Being a Probability","stem":"Use declared mean=10/11, positive finite variance=11/12, and iid_certificate=symbolic_all_finite_subfamilies. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive epsilon=1/12 and sample size n=30. Report the raw upper bound without clipping it to1. Give (mean, epsilon, n, probability_upper_bound, conditional_upper_bound_valid, certificate).","answer":{"kind":"expression","value":"(10/11, 1/12, 30, 22/5, Yes, probability_upper_bound_not_finite_n_equality)","explanation":"Compute variance/(n*epsilon^2). It is an upper bound, not the event probability or a finite-n equality claim; a bound above1 is loose but not an invalid probability value. The exact result, retaining the requested order, is (10/11, 1/12, 30, 22/5, Yes, probability_upper_bound_not_finite_n_equality)."},"skills":["A Chebyshev Upper Bound May Exceed One without Being a Probability"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261105-r07-weak-law-large-numbers-0037","batch_id":"batch-20261014-advance-20261105-r07-weak-law-large-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3c135fbfce6935dd29f34f85be22365b5c7d55f07c5a55a620c213914d36ca43","family_id":"qf-mod36-wlln-chebyshev-bound","seed":202611057037,"semantic_template":"qf-mod36-wlln-chebyshev-bound/{\"epsilon\":[1,13],\"iid_certificate\":\"symbolic_all_finite_subfamilies\",\"mean\":[11,12],\"n\":31,\"variance\":[12,13]}"},"fields":{"title":"A Chebyshev Upper Bound May Exceed One without Being a Probability","stem":"Use declared mean=11/12, positive finite variance=12/13, and iid_certificate=symbolic_all_finite_subfamilies. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive epsilon=1/13 and sample size n=31. Report the raw upper bound without clipping it to1. Give (mean, epsilon, n, probability_upper_bound, conditional_upper_bound_valid, certificate).","answer":{"kind":"expression","value":"(11/12, 1/13, 31, 156/31, Yes, probability_upper_bound_not_finite_n_equality)","explanation":"Compute variance/(n*epsilon^2). It is an upper bound, not the event probability or a finite-n equality claim; a bound above1 is loose but not an invalid probability value. The exact result, retaining the requested order, is (11/12, 1/13, 31, 156/31, Yes, probability_upper_bound_not_finite_n_equality)."},"skills":["A Chebyshev Upper Bound May Exceed One without Being a Probability"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261105-r07-weak-law-large-numbers-0038","batch_id":"batch-20261014-advance-20261105-r07-weak-law-large-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"70669f93893fa8de25a974c1d91109df26d65f96245bb5dafbb9396e248db0e2","family_id":"qf-mod36-wlln-chebyshev-bound","seed":202611057038,"semantic_template":"qf-mod36-wlln-chebyshev-bound/{\"epsilon\":[1,14],\"iid_certificate\":\"symbolic_all_finite_subfamilies\",\"mean\":[12,13],\"n\":32,\"variance\":[13,14]}"},"fields":{"title":"A Chebyshev Upper Bound May Exceed One without Being a Probability","stem":"Use declared mean=12/13, positive finite variance=13/14, and iid_certificate=symbolic_all_finite_subfamilies. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive epsilon=1/14 and sample size n=32. Report the raw upper bound without clipping it to1. Give (mean, epsilon, n, probability_upper_bound, conditional_upper_bound_valid, certificate).","answer":{"kind":"expression","value":"(12/13, 1/14, 32, 91/16, Yes, probability_upper_bound_not_finite_n_equality)","explanation":"Compute variance/(n*epsilon^2). It is an upper bound, not the event probability or a finite-n equality claim; a bound above1 is loose but not an invalid probability value. The exact result, retaining the requested order, is (12/13, 1/14, 32, 91/16, Yes, probability_upper_bound_not_finite_n_equality)."},"skills":["A Chebyshev Upper Bound May Exceed One without Being a Probability"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261105-r07-weak-law-large-numbers-0039","batch_id":"batch-20261014-advance-20261105-r07-weak-law-large-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3dfdd2c4a148d00e80b200c10de7eef5a2382c94c19426ec061fc11b5edaabb8","family_id":"qf-mod36-wlln-chebyshev-bound","seed":202611057039,"semantic_template":"qf-mod36-wlln-chebyshev-bound/{\"epsilon\":[1,15],\"iid_certificate\":\"symbolic_all_finite_subfamilies\",\"mean\":[13,14],\"n\":33,\"variance\":[14,15]}"},"fields":{"title":"A Chebyshev Upper Bound May Exceed One without Being a Probability","stem":"Use declared mean=13/14, positive finite variance=14/15, and iid_certificate=symbolic_all_finite_subfamilies. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive epsilon=1/15 and sample size n=33. Report the raw upper bound without clipping it to1. Give (mean, epsilon, n, probability_upper_bound, conditional_upper_bound_valid, certificate).","answer":{"kind":"expression","value":"(13/14, 1/15, 33, 70/11, Yes, probability_upper_bound_not_finite_n_equality)","explanation":"Compute variance/(n*epsilon^2). It is an upper bound, not the event probability or a finite-n equality claim; a bound above1 is loose but not an invalid probability value. The exact result, retaining the requested order, is (13/14, 1/15, 33, 70/11, Yes, probability_upper_bound_not_finite_n_equality)."},"skills":["A Chebyshev Upper Bound May Exceed One without Being a Probability"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261105-r07-weak-law-large-numbers-0040","batch_id":"batch-20261014-advance-20261105-r07-weak-law-large-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1c3dbb815a9733503b17a0712471750366ed341164f85d21b5567216d208e8c9","family_id":"qf-mod36-wlln-chebyshev-bound","seed":202611057040,"semantic_template":"qf-mod36-wlln-chebyshev-bound/{\"epsilon\":[1,16],\"iid_certificate\":\"symbolic_all_finite_subfamilies\",\"mean\":[14,15],\"n\":34,\"variance\":[15,16]}"},"fields":{"title":"A Chebyshev Upper Bound May Exceed One without Being a Probability","stem":"Use declared mean=14/15, positive finite variance=15/16, and iid_certificate=symbolic_all_finite_subfamilies. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive epsilon=1/16 and sample size n=34. Report the raw upper bound without clipping it to1. Give (mean, epsilon, n, probability_upper_bound, conditional_upper_bound_valid, certificate).","answer":{"kind":"expression","value":"(14/15, 1/16, 34, 120/17, Yes, probability_upper_bound_not_finite_n_equality)","explanation":"Compute variance/(n*epsilon^2). It is an upper bound, not the event probability or a finite-n equality claim; a bound above1 is loose but not an invalid probability value. The exact result, retaining the requested order, is (14/15, 1/16, 34, 120/17, Yes, probability_upper_bound_not_finite_n_equality)."},"skills":["A Chebyshev Upper Bound May Exceed One without Being a Probability"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261105-r07-weak-law-large-numbers-0041","batch_id":"batch-20261014-advance-20261105-r07-weak-law-large-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dbc5c2535cf15a34fab626a5b8b780cad74026e07a5a8bbbc617705e7b6886dc","family_id":"qf-mod36-wlln-chebyshev-bound","seed":202611057041,"semantic_template":"qf-mod36-wlln-chebyshev-bound/{\"epsilon\":[1,17],\"iid_certificate\":\"symbolic_all_finite_subfamilies\",\"mean\":[15,16],\"n\":35,\"variance\":[16,17]}"},"fields":{"title":"A Chebyshev Upper Bound May Exceed One without Being a Probability","stem":"Use declared mean=15/16, positive finite variance=16/17, and iid_certificate=symbolic_all_finite_subfamilies. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive epsilon=1/17 and sample size n=35. Report the raw upper bound without clipping it to1. Give (mean, epsilon, n, probability_upper_bound, conditional_upper_bound_valid, certificate).","answer":{"kind":"expression","value":"(15/16, 1/17, 35, 272/35, Yes, probability_upper_bound_not_finite_n_equality)","explanation":"Compute variance/(n*epsilon^2). It is an upper bound, not the event probability or a finite-n equality claim; a bound above1 is loose but not an invalid probability value. The exact result, retaining the requested order, is (15/16, 1/17, 35, 272/35, Yes, probability_upper_bound_not_finite_n_equality)."},"skills":["A Chebyshev Upper Bound May Exceed One without Being a Probability"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261105-r07-weak-law-large-numbers-0042","batch_id":"batch-20261014-advance-20261105-r07-weak-law-large-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e63a447412fe762e54e0f6e8a7d04330eb09647bc64dd5636fa8e8f1c9e8b97c","family_id":"qf-mod36-wlln-chebyshev-bound","seed":202611057042,"semantic_template":"qf-mod36-wlln-chebyshev-bound/{\"epsilon\":[1,18],\"iid_certificate\":\"symbolic_all_finite_subfamilies\",\"mean\":[16,17],\"n\":36,\"variance\":[17,18]}"},"fields":{"title":"A Chebyshev Upper Bound May Exceed One without Being a Probability","stem":"Use declared mean=16/17, positive finite variance=17/18, and iid_certificate=symbolic_all_finite_subfamilies. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive epsilon=1/18 and sample size n=36. Report the raw upper bound without clipping it to1. Give (mean, epsilon, n, probability_upper_bound, conditional_upper_bound_valid, certificate).","answer":{"kind":"expression","value":"(16/17, 1/18, 36, 17/2, Yes, probability_upper_bound_not_finite_n_equality)","explanation":"Compute variance/(n*epsilon^2). It is an upper bound, not the event probability or a finite-n equality claim; a bound above1 is loose but not an invalid probability value. The exact result, retaining the requested order, is (16/17, 1/18, 36, 17/2, Yes, probability_upper_bound_not_finite_n_equality)."},"skills":["A Chebyshev Upper Bound May Exceed One without Being a Probability"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261105-r07-weak-law-large-numbers-0043","batch_id":"batch-20261014-advance-20261105-r07-weak-law-large-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"292692e4ebd15792369f47bf434bf19a0ca657cf1e28dab6c3aa8d13bfe16ace","family_id":"qf-mod36-wlln-chebyshev-bound","seed":202611057043,"semantic_template":"qf-mod36-wlln-chebyshev-bound/{\"epsilon\":[1,19],\"iid_certificate\":\"symbolic_all_finite_subfamilies\",\"mean\":[17,18],\"n\":37,\"variance\":[18,19]}"},"fields":{"title":"A Chebyshev Upper Bound May Exceed One without Being a Probability","stem":"Use declared mean=17/18, positive finite variance=18/19, and iid_certificate=symbolic_all_finite_subfamilies. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive epsilon=1/19 and sample size n=37. Report the raw upper bound without clipping it to1. Give (mean, epsilon, n, probability_upper_bound, conditional_upper_bound_valid, certificate).","answer":{"kind":"expression","value":"(17/18, 1/19, 37, 342/37, Yes, probability_upper_bound_not_finite_n_equality)","explanation":"Compute variance/(n*epsilon^2). It is an upper bound, not the event probability or a finite-n equality claim; a bound above1 is loose but not an invalid probability value. The exact result, retaining the requested order, is (17/18, 1/19, 37, 342/37, Yes, probability_upper_bound_not_finite_n_equality)."},"skills":["A Chebyshev Upper Bound May Exceed One without Being a Probability"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261105-r07-weak-law-large-numbers-0044","batch_id":"batch-20261014-advance-20261105-r07-weak-law-large-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b56a96779de137ca16cef088d88159c9ee64aa2ae85635312745a04e2122ee5e","family_id":"qf-mod36-wlln-chebyshev-bound","seed":202611057044,"semantic_template":"qf-mod36-wlln-chebyshev-bound/{\"epsilon\":[1,20],\"iid_certificate\":\"symbolic_all_finite_subfamilies\",\"mean\":[18,19],\"n\":38,\"variance\":[19,20]}"},"fields":{"title":"A Chebyshev Upper Bound May Exceed One without Being a Probability","stem":"Use declared mean=18/19, positive finite variance=19/20, and iid_certificate=symbolic_all_finite_subfamilies. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive epsilon=1/20 and sample size n=38. Report the raw upper bound without clipping it to1. Give (mean, epsilon, n, probability_upper_bound, conditional_upper_bound_valid, certificate).","answer":{"kind":"expression","value":"(18/19, 1/20, 38, 10, Yes, probability_upper_bound_not_finite_n_equality)","explanation":"Compute variance/(n*epsilon^2). It is an upper bound, not the event probability or a finite-n equality claim; a bound above1 is loose but not an invalid probability value. The exact result, retaining the requested order, is (18/19, 1/20, 38, 10, Yes, probability_upper_bound_not_finite_n_equality)."},"skills":["A Chebyshev Upper Bound May Exceed One without Being a Probability"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261105-r07-weak-law-large-numbers-0045","batch_id":"batch-20261014-advance-20261105-r07-weak-law-large-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bb9e573a0b9a9bb74b9993e88becc22e9778c7405e872852d7f5e8e58527f762","family_id":"qf-mod36-wlln-chebyshev-bound","seed":202611057045,"semantic_template":"qf-mod36-wlln-chebyshev-bound/{\"epsilon\":[1,21],\"iid_certificate\":\"symbolic_all_finite_subfamilies\",\"mean\":[19,20],\"n\":39,\"variance\":[20,21]}"},"fields":{"title":"A Chebyshev Upper Bound May Exceed One without Being a Probability","stem":"Use declared mean=19/20, positive finite variance=20/21, and iid_certificate=symbolic_all_finite_subfamilies. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive epsilon=1/21 and sample size n=39. Report the raw upper bound without clipping it to1. Give (mean, epsilon, n, probability_upper_bound, conditional_upper_bound_valid, certificate).","answer":{"kind":"expression","value":"(19/20, 1/21, 39, 140/13, Yes, probability_upper_bound_not_finite_n_equality)","explanation":"Compute variance/(n*epsilon^2). It is an upper bound, not the event probability or a finite-n equality claim; a bound above1 is loose but not an invalid probability value. The exact result, retaining the requested order, is (19/20, 1/21, 39, 140/13, Yes, probability_upper_bound_not_finite_n_equality)."},"skills":["A Chebyshev Upper Bound May Exceed One without Being a Probability"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261105-r07-weak-law-large-numbers-0046","batch_id":"batch-20261014-advance-20261105-r07-weak-law-large-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f4e4bf464c58994f7b8a5ec4d586237c4471655a6707b0826e41cf1d4ec3c206","family_id":"qf-mod36-wlln-chebyshev-bound","seed":202611057046,"semantic_template":"qf-mod36-wlln-chebyshev-bound/{\"epsilon\":[1,22],\"iid_certificate\":\"symbolic_all_finite_subfamilies\",\"mean\":[20,21],\"n\":40,\"variance\":[21,22]}"},"fields":{"title":"A Chebyshev Upper Bound May Exceed One without Being a Probability","stem":"Use declared mean=20/21, positive finite variance=21/22, and iid_certificate=symbolic_all_finite_subfamilies. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive epsilon=1/22 and sample size n=40. Report the raw upper bound without clipping it to1. Give (mean, epsilon, n, probability_upper_bound, conditional_upper_bound_valid, certificate).","answer":{"kind":"expression","value":"(20/21, 1/22, 40, 231/20, Yes, probability_upper_bound_not_finite_n_equality)","explanation":"Compute variance/(n*epsilon^2). It is an upper bound, not the event probability or a finite-n equality claim; a bound above1 is loose but not an invalid probability value. The exact result, retaining the requested order, is (20/21, 1/22, 40, 231/20, Yes, probability_upper_bound_not_finite_n_equality)."},"skills":["A Chebyshev Upper Bound May Exceed One without Being a Probability"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261105-r07-weak-law-large-numbers-0047","batch_id":"batch-20261014-advance-20261105-r07-weak-law-large-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b25fa412f37abc21e0d1231d5d0f55ae89546250d2aa91331c7d4521873e69e0","family_id":"qf-mod36-wlln-chebyshev-bound","seed":202611057047,"semantic_template":"qf-mod36-wlln-chebyshev-bound/{\"epsilon\":[1,23],\"iid_certificate\":\"symbolic_all_finite_subfamilies\",\"mean\":[21,22],\"n\":41,\"variance\":[22,23]}"},"fields":{"title":"A Chebyshev Upper Bound May Exceed One without Being a Probability","stem":"Use declared mean=21/22, positive finite variance=22/23, and iid_certificate=symbolic_all_finite_subfamilies. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive epsilon=1/23 and sample size n=41. Report the raw upper bound without clipping it to1. Give (mean, epsilon, n, probability_upper_bound, conditional_upper_bound_valid, certificate).","answer":{"kind":"expression","value":"(21/22, 1/23, 41, 506/41, Yes, probability_upper_bound_not_finite_n_equality)","explanation":"Compute variance/(n*epsilon^2). It is an upper bound, not the event probability or a finite-n equality claim; a bound above1 is loose but not an invalid probability value. The exact result, retaining the requested order, is (21/22, 1/23, 41, 506/41, Yes, probability_upper_bound_not_finite_n_equality)."},"skills":["A Chebyshev Upper Bound May Exceed One without Being a Probability"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261105-r07-weak-law-large-numbers-0048","batch_id":"batch-20261014-advance-20261105-r07-weak-law-large-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"92c760e011a09dda0f96b437ab5d5d7282038980bbcf986a1790b9e48a62dd94","family_id":"qf-mod36-wlln-chebyshev-bound","seed":202611057048,"semantic_template":"qf-mod36-wlln-chebyshev-bound/{\"epsilon\":[1,24],\"iid_certificate\":\"symbolic_all_finite_subfamilies\",\"mean\":[22,23],\"n\":42,\"variance\":[23,24]}"},"fields":{"title":"A Chebyshev Upper Bound May Exceed One without Being a Probability","stem":"Use declared mean=22/23, positive finite variance=23/24, and iid_certificate=symbolic_all_finite_subfamilies. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive epsilon=1/24 and sample size n=42. Report the raw upper bound without clipping it to1. Give (mean, epsilon, n, probability_upper_bound, conditional_upper_bound_valid, certificate).","answer":{"kind":"expression","value":"(22/23, 1/24, 42, 92/7, Yes, probability_upper_bound_not_finite_n_equality)","explanation":"Compute variance/(n*epsilon^2). It is an upper bound, not the event probability or a finite-n equality claim; a bound above1 is loose but not an invalid probability value. The exact result, retaining the requested order, is (22/23, 1/24, 42, 92/7, Yes, probability_upper_bound_not_finite_n_equality)."},"skills":["A Chebyshev Upper Bound May Exceed One without Being a Probability"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261105-r07-weak-law-large-numbers-0049","batch_id":"batch-20261014-advance-20261105-r07-weak-law-large-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"92c6d1bdd3d1da8de4c143fca195baff2c161182e5755c8d70cbb426de89dfd5","family_id":"qf-mod36-wlln-chebyshev-bound","seed":202611057049,"semantic_template":"qf-mod36-wlln-chebyshev-bound/{\"epsilon\":[1,25],\"iid_certificate\":\"symbolic_all_finite_subfamilies\",\"mean\":[23,24],\"n\":43,\"variance\":[24,25]}"},"fields":{"title":"A Chebyshev Upper Bound May Exceed One without Being a Probability","stem":"Use declared mean=23/24, positive finite variance=24/25, and iid_certificate=symbolic_all_finite_subfamilies. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive epsilon=1/25 and sample size n=43. Report the raw upper bound without clipping it to1. Give (mean, epsilon, n, probability_upper_bound, conditional_upper_bound_valid, certificate).","answer":{"kind":"expression","value":"(23/24, 1/25, 43, 600/43, Yes, probability_upper_bound_not_finite_n_equality)","explanation":"Compute variance/(n*epsilon^2). It is an upper bound, not the event probability or a finite-n equality claim; a bound above1 is loose but not an invalid probability value. The exact result, retaining the requested order, is (23/24, 1/25, 43, 600/43, Yes, probability_upper_bound_not_finite_n_equality)."},"skills":["A Chebyshev Upper Bound May Exceed One without Being a Probability"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261105-r07-weak-law-large-numbers-0050","batch_id":"batch-20261014-advance-20261105-r07-weak-law-large-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c09b810d155317056c81fed8ffb1abc8b9a00cb22c12c2df7594132f703f98ed","family_id":"qf-mod36-wlln-chebyshev-bound","seed":202611057050,"semantic_template":"qf-mod36-wlln-chebyshev-bound/{\"epsilon\":[1,26],\"iid_certificate\":\"symbolic_all_finite_subfamilies\",\"mean\":[24,25],\"n\":44,\"variance\":[25,26]}"},"fields":{"title":"A Chebyshev Upper Bound May Exceed One without Being a Probability","stem":"Use declared mean=24/25, positive finite variance=25/26, and iid_certificate=symbolic_all_finite_subfamilies. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive epsilon=1/26 and sample size n=44. Report the raw upper bound without clipping it to1. Give (mean, epsilon, n, probability_upper_bound, conditional_upper_bound_valid, certificate).","answer":{"kind":"expression","value":"(24/25, 1/26, 44, 325/22, Yes, probability_upper_bound_not_finite_n_equality)","explanation":"Compute variance/(n*epsilon^2). It is an upper bound, not the event probability or a finite-n equality claim; a bound above1 is loose but not an invalid probability value. The exact result, retaining the requested order, is (24/25, 1/26, 44, 325/22, Yes, probability_upper_bound_not_finite_n_equality)."},"skills":["A Chebyshev Upper Bound May Exceed One without Being a Probability"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261105-r07-weak-law-large-numbers-0051","batch_id":"batch-20261014-advance-20261105-r07-weak-law-large-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c47a509ea25cf4686a1b916c7188a0d1d32e52ae501c52640b07b09ffc0cb91d","family_id":"qf-mod36-wlln-epsilon-delta-witness","seed":202611057051,"semantic_template":"qf-mod36-wlln-epsilon-delta-witness/{\"delta\":[1,10],\"epsilon\":[1,2],\"iid_certificate\":\"symbolic_all_finite_subfamilies\",\"mean\":[0,1],\"variance\":[1,2]}"},"fields":{"title":"Choose a Strict All-n Chebyshev Tail Witness","stem":"Use declared mean=0/1, positive finite variance=1/2, and iid_certificate=symbolic_all_finite_subfamilies. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive epsilon=1/2, delta=1/10. Give (epsilon, delta, N, chebyshev_bound_at_N, strict_delta_tail, certificate).","answer":{"kind":"expression","value":"(1/2, 1/10, 21, 2/21, Yes, forall_n_ge_N_monotone_Chebyshev_tail)","explanation":"Choose N=floor(variance/(delta*epsilon^2))+1 and compute variance/(N*epsilon^2). The strict bound is below delta, including at integral thresholds; monotonicity extends it to every n≥N under the same declared IID premise. The exact result, retaining the requested order, is (1/2, 1/10, 21, 2/21, Yes, forall_n_ge_N_monotone_Chebyshev_tail)."},"skills":["Choose a Strict All-n Chebyshev Tail Witness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r07-weak-law-large-numbers-0052","batch_id":"batch-20261014-advance-20261105-r07-weak-law-large-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5faf3f77c16682f50f0e2cb7451574ff3cd31a6ae4ff875c89c99236d0024fcd","family_id":"qf-mod36-wlln-epsilon-delta-witness","seed":202611057052,"semantic_template":"qf-mod36-wlln-epsilon-delta-witness/{\"delta\":[1,11],\"epsilon\":[1,3],\"iid_certificate\":\"symbolic_all_finite_subfamilies\",\"mean\":[1,2],\"variance\":[2,3]}"},"fields":{"title":"Choose a Strict All-n Chebyshev Tail Witness","stem":"Use declared mean=1/2, positive finite variance=2/3, and iid_certificate=symbolic_all_finite_subfamilies. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive epsilon=1/3, delta=1/11. Give (epsilon, delta, N, chebyshev_bound_at_N, strict_delta_tail, certificate).","answer":{"kind":"expression","value":"(1/3, 1/11, 67, 6/67, Yes, forall_n_ge_N_monotone_Chebyshev_tail)","explanation":"Choose N=floor(variance/(delta*epsilon^2))+1 and compute variance/(N*epsilon^2). The strict bound is below delta, including at integral thresholds; monotonicity extends it to every n≥N under the same declared IID premise. The exact result, retaining the requested order, is (1/3, 1/11, 67, 6/67, Yes, forall_n_ge_N_monotone_Chebyshev_tail)."},"skills":["Choose a Strict All-n Chebyshev Tail Witness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r07-weak-law-large-numbers-0053","batch_id":"batch-20261014-advance-20261105-r07-weak-law-large-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0a4cd66302ad82107e4f792bf434c12da253a208684a3eca7ae346feb14ae5a1","family_id":"qf-mod36-wlln-epsilon-delta-witness","seed":202611057053,"semantic_template":"qf-mod36-wlln-epsilon-delta-witness/{\"delta\":[1,12],\"epsilon\":[1,4],\"iid_certificate\":\"symbolic_all_finite_subfamilies\",\"mean\":[2,3],\"variance\":[3,4]}"},"fields":{"title":"Choose a Strict All-n Chebyshev Tail Witness","stem":"Use declared mean=2/3, positive finite variance=3/4, and iid_certificate=symbolic_all_finite_subfamilies. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive epsilon=1/4, delta=1/12. Give (epsilon, delta, N, chebyshev_bound_at_N, strict_delta_tail, certificate).","answer":{"kind":"expression","value":"(1/4, 1/12, 145, 12/145, Yes, forall_n_ge_N_monotone_Chebyshev_tail)","explanation":"Choose N=floor(variance/(delta*epsilon^2))+1 and compute variance/(N*epsilon^2). The strict bound is below delta, including at integral thresholds; monotonicity extends it to every n≥N under the same declared IID premise. The exact result, retaining the requested order, is (1/4, 1/12, 145, 12/145, Yes, forall_n_ge_N_monotone_Chebyshev_tail)."},"skills":["Choose a Strict All-n Chebyshev Tail Witness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r07-weak-law-large-numbers-0054","batch_id":"batch-20261014-advance-20261105-r07-weak-law-large-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d20a3c8945d03032aa917e437723aca0fcfc04608a09f7c1849ff8ff9966a39e","family_id":"qf-mod36-wlln-epsilon-delta-witness","seed":202611057054,"semantic_template":"qf-mod36-wlln-epsilon-delta-witness/{\"delta\":[1,13],\"epsilon\":[1,5],\"iid_certificate\":\"symbolic_all_finite_subfamilies\",\"mean\":[3,4],\"variance\":[4,5]}"},"fields":{"title":"Choose a Strict All-n Chebyshev Tail Witness","stem":"Use declared mean=3/4, positive finite variance=4/5, and iid_certificate=symbolic_all_finite_subfamilies. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive epsilon=1/5, delta=1/13. Give (epsilon, delta, N, chebyshev_bound_at_N, strict_delta_tail, certificate).","answer":{"kind":"expression","value":"(1/5, 1/13, 261, 20/261, Yes, forall_n_ge_N_monotone_Chebyshev_tail)","explanation":"Choose N=floor(variance/(delta*epsilon^2))+1 and compute variance/(N*epsilon^2). The strict bound is below delta, including at integral thresholds; monotonicity extends it to every n≥N under the same declared IID premise. The exact result, retaining the requested order, is (1/5, 1/13, 261, 20/261, Yes, forall_n_ge_N_monotone_Chebyshev_tail)."},"skills":["Choose a Strict All-n Chebyshev Tail Witness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r07-weak-law-large-numbers-0055","batch_id":"batch-20261014-advance-20261105-r07-weak-law-large-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"29a28e8a59ecca51ed19a88109886a2bddc2bf56a1c0c34df7fc1fcc1f612d30","family_id":"qf-mod36-wlln-epsilon-delta-witness","seed":202611057055,"semantic_template":"qf-mod36-wlln-epsilon-delta-witness/{\"delta\":[1,14],\"epsilon\":[1,6],\"iid_certificate\":\"symbolic_all_finite_subfamilies\",\"mean\":[4,5],\"variance\":[5,6]}"},"fields":{"title":"Choose a Strict All-n Chebyshev Tail Witness","stem":"Use declared mean=4/5, positive finite variance=5/6, and iid_certificate=symbolic_all_finite_subfamilies. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive epsilon=1/6, delta=1/14. Give (epsilon, delta, N, chebyshev_bound_at_N, strict_delta_tail, certificate).","answer":{"kind":"expression","value":"(1/6, 1/14, 421, 30/421, Yes, forall_n_ge_N_monotone_Chebyshev_tail)","explanation":"Choose N=floor(variance/(delta*epsilon^2))+1 and compute variance/(N*epsilon^2). The strict bound is below delta, including at integral thresholds; monotonicity extends it to every n≥N under the same declared IID premise. The exact result, retaining the requested order, is (1/6, 1/14, 421, 30/421, Yes, forall_n_ge_N_monotone_Chebyshev_tail)."},"skills":["Choose a Strict All-n Chebyshev Tail Witness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r07-weak-law-large-numbers-0056","batch_id":"batch-20261014-advance-20261105-r07-weak-law-large-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3595e024e1367c6a482d8685b1e73f5d2885d9856bb5aab8e3b508539fa01b1a","family_id":"qf-mod36-wlln-epsilon-delta-witness","seed":202611057056,"semantic_template":"qf-mod36-wlln-epsilon-delta-witness/{\"delta\":[1,15],\"epsilon\":[1,7],\"iid_certificate\":\"symbolic_all_finite_subfamilies\",\"mean\":[5,6],\"variance\":[6,7]}"},"fields":{"title":"Choose a Strict All-n Chebyshev Tail Witness","stem":"Use declared mean=5/6, positive finite variance=6/7, and iid_certificate=symbolic_all_finite_subfamilies. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive epsilon=1/7, delta=1/15. Give (epsilon, delta, N, chebyshev_bound_at_N, strict_delta_tail, certificate).","answer":{"kind":"expression","value":"(1/7, 1/15, 631, 42/631, Yes, forall_n_ge_N_monotone_Chebyshev_tail)","explanation":"Choose N=floor(variance/(delta*epsilon^2))+1 and compute variance/(N*epsilon^2). The strict bound is below delta, including at integral thresholds; monotonicity extends it to every n≥N under the same declared IID premise. The exact result, retaining the requested order, is (1/7, 1/15, 631, 42/631, Yes, forall_n_ge_N_monotone_Chebyshev_tail)."},"skills":["Choose a Strict All-n Chebyshev Tail Witness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r07-weak-law-large-numbers-0057","batch_id":"batch-20261014-advance-20261105-r07-weak-law-large-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"62892669d7471882c673559f5b465eac61fa213a3a37701b5a9652d81862e4e9","family_id":"qf-mod36-wlln-epsilon-delta-witness","seed":202611057057,"semantic_template":"qf-mod36-wlln-epsilon-delta-witness/{\"delta\":[1,16],\"epsilon\":[1,8],\"iid_certificate\":\"symbolic_all_finite_subfamilies\",\"mean\":[6,7],\"variance\":[7,8]}"},"fields":{"title":"Choose a Strict All-n Chebyshev Tail Witness","stem":"Use declared mean=6/7, positive finite variance=7/8, and iid_certificate=symbolic_all_finite_subfamilies. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive epsilon=1/8, delta=1/16. Give (epsilon, delta, N, chebyshev_bound_at_N, strict_delta_tail, certificate).","answer":{"kind":"expression","value":"(1/8, 1/16, 897, 56/897, Yes, forall_n_ge_N_monotone_Chebyshev_tail)","explanation":"Choose N=floor(variance/(delta*epsilon^2))+1 and compute variance/(N*epsilon^2). The strict bound is below delta, including at integral thresholds; monotonicity extends it to every n≥N under the same declared IID premise. The exact result, retaining the requested order, is (1/8, 1/16, 897, 56/897, Yes, forall_n_ge_N_monotone_Chebyshev_tail)."},"skills":["Choose a Strict All-n Chebyshev Tail Witness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r07-weak-law-large-numbers-0058","batch_id":"batch-20261014-advance-20261105-r07-weak-law-large-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ac92fb68bf08124a36467fab31c3c977fa41ae6477615e8353bfc43c0833d477","family_id":"qf-mod36-wlln-epsilon-delta-witness","seed":202611057058,"semantic_template":"qf-mod36-wlln-epsilon-delta-witness/{\"delta\":[1,17],\"epsilon\":[1,9],\"iid_certificate\":\"symbolic_all_finite_subfamilies\",\"mean\":[7,8],\"variance\":[8,9]}"},"fields":{"title":"Choose a Strict All-n Chebyshev Tail Witness","stem":"Use declared mean=7/8, positive finite variance=8/9, and iid_certificate=symbolic_all_finite_subfamilies. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive epsilon=1/9, delta=1/17. Give (epsilon, delta, N, chebyshev_bound_at_N, strict_delta_tail, certificate).","answer":{"kind":"expression","value":"(1/9, 1/17, 1225, 72/1225, Yes, forall_n_ge_N_monotone_Chebyshev_tail)","explanation":"Choose N=floor(variance/(delta*epsilon^2))+1 and compute variance/(N*epsilon^2). The strict bound is below delta, including at integral thresholds; monotonicity extends it to every n≥N under the same declared IID premise. The exact result, retaining the requested order, is (1/9, 1/17, 1225, 72/1225, Yes, forall_n_ge_N_monotone_Chebyshev_tail)."},"skills":["Choose a Strict All-n Chebyshev Tail Witness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r07-weak-law-large-numbers-0059","batch_id":"batch-20261014-advance-20261105-r07-weak-law-large-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"49f48538a8c12143576f1b8071f5c4d1ab3c513c0e35741440c3246b904acc33","family_id":"qf-mod36-wlln-epsilon-delta-witness","seed":202611057059,"semantic_template":"qf-mod36-wlln-epsilon-delta-witness/{\"delta\":[1,18],\"epsilon\":[1,10],\"iid_certificate\":\"symbolic_all_finite_subfamilies\",\"mean\":[8,9],\"variance\":[9,10]}"},"fields":{"title":"Choose a Strict All-n Chebyshev Tail Witness","stem":"Use declared mean=8/9, positive finite variance=9/10, and iid_certificate=symbolic_all_finite_subfamilies. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive epsilon=1/10, delta=1/18. Give (epsilon, delta, N, chebyshev_bound_at_N, strict_delta_tail, certificate).","answer":{"kind":"expression","value":"(1/10, 1/18, 1621, 90/1621, Yes, forall_n_ge_N_monotone_Chebyshev_tail)","explanation":"Choose N=floor(variance/(delta*epsilon^2))+1 and compute variance/(N*epsilon^2). The strict bound is below delta, including at integral thresholds; monotonicity extends it to every n≥N under the same declared IID premise. The exact result, retaining the requested order, is (1/10, 1/18, 1621, 90/1621, Yes, forall_n_ge_N_monotone_Chebyshev_tail)."},"skills":["Choose a Strict All-n Chebyshev Tail Witness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r07-weak-law-large-numbers-0060","batch_id":"batch-20261014-advance-20261105-r07-weak-law-large-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7db81ce6a18d2215858e7512f8daf7f775784e373d3b94bf132f051ba1c28053","family_id":"qf-mod36-wlln-epsilon-delta-witness","seed":202611057060,"semantic_template":"qf-mod36-wlln-epsilon-delta-witness/{\"delta\":[1,19],\"epsilon\":[1,11],\"iid_certificate\":\"symbolic_all_finite_subfamilies\",\"mean\":[9,10],\"variance\":[10,11]}"},"fields":{"title":"Choose a Strict All-n Chebyshev Tail Witness","stem":"Use declared mean=9/10, positive finite variance=10/11, and iid_certificate=symbolic_all_finite_subfamilies. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive epsilon=1/11, delta=1/19. Give (epsilon, delta, N, chebyshev_bound_at_N, strict_delta_tail, certificate).","answer":{"kind":"expression","value":"(1/11, 1/19, 2091, 110/2091, Yes, forall_n_ge_N_monotone_Chebyshev_tail)","explanation":"Choose N=floor(variance/(delta*epsilon^2))+1 and compute variance/(N*epsilon^2). The strict bound is below delta, including at integral thresholds; monotonicity extends it to every n≥N under the same declared IID premise. The exact result, retaining the requested order, is (1/11, 1/19, 2091, 110/2091, Yes, forall_n_ge_N_monotone_Chebyshev_tail)."},"skills":["Choose a Strict All-n Chebyshev Tail Witness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r07-weak-law-large-numbers-0061","batch_id":"batch-20261014-advance-20261105-r07-weak-law-large-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"17ee624eea66d96d12ebd84d18757b0a460440155f776774c93a4a27d07a6aa5","family_id":"qf-mod36-wlln-epsilon-delta-witness","seed":202611057061,"semantic_template":"qf-mod36-wlln-epsilon-delta-witness/{\"delta\":[1,20],\"epsilon\":[1,12],\"iid_certificate\":\"symbolic_all_finite_subfamilies\",\"mean\":[10,11],\"variance\":[11,12]}"},"fields":{"title":"Choose a Strict All-n Chebyshev Tail Witness","stem":"Use declared mean=10/11, positive finite variance=11/12, and iid_certificate=symbolic_all_finite_subfamilies. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive epsilon=1/12, delta=1/20. Give (epsilon, delta, N, chebyshev_bound_at_N, strict_delta_tail, certificate).","answer":{"kind":"expression","value":"(1/12, 1/20, 2641, 132/2641, Yes, forall_n_ge_N_monotone_Chebyshev_tail)","explanation":"Choose N=floor(variance/(delta*epsilon^2))+1 and compute variance/(N*epsilon^2). The strict bound is below delta, including at integral thresholds; monotonicity extends it to every n≥N under the same declared IID premise. The exact result, retaining the requested order, is (1/12, 1/20, 2641, 132/2641, Yes, forall_n_ge_N_monotone_Chebyshev_tail)."},"skills":["Choose a Strict All-n Chebyshev Tail Witness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r07-weak-law-large-numbers-0062","batch_id":"batch-20261014-advance-20261105-r07-weak-law-large-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4287a3e825408935f55f2504c04b945a1853e9fa9deb1613e01178aa2d2eaa50","family_id":"qf-mod36-wlln-epsilon-delta-witness","seed":202611057062,"semantic_template":"qf-mod36-wlln-epsilon-delta-witness/{\"delta\":[1,21],\"epsilon\":[1,13],\"iid_certificate\":\"symbolic_all_finite_subfamilies\",\"mean\":[11,12],\"variance\":[12,13]}"},"fields":{"title":"Choose a Strict All-n Chebyshev Tail Witness","stem":"Use declared mean=11/12, positive finite variance=12/13, and iid_certificate=symbolic_all_finite_subfamilies. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive epsilon=1/13, delta=1/21. Give (epsilon, delta, N, chebyshev_bound_at_N, strict_delta_tail, certificate).","answer":{"kind":"expression","value":"(1/13, 1/21, 3277, 156/3277, Yes, forall_n_ge_N_monotone_Chebyshev_tail)","explanation":"Choose N=floor(variance/(delta*epsilon^2))+1 and compute variance/(N*epsilon^2). The strict bound is below delta, including at integral thresholds; monotonicity extends it to every n≥N under the same declared IID premise. The exact result, retaining the requested order, is (1/13, 1/21, 3277, 156/3277, Yes, forall_n_ge_N_monotone_Chebyshev_tail)."},"skills":["Choose a Strict All-n Chebyshev Tail Witness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r07-weak-law-large-numbers-0063","batch_id":"batch-20261014-advance-20261105-r07-weak-law-large-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"02f3f169c56c8495b508dd9000a8fdfb8822acee7574ffe95b1e9bed95cf5ac4","family_id":"qf-mod36-wlln-epsilon-delta-witness","seed":202611057063,"semantic_template":"qf-mod36-wlln-epsilon-delta-witness/{\"delta\":[1,22],\"epsilon\":[1,14],\"iid_certificate\":\"symbolic_all_finite_subfamilies\",\"mean\":[12,13],\"variance\":[13,14]}"},"fields":{"title":"Choose a Strict All-n Chebyshev Tail Witness","stem":"Use declared mean=12/13, positive finite variance=13/14, and iid_certificate=symbolic_all_finite_subfamilies. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive epsilon=1/14, delta=1/22. Give (epsilon, delta, N, chebyshev_bound_at_N, strict_delta_tail, certificate).","answer":{"kind":"expression","value":"(1/14, 1/22, 4005, 182/4005, Yes, forall_n_ge_N_monotone_Chebyshev_tail)","explanation":"Choose N=floor(variance/(delta*epsilon^2))+1 and compute variance/(N*epsilon^2). The strict bound is below delta, including at integral thresholds; monotonicity extends it to every n≥N under the same declared IID premise. The exact result, retaining the requested order, is (1/14, 1/22, 4005, 182/4005, Yes, forall_n_ge_N_monotone_Chebyshev_tail)."},"skills":["Choose a Strict All-n Chebyshev Tail Witness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r07-weak-law-large-numbers-0064","batch_id":"batch-20261014-advance-20261105-r07-weak-law-large-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e5e67412a787d291388af956a5cd5d41206fd21f4c45797720ab286ed341dfe4","family_id":"qf-mod36-wlln-epsilon-delta-witness","seed":202611057064,"semantic_template":"qf-mod36-wlln-epsilon-delta-witness/{\"delta\":[1,23],\"epsilon\":[1,15],\"iid_certificate\":\"symbolic_all_finite_subfamilies\",\"mean\":[13,14],\"variance\":[14,15]}"},"fields":{"title":"Choose a Strict All-n Chebyshev Tail Witness","stem":"Use declared mean=13/14, positive finite variance=14/15, and iid_certificate=symbolic_all_finite_subfamilies. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive epsilon=1/15, delta=1/23. Give (epsilon, delta, N, chebyshev_bound_at_N, strict_delta_tail, certificate).","answer":{"kind":"expression","value":"(1/15, 1/23, 4831, 210/4831, Yes, forall_n_ge_N_monotone_Chebyshev_tail)","explanation":"Choose N=floor(variance/(delta*epsilon^2))+1 and compute variance/(N*epsilon^2). The strict bound is below delta, including at integral thresholds; monotonicity extends it to every n≥N under the same declared IID premise. The exact result, retaining the requested order, is (1/15, 1/23, 4831, 210/4831, Yes, forall_n_ge_N_monotone_Chebyshev_tail)."},"skills":["Choose a Strict All-n Chebyshev Tail Witness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r07-weak-law-large-numbers-0065","batch_id":"batch-20261014-advance-20261105-r07-weak-law-large-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"74b0cc8538a95848bb5bf9c39c453f85dc2217c7a9720a5a2206e0baf8a72932","family_id":"qf-mod36-wlln-epsilon-delta-witness","seed":202611057065,"semantic_template":"qf-mod36-wlln-epsilon-delta-witness/{\"delta\":[1,24],\"epsilon\":[1,16],\"iid_certificate\":\"symbolic_all_finite_subfamilies\",\"mean\":[14,15],\"variance\":[15,16]}"},"fields":{"title":"Choose a Strict All-n Chebyshev Tail Witness","stem":"Use declared mean=14/15, positive finite variance=15/16, and iid_certificate=symbolic_all_finite_subfamilies. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive epsilon=1/16, delta=1/24. Give (epsilon, delta, N, chebyshev_bound_at_N, strict_delta_tail, certificate).","answer":{"kind":"expression","value":"(1/16, 1/24, 5761, 240/5761, Yes, forall_n_ge_N_monotone_Chebyshev_tail)","explanation":"Choose N=floor(variance/(delta*epsilon^2))+1 and compute variance/(N*epsilon^2). The strict bound is below delta, including at integral thresholds; monotonicity extends it to every n≥N under the same declared IID premise. The exact result, retaining the requested order, is (1/16, 1/24, 5761, 240/5761, Yes, forall_n_ge_N_monotone_Chebyshev_tail)."},"skills":["Choose a Strict All-n Chebyshev Tail Witness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r07-weak-law-large-numbers-0066","batch_id":"batch-20261014-advance-20261105-r07-weak-law-large-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"50a02801a357e51c02c91487b5d3ac5d3fe23dcc06bfa450bae22e168c506bde","family_id":"qf-mod36-wlln-epsilon-delta-witness","seed":202611057066,"semantic_template":"qf-mod36-wlln-epsilon-delta-witness/{\"delta\":[1,25],\"epsilon\":[1,17],\"iid_certificate\":\"symbolic_all_finite_subfamilies\",\"mean\":[15,16],\"variance\":[16,17]}"},"fields":{"title":"Choose a Strict All-n Chebyshev Tail Witness","stem":"Use declared mean=15/16, positive finite variance=16/17, and iid_certificate=symbolic_all_finite_subfamilies. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive epsilon=1/17, delta=1/25. Give (epsilon, delta, N, chebyshev_bound_at_N, strict_delta_tail, certificate).","answer":{"kind":"expression","value":"(1/17, 1/25, 6801, 272/6801, Yes, forall_n_ge_N_monotone_Chebyshev_tail)","explanation":"Choose N=floor(variance/(delta*epsilon^2))+1 and compute variance/(N*epsilon^2). The strict bound is below delta, including at integral thresholds; monotonicity extends it to every n≥N under the same declared IID premise. The exact result, retaining the requested order, is (1/17, 1/25, 6801, 272/6801, Yes, forall_n_ge_N_monotone_Chebyshev_tail)."},"skills":["Choose a Strict All-n Chebyshev Tail Witness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r07-weak-law-large-numbers-0067","batch_id":"batch-20261014-advance-20261105-r07-weak-law-large-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f056d9cfa9bcf27ee47e977fdfee02b2c2d6776286b9033f6253be97bcfd8525","family_id":"qf-mod36-wlln-epsilon-delta-witness","seed":202611057067,"semantic_template":"qf-mod36-wlln-epsilon-delta-witness/{\"delta\":[1,26],\"epsilon\":[1,18],\"iid_certificate\":\"symbolic_all_finite_subfamilies\",\"mean\":[16,17],\"variance\":[17,18]}"},"fields":{"title":"Choose a Strict All-n Chebyshev Tail Witness","stem":"Use declared mean=16/17, positive finite variance=17/18, and iid_certificate=symbolic_all_finite_subfamilies. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive epsilon=1/18, delta=1/26. Give (epsilon, delta, N, chebyshev_bound_at_N, strict_delta_tail, certificate).","answer":{"kind":"expression","value":"(1/18, 1/26, 7957, 306/7957, Yes, forall_n_ge_N_monotone_Chebyshev_tail)","explanation":"Choose N=floor(variance/(delta*epsilon^2))+1 and compute variance/(N*epsilon^2). The strict bound is below delta, including at integral thresholds; monotonicity extends it to every n≥N under the same declared IID premise. The exact result, retaining the requested order, is (1/18, 1/26, 7957, 306/7957, Yes, forall_n_ge_N_monotone_Chebyshev_tail)."},"skills":["Choose a Strict All-n Chebyshev Tail Witness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r07-weak-law-large-numbers-0068","batch_id":"batch-20261014-advance-20261105-r07-weak-law-large-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1ac2b7124d70c9e6a4eabb2c45ae8f1865eed719e57dd64f022c835bdba96a1c","family_id":"qf-mod36-wlln-epsilon-delta-witness","seed":202611057068,"semantic_template":"qf-mod36-wlln-epsilon-delta-witness/{\"delta\":[1,27],\"epsilon\":[1,19],\"iid_certificate\":\"symbolic_all_finite_subfamilies\",\"mean\":[17,18],\"variance\":[18,19]}"},"fields":{"title":"Choose a Strict All-n Chebyshev Tail Witness","stem":"Use declared mean=17/18, positive finite variance=18/19, and iid_certificate=symbolic_all_finite_subfamilies. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive epsilon=1/19, delta=1/27. Give (epsilon, delta, N, chebyshev_bound_at_N, strict_delta_tail, certificate).","answer":{"kind":"expression","value":"(1/19, 1/27, 9235, 342/9235, Yes, forall_n_ge_N_monotone_Chebyshev_tail)","explanation":"Choose N=floor(variance/(delta*epsilon^2))+1 and compute variance/(N*epsilon^2). The strict bound is below delta, including at integral thresholds; monotonicity extends it to every n≥N under the same declared IID premise. The exact result, retaining the requested order, is (1/19, 1/27, 9235, 342/9235, Yes, forall_n_ge_N_monotone_Chebyshev_tail)."},"skills":["Choose a Strict All-n Chebyshev Tail Witness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r07-weak-law-large-numbers-0069","batch_id":"batch-20261014-advance-20261105-r07-weak-law-large-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f664c7e5aa78e40503e806358c8bd9c95acbaa80778f241fa4d336f80e8f0e23","family_id":"qf-mod36-wlln-epsilon-delta-witness","seed":202611057069,"semantic_template":"qf-mod36-wlln-epsilon-delta-witness/{\"delta\":[1,28],\"epsilon\":[1,20],\"iid_certificate\":\"symbolic_all_finite_subfamilies\",\"mean\":[18,19],\"variance\":[19,20]}"},"fields":{"title":"Choose a Strict All-n Chebyshev Tail Witness","stem":"Use declared mean=18/19, positive finite variance=19/20, and iid_certificate=symbolic_all_finite_subfamilies. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive epsilon=1/20, delta=1/28. Give (epsilon, delta, N, chebyshev_bound_at_N, strict_delta_tail, certificate).","answer":{"kind":"expression","value":"(1/20, 1/28, 10641, 380/10641, Yes, forall_n_ge_N_monotone_Chebyshev_tail)","explanation":"Choose N=floor(variance/(delta*epsilon^2))+1 and compute variance/(N*epsilon^2). The strict bound is below delta, including at integral thresholds; monotonicity extends it to every n≥N under the same declared IID premise. The exact result, retaining the requested order, is (1/20, 1/28, 10641, 380/10641, Yes, forall_n_ge_N_monotone_Chebyshev_tail)."},"skills":["Choose a Strict All-n Chebyshev Tail Witness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r07-weak-law-large-numbers-0070","batch_id":"batch-20261014-advance-20261105-r07-weak-law-large-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9149c2266abe1fe16a2f5c19009b93baae681ad15c395ad6ab74542d98c5645b","family_id":"qf-mod36-wlln-epsilon-delta-witness","seed":202611057070,"semantic_template":"qf-mod36-wlln-epsilon-delta-witness/{\"delta\":[1,29],\"epsilon\":[1,21],\"iid_certificate\":\"symbolic_all_finite_subfamilies\",\"mean\":[19,20],\"variance\":[20,21]}"},"fields":{"title":"Choose a Strict All-n Chebyshev Tail Witness","stem":"Use declared mean=19/20, positive finite variance=20/21, and iid_certificate=symbolic_all_finite_subfamilies. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive epsilon=1/21, delta=1/29. Give (epsilon, delta, N, chebyshev_bound_at_N, strict_delta_tail, certificate).","answer":{"kind":"expression","value":"(1/21, 1/29, 12181, 420/12181, Yes, forall_n_ge_N_monotone_Chebyshev_tail)","explanation":"Choose N=floor(variance/(delta*epsilon^2))+1 and compute variance/(N*epsilon^2). The strict bound is below delta, including at integral thresholds; monotonicity extends it to every n≥N under the same declared IID premise. The exact result, retaining the requested order, is (1/21, 1/29, 12181, 420/12181, Yes, forall_n_ge_N_monotone_Chebyshev_tail)."},"skills":["Choose a Strict All-n Chebyshev Tail Witness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r07-weak-law-large-numbers-0071","batch_id":"batch-20261014-advance-20261105-r07-weak-law-large-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6995a85d513d12e0ec6b1b8947c871b840457bcd82ecd178d94df60a94c7ebec","family_id":"qf-mod36-wlln-epsilon-delta-witness","seed":202611057071,"semantic_template":"qf-mod36-wlln-epsilon-delta-witness/{\"delta\":[1,30],\"epsilon\":[1,22],\"iid_certificate\":\"symbolic_all_finite_subfamilies\",\"mean\":[20,21],\"variance\":[21,22]}"},"fields":{"title":"Choose a Strict All-n Chebyshev Tail Witness","stem":"Use declared mean=20/21, positive finite variance=21/22, and iid_certificate=symbolic_all_finite_subfamilies. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive epsilon=1/22, delta=1/30. Give (epsilon, delta, N, chebyshev_bound_at_N, strict_delta_tail, certificate).","answer":{"kind":"expression","value":"(1/22, 1/30, 13861, 462/13861, Yes, forall_n_ge_N_monotone_Chebyshev_tail)","explanation":"Choose N=floor(variance/(delta*epsilon^2))+1 and compute variance/(N*epsilon^2). The strict bound is below delta, including at integral thresholds; monotonicity extends it to every n≥N under the same declared IID premise. The exact result, retaining the requested order, is (1/22, 1/30, 13861, 462/13861, Yes, forall_n_ge_N_monotone_Chebyshev_tail)."},"skills":["Choose a Strict All-n Chebyshev Tail Witness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r07-weak-law-large-numbers-0072","batch_id":"batch-20261014-advance-20261105-r07-weak-law-large-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d2feb67afd9315deed7eeb1b240d6ca534a13be2f52bd1a332e7bb8fb45d3730","family_id":"qf-mod36-wlln-epsilon-delta-witness","seed":202611057072,"semantic_template":"qf-mod36-wlln-epsilon-delta-witness/{\"delta\":[1,31],\"epsilon\":[1,23],\"iid_certificate\":\"symbolic_all_finite_subfamilies\",\"mean\":[21,22],\"variance\":[22,23]}"},"fields":{"title":"Choose a Strict All-n Chebyshev Tail Witness","stem":"Use declared mean=21/22, positive finite variance=22/23, and iid_certificate=symbolic_all_finite_subfamilies. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive epsilon=1/23, delta=1/31. Give (epsilon, delta, N, chebyshev_bound_at_N, strict_delta_tail, certificate).","answer":{"kind":"expression","value":"(1/23, 1/31, 15687, 506/15687, Yes, forall_n_ge_N_monotone_Chebyshev_tail)","explanation":"Choose N=floor(variance/(delta*epsilon^2))+1 and compute variance/(N*epsilon^2). The strict bound is below delta, including at integral thresholds; monotonicity extends it to every n≥N under the same declared IID premise. The exact result, retaining the requested order, is (1/23, 1/31, 15687, 506/15687, Yes, forall_n_ge_N_monotone_Chebyshev_tail)."},"skills":["Choose a Strict All-n Chebyshev Tail Witness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r07-weak-law-large-numbers-0073","batch_id":"batch-20261014-advance-20261105-r07-weak-law-large-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"890036e5323f4fc5981933e545ae67fa77092104b564a2da4079ad2a20935de1","family_id":"qf-mod36-wlln-epsilon-delta-witness","seed":202611057073,"semantic_template":"qf-mod36-wlln-epsilon-delta-witness/{\"delta\":[1,32],\"epsilon\":[1,24],\"iid_certificate\":\"symbolic_all_finite_subfamilies\",\"mean\":[22,23],\"variance\":[23,24]}"},"fields":{"title":"Choose a Strict All-n Chebyshev Tail Witness","stem":"Use declared mean=22/23, positive finite variance=23/24, and iid_certificate=symbolic_all_finite_subfamilies. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive epsilon=1/24, delta=1/32. Give (epsilon, delta, N, chebyshev_bound_at_N, strict_delta_tail, certificate).","answer":{"kind":"expression","value":"(1/24, 1/32, 17665, 552/17665, Yes, forall_n_ge_N_monotone_Chebyshev_tail)","explanation":"Choose N=floor(variance/(delta*epsilon^2))+1 and compute variance/(N*epsilon^2). The strict bound is below delta, including at integral thresholds; monotonicity extends it to every n≥N under the same declared IID premise. The exact result, retaining the requested order, is (1/24, 1/32, 17665, 552/17665, Yes, forall_n_ge_N_monotone_Chebyshev_tail)."},"skills":["Choose a Strict All-n Chebyshev Tail Witness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r07-weak-law-large-numbers-0074","batch_id":"batch-20261014-advance-20261105-r07-weak-law-large-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cf9741fa226553744d8498e6394711b7180ac8f283a7712d6e6b34edc08b64eb","family_id":"qf-mod36-wlln-epsilon-delta-witness","seed":202611057074,"semantic_template":"qf-mod36-wlln-epsilon-delta-witness/{\"delta\":[1,33],\"epsilon\":[1,25],\"iid_certificate\":\"symbolic_all_finite_subfamilies\",\"mean\":[23,24],\"variance\":[24,25]}"},"fields":{"title":"Choose a Strict All-n Chebyshev Tail Witness","stem":"Use declared mean=23/24, positive finite variance=24/25, and iid_certificate=symbolic_all_finite_subfamilies. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive epsilon=1/25, delta=1/33. Give (epsilon, delta, N, chebyshev_bound_at_N, strict_delta_tail, certificate).","answer":{"kind":"expression","value":"(1/25, 1/33, 19801, 600/19801, Yes, forall_n_ge_N_monotone_Chebyshev_tail)","explanation":"Choose N=floor(variance/(delta*epsilon^2))+1 and compute variance/(N*epsilon^2). The strict bound is below delta, including at integral thresholds; monotonicity extends it to every n≥N under the same declared IID premise. The exact result, retaining the requested order, is (1/25, 1/33, 19801, 600/19801, Yes, forall_n_ge_N_monotone_Chebyshev_tail)."},"skills":["Choose a Strict All-n Chebyshev Tail Witness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r07-weak-law-large-numbers-0075","batch_id":"batch-20261014-advance-20261105-r07-weak-law-large-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a7c61e917ac6e440340aba9a1dd5cc37908ad8d53081e0e46ef688fffb56c1ff","family_id":"qf-mod36-wlln-epsilon-delta-witness","seed":202611057075,"semantic_template":"qf-mod36-wlln-epsilon-delta-witness/{\"delta\":[1,34],\"epsilon\":[1,26],\"iid_certificate\":\"symbolic_all_finite_subfamilies\",\"mean\":[24,25],\"variance\":[25,26]}"},"fields":{"title":"Choose a Strict All-n Chebyshev Tail Witness","stem":"Use declared mean=24/25, positive finite variance=25/26, and iid_certificate=symbolic_all_finite_subfamilies. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive epsilon=1/26, delta=1/34. Give (epsilon, delta, N, chebyshev_bound_at_N, strict_delta_tail, certificate).","answer":{"kind":"expression","value":"(1/26, 1/34, 22101, 650/22101, Yes, forall_n_ge_N_monotone_Chebyshev_tail)","explanation":"Choose N=floor(variance/(delta*epsilon^2))+1 and compute variance/(N*epsilon^2). The strict bound is below delta, including at integral thresholds; monotonicity extends it to every n≥N under the same declared IID premise. The exact result, retaining the requested order, is (1/26, 1/34, 22101, 650/22101, Yes, forall_n_ge_N_monotone_Chebyshev_tail)."},"skills":["Choose a Strict All-n Chebyshev Tail Witness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r07-weak-law-large-numbers-0076","batch_id":"batch-20261014-advance-20261105-r07-weak-law-large-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"087ddc787beb69e78436f2cb0989bf2d3b3d6bebe2d5b06811c53145ac83a629","family_id":"qf-mod36-wlln-finite-simulation-boundary","seed":202611057076,"semantic_template":"qf-mod36-wlln-finite-simulation-boundary/{\"epsilon\":[1,2],\"iid_certificate\":\"finite_simulation_only\",\"mean\":[0,1],\"prefix_sample_means\":[[1,2],[2,3],[3,4]],\"variance\":[1,2]}"},"fields":{"title":"A Finite Simulation Does Not Certify the Whole IID Sequence","stem":"Use declared mean=0/1, positive finite variance=1/2, and iid_certificate=finite_simulation_only. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive epsilon=1/2, finite prefix of sample means=[1/2, 2/3, 3/4]. The supplied marker is finite_simulation_only, not the symbolic whole-sequence premise. Give (mean, epsilon, prefix_sample_means, sufficient, boundary).","answer":{"kind":"expression","value":"(0, 1/2, [1/2, 2/3, 3/4], No, finite_simulation_not_IID_certificate_or_WLLN_proof)","explanation":"Retain the exact observed prefix and No for sufficiency. No finite prefix alone proves IID for arbitrary finite subfamilies, the WLLN tail or a stronger convergence mode. The exact result, retaining the requested order, is (0, 1/2, [1/2, 2/3, 3/4], No, finite_simulation_not_IID_certificate_or_WLLN_proof)."},"skills":["A Finite Simulation Does Not Certify the Whole IID Sequence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261105-r07-weak-law-large-numbers-0077","batch_id":"batch-20261014-advance-20261105-r07-weak-law-large-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e756b354829ab4828701cbc28bd7573c819916b2c20be7459db96dcd3dc82409","family_id":"qf-mod36-wlln-finite-simulation-boundary","seed":202611057077,"semantic_template":"qf-mod36-wlln-finite-simulation-boundary/{\"epsilon\":[1,3],\"iid_certificate\":\"finite_simulation_only\",\"mean\":[1,2],\"prefix_sample_means\":[[2,3],[3,4],[4,5]],\"variance\":[2,3]}"},"fields":{"title":"A Finite Simulation Does Not Certify the Whole IID Sequence","stem":"Use declared mean=1/2, positive finite variance=2/3, and iid_certificate=finite_simulation_only. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive epsilon=1/3, finite prefix of sample means=[2/3, 3/4, 4/5]. The supplied marker is finite_simulation_only, not the symbolic whole-sequence premise. Give (mean, epsilon, prefix_sample_means, sufficient, boundary).","answer":{"kind":"expression","value":"(1/2, 1/3, [2/3, 3/4, 4/5], No, finite_simulation_not_IID_certificate_or_WLLN_proof)","explanation":"Retain the exact observed prefix and No for sufficiency. No finite prefix alone proves IID for arbitrary finite subfamilies, the WLLN tail or a stronger convergence mode. The exact result, retaining the requested order, is (1/2, 1/3, [2/3, 3/4, 4/5], No, finite_simulation_not_IID_certificate_or_WLLN_proof)."},"skills":["A Finite Simulation Does Not Certify the Whole IID Sequence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261105-r07-weak-law-large-numbers-0078","batch_id":"batch-20261014-advance-20261105-r07-weak-law-large-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bab1b3583b05a352dcc2acb0c0dcc528f6aeb8e4e6c34a42efe3f25427176d2a","family_id":"qf-mod36-wlln-finite-simulation-boundary","seed":202611057078,"semantic_template":"qf-mod36-wlln-finite-simulation-boundary/{\"epsilon\":[1,4],\"iid_certificate\":\"finite_simulation_only\",\"mean\":[2,3],\"prefix_sample_means\":[[3,4],[4,5],[5,6]],\"variance\":[3,4]}"},"fields":{"title":"A Finite Simulation Does Not Certify the Whole IID Sequence","stem":"Use declared mean=2/3, positive finite variance=3/4, and iid_certificate=finite_simulation_only. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive epsilon=1/4, finite prefix of sample means=[3/4, 4/5, 5/6]. The supplied marker is finite_simulation_only, not the symbolic whole-sequence premise. Give (mean, epsilon, prefix_sample_means, sufficient, boundary).","answer":{"kind":"expression","value":"(2/3, 1/4, [3/4, 4/5, 5/6], No, finite_simulation_not_IID_certificate_or_WLLN_proof)","explanation":"Retain the exact observed prefix and No for sufficiency. No finite prefix alone proves IID for arbitrary finite subfamilies, the WLLN tail or a stronger convergence mode. The exact result, retaining the requested order, is (2/3, 1/4, [3/4, 4/5, 5/6], No, finite_simulation_not_IID_certificate_or_WLLN_proof)."},"skills":["A Finite Simulation Does Not Certify the Whole IID Sequence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261105-r07-weak-law-large-numbers-0079","batch_id":"batch-20261014-advance-20261105-r07-weak-law-large-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"68c729cbc2eb917272edf9c542aa4ba7e0b452e8ae3926a234372b4f8bb30554","family_id":"qf-mod36-wlln-finite-simulation-boundary","seed":202611057079,"semantic_template":"qf-mod36-wlln-finite-simulation-boundary/{\"epsilon\":[1,5],\"iid_certificate\":\"finite_simulation_only\",\"mean\":[3,4],\"prefix_sample_means\":[[4,5],[5,6],[6,7]],\"variance\":[4,5]}"},"fields":{"title":"A Finite Simulation Does Not Certify the Whole IID Sequence","stem":"Use declared mean=3/4, positive finite variance=4/5, and iid_certificate=finite_simulation_only. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive epsilon=1/5, finite prefix of sample means=[4/5, 5/6, 6/7]. The supplied marker is finite_simulation_only, not the symbolic whole-sequence premise. Give (mean, epsilon, prefix_sample_means, sufficient, boundary).","answer":{"kind":"expression","value":"(3/4, 1/5, [4/5, 5/6, 6/7], No, finite_simulation_not_IID_certificate_or_WLLN_proof)","explanation":"Retain the exact observed prefix and No for sufficiency. No finite prefix alone proves IID for arbitrary finite subfamilies, the WLLN tail or a stronger convergence mode. The exact result, retaining the requested order, is (3/4, 1/5, [4/5, 5/6, 6/7], No, finite_simulation_not_IID_certificate_or_WLLN_proof)."},"skills":["A Finite Simulation Does Not Certify the Whole IID Sequence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261105-r07-weak-law-large-numbers-0080","batch_id":"batch-20261014-advance-20261105-r07-weak-law-large-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"959c643bc04999e0cabaab8d693215658c483314ca378b1f5f562fd86823213a","family_id":"qf-mod36-wlln-finite-simulation-boundary","seed":202611057080,"semantic_template":"qf-mod36-wlln-finite-simulation-boundary/{\"epsilon\":[1,6],\"iid_certificate\":\"finite_simulation_only\",\"mean\":[4,5],\"prefix_sample_means\":[[5,6],[6,7],[7,8]],\"variance\":[5,6]}"},"fields":{"title":"A Finite Simulation Does Not Certify the Whole IID Sequence","stem":"Use declared mean=4/5, positive finite variance=5/6, and iid_certificate=finite_simulation_only. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive epsilon=1/6, finite prefix of sample means=[5/6, 6/7, 7/8]. The supplied marker is finite_simulation_only, not the symbolic whole-sequence premise. Give (mean, epsilon, prefix_sample_means, sufficient, boundary).","answer":{"kind":"expression","value":"(4/5, 1/6, [5/6, 6/7, 7/8], No, finite_simulation_not_IID_certificate_or_WLLN_proof)","explanation":"Retain the exact observed prefix and No for sufficiency. No finite prefix alone proves IID for arbitrary finite subfamilies, the WLLN tail or a stronger convergence mode. The exact result, retaining the requested order, is (4/5, 1/6, [5/6, 6/7, 7/8], No, finite_simulation_not_IID_certificate_or_WLLN_proof)."},"skills":["A Finite Simulation Does Not Certify the Whole IID Sequence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261105-r07-weak-law-large-numbers-0081","batch_id":"batch-20261014-advance-20261105-r07-weak-law-large-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a65c94f11c15a0a949edbd6a65d2e0bb31f21d91661e936369d2400f7db1ec56","family_id":"qf-mod36-wlln-finite-simulation-boundary","seed":202611057081,"semantic_template":"qf-mod36-wlln-finite-simulation-boundary/{\"epsilon\":[1,7],\"iid_certificate\":\"finite_simulation_only\",\"mean\":[5,6],\"prefix_sample_means\":[[6,7],[7,8],[8,9]],\"variance\":[6,7]}"},"fields":{"title":"A Finite Simulation Does Not Certify the Whole IID Sequence","stem":"Use declared mean=5/6, positive finite variance=6/7, and iid_certificate=finite_simulation_only. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive epsilon=1/7, finite prefix of sample means=[6/7, 7/8, 8/9]. The supplied marker is finite_simulation_only, not the symbolic whole-sequence premise. Give (mean, epsilon, prefix_sample_means, sufficient, boundary).","answer":{"kind":"expression","value":"(5/6, 1/7, [6/7, 7/8, 8/9], No, finite_simulation_not_IID_certificate_or_WLLN_proof)","explanation":"Retain the exact observed prefix and No for sufficiency. No finite prefix alone proves IID for arbitrary finite subfamilies, the WLLN tail or a stronger convergence mode. The exact result, retaining the requested order, is (5/6, 1/7, [6/7, 7/8, 8/9], No, finite_simulation_not_IID_certificate_or_WLLN_proof)."},"skills":["A Finite Simulation Does Not Certify the Whole IID Sequence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261105-r07-weak-law-large-numbers-0082","batch_id":"batch-20261014-advance-20261105-r07-weak-law-large-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"58a09fefb9f9645f1170531946ec7825a0f87b5098e4b60dde4de842309f9a6d","family_id":"qf-mod36-wlln-finite-simulation-boundary","seed":202611057082,"semantic_template":"qf-mod36-wlln-finite-simulation-boundary/{\"epsilon\":[1,8],\"iid_certificate\":\"finite_simulation_only\",\"mean\":[6,7],\"prefix_sample_means\":[[7,8],[8,9],[9,10]],\"variance\":[7,8]}"},"fields":{"title":"A Finite Simulation Does Not Certify the Whole IID Sequence","stem":"Use declared mean=6/7, positive finite variance=7/8, and iid_certificate=finite_simulation_only. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive epsilon=1/8, finite prefix of sample means=[7/8, 8/9, 9/10]. The supplied marker is finite_simulation_only, not the symbolic whole-sequence premise. Give (mean, epsilon, prefix_sample_means, sufficient, boundary).","answer":{"kind":"expression","value":"(6/7, 1/8, [7/8, 8/9, 9/10], No, finite_simulation_not_IID_certificate_or_WLLN_proof)","explanation":"Retain the exact observed prefix and No for sufficiency. No finite prefix alone proves IID for arbitrary finite subfamilies, the WLLN tail or a stronger convergence mode. The exact result, retaining the requested order, is (6/7, 1/8, [7/8, 8/9, 9/10], No, finite_simulation_not_IID_certificate_or_WLLN_proof)."},"skills":["A Finite Simulation Does Not Certify the Whole IID Sequence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261105-r07-weak-law-large-numbers-0083","batch_id":"batch-20261014-advance-20261105-r07-weak-law-large-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3300393e7435f6138bbc4cb7ce7680446a26b89a207df66c35470bcccf3762de","family_id":"qf-mod36-wlln-finite-simulation-boundary","seed":202611057083,"semantic_template":"qf-mod36-wlln-finite-simulation-boundary/{\"epsilon\":[1,9],\"iid_certificate\":\"finite_simulation_only\",\"mean\":[7,8],\"prefix_sample_means\":[[8,9],[9,10],[10,11]],\"variance\":[8,9]}"},"fields":{"title":"A Finite Simulation Does Not Certify the Whole IID Sequence","stem":"Use declared mean=7/8, positive finite variance=8/9, and iid_certificate=finite_simulation_only. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive epsilon=1/9, finite prefix of sample means=[8/9, 9/10, 10/11]. The supplied marker is finite_simulation_only, not the symbolic whole-sequence premise. Give (mean, epsilon, prefix_sample_means, sufficient, boundary).","answer":{"kind":"expression","value":"(7/8, 1/9, [8/9, 9/10, 10/11], No, finite_simulation_not_IID_certificate_or_WLLN_proof)","explanation":"Retain the exact observed prefix and No for sufficiency. No finite prefix alone proves IID for arbitrary finite subfamilies, the WLLN tail or a stronger convergence mode. The exact result, retaining the requested order, is (7/8, 1/9, [8/9, 9/10, 10/11], No, finite_simulation_not_IID_certificate_or_WLLN_proof)."},"skills":["A Finite Simulation Does Not Certify the Whole IID Sequence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261105-r07-weak-law-large-numbers-0084","batch_id":"batch-20261014-advance-20261105-r07-weak-law-large-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"849d8d94d97d0dabf2b29d2f2086fe0ba0ca94e31f648201a0ac93b738ce4d4d","family_id":"qf-mod36-wlln-finite-simulation-boundary","seed":202611057084,"semantic_template":"qf-mod36-wlln-finite-simulation-boundary/{\"epsilon\":[1,10],\"iid_certificate\":\"finite_simulation_only\",\"mean\":[8,9],\"prefix_sample_means\":[[9,10],[10,11],[11,12]],\"variance\":[9,10]}"},"fields":{"title":"A Finite Simulation Does Not Certify the Whole IID Sequence","stem":"Use declared mean=8/9, positive finite variance=9/10, and iid_certificate=finite_simulation_only. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive epsilon=1/10, finite prefix of sample means=[9/10, 10/11, 11/12]. The supplied marker is finite_simulation_only, not the symbolic whole-sequence premise. Give (mean, epsilon, prefix_sample_means, sufficient, boundary).","answer":{"kind":"expression","value":"(8/9, 1/10, [9/10, 10/11, 11/12], No, finite_simulation_not_IID_certificate_or_WLLN_proof)","explanation":"Retain the exact observed prefix and No for sufficiency. No finite prefix alone proves IID for arbitrary finite subfamilies, the WLLN tail or a stronger convergence mode. The exact result, retaining the requested order, is (8/9, 1/10, [9/10, 10/11, 11/12], No, finite_simulation_not_IID_certificate_or_WLLN_proof)."},"skills":["A Finite Simulation Does Not Certify the Whole IID Sequence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261105-r07-weak-law-large-numbers-0085","batch_id":"batch-20261014-advance-20261105-r07-weak-law-large-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8cff1ae625ade00a4e6e2ca2aa1c7ed6ee1e33c354d06ecadfcebeede5a0090a","family_id":"qf-mod36-wlln-finite-simulation-boundary","seed":202611057085,"semantic_template":"qf-mod36-wlln-finite-simulation-boundary/{\"epsilon\":[1,11],\"iid_certificate\":\"finite_simulation_only\",\"mean\":[9,10],\"prefix_sample_means\":[[10,11],[11,12],[12,13]],\"variance\":[10,11]}"},"fields":{"title":"A Finite Simulation Does Not Certify the Whole IID Sequence","stem":"Use declared mean=9/10, positive finite variance=10/11, and iid_certificate=finite_simulation_only. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive epsilon=1/11, finite prefix of sample means=[10/11, 11/12, 12/13]. The supplied marker is finite_simulation_only, not the symbolic whole-sequence premise. Give (mean, epsilon, prefix_sample_means, sufficient, boundary).","answer":{"kind":"expression","value":"(9/10, 1/11, [10/11, 11/12, 12/13], No, finite_simulation_not_IID_certificate_or_WLLN_proof)","explanation":"Retain the exact observed prefix and No for sufficiency. No finite prefix alone proves IID for arbitrary finite subfamilies, the WLLN tail or a stronger convergence mode. The exact result, retaining the requested order, is (9/10, 1/11, [10/11, 11/12, 12/13], No, finite_simulation_not_IID_certificate_or_WLLN_proof)."},"skills":["A Finite Simulation Does Not Certify the Whole IID Sequence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261105-r07-weak-law-large-numbers-0086","batch_id":"batch-20261014-advance-20261105-r07-weak-law-large-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2f0f683676c58f3fb8c6c3e7ee2a635e9ca74fd4949532bf2416b6b7c60169bc","family_id":"qf-mod36-wlln-finite-simulation-boundary","seed":202611057086,"semantic_template":"qf-mod36-wlln-finite-simulation-boundary/{\"epsilon\":[1,12],\"iid_certificate\":\"finite_simulation_only\",\"mean\":[10,11],\"prefix_sample_means\":[[11,12],[12,13],[13,14]],\"variance\":[11,12]}"},"fields":{"title":"A Finite Simulation Does Not Certify the Whole IID Sequence","stem":"Use declared mean=10/11, positive finite variance=11/12, and iid_certificate=finite_simulation_only. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive epsilon=1/12, finite prefix of sample means=[11/12, 12/13, 13/14]. The supplied marker is finite_simulation_only, not the symbolic whole-sequence premise. Give (mean, epsilon, prefix_sample_means, sufficient, boundary).","answer":{"kind":"expression","value":"(10/11, 1/12, [11/12, 12/13, 13/14], No, finite_simulation_not_IID_certificate_or_WLLN_proof)","explanation":"Retain the exact observed prefix and No for sufficiency. No finite prefix alone proves IID for arbitrary finite subfamilies, the WLLN tail or a stronger convergence mode. The exact result, retaining the requested order, is (10/11, 1/12, [11/12, 12/13, 13/14], No, finite_simulation_not_IID_certificate_or_WLLN_proof)."},"skills":["A Finite Simulation Does Not Certify the Whole IID Sequence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261105-r07-weak-law-large-numbers-0087","batch_id":"batch-20261014-advance-20261105-r07-weak-law-large-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"02c57283eacd7984ec14778a4cabdf7cc7cb31f3c87c71e7b67f4688f412ccb1","family_id":"qf-mod36-wlln-finite-simulation-boundary","seed":202611057087,"semantic_template":"qf-mod36-wlln-finite-simulation-boundary/{\"epsilon\":[1,13],\"iid_certificate\":\"finite_simulation_only\",\"mean\":[11,12],\"prefix_sample_means\":[[12,13],[13,14],[14,15]],\"variance\":[12,13]}"},"fields":{"title":"A Finite Simulation Does Not Certify the Whole IID Sequence","stem":"Use declared mean=11/12, positive finite variance=12/13, and iid_certificate=finite_simulation_only. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive epsilon=1/13, finite prefix of sample means=[12/13, 13/14, 14/15]. The supplied marker is finite_simulation_only, not the symbolic whole-sequence premise. Give (mean, epsilon, prefix_sample_means, sufficient, boundary).","answer":{"kind":"expression","value":"(11/12, 1/13, [12/13, 13/14, 14/15], No, finite_simulation_not_IID_certificate_or_WLLN_proof)","explanation":"Retain the exact observed prefix and No for sufficiency. No finite prefix alone proves IID for arbitrary finite subfamilies, the WLLN tail or a stronger convergence mode. The exact result, retaining the requested order, is (11/12, 1/13, [12/13, 13/14, 14/15], No, finite_simulation_not_IID_certificate_or_WLLN_proof)."},"skills":["A Finite Simulation Does Not Certify the Whole IID Sequence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261105-r07-weak-law-large-numbers-0088","batch_id":"batch-20261014-advance-20261105-r07-weak-law-large-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ea1dd9a1bec7879dd150ed5655e4f34d25ecf5af8ee223348036e68067a895ce","family_id":"qf-mod36-wlln-finite-simulation-boundary","seed":202611057088,"semantic_template":"qf-mod36-wlln-finite-simulation-boundary/{\"epsilon\":[1,14],\"iid_certificate\":\"finite_simulation_only\",\"mean\":[12,13],\"prefix_sample_means\":[[13,14],[14,15],[15,16]],\"variance\":[13,14]}"},"fields":{"title":"A Finite Simulation Does Not Certify the Whole IID Sequence","stem":"Use declared mean=12/13, positive finite variance=13/14, and iid_certificate=finite_simulation_only. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive epsilon=1/14, finite prefix of sample means=[13/14, 14/15, 15/16]. The supplied marker is finite_simulation_only, not the symbolic whole-sequence premise. Give (mean, epsilon, prefix_sample_means, sufficient, boundary).","answer":{"kind":"expression","value":"(12/13, 1/14, [13/14, 14/15, 15/16], No, finite_simulation_not_IID_certificate_or_WLLN_proof)","explanation":"Retain the exact observed prefix and No for sufficiency. No finite prefix alone proves IID for arbitrary finite subfamilies, the WLLN tail or a stronger convergence mode. The exact result, retaining the requested order, is (12/13, 1/14, [13/14, 14/15, 15/16], No, finite_simulation_not_IID_certificate_or_WLLN_proof)."},"skills":["A Finite Simulation Does Not Certify the Whole IID Sequence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261105-r07-weak-law-large-numbers-0089","batch_id":"batch-20261014-advance-20261105-r07-weak-law-large-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7f3d69299f2703cb1bc2a2e78420385acb39d381a19146b40738d1e400682528","family_id":"qf-mod36-wlln-finite-simulation-boundary","seed":202611057089,"semantic_template":"qf-mod36-wlln-finite-simulation-boundary/{\"epsilon\":[1,15],\"iid_certificate\":\"finite_simulation_only\",\"mean\":[13,14],\"prefix_sample_means\":[[14,15],[15,16],[16,17]],\"variance\":[14,15]}"},"fields":{"title":"A Finite Simulation Does Not Certify the Whole IID Sequence","stem":"Use declared mean=13/14, positive finite variance=14/15, and iid_certificate=finite_simulation_only. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive epsilon=1/15, finite prefix of sample means=[14/15, 15/16, 16/17]. The supplied marker is finite_simulation_only, not the symbolic whole-sequence premise. Give (mean, epsilon, prefix_sample_means, sufficient, boundary).","answer":{"kind":"expression","value":"(13/14, 1/15, [14/15, 15/16, 16/17], No, finite_simulation_not_IID_certificate_or_WLLN_proof)","explanation":"Retain the exact observed prefix and No for sufficiency. No finite prefix alone proves IID for arbitrary finite subfamilies, the WLLN tail or a stronger convergence mode. The exact result, retaining the requested order, is (13/14, 1/15, [14/15, 15/16, 16/17], No, finite_simulation_not_IID_certificate_or_WLLN_proof)."},"skills":["A Finite Simulation Does Not Certify the Whole IID Sequence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261105-r07-weak-law-large-numbers-0090","batch_id":"batch-20261014-advance-20261105-r07-weak-law-large-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fe440c9e5ea3c14c0fee4d8f83e563e2be2b3aaf7270d38c60ae7e4605de9692","family_id":"qf-mod36-wlln-finite-simulation-boundary","seed":202611057090,"semantic_template":"qf-mod36-wlln-finite-simulation-boundary/{\"epsilon\":[1,16],\"iid_certificate\":\"finite_simulation_only\",\"mean\":[14,15],\"prefix_sample_means\":[[15,16],[16,17],[17,18]],\"variance\":[15,16]}"},"fields":{"title":"A Finite Simulation Does Not Certify the Whole IID Sequence","stem":"Use declared mean=14/15, positive finite variance=15/16, and iid_certificate=finite_simulation_only. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive epsilon=1/16, finite prefix of sample means=[15/16, 16/17, 17/18]. The supplied marker is finite_simulation_only, not the symbolic whole-sequence premise. Give (mean, epsilon, prefix_sample_means, sufficient, boundary).","answer":{"kind":"expression","value":"(14/15, 1/16, [15/16, 16/17, 17/18], No, finite_simulation_not_IID_certificate_or_WLLN_proof)","explanation":"Retain the exact observed prefix and No for sufficiency. No finite prefix alone proves IID for arbitrary finite subfamilies, the WLLN tail or a stronger convergence mode. The exact result, retaining the requested order, is (14/15, 1/16, [15/16, 16/17, 17/18], No, finite_simulation_not_IID_certificate_or_WLLN_proof)."},"skills":["A Finite Simulation Does Not Certify the Whole IID Sequence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261105-r07-weak-law-large-numbers-0091","batch_id":"batch-20261014-advance-20261105-r07-weak-law-large-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b8deac9566b721573a936e449c5b8660f655ae068a7dfe94c54eef7cedc938be","family_id":"qf-mod36-wlln-finite-simulation-boundary","seed":202611057091,"semantic_template":"qf-mod36-wlln-finite-simulation-boundary/{\"epsilon\":[1,17],\"iid_certificate\":\"finite_simulation_only\",\"mean\":[15,16],\"prefix_sample_means\":[[16,17],[17,18],[18,19]],\"variance\":[16,17]}"},"fields":{"title":"A Finite Simulation Does Not Certify the Whole IID Sequence","stem":"Use declared mean=15/16, positive finite variance=16/17, and iid_certificate=finite_simulation_only. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive epsilon=1/17, finite prefix of sample means=[16/17, 17/18, 18/19]. The supplied marker is finite_simulation_only, not the symbolic whole-sequence premise. Give (mean, epsilon, prefix_sample_means, sufficient, boundary).","answer":{"kind":"expression","value":"(15/16, 1/17, [16/17, 17/18, 18/19], No, finite_simulation_not_IID_certificate_or_WLLN_proof)","explanation":"Retain the exact observed prefix and No for sufficiency. No finite prefix alone proves IID for arbitrary finite subfamilies, the WLLN tail or a stronger convergence mode. The exact result, retaining the requested order, is (15/16, 1/17, [16/17, 17/18, 18/19], No, finite_simulation_not_IID_certificate_or_WLLN_proof)."},"skills":["A Finite Simulation Does Not Certify the Whole IID Sequence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261105-r07-weak-law-large-numbers-0092","batch_id":"batch-20261014-advance-20261105-r07-weak-law-large-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8c794153700a05fd3b8a5864a5b4039ad303c339a4473660791038cefc5bf473","family_id":"qf-mod36-wlln-finite-simulation-boundary","seed":202611057092,"semantic_template":"qf-mod36-wlln-finite-simulation-boundary/{\"epsilon\":[1,18],\"iid_certificate\":\"finite_simulation_only\",\"mean\":[16,17],\"prefix_sample_means\":[[17,18],[18,19],[19,20]],\"variance\":[17,18]}"},"fields":{"title":"A Finite Simulation Does Not Certify the Whole IID Sequence","stem":"Use declared mean=16/17, positive finite variance=17/18, and iid_certificate=finite_simulation_only. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive epsilon=1/18, finite prefix of sample means=[17/18, 18/19, 19/20]. The supplied marker is finite_simulation_only, not the symbolic whole-sequence premise. Give (mean, epsilon, prefix_sample_means, sufficient, boundary).","answer":{"kind":"expression","value":"(16/17, 1/18, [17/18, 18/19, 19/20], No, finite_simulation_not_IID_certificate_or_WLLN_proof)","explanation":"Retain the exact observed prefix and No for sufficiency. No finite prefix alone proves IID for arbitrary finite subfamilies, the WLLN tail or a stronger convergence mode. The exact result, retaining the requested order, is (16/17, 1/18, [17/18, 18/19, 19/20], No, finite_simulation_not_IID_certificate_or_WLLN_proof)."},"skills":["A Finite Simulation Does Not Certify the Whole IID Sequence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261105-r07-weak-law-large-numbers-0093","batch_id":"batch-20261014-advance-20261105-r07-weak-law-large-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6fc31027612e7f8c6b27a5aa6495dd023b57529b6b509e25665c7e39bedf5f93","family_id":"qf-mod36-wlln-finite-simulation-boundary","seed":202611057093,"semantic_template":"qf-mod36-wlln-finite-simulation-boundary/{\"epsilon\":[1,19],\"iid_certificate\":\"finite_simulation_only\",\"mean\":[17,18],\"prefix_sample_means\":[[18,19],[19,20],[20,21]],\"variance\":[18,19]}"},"fields":{"title":"A Finite Simulation Does Not Certify the Whole IID Sequence","stem":"Use declared mean=17/18, positive finite variance=18/19, and iid_certificate=finite_simulation_only. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive epsilon=1/19, finite prefix of sample means=[18/19, 19/20, 20/21]. The supplied marker is finite_simulation_only, not the symbolic whole-sequence premise. Give (mean, epsilon, prefix_sample_means, sufficient, boundary).","answer":{"kind":"expression","value":"(17/18, 1/19, [18/19, 19/20, 20/21], No, finite_simulation_not_IID_certificate_or_WLLN_proof)","explanation":"Retain the exact observed prefix and No for sufficiency. No finite prefix alone proves IID for arbitrary finite subfamilies, the WLLN tail or a stronger convergence mode. The exact result, retaining the requested order, is (17/18, 1/19, [18/19, 19/20, 20/21], No, finite_simulation_not_IID_certificate_or_WLLN_proof)."},"skills":["A Finite Simulation Does Not Certify the Whole IID Sequence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261105-r07-weak-law-large-numbers-0094","batch_id":"batch-20261014-advance-20261105-r07-weak-law-large-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6b897ec6d32ad7e93ac4280a82937ee60d9608df1dc48359bab3e03fbed13adf","family_id":"qf-mod36-wlln-finite-simulation-boundary","seed":202611057094,"semantic_template":"qf-mod36-wlln-finite-simulation-boundary/{\"epsilon\":[1,20],\"iid_certificate\":\"finite_simulation_only\",\"mean\":[18,19],\"prefix_sample_means\":[[19,20],[20,21],[21,22]],\"variance\":[19,20]}"},"fields":{"title":"A Finite Simulation Does Not Certify the Whole IID Sequence","stem":"Use declared mean=18/19, positive finite variance=19/20, and iid_certificate=finite_simulation_only. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive epsilon=1/20, finite prefix of sample means=[19/20, 20/21, 21/22]. The supplied marker is finite_simulation_only, not the symbolic whole-sequence premise. Give (mean, epsilon, prefix_sample_means, sufficient, boundary).","answer":{"kind":"expression","value":"(18/19, 1/20, [19/20, 20/21, 21/22], No, finite_simulation_not_IID_certificate_or_WLLN_proof)","explanation":"Retain the exact observed prefix and No for sufficiency. No finite prefix alone proves IID for arbitrary finite subfamilies, the WLLN tail or a stronger convergence mode. The exact result, retaining the requested order, is (18/19, 1/20, [19/20, 20/21, 21/22], No, finite_simulation_not_IID_certificate_or_WLLN_proof)."},"skills":["A Finite Simulation Does Not Certify the Whole IID Sequence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261105-r07-weak-law-large-numbers-0095","batch_id":"batch-20261014-advance-20261105-r07-weak-law-large-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"01e56e55cc6106ecbf56715ba2a3dba5bc45b578bb65f92e34fa136348849923","family_id":"qf-mod36-wlln-finite-simulation-boundary","seed":202611057095,"semantic_template":"qf-mod36-wlln-finite-simulation-boundary/{\"epsilon\":[1,21],\"iid_certificate\":\"finite_simulation_only\",\"mean\":[19,20],\"prefix_sample_means\":[[20,21],[21,22],[22,23]],\"variance\":[20,21]}"},"fields":{"title":"A Finite Simulation Does Not Certify the Whole IID Sequence","stem":"Use declared mean=19/20, positive finite variance=20/21, and iid_certificate=finite_simulation_only. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive epsilon=1/21, finite prefix of sample means=[20/21, 21/22, 22/23]. The supplied marker is finite_simulation_only, not the symbolic whole-sequence premise. Give (mean, epsilon, prefix_sample_means, sufficient, boundary).","answer":{"kind":"expression","value":"(19/20, 1/21, [20/21, 21/22, 22/23], No, finite_simulation_not_IID_certificate_or_WLLN_proof)","explanation":"Retain the exact observed prefix and No for sufficiency. No finite prefix alone proves IID for arbitrary finite subfamilies, the WLLN tail or a stronger convergence mode. The exact result, retaining the requested order, is (19/20, 1/21, [20/21, 21/22, 22/23], No, finite_simulation_not_IID_certificate_or_WLLN_proof)."},"skills":["A Finite Simulation Does Not Certify the Whole IID Sequence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261105-r07-weak-law-large-numbers-0096","batch_id":"batch-20261014-advance-20261105-r07-weak-law-large-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c183ca41187702c97809d93a0c8936edf270aa2adb0d140e4bdd37f665beff0c","family_id":"qf-mod36-wlln-finite-simulation-boundary","seed":202611057096,"semantic_template":"qf-mod36-wlln-finite-simulation-boundary/{\"epsilon\":[1,22],\"iid_certificate\":\"finite_simulation_only\",\"mean\":[20,21],\"prefix_sample_means\":[[21,22],[22,23],[23,24]],\"variance\":[21,22]}"},"fields":{"title":"A Finite Simulation Does Not Certify the Whole IID Sequence","stem":"Use declared mean=20/21, positive finite variance=21/22, and iid_certificate=finite_simulation_only. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive epsilon=1/22, finite prefix of sample means=[21/22, 22/23, 23/24]. The supplied marker is finite_simulation_only, not the symbolic whole-sequence premise. Give (mean, epsilon, prefix_sample_means, sufficient, boundary).","answer":{"kind":"expression","value":"(20/21, 1/22, [21/22, 22/23, 23/24], No, finite_simulation_not_IID_certificate_or_WLLN_proof)","explanation":"Retain the exact observed prefix and No for sufficiency. No finite prefix alone proves IID for arbitrary finite subfamilies, the WLLN tail or a stronger convergence mode. The exact result, retaining the requested order, is (20/21, 1/22, [21/22, 22/23, 23/24], No, finite_simulation_not_IID_certificate_or_WLLN_proof)."},"skills":["A Finite Simulation Does Not Certify the Whole IID Sequence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261105-r07-weak-law-large-numbers-0097","batch_id":"batch-20261014-advance-20261105-r07-weak-law-large-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8013fbd4b0d2c76f0d61c39e77a56fb387e47a7878118876e16f7f757c000a36","family_id":"qf-mod36-wlln-finite-simulation-boundary","seed":202611057097,"semantic_template":"qf-mod36-wlln-finite-simulation-boundary/{\"epsilon\":[1,23],\"iid_certificate\":\"finite_simulation_only\",\"mean\":[21,22],\"prefix_sample_means\":[[22,23],[23,24],[24,25]],\"variance\":[22,23]}"},"fields":{"title":"A Finite Simulation Does Not Certify the Whole IID Sequence","stem":"Use declared mean=21/22, positive finite variance=22/23, and iid_certificate=finite_simulation_only. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive epsilon=1/23, finite prefix of sample means=[22/23, 23/24, 24/25]. The supplied marker is finite_simulation_only, not the symbolic whole-sequence premise. Give (mean, epsilon, prefix_sample_means, sufficient, boundary).","answer":{"kind":"expression","value":"(21/22, 1/23, [22/23, 23/24, 24/25], No, finite_simulation_not_IID_certificate_or_WLLN_proof)","explanation":"Retain the exact observed prefix and No for sufficiency. No finite prefix alone proves IID for arbitrary finite subfamilies, the WLLN tail or a stronger convergence mode. The exact result, retaining the requested order, is (21/22, 1/23, [22/23, 23/24, 24/25], No, finite_simulation_not_IID_certificate_or_WLLN_proof)."},"skills":["A Finite Simulation Does Not Certify the Whole IID Sequence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261105-r07-weak-law-large-numbers-0098","batch_id":"batch-20261014-advance-20261105-r07-weak-law-large-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5ac4a7d99863e3cea43f5f283c4ce51c07b634e20c6e289148418b08527e3e98","family_id":"qf-mod36-wlln-finite-simulation-boundary","seed":202611057098,"semantic_template":"qf-mod36-wlln-finite-simulation-boundary/{\"epsilon\":[1,24],\"iid_certificate\":\"finite_simulation_only\",\"mean\":[22,23],\"prefix_sample_means\":[[23,24],[24,25],[25,26]],\"variance\":[23,24]}"},"fields":{"title":"A Finite Simulation Does Not Certify the Whole IID Sequence","stem":"Use declared mean=22/23, positive finite variance=23/24, and iid_certificate=finite_simulation_only. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive epsilon=1/24, finite prefix of sample means=[23/24, 24/25, 25/26]. The supplied marker is finite_simulation_only, not the symbolic whole-sequence premise. Give (mean, epsilon, prefix_sample_means, sufficient, boundary).","answer":{"kind":"expression","value":"(22/23, 1/24, [23/24, 24/25, 25/26], No, finite_simulation_not_IID_certificate_or_WLLN_proof)","explanation":"Retain the exact observed prefix and No for sufficiency. No finite prefix alone proves IID for arbitrary finite subfamilies, the WLLN tail or a stronger convergence mode. The exact result, retaining the requested order, is (22/23, 1/24, [23/24, 24/25, 25/26], No, finite_simulation_not_IID_certificate_or_WLLN_proof)."},"skills":["A Finite Simulation Does Not Certify the Whole IID Sequence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261105-r07-weak-law-large-numbers-0099","batch_id":"batch-20261014-advance-20261105-r07-weak-law-large-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"17c93d7beceee5d1ef242b87fb1fce698652c3118d996b9cb741d05d9b4a0745","family_id":"qf-mod36-wlln-finite-simulation-boundary","seed":202611057099,"semantic_template":"qf-mod36-wlln-finite-simulation-boundary/{\"epsilon\":[1,25],\"iid_certificate\":\"finite_simulation_only\",\"mean\":[23,24],\"prefix_sample_means\":[[24,25],[25,26],[26,27]],\"variance\":[24,25]}"},"fields":{"title":"A Finite Simulation Does Not Certify the Whole IID Sequence","stem":"Use declared mean=23/24, positive finite variance=24/25, and iid_certificate=finite_simulation_only. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive epsilon=1/25, finite prefix of sample means=[24/25, 25/26, 26/27]. The supplied marker is finite_simulation_only, not the symbolic whole-sequence premise. Give (mean, epsilon, prefix_sample_means, sufficient, boundary).","answer":{"kind":"expression","value":"(23/24, 1/25, [24/25, 25/26, 26/27], No, finite_simulation_not_IID_certificate_or_WLLN_proof)","explanation":"Retain the exact observed prefix and No for sufficiency. No finite prefix alone proves IID for arbitrary finite subfamilies, the WLLN tail or a stronger convergence mode. The exact result, retaining the requested order, is (23/24, 1/25, [24/25, 25/26, 26/27], No, finite_simulation_not_IID_certificate_or_WLLN_proof)."},"skills":["A Finite Simulation Does Not Certify the Whole IID Sequence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261105-r07-weak-law-large-numbers-0100","batch_id":"batch-20261014-advance-20261105-r07-weak-law-large-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cb6636c69b77037a4c612ebcf3a10598f4a23fe7500be6f819f5f201b0116feb","family_id":"qf-mod36-wlln-finite-simulation-boundary","seed":202611057100,"semantic_template":"qf-mod36-wlln-finite-simulation-boundary/{\"epsilon\":[1,26],\"iid_certificate\":\"finite_simulation_only\",\"mean\":[24,25],\"prefix_sample_means\":[[25,26],[26,27],[27,28]],\"variance\":[25,26]}"},"fields":{"title":"A Finite Simulation Does Not Certify the Whole IID Sequence","stem":"Use declared mean=24/25, positive finite variance=25/26, and iid_certificate=finite_simulation_only. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive epsilon=1/26, finite prefix of sample means=[25/26, 26/27, 27/28]. The supplied marker is finite_simulation_only, not the symbolic whole-sequence premise. Give (mean, epsilon, prefix_sample_means, sufficient, boundary).","answer":{"kind":"expression","value":"(24/25, 1/26, [25/26, 26/27, 27/28], No, finite_simulation_not_IID_certificate_or_WLLN_proof)","explanation":"Retain the exact observed prefix and No for sufficiency. No finite prefix alone proves IID for arbitrary finite subfamilies, the WLLN tail or a stronger convergence mode. The exact result, retaining the requested order, is (24/25, 1/26, [25/26, 26/27, 27/28], No, finite_simulation_not_IID_certificate_or_WLLN_proof)."},"skills":["A Finite Simulation Does Not Certify the Whole IID Sequence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}