{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-0001","batch_id":"batch-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"73ff4f2977bddd6da635550d56090c25476e7e159e50a3e6dbd499f232bd2139","family_id":"qf-mod49-slln-process-law-qualification","seed":202611182001,"semantic_template":"slln-process-law-qualification|{law:[[-1,[1,2]],[-2,[1,2]]],process_certificate:\"infinite_iid\"}"},"fields":{"title":"The Almost-Sure Limit Uses an Explicit Infinite IID Premise","stem":"Use the aligned finite law values=[-2, -1], probabilities=[1/2, 1/2], with declared process certificate=infinite_iid. The token is an explicit infinite IID premise, not a process construction or proof by this finite calculation. Give (law_rows, mean, variance, declared_process_certificate, named_theorem_conclusion, conditional_application_qualified).","answer":{"kind":"expression","value":"([[-2, 1/2], [-1, 1/2]], -3/2, 1/4, infinite_iid, P(lim_sample_mean=mu)=1, Yes)","explanation":"Compute the common mean and variance of the finite law. Zero variance is allowed for this theorem. State P(lim sample_mean=mu)=1 as the conditional named-theorem conclusion, not an inference from finite observations or a proof of the infinite process. The exact result, retaining the requested order, is ([[-2, 1/2], [-1, 1/2]], -3/2, 1/4, infinite_iid, P(lim_sample_mean=mu)=1, Yes)."},"skills":["The Almost-Sure Limit Uses an Explicit Infinite IID Premise"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-0002","batch_id":"batch-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"906f07f5deab4dd09cad10a3968df25b5460ef0b8cf033e37dfe1967fec415ce","family_id":"qf-mod49-slln-process-law-qualification","seed":202611182002,"semantic_template":"slln-process-law-qualification|{law:[[-1,[1,2]],[0,[1,2]]],process_certificate:\"infinite_iid\"}"},"fields":{"title":"The Almost-Sure Limit Uses an Explicit Infinite IID Premise","stem":"Use the aligned finite law values=[-1, 0], probabilities=[1/2, 1/2], with declared process certificate=infinite_iid. The token is an explicit infinite IID premise, not a process construction or proof by this finite calculation. Give (law_rows, mean, variance, declared_process_certificate, named_theorem_conclusion, conditional_application_qualified).","answer":{"kind":"expression","value":"([[-1, 1/2], [0, 1/2]], -1/2, 1/4, infinite_iid, P(lim_sample_mean=mu)=1, Yes)","explanation":"Compute the common mean and variance of the finite law. Zero variance is allowed for this theorem. State P(lim sample_mean=mu)=1 as the conditional named-theorem conclusion, not an inference from finite observations or a proof of the infinite process. The exact result, retaining the requested order, is ([[-1, 1/2], [0, 1/2]], -1/2, 1/4, infinite_iid, P(lim_sample_mean=mu)=1, Yes)."},"skills":["The Almost-Sure Limit Uses an Explicit Infinite IID Premise"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-0003","batch_id":"batch-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b8c4550ab8d51207d5ef02398628e8bd70fd25e5e7ae20b9c72f3a03cf0fb05d","family_id":"qf-mod49-slln-process-law-qualification","seed":202611182003,"semantic_template":"slln-process-law-qualification|{law:[[0,[1,2]],[1,[1,2]]],process_certificate:\"infinite_iid\"}"},"fields":{"title":"The Almost-Sure Limit Uses an Explicit Infinite IID Premise","stem":"Use the aligned finite law values=[0, 1], probabilities=[1/2, 1/2], with declared process certificate=infinite_iid. The token is an explicit infinite IID premise, not a process construction or proof by this finite calculation. Give (law_rows, mean, variance, declared_process_certificate, named_theorem_conclusion, conditional_application_qualified).","answer":{"kind":"expression","value":"([[0, 1/2], [1, 1/2]], 1/2, 1/4, infinite_iid, P(lim_sample_mean=mu)=1, Yes)","explanation":"Compute the common mean and variance of the finite law. Zero variance is allowed for this theorem. State P(lim sample_mean=mu)=1 as the conditional named-theorem conclusion, not an inference from finite observations or a proof of the infinite process. The exact result, retaining the requested order, is ([[0, 1/2], [1, 1/2]], 1/2, 1/4, infinite_iid, P(lim_sample_mean=mu)=1, Yes)."},"skills":["The Almost-Sure Limit Uses an Explicit Infinite IID Premise"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-0004","batch_id":"batch-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9db22d862797cb4c9734313094e2b9a51149efebedf98f38e4b5f65b23077fdb","family_id":"qf-mod49-slln-process-law-qualification","seed":202611182004,"semantic_template":"slln-process-law-qualification|{law:[[1,[1,2]],[2,[1,2]]],process_certificate:\"infinite_iid\"}"},"fields":{"title":"The Almost-Sure Limit Uses an Explicit Infinite IID Premise","stem":"Use the aligned finite law values=[1, 2], probabilities=[1/2, 1/2], with declared process certificate=infinite_iid. The token is an explicit infinite IID premise, not a process construction or proof by this finite calculation. Give (law_rows, mean, variance, declared_process_certificate, named_theorem_conclusion, conditional_application_qualified).","answer":{"kind":"expression","value":"([[1, 1/2], [2, 1/2]], 3/2, 1/4, infinite_iid, P(lim_sample_mean=mu)=1, Yes)","explanation":"Compute the common mean and variance of the finite law. Zero variance is allowed for this theorem. State P(lim sample_mean=mu)=1 as the conditional named-theorem conclusion, not an inference from finite observations or a proof of the infinite process. The exact result, retaining the requested order, is ([[1, 1/2], [2, 1/2]], 3/2, 1/4, infinite_iid, P(lim_sample_mean=mu)=1, Yes)."},"skills":["The Almost-Sure Limit Uses an Explicit Infinite IID Premise"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-0005","batch_id":"batch-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0fa3a554c2ea2a6162d28e7212e2a2af90ffb924dbda45a1e9e2fffc1a07f785","family_id":"qf-mod49-slln-process-law-qualification","seed":202611182005,"semantic_template":"slln-process-law-qualification|{law:[[2,[1,2]],[3,[1,2]]],process_certificate:\"infinite_iid\"}"},"fields":{"title":"The Almost-Sure Limit Uses an Explicit Infinite IID Premise","stem":"Use the aligned finite law values=[2, 3], probabilities=[1/2, 1/2], with declared process certificate=infinite_iid. The token is an explicit infinite IID premise, not a process construction or proof by this finite calculation. Give (law_rows, mean, variance, declared_process_certificate, named_theorem_conclusion, conditional_application_qualified).","answer":{"kind":"expression","value":"([[2, 1/2], [3, 1/2]], 5/2, 1/4, infinite_iid, P(lim_sample_mean=mu)=1, Yes)","explanation":"Compute the common mean and variance of the finite law. Zero variance is allowed for this theorem. State P(lim sample_mean=mu)=1 as the conditional named-theorem conclusion, not an inference from finite observations or a proof of the infinite process. The exact result, retaining the requested order, is ([[2, 1/2], [3, 1/2]], 5/2, 1/4, infinite_iid, P(lim_sample_mean=mu)=1, Yes)."},"skills":["The Almost-Sure Limit Uses an Explicit Infinite IID Premise"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-0006","batch_id":"batch-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"37094873cd06a3bcd8985005085c4b932cda6485045d501481adee1accbb1f67","family_id":"qf-mod49-slln-process-law-qualification","seed":202611182006,"semantic_template":"slln-process-law-qualification|{law:[[-1,[2,3]],[-2,[1,3]]],process_certificate:\"infinite_iid\"}"},"fields":{"title":"The Almost-Sure Limit Uses an Explicit Infinite IID Premise","stem":"Use the aligned finite law values=[-2, -1], probabilities=[1/3, 2/3], with declared process certificate=infinite_iid. The token is an explicit infinite IID premise, not a process construction or proof by this finite calculation. Give (law_rows, mean, variance, declared_process_certificate, named_theorem_conclusion, conditional_application_qualified).","answer":{"kind":"expression","value":"([[-2, 1/3], [-1, 2/3]], -4/3, 2/9, infinite_iid, P(lim_sample_mean=mu)=1, Yes)","explanation":"Compute the common mean and variance of the finite law. Zero variance is allowed for this theorem. State P(lim sample_mean=mu)=1 as the conditional named-theorem conclusion, not an inference from finite observations or a proof of the infinite process. The exact result, retaining the requested order, is ([[-2, 1/3], [-1, 2/3]], -4/3, 2/9, infinite_iid, P(lim_sample_mean=mu)=1, Yes)."},"skills":["The Almost-Sure Limit Uses an Explicit Infinite IID Premise"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-0007","batch_id":"batch-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c1d7625a26315cf5427500fc3cbc52152f06ada9d86f71a7eae373e383341207","family_id":"qf-mod49-slln-process-law-qualification","seed":202611182007,"semantic_template":"slln-process-law-qualification|{law:[[-1,[1,3]],[0,[2,3]]],process_certificate:\"infinite_iid\"}"},"fields":{"title":"The Almost-Sure Limit Uses an Explicit Infinite IID Premise","stem":"Use the aligned finite law values=[-1, 0], probabilities=[1/3, 2/3], with declared process certificate=infinite_iid. The token is an explicit infinite IID premise, not a process construction or proof by this finite calculation. Give (law_rows, mean, variance, declared_process_certificate, named_theorem_conclusion, conditional_application_qualified).","answer":{"kind":"expression","value":"([[-1, 1/3], [0, 2/3]], -1/3, 2/9, infinite_iid, P(lim_sample_mean=mu)=1, Yes)","explanation":"Compute the common mean and variance of the finite law. Zero variance is allowed for this theorem. State P(lim sample_mean=mu)=1 as the conditional named-theorem conclusion, not an inference from finite observations or a proof of the infinite process. The exact result, retaining the requested order, is ([[-1, 1/3], [0, 2/3]], -1/3, 2/9, infinite_iid, P(lim_sample_mean=mu)=1, Yes)."},"skills":["The Almost-Sure Limit Uses an Explicit Infinite IID Premise"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-0008","batch_id":"batch-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"eeecb0b33e0f0cb03cf972f3f140a43cf2674e40d56e29f5e3265ad9bc41d80b","family_id":"qf-mod49-slln-process-law-qualification","seed":202611182008,"semantic_template":"slln-process-law-qualification|{law:[[0,[1,3]],[1,[2,3]]],process_certificate:\"infinite_iid\"}"},"fields":{"title":"The Almost-Sure Limit Uses an Explicit Infinite IID Premise","stem":"Use the aligned finite law values=[0, 1], probabilities=[1/3, 2/3], with declared process certificate=infinite_iid. The token is an explicit infinite IID premise, not a process construction or proof by this finite calculation. Give (law_rows, mean, variance, declared_process_certificate, named_theorem_conclusion, conditional_application_qualified).","answer":{"kind":"expression","value":"([[0, 1/3], [1, 2/3]], 2/3, 2/9, infinite_iid, P(lim_sample_mean=mu)=1, Yes)","explanation":"Compute the common mean and variance of the finite law. Zero variance is allowed for this theorem. State P(lim sample_mean=mu)=1 as the conditional named-theorem conclusion, not an inference from finite observations or a proof of the infinite process. The exact result, retaining the requested order, is ([[0, 1/3], [1, 2/3]], 2/3, 2/9, infinite_iid, P(lim_sample_mean=mu)=1, Yes)."},"skills":["The Almost-Sure Limit Uses an Explicit Infinite IID Premise"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-0009","batch_id":"batch-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ef46723d764347634dcf0f0eaff33d040dc711578156e04df5016fd2c9769dc0","family_id":"qf-mod49-slln-process-law-qualification","seed":202611182009,"semantic_template":"slln-process-law-qualification|{law:[[1,[1,3]],[2,[2,3]]],process_certificate:\"infinite_iid\"}"},"fields":{"title":"The Almost-Sure Limit Uses an Explicit Infinite IID Premise","stem":"Use the aligned finite law values=[1, 2], probabilities=[1/3, 2/3], with declared process certificate=infinite_iid. The token is an explicit infinite IID premise, not a process construction or proof by this finite calculation. Give (law_rows, mean, variance, declared_process_certificate, named_theorem_conclusion, conditional_application_qualified).","answer":{"kind":"expression","value":"([[1, 1/3], [2, 2/3]], 5/3, 2/9, infinite_iid, P(lim_sample_mean=mu)=1, Yes)","explanation":"Compute the common mean and variance of the finite law. Zero variance is allowed for this theorem. State P(lim sample_mean=mu)=1 as the conditional named-theorem conclusion, not an inference from finite observations or a proof of the infinite process. The exact result, retaining the requested order, is ([[1, 1/3], [2, 2/3]], 5/3, 2/9, infinite_iid, P(lim_sample_mean=mu)=1, Yes)."},"skills":["The Almost-Sure Limit Uses an Explicit Infinite IID Premise"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-0010","batch_id":"batch-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dd6f7346c6589387588abb889fc89991f9c45fe385575f995bec91e20c2104a1","family_id":"qf-mod49-slln-process-law-qualification","seed":202611182010,"semantic_template":"slln-process-law-qualification|{law:[[2,[1,3]],[3,[2,3]]],process_certificate:\"infinite_iid\"}"},"fields":{"title":"The Almost-Sure Limit Uses an Explicit Infinite IID Premise","stem":"Use the aligned finite law values=[2, 3], probabilities=[1/3, 2/3], with declared process certificate=infinite_iid. The token is an explicit infinite IID premise, not a process construction or proof by this finite calculation. Give (law_rows, mean, variance, declared_process_certificate, named_theorem_conclusion, conditional_application_qualified).","answer":{"kind":"expression","value":"([[2, 1/3], [3, 2/3]], 8/3, 2/9, infinite_iid, P(lim_sample_mean=mu)=1, Yes)","explanation":"Compute the common mean and variance of the finite law. Zero variance is allowed for this theorem. State P(lim sample_mean=mu)=1 as the conditional named-theorem conclusion, not an inference from finite observations or a proof of the infinite process. The exact result, retaining the requested order, is ([[2, 1/3], [3, 2/3]], 8/3, 2/9, infinite_iid, P(lim_sample_mean=mu)=1, Yes)."},"skills":["The Almost-Sure Limit Uses an Explicit Infinite IID Premise"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-0011","batch_id":"batch-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"36b20c8e332615e3134e1eba75aca8537b078c76a2ed1cd9e1ed831bc6031b12","family_id":"qf-mod49-slln-process-law-qualification","seed":202611182011,"semantic_template":"slln-process-law-qualification|{law:[[-1,[3,4]],[-2,[1,4]]],process_certificate:\"infinite_iid\"}"},"fields":{"title":"The Almost-Sure Limit Uses an Explicit Infinite IID Premise","stem":"Use the aligned finite law values=[-2, -1], probabilities=[1/4, 3/4], with declared process certificate=infinite_iid. The token is an explicit infinite IID premise, not a process construction or proof by this finite calculation. Give (law_rows, mean, variance, declared_process_certificate, named_theorem_conclusion, conditional_application_qualified).","answer":{"kind":"expression","value":"([[-2, 1/4], [-1, 3/4]], -5/4, 3/16, infinite_iid, P(lim_sample_mean=mu)=1, Yes)","explanation":"Compute the common mean and variance of the finite law. Zero variance is allowed for this theorem. State P(lim sample_mean=mu)=1 as the conditional named-theorem conclusion, not an inference from finite observations or a proof of the infinite process. The exact result, retaining the requested order, is ([[-2, 1/4], [-1, 3/4]], -5/4, 3/16, infinite_iid, P(lim_sample_mean=mu)=1, Yes)."},"skills":["The Almost-Sure Limit Uses an Explicit Infinite IID Premise"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-0012","batch_id":"batch-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c617c69e45c28503d96c3d8e529922cc9245003c3de0f024a92a5ae8467cb122","family_id":"qf-mod49-slln-process-law-qualification","seed":202611182012,"semantic_template":"slln-process-law-qualification|{law:[[-1,[1,4]],[0,[3,4]]],process_certificate:\"infinite_iid\"}"},"fields":{"title":"The Almost-Sure Limit Uses an Explicit Infinite IID Premise","stem":"Use the aligned finite law values=[-1, 0], probabilities=[1/4, 3/4], with declared process certificate=infinite_iid. The token is an explicit infinite IID premise, not a process construction or proof by this finite calculation. Give (law_rows, mean, variance, declared_process_certificate, named_theorem_conclusion, conditional_application_qualified).","answer":{"kind":"expression","value":"([[-1, 1/4], [0, 3/4]], -1/4, 3/16, infinite_iid, P(lim_sample_mean=mu)=1, Yes)","explanation":"Compute the common mean and variance of the finite law. Zero variance is allowed for this theorem. State P(lim sample_mean=mu)=1 as the conditional named-theorem conclusion, not an inference from finite observations or a proof of the infinite process. The exact result, retaining the requested order, is ([[-1, 1/4], [0, 3/4]], -1/4, 3/16, infinite_iid, P(lim_sample_mean=mu)=1, Yes)."},"skills":["The Almost-Sure Limit Uses an Explicit Infinite IID Premise"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-0013","batch_id":"batch-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7541e3459b234a9c646447baa1c522a411a2502bfab418079a83c9ab80b9978c","family_id":"qf-mod49-slln-process-law-qualification","seed":202611182013,"semantic_template":"slln-process-law-qualification|{law:[[0,[1,4]],[1,[3,4]]],process_certificate:\"infinite_iid\"}"},"fields":{"title":"The Almost-Sure Limit Uses an Explicit Infinite IID Premise","stem":"Use the aligned finite law values=[0, 1], probabilities=[1/4, 3/4], with declared process certificate=infinite_iid. The token is an explicit infinite IID premise, not a process construction or proof by this finite calculation. Give (law_rows, mean, variance, declared_process_certificate, named_theorem_conclusion, conditional_application_qualified).","answer":{"kind":"expression","value":"([[0, 1/4], [1, 3/4]], 3/4, 3/16, infinite_iid, P(lim_sample_mean=mu)=1, Yes)","explanation":"Compute the common mean and variance of the finite law. Zero variance is allowed for this theorem. State P(lim sample_mean=mu)=1 as the conditional named-theorem conclusion, not an inference from finite observations or a proof of the infinite process. The exact result, retaining the requested order, is ([[0, 1/4], [1, 3/4]], 3/4, 3/16, infinite_iid, P(lim_sample_mean=mu)=1, Yes)."},"skills":["The Almost-Sure Limit Uses an Explicit Infinite IID Premise"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-0014","batch_id":"batch-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"df1a93e691505fc4f17f3d4e5d13a15a89e7cd63559894a0400e53f40b6fc533","family_id":"qf-mod49-slln-process-law-qualification","seed":202611182014,"semantic_template":"slln-process-law-qualification|{law:[[1,[1,4]],[2,[3,4]]],process_certificate:\"infinite_iid\"}"},"fields":{"title":"The Almost-Sure Limit Uses an Explicit Infinite IID Premise","stem":"Use the aligned finite law values=[1, 2], probabilities=[1/4, 3/4], with declared process certificate=infinite_iid. The token is an explicit infinite IID premise, not a process construction or proof by this finite calculation. Give (law_rows, mean, variance, declared_process_certificate, named_theorem_conclusion, conditional_application_qualified).","answer":{"kind":"expression","value":"([[1, 1/4], [2, 3/4]], 7/4, 3/16, infinite_iid, P(lim_sample_mean=mu)=1, Yes)","explanation":"Compute the common mean and variance of the finite law. Zero variance is allowed for this theorem. State P(lim sample_mean=mu)=1 as the conditional named-theorem conclusion, not an inference from finite observations or a proof of the infinite process. The exact result, retaining the requested order, is ([[1, 1/4], [2, 3/4]], 7/4, 3/16, infinite_iid, P(lim_sample_mean=mu)=1, Yes)."},"skills":["The Almost-Sure Limit Uses an Explicit Infinite IID Premise"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-0015","batch_id":"batch-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a60624652a55fd31775db8a44e13564fe301adeeeba6f1c356258c5477b9b363","family_id":"qf-mod49-slln-process-law-qualification","seed":202611182015,"semantic_template":"slln-process-law-qualification|{law:[[2,[1,4]],[3,[3,4]]],process_certificate:\"infinite_iid\"}"},"fields":{"title":"The Almost-Sure Limit Uses an Explicit Infinite IID Premise","stem":"Use the aligned finite law values=[2, 3], probabilities=[1/4, 3/4], with declared process certificate=infinite_iid. The token is an explicit infinite IID premise, not a process construction or proof by this finite calculation. Give (law_rows, mean, variance, declared_process_certificate, named_theorem_conclusion, conditional_application_qualified).","answer":{"kind":"expression","value":"([[2, 1/4], [3, 3/4]], 11/4, 3/16, infinite_iid, P(lim_sample_mean=mu)=1, Yes)","explanation":"Compute the common mean and variance of the finite law. Zero variance is allowed for this theorem. State P(lim sample_mean=mu)=1 as the conditional named-theorem conclusion, not an inference from finite observations or a proof of the infinite process. The exact result, retaining the requested order, is ([[2, 1/4], [3, 3/4]], 11/4, 3/16, infinite_iid, P(lim_sample_mean=mu)=1, Yes)."},"skills":["The Almost-Sure Limit Uses an Explicit Infinite IID Premise"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-0016","batch_id":"batch-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d570472894e5f179fc61adc38c50d40b7a926b546fd1d632db7d9d156e7a8096","family_id":"qf-mod49-slln-process-law-qualification","seed":202611182016,"semantic_template":"slln-process-law-qualification|{law:[[-1,[3,5]],[-2,[2,5]]],process_certificate:\"infinite_iid\"}"},"fields":{"title":"The Almost-Sure Limit Uses an Explicit Infinite IID Premise","stem":"Use the aligned finite law values=[-2, -1], probabilities=[2/5, 3/5], with declared process certificate=infinite_iid. The token is an explicit infinite IID premise, not a process construction or proof by this finite calculation. Give (law_rows, mean, variance, declared_process_certificate, named_theorem_conclusion, conditional_application_qualified).","answer":{"kind":"expression","value":"([[-2, 2/5], [-1, 3/5]], -7/5, 6/25, infinite_iid, P(lim_sample_mean=mu)=1, Yes)","explanation":"Compute the common mean and variance of the finite law. Zero variance is allowed for this theorem. State P(lim sample_mean=mu)=1 as the conditional named-theorem conclusion, not an inference from finite observations or a proof of the infinite process. The exact result, retaining the requested order, is ([[-2, 2/5], [-1, 3/5]], -7/5, 6/25, infinite_iid, P(lim_sample_mean=mu)=1, Yes)."},"skills":["The Almost-Sure Limit Uses an Explicit Infinite IID Premise"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-0017","batch_id":"batch-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2eed57be650058ed718f8b3380298af0e174a411709db183834da83dfbc6dc51","family_id":"qf-mod49-slln-process-law-qualification","seed":202611182017,"semantic_template":"slln-process-law-qualification|{law:[[-1,[2,5]],[0,[3,5]]],process_certificate:\"infinite_iid\"}"},"fields":{"title":"The Almost-Sure Limit Uses an Explicit Infinite IID Premise","stem":"Use the aligned finite law values=[-1, 0], probabilities=[2/5, 3/5], with declared process certificate=infinite_iid. The token is an explicit infinite IID premise, not a process construction or proof by this finite calculation. Give (law_rows, mean, variance, declared_process_certificate, named_theorem_conclusion, conditional_application_qualified).","answer":{"kind":"expression","value":"([[-1, 2/5], [0, 3/5]], -2/5, 6/25, infinite_iid, P(lim_sample_mean=mu)=1, Yes)","explanation":"Compute the common mean and variance of the finite law. Zero variance is allowed for this theorem. State P(lim sample_mean=mu)=1 as the conditional named-theorem conclusion, not an inference from finite observations or a proof of the infinite process. The exact result, retaining the requested order, is ([[-1, 2/5], [0, 3/5]], -2/5, 6/25, infinite_iid, P(lim_sample_mean=mu)=1, Yes)."},"skills":["The Almost-Sure Limit Uses an Explicit Infinite IID Premise"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-0018","batch_id":"batch-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"11947498eef68c536c67b164bd635a03f294e40864c06d81fe20a6ab01d2ec18","family_id":"qf-mod49-slln-process-law-qualification","seed":202611182018,"semantic_template":"slln-process-law-qualification|{law:[[0,[2,5]],[1,[3,5]]],process_certificate:\"infinite_iid\"}"},"fields":{"title":"The Almost-Sure Limit Uses an Explicit Infinite IID Premise","stem":"Use the aligned finite law values=[0, 1], probabilities=[2/5, 3/5], with declared process certificate=infinite_iid. The token is an explicit infinite IID premise, not a process construction or proof by this finite calculation. Give (law_rows, mean, variance, declared_process_certificate, named_theorem_conclusion, conditional_application_qualified).","answer":{"kind":"expression","value":"([[0, 2/5], [1, 3/5]], 3/5, 6/25, infinite_iid, P(lim_sample_mean=mu)=1, Yes)","explanation":"Compute the common mean and variance of the finite law. Zero variance is allowed for this theorem. State P(lim sample_mean=mu)=1 as the conditional named-theorem conclusion, not an inference from finite observations or a proof of the infinite process. The exact result, retaining the requested order, is ([[0, 2/5], [1, 3/5]], 3/5, 6/25, infinite_iid, P(lim_sample_mean=mu)=1, Yes)."},"skills":["The Almost-Sure Limit Uses an Explicit Infinite IID Premise"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-0019","batch_id":"batch-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f2a04a2dab69f9c6b89cc6cf3cd3981118a4d1c40450d9621ede5a0a001c40d2","family_id":"qf-mod49-slln-process-law-qualification","seed":202611182019,"semantic_template":"slln-process-law-qualification|{law:[[1,[2,5]],[2,[3,5]]],process_certificate:\"infinite_iid\"}"},"fields":{"title":"The Almost-Sure Limit Uses an Explicit Infinite IID Premise","stem":"Use the aligned finite law values=[1, 2], probabilities=[2/5, 3/5], with declared process certificate=infinite_iid. The token is an explicit infinite IID premise, not a process construction or proof by this finite calculation. Give (law_rows, mean, variance, declared_process_certificate, named_theorem_conclusion, conditional_application_qualified).","answer":{"kind":"expression","value":"([[1, 2/5], [2, 3/5]], 8/5, 6/25, infinite_iid, P(lim_sample_mean=mu)=1, Yes)","explanation":"Compute the common mean and variance of the finite law. Zero variance is allowed for this theorem. State P(lim sample_mean=mu)=1 as the conditional named-theorem conclusion, not an inference from finite observations or a proof of the infinite process. The exact result, retaining the requested order, is ([[1, 2/5], [2, 3/5]], 8/5, 6/25, infinite_iid, P(lim_sample_mean=mu)=1, Yes)."},"skills":["The Almost-Sure Limit Uses an Explicit Infinite IID Premise"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-0020","batch_id":"batch-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"344d9ab6f3a743fba22091eded11f9215a835204a936493e90dd84e9c5b26e2a","family_id":"qf-mod49-slln-process-law-qualification","seed":202611182020,"semantic_template":"slln-process-law-qualification|{law:[[2,[2,5]],[3,[3,5]]],process_certificate:\"infinite_iid\"}"},"fields":{"title":"The Almost-Sure Limit Uses an Explicit Infinite IID Premise","stem":"Use the aligned finite law values=[2, 3], probabilities=[2/5, 3/5], with declared process certificate=infinite_iid. The token is an explicit infinite IID premise, not a process construction or proof by this finite calculation. Give (law_rows, mean, variance, declared_process_certificate, named_theorem_conclusion, conditional_application_qualified).","answer":{"kind":"expression","value":"([[2, 2/5], [3, 3/5]], 13/5, 6/25, infinite_iid, P(lim_sample_mean=mu)=1, Yes)","explanation":"Compute the common mean and variance of the finite law. Zero variance is allowed for this theorem. State P(lim sample_mean=mu)=1 as the conditional named-theorem conclusion, not an inference from finite observations or a proof of the infinite process. The exact result, retaining the requested order, is ([[2, 2/5], [3, 3/5]], 13/5, 6/25, infinite_iid, P(lim_sample_mean=mu)=1, Yes)."},"skills":["The Almost-Sure Limit Uses an Explicit Infinite IID Premise"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-0021","batch_id":"batch-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"83523966c1629a00084a3a063f88cd0344cb29a97d31e2bfe3cc0c5b3600b831","family_id":"qf-mod49-slln-process-law-qualification","seed":202611182021,"semantic_template":"slln-process-law-qualification|{law:[[-1,[4,5]],[-2,[1,5]]],process_certificate:\"infinite_iid\"}"},"fields":{"title":"The Almost-Sure Limit Uses an Explicit Infinite IID Premise","stem":"Use the aligned finite law values=[-2, -1], probabilities=[1/5, 4/5], with declared process certificate=infinite_iid. The token is an explicit infinite IID premise, not a process construction or proof by this finite calculation. Give (law_rows, mean, variance, declared_process_certificate, named_theorem_conclusion, conditional_application_qualified).","answer":{"kind":"expression","value":"([[-2, 1/5], [-1, 4/5]], -6/5, 4/25, infinite_iid, P(lim_sample_mean=mu)=1, Yes)","explanation":"Compute the common mean and variance of the finite law. Zero variance is allowed for this theorem. State P(lim sample_mean=mu)=1 as the conditional named-theorem conclusion, not an inference from finite observations or a proof of the infinite process. The exact result, retaining the requested order, is ([[-2, 1/5], [-1, 4/5]], -6/5, 4/25, infinite_iid, P(lim_sample_mean=mu)=1, Yes)."},"skills":["The Almost-Sure Limit Uses an Explicit Infinite IID Premise"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-0022","batch_id":"batch-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e694fd50902d2252bd81fcf10d965d59b36aba6b489bf81bfaa6dd4d699d6077","family_id":"qf-mod49-slln-process-law-qualification","seed":202611182022,"semantic_template":"slln-process-law-qualification|{law:[[-1,[1,5]],[0,[4,5]]],process_certificate:\"infinite_iid\"}"},"fields":{"title":"The Almost-Sure Limit Uses an Explicit Infinite IID Premise","stem":"Use the aligned finite law values=[-1, 0], probabilities=[1/5, 4/5], with declared process certificate=infinite_iid. The token is an explicit infinite IID premise, not a process construction or proof by this finite calculation. Give (law_rows, mean, variance, declared_process_certificate, named_theorem_conclusion, conditional_application_qualified).","answer":{"kind":"expression","value":"([[-1, 1/5], [0, 4/5]], -1/5, 4/25, infinite_iid, P(lim_sample_mean=mu)=1, Yes)","explanation":"Compute the common mean and variance of the finite law. Zero variance is allowed for this theorem. State P(lim sample_mean=mu)=1 as the conditional named-theorem conclusion, not an inference from finite observations or a proof of the infinite process. The exact result, retaining the requested order, is ([[-1, 1/5], [0, 4/5]], -1/5, 4/25, infinite_iid, P(lim_sample_mean=mu)=1, Yes)."},"skills":["The Almost-Sure Limit Uses an Explicit Infinite IID Premise"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-0023","batch_id":"batch-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"645278574c19edb3d79eea7369dd36204ac40ba6ae11da222d1ce64def220629","family_id":"qf-mod49-slln-process-law-qualification","seed":202611182023,"semantic_template":"slln-process-law-qualification|{law:[[0,[1,5]],[1,[4,5]]],process_certificate:\"infinite_iid\"}"},"fields":{"title":"The Almost-Sure Limit Uses an Explicit Infinite IID Premise","stem":"Use the aligned finite law values=[0, 1], probabilities=[1/5, 4/5], with declared process certificate=infinite_iid. The token is an explicit infinite IID premise, not a process construction or proof by this finite calculation. Give (law_rows, mean, variance, declared_process_certificate, named_theorem_conclusion, conditional_application_qualified).","answer":{"kind":"expression","value":"([[0, 1/5], [1, 4/5]], 4/5, 4/25, infinite_iid, P(lim_sample_mean=mu)=1, Yes)","explanation":"Compute the common mean and variance of the finite law. Zero variance is allowed for this theorem. State P(lim sample_mean=mu)=1 as the conditional named-theorem conclusion, not an inference from finite observations or a proof of the infinite process. The exact result, retaining the requested order, is ([[0, 1/5], [1, 4/5]], 4/5, 4/25, infinite_iid, P(lim_sample_mean=mu)=1, Yes)."},"skills":["The Almost-Sure Limit Uses an Explicit Infinite IID Premise"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-0024","batch_id":"batch-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3618c642147da67cd5609d894cf69d01c2e252e91af3c5b8c84d73526c4bd2fc","family_id":"qf-mod49-slln-process-law-qualification","seed":202611182024,"semantic_template":"slln-process-law-qualification|{law:[[1,[1,5]],[2,[4,5]]],process_certificate:\"infinite_iid\"}"},"fields":{"title":"The Almost-Sure Limit Uses an Explicit Infinite IID Premise","stem":"Use the aligned finite law values=[1, 2], probabilities=[1/5, 4/5], with declared process certificate=infinite_iid. The token is an explicit infinite IID premise, not a process construction or proof by this finite calculation. Give (law_rows, mean, variance, declared_process_certificate, named_theorem_conclusion, conditional_application_qualified).","answer":{"kind":"expression","value":"([[1, 1/5], [2, 4/5]], 9/5, 4/25, infinite_iid, P(lim_sample_mean=mu)=1, Yes)","explanation":"Compute the common mean and variance of the finite law. Zero variance is allowed for this theorem. State P(lim sample_mean=mu)=1 as the conditional named-theorem conclusion, not an inference from finite observations or a proof of the infinite process. The exact result, retaining the requested order, is ([[1, 1/5], [2, 4/5]], 9/5, 4/25, infinite_iid, P(lim_sample_mean=mu)=1, Yes)."},"skills":["The Almost-Sure Limit Uses an Explicit Infinite IID Premise"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-0025","batch_id":"batch-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ea367ab2aee8c8eec308a495467fa05321af0abe8e7dc6a6f60ce91895c3bf63","family_id":"qf-mod49-slln-process-law-qualification","seed":202611182025,"semantic_template":"slln-process-law-qualification|{law:[[2,[1,5]],[3,[4,5]]],process_certificate:\"infinite_iid\"}"},"fields":{"title":"The Almost-Sure Limit Uses an Explicit Infinite IID Premise","stem":"Use the aligned finite law values=[2, 3], probabilities=[1/5, 4/5], with declared process certificate=infinite_iid. The token is an explicit infinite IID premise, not a process construction or proof by this finite calculation. Give (law_rows, mean, variance, declared_process_certificate, named_theorem_conclusion, conditional_application_qualified).","answer":{"kind":"expression","value":"([[2, 1/5], [3, 4/5]], 14/5, 4/25, infinite_iid, P(lim_sample_mean=mu)=1, Yes)","explanation":"Compute the common mean and variance of the finite law. Zero variance is allowed for this theorem. State P(lim sample_mean=mu)=1 as the conditional named-theorem conclusion, not an inference from finite observations or a proof of the infinite process. The exact result, retaining the requested order, is ([[2, 1/5], [3, 4/5]], 14/5, 4/25, infinite_iid, P(lim_sample_mean=mu)=1, Yes)."},"skills":["The Almost-Sure Limit Uses an Explicit Infinite IID Premise"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-0026","batch_id":"batch-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ac814996ae1f4b571228c45de3069084c90cac4708109263f29f3d8782bd52c4","family_id":"qf-mod49-slln-prefix-sample-mean","seed":202611182026,"semantic_template":"slln-prefix-sample-mean|{law:[[-1,[1,2]],[-2,[1,2]]],observations:[-2],process_certificate:\"infinite_iid\"}"},"fields":{"title":"A Fixed Prefix Mean Is a Finite Audit, Not an Almost-Sure Limit Certificate","stem":"Use the aligned finite law values=[-2, -1], probabilities=[1/2, 1/2], with declared process certificate=infinite_iid. The token is an explicit infinite IID premise, not a process construction or proof by this finite calculation. The observed finite prefix is=[-2]. Give (observations, common_mean, prefix_sample_mean, absolute_error, evidence_scope, almost_sure_limit_proof_claim, boundary).","answer":{"kind":"expression","value":"([-2], -3/2, -2, 1/2, fixed_prefix_only, No, finite_prefix_does_not_prove_almost_sure_convergence)","explanation":"Compute its exact sample mean and absolute error from the common mean. All observations must belong to the declared carrier. A close prefix, or any one finite prefix, does not establish the infinite IID premise or a probability-one limit. The exact result, retaining the requested order, is ([-2], -3/2, -2, 1/2, fixed_prefix_only, No, finite_prefix_does_not_prove_almost_sure_convergence)."},"skills":["A Fixed Prefix Mean Is a Finite Audit, Not an Almost-Sure Limit Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-0027","batch_id":"batch-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d91a95427c893d6ebb0f0547d87afdb3081461ce51f41d4bfe0d0f64df600d5f","family_id":"qf-mod49-slln-prefix-sample-mean","seed":202611182027,"semantic_template":"slln-prefix-sample-mean|{law:[[-1,[1,2]],[0,[1,2]]],observations:[0,-1],process_certificate:\"infinite_iid\"}"},"fields":{"title":"A Fixed Prefix Mean Is a Finite Audit, Not an Almost-Sure Limit Certificate","stem":"Use the aligned finite law values=[-1, 0], probabilities=[1/2, 1/2], with declared process certificate=infinite_iid. The token is an explicit infinite IID premise, not a process construction or proof by this finite calculation. The observed finite prefix is=[0, -1]. Give (observations, common_mean, prefix_sample_mean, absolute_error, evidence_scope, almost_sure_limit_proof_claim, boundary).","answer":{"kind":"expression","value":"([0, -1], -1/2, -1/2, 0, fixed_prefix_only, No, finite_prefix_does_not_prove_almost_sure_convergence)","explanation":"Compute its exact sample mean and absolute error from the common mean. All observations must belong to the declared carrier. A close prefix, or any one finite prefix, does not establish the infinite IID premise or a probability-one limit. The exact result, retaining the requested order, is ([0, -1], -1/2, -1/2, 0, fixed_prefix_only, No, finite_prefix_does_not_prove_almost_sure_convergence)."},"skills":["A Fixed Prefix Mean Is a Finite Audit, Not an Almost-Sure Limit Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-0028","batch_id":"batch-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d0819fe72171a07be70c42c1e8017f64cefd3c6b15e20930563e4696986df97e","family_id":"qf-mod49-slln-prefix-sample-mean","seed":202611182028,"semantic_template":"slln-prefix-sample-mean|{law:[[0,[1,2]],[1,[1,2]]],observations:[0,1,0],process_certificate:\"infinite_iid\"}"},"fields":{"title":"A Fixed Prefix Mean Is a Finite Audit, Not an Almost-Sure Limit Certificate","stem":"Use the aligned finite law values=[0, 1], probabilities=[1/2, 1/2], with declared process certificate=infinite_iid. The token is an explicit infinite IID premise, not a process construction or proof by this finite calculation. The observed finite prefix is=[0, 1, 0]. Give (observations, common_mean, prefix_sample_mean, absolute_error, evidence_scope, almost_sure_limit_proof_claim, boundary).","answer":{"kind":"expression","value":"([0, 1, 0], 1/2, 1/3, 1/6, fixed_prefix_only, No, finite_prefix_does_not_prove_almost_sure_convergence)","explanation":"Compute its exact sample mean and absolute error from the common mean. All observations must belong to the declared carrier. A close prefix, or any one finite prefix, does not establish the infinite IID premise or a probability-one limit. The exact result, retaining the requested order, is ([0, 1, 0], 1/2, 1/3, 1/6, fixed_prefix_only, No, finite_prefix_does_not_prove_almost_sure_convergence)."},"skills":["A Fixed Prefix Mean Is a Finite Audit, Not an Almost-Sure Limit Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-0029","batch_id":"batch-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"482a6ff1dc66369af0bb477b02923924be30dbc4281f66d62cef01e79c4aefee","family_id":"qf-mod49-slln-prefix-sample-mean","seed":202611182029,"semantic_template":"slln-prefix-sample-mean|{law:[[1,[1,2]],[2,[1,2]]],observations:[2,1,2,1],process_certificate:\"infinite_iid\"}"},"fields":{"title":"A Fixed Prefix Mean Is a Finite Audit, Not an Almost-Sure Limit Certificate","stem":"Use the aligned finite law values=[1, 2], probabilities=[1/2, 1/2], with declared process certificate=infinite_iid. The token is an explicit infinite IID premise, not a process construction or proof by this finite calculation. The observed finite prefix is=[2, 1, 2, 1]. Give (observations, common_mean, prefix_sample_mean, absolute_error, evidence_scope, almost_sure_limit_proof_claim, boundary).","answer":{"kind":"expression","value":"([2, 1, 2, 1], 3/2, 3/2, 0, fixed_prefix_only, No, finite_prefix_does_not_prove_almost_sure_convergence)","explanation":"Compute its exact sample mean and absolute error from the common mean. All observations must belong to the declared carrier. A close prefix, or any one finite prefix, does not establish the infinite IID premise or a probability-one limit. The exact result, retaining the requested order, is ([2, 1, 2, 1], 3/2, 3/2, 0, fixed_prefix_only, No, finite_prefix_does_not_prove_almost_sure_convergence)."},"skills":["A Fixed Prefix Mean Is a Finite Audit, Not an Almost-Sure Limit Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-0030","batch_id":"batch-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c1d8b61ef19485cb2a81de6d7d6a560ea712b6de4defd78c0667abfb7bc18e6a","family_id":"qf-mod49-slln-prefix-sample-mean","seed":202611182030,"semantic_template":"slln-prefix-sample-mean|{law:[[2,[1,2]],[3,[1,2]]],observations:[2,3,2,3,2],process_certificate:\"infinite_iid\"}"},"fields":{"title":"A Fixed Prefix Mean Is a Finite Audit, Not an Almost-Sure Limit Certificate","stem":"Use the aligned finite law values=[2, 3], probabilities=[1/2, 1/2], with declared process certificate=infinite_iid. The token is an explicit infinite IID premise, not a process construction or proof by this finite calculation. The observed finite prefix is=[2, 3, 2, 3, 2]. Give (observations, common_mean, prefix_sample_mean, absolute_error, evidence_scope, almost_sure_limit_proof_claim, boundary).","answer":{"kind":"expression","value":"([2, 3, 2, 3, 2], 5/2, 12/5, 1/10, fixed_prefix_only, No, finite_prefix_does_not_prove_almost_sure_convergence)","explanation":"Compute its exact sample mean and absolute error from the common mean. All observations must belong to the declared carrier. A close prefix, or any one finite prefix, does not establish the infinite IID premise or a probability-one limit. The exact result, retaining the requested order, is ([2, 3, 2, 3, 2], 5/2, 12/5, 1/10, fixed_prefix_only, No, finite_prefix_does_not_prove_almost_sure_convergence)."},"skills":["A Fixed Prefix Mean Is a Finite Audit, Not an Almost-Sure Limit Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-0031","batch_id":"batch-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7aa97c5d42740a5acc6ceb26c0046d3e299b16c59e9b82cf0ec49f84c6aaa2b8","family_id":"qf-mod49-slln-prefix-sample-mean","seed":202611182031,"semantic_template":"slln-prefix-sample-mean|{law:[[-1,[2,3]],[-2,[1,3]]],observations:[-1,-2,-1,-2,-1,-2],process_certificate:\"infinite_iid\"}"},"fields":{"title":"A Fixed Prefix Mean Is a Finite Audit, Not an Almost-Sure Limit Certificate","stem":"Use the aligned finite law values=[-2, -1], probabilities=[1/3, 2/3], with declared process certificate=infinite_iid. The token is an explicit infinite IID premise, not a process construction or proof by this finite calculation. The observed finite prefix is=[-1, -2, -1, -2, -1, -2]. Give (observations, common_mean, prefix_sample_mean, absolute_error, evidence_scope, almost_sure_limit_proof_claim, boundary).","answer":{"kind":"expression","value":"([-1, -2, -1, -2, -1, -2], -4/3, -3/2, 1/6, fixed_prefix_only, No, finite_prefix_does_not_prove_almost_sure_convergence)","explanation":"Compute its exact sample mean and absolute error from the common mean. All observations must belong to the declared carrier. A close prefix, or any one finite prefix, does not establish the infinite IID premise or a probability-one limit. The exact result, retaining the requested order, is ([-1, -2, -1, -2, -1, -2], -4/3, -3/2, 1/6, fixed_prefix_only, No, finite_prefix_does_not_prove_almost_sure_convergence)."},"skills":["A Fixed Prefix Mean Is a Finite Audit, Not an Almost-Sure Limit Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-0032","batch_id":"batch-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4601624cb61e6985b3f52c7f0d67ba5706a15294e30636f1167693ed45ee2a4b","family_id":"qf-mod49-slln-prefix-sample-mean","seed":202611182032,"semantic_template":"slln-prefix-sample-mean|{law:[[-1,[1,3]],[0,[2,3]]],observations:[-1],process_certificate:\"infinite_iid\"}"},"fields":{"title":"A Fixed Prefix Mean Is a Finite Audit, Not an Almost-Sure Limit Certificate","stem":"Use the aligned finite law values=[-1, 0], probabilities=[1/3, 2/3], with declared process certificate=infinite_iid. The token is an explicit infinite IID premise, not a process construction or proof by this finite calculation. The observed finite prefix is=[-1]. Give (observations, common_mean, prefix_sample_mean, absolute_error, evidence_scope, almost_sure_limit_proof_claim, boundary).","answer":{"kind":"expression","value":"([-1], -1/3, -1, 2/3, fixed_prefix_only, No, finite_prefix_does_not_prove_almost_sure_convergence)","explanation":"Compute its exact sample mean and absolute error from the common mean. All observations must belong to the declared carrier. A close prefix, or any one finite prefix, does not establish the infinite IID premise or a probability-one limit. The exact result, retaining the requested order, is ([-1], -1/3, -1, 2/3, fixed_prefix_only, No, finite_prefix_does_not_prove_almost_sure_convergence)."},"skills":["A Fixed Prefix Mean Is a Finite Audit, Not an Almost-Sure Limit Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-0033","batch_id":"batch-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7a8f092a7002c4342d64f0be9bb7222121b3fb53f2f9b7b7395d205939d0d3c2","family_id":"qf-mod49-slln-prefix-sample-mean","seed":202611182033,"semantic_template":"slln-prefix-sample-mean|{law:[[0,[1,3]],[1,[2,3]]],observations:[1,0],process_certificate:\"infinite_iid\"}"},"fields":{"title":"A Fixed Prefix Mean Is a Finite Audit, Not an Almost-Sure Limit Certificate","stem":"Use the aligned finite law values=[0, 1], probabilities=[1/3, 2/3], with declared process certificate=infinite_iid. The token is an explicit infinite IID premise, not a process construction or proof by this finite calculation. The observed finite prefix is=[1, 0]. Give (observations, common_mean, prefix_sample_mean, absolute_error, evidence_scope, almost_sure_limit_proof_claim, boundary).","answer":{"kind":"expression","value":"([1, 0], 2/3, 1/2, 1/6, fixed_prefix_only, No, finite_prefix_does_not_prove_almost_sure_convergence)","explanation":"Compute its exact sample mean and absolute error from the common mean. All observations must belong to the declared carrier. A close prefix, or any one finite prefix, does not establish the infinite IID premise or a probability-one limit. The exact result, retaining the requested order, is ([1, 0], 2/3, 1/2, 1/6, fixed_prefix_only, No, finite_prefix_does_not_prove_almost_sure_convergence)."},"skills":["A Fixed Prefix Mean Is a Finite Audit, Not an Almost-Sure Limit Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-0034","batch_id":"batch-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e56edb344ce8eaa03f9aa7e384ae0156c10a23483d05dcb724a17f4da297f1bf","family_id":"qf-mod49-slln-prefix-sample-mean","seed":202611182034,"semantic_template":"slln-prefix-sample-mean|{law:[[1,[1,3]],[2,[2,3]]],observations:[1,2,1],process_certificate:\"infinite_iid\"}"},"fields":{"title":"A Fixed Prefix Mean Is a Finite Audit, Not an Almost-Sure Limit Certificate","stem":"Use the aligned finite law values=[1, 2], probabilities=[1/3, 2/3], with declared process certificate=infinite_iid. The token is an explicit infinite IID premise, not a process construction or proof by this finite calculation. The observed finite prefix is=[1, 2, 1]. Give (observations, common_mean, prefix_sample_mean, absolute_error, evidence_scope, almost_sure_limit_proof_claim, boundary).","answer":{"kind":"expression","value":"([1, 2, 1], 5/3, 4/3, 1/3, fixed_prefix_only, No, finite_prefix_does_not_prove_almost_sure_convergence)","explanation":"Compute its exact sample mean and absolute error from the common mean. All observations must belong to the declared carrier. A close prefix, or any one finite prefix, does not establish the infinite IID premise or a probability-one limit. The exact result, retaining the requested order, is ([1, 2, 1], 5/3, 4/3, 1/3, fixed_prefix_only, No, finite_prefix_does_not_prove_almost_sure_convergence)."},"skills":["A Fixed Prefix Mean Is a Finite Audit, Not an Almost-Sure Limit Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-0035","batch_id":"batch-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1242325b25fd56f043381d4c4cefb53382293b148bf6cda3625bfbc723e41595","family_id":"qf-mod49-slln-prefix-sample-mean","seed":202611182035,"semantic_template":"slln-prefix-sample-mean|{law:[[2,[1,3]],[3,[2,3]]],observations:[3,2,3,2],process_certificate:\"infinite_iid\"}"},"fields":{"title":"A Fixed Prefix Mean Is a Finite Audit, Not an Almost-Sure Limit Certificate","stem":"Use the aligned finite law values=[2, 3], probabilities=[1/3, 2/3], with declared process certificate=infinite_iid. The token is an explicit infinite IID premise, not a process construction or proof by this finite calculation. The observed finite prefix is=[3, 2, 3, 2]. Give (observations, common_mean, prefix_sample_mean, absolute_error, evidence_scope, almost_sure_limit_proof_claim, boundary).","answer":{"kind":"expression","value":"([3, 2, 3, 2], 8/3, 5/2, 1/6, fixed_prefix_only, No, finite_prefix_does_not_prove_almost_sure_convergence)","explanation":"Compute its exact sample mean and absolute error from the common mean. All observations must belong to the declared carrier. A close prefix, or any one finite prefix, does not establish the infinite IID premise or a probability-one limit. The exact result, retaining the requested order, is ([3, 2, 3, 2], 8/3, 5/2, 1/6, fixed_prefix_only, No, finite_prefix_does_not_prove_almost_sure_convergence)."},"skills":["A Fixed Prefix Mean Is a Finite Audit, Not an Almost-Sure Limit Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-0036","batch_id":"batch-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c08500116d17c82e58d94a3a0c3cc5d6fa35d2d157ba2d28778b8871e59b8fb9","family_id":"qf-mod49-slln-prefix-sample-mean","seed":202611182036,"semantic_template":"slln-prefix-sample-mean|{law:[[-1,[3,4]],[-2,[1,4]]],observations:[-2,-1,-2,-1,-2],process_certificate:\"infinite_iid\"}"},"fields":{"title":"A Fixed Prefix Mean Is a Finite Audit, Not an Almost-Sure Limit Certificate","stem":"Use the aligned finite law values=[-2, -1], probabilities=[1/4, 3/4], with declared process certificate=infinite_iid. The token is an explicit infinite IID premise, not a process construction or proof by this finite calculation. The observed finite prefix is=[-2, -1, -2, -1, -2]. Give (observations, common_mean, prefix_sample_mean, absolute_error, evidence_scope, almost_sure_limit_proof_claim, boundary).","answer":{"kind":"expression","value":"([-2, -1, -2, -1, -2], -5/4, -8/5, 7/20, fixed_prefix_only, No, finite_prefix_does_not_prove_almost_sure_convergence)","explanation":"Compute its exact sample mean and absolute error from the common mean. All observations must belong to the declared carrier. A close prefix, or any one finite prefix, does not establish the infinite IID premise or a probability-one limit. The exact result, retaining the requested order, is ([-2, -1, -2, -1, -2], -5/4, -8/5, 7/20, fixed_prefix_only, No, finite_prefix_does_not_prove_almost_sure_convergence)."},"skills":["A Fixed Prefix Mean Is a Finite Audit, Not an Almost-Sure Limit Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-0037","batch_id":"batch-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7381cb09dbdd9e4c6e6562812bd0dba60dadef19ef0990494352377eaeedb61b","family_id":"qf-mod49-slln-prefix-sample-mean","seed":202611182037,"semantic_template":"slln-prefix-sample-mean|{law:[[-1,[1,4]],[0,[3,4]]],observations:[0,-1,0,-1,0,-1],process_certificate:\"infinite_iid\"}"},"fields":{"title":"A Fixed Prefix Mean Is a Finite Audit, Not an Almost-Sure Limit Certificate","stem":"Use the aligned finite law values=[-1, 0], probabilities=[1/4, 3/4], with declared process certificate=infinite_iid. The token is an explicit infinite IID premise, not a process construction or proof by this finite calculation. The observed finite prefix is=[0, -1, 0, -1, 0, -1]. Give (observations, common_mean, prefix_sample_mean, absolute_error, evidence_scope, almost_sure_limit_proof_claim, boundary).","answer":{"kind":"expression","value":"([0, -1, 0, -1, 0, -1], -1/4, -1/2, 1/4, fixed_prefix_only, No, finite_prefix_does_not_prove_almost_sure_convergence)","explanation":"Compute its exact sample mean and absolute error from the common mean. All observations must belong to the declared carrier. A close prefix, or any one finite prefix, does not establish the infinite IID premise or a probability-one limit. The exact result, retaining the requested order, is ([0, -1, 0, -1, 0, -1], -1/4, -1/2, 1/4, fixed_prefix_only, No, finite_prefix_does_not_prove_almost_sure_convergence)."},"skills":["A Fixed Prefix Mean Is a Finite Audit, Not an Almost-Sure Limit Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-0038","batch_id":"batch-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7b301805ab6511a43f7262de258029396e051569cc5d1355bf2fed2e34b0ae86","family_id":"qf-mod49-slln-prefix-sample-mean","seed":202611182038,"semantic_template":"slln-prefix-sample-mean|{law:[[0,[1,4]],[1,[3,4]]],observations:[0],process_certificate:\"infinite_iid\"}"},"fields":{"title":"A Fixed Prefix Mean Is a Finite Audit, Not an Almost-Sure Limit Certificate","stem":"Use the aligned finite law values=[0, 1], probabilities=[1/4, 3/4], with declared process certificate=infinite_iid. The token is an explicit infinite IID premise, not a process construction or proof by this finite calculation. The observed finite prefix is=[0]. Give (observations, common_mean, prefix_sample_mean, absolute_error, evidence_scope, almost_sure_limit_proof_claim, boundary).","answer":{"kind":"expression","value":"([0], 3/4, 0, 3/4, fixed_prefix_only, No, finite_prefix_does_not_prove_almost_sure_convergence)","explanation":"Compute its exact sample mean and absolute error from the common mean. All observations must belong to the declared carrier. A close prefix, or any one finite prefix, does not establish the infinite IID premise or a probability-one limit. The exact result, retaining the requested order, is ([0], 3/4, 0, 3/4, fixed_prefix_only, No, finite_prefix_does_not_prove_almost_sure_convergence)."},"skills":["A Fixed Prefix Mean Is a Finite Audit, Not an Almost-Sure Limit Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-0039","batch_id":"batch-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9b26fd681342521399e95dc5f2e8e672fe099ff698fac742c702b3fe61dc144a","family_id":"qf-mod49-slln-prefix-sample-mean","seed":202611182039,"semantic_template":"slln-prefix-sample-mean|{law:[[1,[1,4]],[2,[3,4]]],observations:[2,1],process_certificate:\"infinite_iid\"}"},"fields":{"title":"A Fixed Prefix Mean Is a Finite Audit, Not an Almost-Sure Limit Certificate","stem":"Use the aligned finite law values=[1, 2], probabilities=[1/4, 3/4], with declared process certificate=infinite_iid. The token is an explicit infinite IID premise, not a process construction or proof by this finite calculation. The observed finite prefix is=[2, 1]. Give (observations, common_mean, prefix_sample_mean, absolute_error, evidence_scope, almost_sure_limit_proof_claim, boundary).","answer":{"kind":"expression","value":"([2, 1], 7/4, 3/2, 1/4, fixed_prefix_only, No, finite_prefix_does_not_prove_almost_sure_convergence)","explanation":"Compute its exact sample mean and absolute error from the common mean. All observations must belong to the declared carrier. A close prefix, or any one finite prefix, does not establish the infinite IID premise or a probability-one limit. The exact result, retaining the requested order, is ([2, 1], 7/4, 3/2, 1/4, fixed_prefix_only, No, finite_prefix_does_not_prove_almost_sure_convergence)."},"skills":["A Fixed Prefix Mean Is a Finite Audit, Not an Almost-Sure Limit Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-0040","batch_id":"batch-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dcdd1973519d8635fb6256bc6b2d87c32faee0d838ebed29d410420132e60843","family_id":"qf-mod49-slln-prefix-sample-mean","seed":202611182040,"semantic_template":"slln-prefix-sample-mean|{law:[[2,[1,4]],[3,[3,4]]],observations:[2,3,2],process_certificate:\"infinite_iid\"}"},"fields":{"title":"A Fixed Prefix Mean Is a Finite Audit, Not an Almost-Sure Limit Certificate","stem":"Use the aligned finite law values=[2, 3], probabilities=[1/4, 3/4], with declared process certificate=infinite_iid. The token is an explicit infinite IID premise, not a process construction or proof by this finite calculation. The observed finite prefix is=[2, 3, 2]. Give (observations, common_mean, prefix_sample_mean, absolute_error, evidence_scope, almost_sure_limit_proof_claim, boundary).","answer":{"kind":"expression","value":"([2, 3, 2], 11/4, 7/3, 5/12, fixed_prefix_only, No, finite_prefix_does_not_prove_almost_sure_convergence)","explanation":"Compute its exact sample mean and absolute error from the common mean. All observations must belong to the declared carrier. A close prefix, or any one finite prefix, does not establish the infinite IID premise or a probability-one limit. The exact result, retaining the requested order, is ([2, 3, 2], 11/4, 7/3, 5/12, fixed_prefix_only, No, finite_prefix_does_not_prove_almost_sure_convergence)."},"skills":["A Fixed Prefix Mean Is a Finite Audit, Not an Almost-Sure Limit Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-0041","batch_id":"batch-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1513290e2525abc4150fb50f4638148ee03b198152d88d5075a538abb2470edf","family_id":"qf-mod49-slln-prefix-sample-mean","seed":202611182041,"semantic_template":"slln-prefix-sample-mean|{law:[[-1,[3,5]],[-2,[2,5]]],observations:[-1,-2,-1,-2],process_certificate:\"infinite_iid\"}"},"fields":{"title":"A Fixed Prefix Mean Is a Finite Audit, Not an Almost-Sure Limit Certificate","stem":"Use the aligned finite law values=[-2, -1], probabilities=[2/5, 3/5], with declared process certificate=infinite_iid. The token is an explicit infinite IID premise, not a process construction or proof by this finite calculation. The observed finite prefix is=[-1, -2, -1, -2]. Give (observations, common_mean, prefix_sample_mean, absolute_error, evidence_scope, almost_sure_limit_proof_claim, boundary).","answer":{"kind":"expression","value":"([-1, -2, -1, -2], -7/5, -3/2, 1/10, fixed_prefix_only, No, finite_prefix_does_not_prove_almost_sure_convergence)","explanation":"Compute its exact sample mean and absolute error from the common mean. All observations must belong to the declared carrier. A close prefix, or any one finite prefix, does not establish the infinite IID premise or a probability-one limit. The exact result, retaining the requested order, is ([-1, -2, -1, -2], -7/5, -3/2, 1/10, fixed_prefix_only, No, finite_prefix_does_not_prove_almost_sure_convergence)."},"skills":["A Fixed Prefix Mean Is a Finite Audit, Not an Almost-Sure Limit Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-0042","batch_id":"batch-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3870a60b42b76ebf4e70d56099fdde869b2f9e405afb0fffb5e1d0cbb154262e","family_id":"qf-mod49-slln-prefix-sample-mean","seed":202611182042,"semantic_template":"slln-prefix-sample-mean|{law:[[-1,[2,5]],[0,[3,5]]],observations:[-1,0,-1,0,-1],process_certificate:\"infinite_iid\"}"},"fields":{"title":"A Fixed Prefix Mean Is a Finite Audit, Not an Almost-Sure Limit Certificate","stem":"Use the aligned finite law values=[-1, 0], probabilities=[2/5, 3/5], with declared process certificate=infinite_iid. The token is an explicit infinite IID premise, not a process construction or proof by this finite calculation. The observed finite prefix is=[-1, 0, -1, 0, -1]. Give (observations, common_mean, prefix_sample_mean, absolute_error, evidence_scope, almost_sure_limit_proof_claim, boundary).","answer":{"kind":"expression","value":"([-1, 0, -1, 0, -1], -2/5, -3/5, 1/5, fixed_prefix_only, No, finite_prefix_does_not_prove_almost_sure_convergence)","explanation":"Compute its exact sample mean and absolute error from the common mean. All observations must belong to the declared carrier. A close prefix, or any one finite prefix, does not establish the infinite IID premise or a probability-one limit. The exact result, retaining the requested order, is ([-1, 0, -1, 0, -1], -2/5, -3/5, 1/5, fixed_prefix_only, No, finite_prefix_does_not_prove_almost_sure_convergence)."},"skills":["A Fixed Prefix Mean Is a Finite Audit, Not an Almost-Sure Limit Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-0043","batch_id":"batch-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a836b46d3be49f6abcdd983d402f97fcd9b9ae3bff66a420e7e0745bb2f51e59","family_id":"qf-mod49-slln-prefix-sample-mean","seed":202611182043,"semantic_template":"slln-prefix-sample-mean|{law:[[0,[2,5]],[1,[3,5]]],observations:[1,0,1,0,1,0],process_certificate:\"infinite_iid\"}"},"fields":{"title":"A Fixed Prefix Mean Is a Finite Audit, Not an Almost-Sure Limit Certificate","stem":"Use the aligned finite law values=[0, 1], probabilities=[2/5, 3/5], with declared process certificate=infinite_iid. The token is an explicit infinite IID premise, not a process construction or proof by this finite calculation. The observed finite prefix is=[1, 0, 1, 0, 1, 0]. Give (observations, common_mean, prefix_sample_mean, absolute_error, evidence_scope, almost_sure_limit_proof_claim, boundary).","answer":{"kind":"expression","value":"([1, 0, 1, 0, 1, 0], 3/5, 1/2, 1/10, fixed_prefix_only, No, finite_prefix_does_not_prove_almost_sure_convergence)","explanation":"Compute its exact sample mean and absolute error from the common mean. All observations must belong to the declared carrier. A close prefix, or any one finite prefix, does not establish the infinite IID premise or a probability-one limit. The exact result, retaining the requested order, is ([1, 0, 1, 0, 1, 0], 3/5, 1/2, 1/10, fixed_prefix_only, No, finite_prefix_does_not_prove_almost_sure_convergence)."},"skills":["A Fixed Prefix Mean Is a Finite Audit, Not an Almost-Sure Limit Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-0044","batch_id":"batch-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8bdc98d77ffeee46b5be983d437755636fbf04ce4f119b49cc58bc329cdc1765","family_id":"qf-mod49-slln-prefix-sample-mean","seed":202611182044,"semantic_template":"slln-prefix-sample-mean|{law:[[1,[2,5]],[2,[3,5]]],observations:[1],process_certificate:\"infinite_iid\"}"},"fields":{"title":"A Fixed Prefix Mean Is a Finite Audit, Not an Almost-Sure Limit Certificate","stem":"Use the aligned finite law values=[1, 2], probabilities=[2/5, 3/5], with declared process certificate=infinite_iid. The token is an explicit infinite IID premise, not a process construction or proof by this finite calculation. The observed finite prefix is=[1]. Give (observations, common_mean, prefix_sample_mean, absolute_error, evidence_scope, almost_sure_limit_proof_claim, boundary).","answer":{"kind":"expression","value":"([1], 8/5, 1, 3/5, fixed_prefix_only, No, finite_prefix_does_not_prove_almost_sure_convergence)","explanation":"Compute its exact sample mean and absolute error from the common mean. All observations must belong to the declared carrier. A close prefix, or any one finite prefix, does not establish the infinite IID premise or a probability-one limit. The exact result, retaining the requested order, is ([1], 8/5, 1, 3/5, fixed_prefix_only, No, finite_prefix_does_not_prove_almost_sure_convergence)."},"skills":["A Fixed Prefix Mean Is a Finite Audit, Not an Almost-Sure Limit Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-0045","batch_id":"batch-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8621eddc8c81de1e23ea41ca73284f62f8eeb668f2d4b45d7dce157db4dcf45f","family_id":"qf-mod49-slln-prefix-sample-mean","seed":202611182045,"semantic_template":"slln-prefix-sample-mean|{law:[[2,[2,5]],[3,[3,5]]],observations:[3,2],process_certificate:\"infinite_iid\"}"},"fields":{"title":"A Fixed Prefix Mean Is a Finite Audit, Not an Almost-Sure Limit Certificate","stem":"Use the aligned finite law values=[2, 3], probabilities=[2/5, 3/5], with declared process certificate=infinite_iid. The token is an explicit infinite IID premise, not a process construction or proof by this finite calculation. The observed finite prefix is=[3, 2]. Give (observations, common_mean, prefix_sample_mean, absolute_error, evidence_scope, almost_sure_limit_proof_claim, boundary).","answer":{"kind":"expression","value":"([3, 2], 13/5, 5/2, 1/10, fixed_prefix_only, No, finite_prefix_does_not_prove_almost_sure_convergence)","explanation":"Compute its exact sample mean and absolute error from the common mean. All observations must belong to the declared carrier. A close prefix, or any one finite prefix, does not establish the infinite IID premise or a probability-one limit. The exact result, retaining the requested order, is ([3, 2], 13/5, 5/2, 1/10, fixed_prefix_only, No, finite_prefix_does_not_prove_almost_sure_convergence)."},"skills":["A Fixed Prefix Mean Is a Finite Audit, Not an Almost-Sure Limit Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-0046","batch_id":"batch-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fd5aec1fb6982a573496fa46d5da58f797e69c823d01bcf0113dde2cc82f1a26","family_id":"qf-mod49-slln-prefix-sample-mean","seed":202611182046,"semantic_template":"slln-prefix-sample-mean|{law:[[-1,[4,5]],[-2,[1,5]]],observations:[-2,-1,-2],process_certificate:\"infinite_iid\"}"},"fields":{"title":"A Fixed Prefix Mean Is a Finite Audit, Not an Almost-Sure Limit Certificate","stem":"Use the aligned finite law values=[-2, -1], probabilities=[1/5, 4/5], with declared process certificate=infinite_iid. The token is an explicit infinite IID premise, not a process construction or proof by this finite calculation. The observed finite prefix is=[-2, -1, -2]. Give (observations, common_mean, prefix_sample_mean, absolute_error, evidence_scope, almost_sure_limit_proof_claim, boundary).","answer":{"kind":"expression","value":"([-2, -1, -2], -6/5, -5/3, 7/15, fixed_prefix_only, No, finite_prefix_does_not_prove_almost_sure_convergence)","explanation":"Compute its exact sample mean and absolute error from the common mean. All observations must belong to the declared carrier. A close prefix, or any one finite prefix, does not establish the infinite IID premise or a probability-one limit. The exact result, retaining the requested order, is ([-2, -1, -2], -6/5, -5/3, 7/15, fixed_prefix_only, No, finite_prefix_does_not_prove_almost_sure_convergence)."},"skills":["A Fixed Prefix Mean Is a Finite Audit, Not an Almost-Sure Limit Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-0047","batch_id":"batch-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"77e67906ac0b6736d542666e9063749d96b340133609caba27691d489e6c8c91","family_id":"qf-mod49-slln-prefix-sample-mean","seed":202611182047,"semantic_template":"slln-prefix-sample-mean|{law:[[-1,[1,5]],[0,[4,5]]],observations:[0,-1,0,-1],process_certificate:\"infinite_iid\"}"},"fields":{"title":"A Fixed Prefix Mean Is a Finite Audit, Not an Almost-Sure Limit Certificate","stem":"Use the aligned finite law values=[-1, 0], probabilities=[1/5, 4/5], with declared process certificate=infinite_iid. The token is an explicit infinite IID premise, not a process construction or proof by this finite calculation. The observed finite prefix is=[0, -1, 0, -1]. Give (observations, common_mean, prefix_sample_mean, absolute_error, evidence_scope, almost_sure_limit_proof_claim, boundary).","answer":{"kind":"expression","value":"([0, -1, 0, -1], -1/5, -1/2, 3/10, fixed_prefix_only, No, finite_prefix_does_not_prove_almost_sure_convergence)","explanation":"Compute its exact sample mean and absolute error from the common mean. All observations must belong to the declared carrier. A close prefix, or any one finite prefix, does not establish the infinite IID premise or a probability-one limit. The exact result, retaining the requested order, is ([0, -1, 0, -1], -1/5, -1/2, 3/10, fixed_prefix_only, No, finite_prefix_does_not_prove_almost_sure_convergence)."},"skills":["A Fixed Prefix Mean Is a Finite Audit, Not an Almost-Sure Limit Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-0048","batch_id":"batch-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"eb53ff3f895cca6645f01495663074a6746993314acd9e779aee36bfe7c43f2f","family_id":"qf-mod49-slln-prefix-sample-mean","seed":202611182048,"semantic_template":"slln-prefix-sample-mean|{law:[[0,[1,5]],[1,[4,5]]],observations:[0,1,0,1,0],process_certificate:\"infinite_iid\"}"},"fields":{"title":"A Fixed Prefix Mean Is a Finite Audit, Not an Almost-Sure Limit Certificate","stem":"Use the aligned finite law values=[0, 1], probabilities=[1/5, 4/5], with declared process certificate=infinite_iid. The token is an explicit infinite IID premise, not a process construction or proof by this finite calculation. The observed finite prefix is=[0, 1, 0, 1, 0]. Give (observations, common_mean, prefix_sample_mean, absolute_error, evidence_scope, almost_sure_limit_proof_claim, boundary).","answer":{"kind":"expression","value":"([0, 1, 0, 1, 0], 4/5, 2/5, 2/5, fixed_prefix_only, No, finite_prefix_does_not_prove_almost_sure_convergence)","explanation":"Compute its exact sample mean and absolute error from the common mean. All observations must belong to the declared carrier. A close prefix, or any one finite prefix, does not establish the infinite IID premise or a probability-one limit. The exact result, retaining the requested order, is ([0, 1, 0, 1, 0], 4/5, 2/5, 2/5, fixed_prefix_only, No, finite_prefix_does_not_prove_almost_sure_convergence)."},"skills":["A Fixed Prefix Mean Is a Finite Audit, Not an Almost-Sure Limit Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-0049","batch_id":"batch-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5c37edf2a2d33f1d6604ff0596246312e5939c32d2fa31bcda6e9b028811212f","family_id":"qf-mod49-slln-prefix-sample-mean","seed":202611182049,"semantic_template":"slln-prefix-sample-mean|{law:[[1,[1,5]],[2,[4,5]]],observations:[2,1,2,1,2,1],process_certificate:\"infinite_iid\"}"},"fields":{"title":"A Fixed Prefix Mean Is a Finite Audit, Not an Almost-Sure Limit Certificate","stem":"Use the aligned finite law values=[1, 2], probabilities=[1/5, 4/5], with declared process certificate=infinite_iid. The token is an explicit infinite IID premise, not a process construction or proof by this finite calculation. The observed finite prefix is=[2, 1, 2, 1, 2, 1]. Give (observations, common_mean, prefix_sample_mean, absolute_error, evidence_scope, almost_sure_limit_proof_claim, boundary).","answer":{"kind":"expression","value":"([2, 1, 2, 1, 2, 1], 9/5, 3/2, 3/10, fixed_prefix_only, No, finite_prefix_does_not_prove_almost_sure_convergence)","explanation":"Compute its exact sample mean and absolute error from the common mean. All observations must belong to the declared carrier. A close prefix, or any one finite prefix, does not establish the infinite IID premise or a probability-one limit. The exact result, retaining the requested order, is ([2, 1, 2, 1, 2, 1], 9/5, 3/2, 3/10, fixed_prefix_only, No, finite_prefix_does_not_prove_almost_sure_convergence)."},"skills":["A Fixed Prefix Mean Is a Finite Audit, Not an Almost-Sure Limit Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-0050","batch_id":"batch-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5a360fd47197188e7521a8ce9c6b162bb5f10aa1a80aab7d49a161acba01960e","family_id":"qf-mod49-slln-prefix-sample-mean","seed":202611182050,"semantic_template":"slln-prefix-sample-mean|{law:[[2,[1,5]],[3,[4,5]]],observations:[2],process_certificate:\"infinite_iid\"}"},"fields":{"title":"A Fixed Prefix Mean Is a Finite Audit, Not an Almost-Sure Limit Certificate","stem":"Use the aligned finite law values=[2, 3], probabilities=[1/5, 4/5], with declared process certificate=infinite_iid. The token is an explicit infinite IID premise, not a process construction or proof by this finite calculation. The observed finite prefix is=[2]. Give (observations, common_mean, prefix_sample_mean, absolute_error, evidence_scope, almost_sure_limit_proof_claim, boundary).","answer":{"kind":"expression","value":"([2], 14/5, 2, 4/5, fixed_prefix_only, No, finite_prefix_does_not_prove_almost_sure_convergence)","explanation":"Compute its exact sample mean and absolute error from the common mean. All observations must belong to the declared carrier. A close prefix, or any one finite prefix, does not establish the infinite IID premise or a probability-one limit. The exact result, retaining the requested order, is ([2], 14/5, 2, 4/5, fixed_prefix_only, No, finite_prefix_does_not_prove_almost_sure_convergence)."},"skills":["A Fixed Prefix Mean Is a Finite Audit, Not an Almost-Sure Limit Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-0051","batch_id":"batch-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c685e26dd70d2f52556aae9f27e5182eaa7c7e85710abb170e30948dcec883b5","family_id":"qf-mod49-slln-fixed-n-deviation","seed":202611182051,"semantic_template":"slln-fixed-n-deviation|{epsilon:[1,3],law:[[-1,[1,2]],[-2,[1,2]]],n:1,process_certificate:\"infinite_iid\"}"},"fields":{"title":"The Complete Fixed-n Deviation Event Is Separate from an Almost-Sure Limit","stem":"Use the aligned finite law values=[-2, -1], probabilities=[1/2, 1/2], with declared process certificate=infinite_iid. The token is an explicit infinite IID premise, not a process construction or proof by this finite calculation. Use n=1 and epsilon=1/3>0; the event is |sample_mean−mu|≥epsilon. Give (n, common_mean, epsilon, deviating_joint_word_count, deviation_probability, event_scope, almost_sure_limit_proof_claim, boundary).","answer":{"kind":"expression","value":"(1, -3/2, 1/3, 2, 1, fixed_n_deviation_event, No, not_an_almost_sure_limit_certificate)","explanation":"Count every qualifying joint word and separately sum its exact product probability. Zero-mass words still enter the combinatorial count. A fixed-n deviation event, even when fully enumerated, is not an almost-sure convergence certificate. The exact result, retaining the requested order, is (1, -3/2, 1/3, 2, 1, fixed_n_deviation_event, No, not_an_almost_sure_limit_certificate)."},"skills":["The Complete Fixed-n Deviation Event Is Separate from an Almost-Sure 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-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-0052","batch_id":"batch-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"afe006be6a251a528d4d4e5f8a31384b8600b0c438fd9ec3c6471ba9f1e53250","family_id":"qf-mod49-slln-fixed-n-deviation","seed":202611182052,"semantic_template":"slln-fixed-n-deviation|{epsilon:[1,2],law:[[-1,[1,2]],[0,[1,2]]],n:1,process_certificate:\"infinite_iid\"}"},"fields":{"title":"The Complete Fixed-n Deviation Event Is Separate from an Almost-Sure Limit","stem":"Use the aligned finite law values=[-1, 0], probabilities=[1/2, 1/2], with declared process certificate=infinite_iid. The token is an explicit infinite IID premise, not a process construction or proof by this finite calculation. Use n=1 and epsilon=2/4>0; the event is |sample_mean−mu|≥epsilon. Give (n, common_mean, epsilon, deviating_joint_word_count, deviation_probability, event_scope, almost_sure_limit_proof_claim, boundary).","answer":{"kind":"expression","value":"(1, -1/2, 1/2, 2, 1, fixed_n_deviation_event, No, not_an_almost_sure_limit_certificate)","explanation":"Count every qualifying joint word and separately sum its exact product probability. Zero-mass words still enter the combinatorial count. A fixed-n deviation event, even when fully enumerated, is not an almost-sure convergence certificate. The exact result, retaining the requested order, is (1, -1/2, 1/2, 2, 1, fixed_n_deviation_event, No, not_an_almost_sure_limit_certificate)."},"skills":["The Complete Fixed-n Deviation Event Is Separate from an Almost-Sure 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-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-0053","batch_id":"batch-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5c0bcd9120bb509cd1c30ab897a4e15d012da9094b89ab1eb42804f320a1dd30","family_id":"qf-mod49-slln-fixed-n-deviation","seed":202611182053,"semantic_template":"slln-fixed-n-deviation|{epsilon:[3,5],law:[[0,[1,2]],[1,[1,2]]],n:1,process_certificate:\"infinite_iid\"}"},"fields":{"title":"The Complete Fixed-n Deviation Event Is Separate from an Almost-Sure Limit","stem":"Use the aligned finite law values=[0, 1], probabilities=[1/2, 1/2], with declared process certificate=infinite_iid. The token is an explicit infinite IID premise, not a process construction or proof by this finite calculation. Use n=1 and epsilon=3/5>0; the event is |sample_mean−mu|≥epsilon. Give (n, common_mean, epsilon, deviating_joint_word_count, deviation_probability, event_scope, almost_sure_limit_proof_claim, boundary).","answer":{"kind":"expression","value":"(1, 1/2, 3/5, 0, 0, fixed_n_deviation_event, No, not_an_almost_sure_limit_certificate)","explanation":"Count every qualifying joint word and separately sum its exact product probability. Zero-mass words still enter the combinatorial count. A fixed-n deviation event, even when fully enumerated, is not an almost-sure convergence certificate. The exact result, retaining the requested order, is (1, 1/2, 3/5, 0, 0, fixed_n_deviation_event, No, not_an_almost_sure_limit_certificate)."},"skills":["The Complete Fixed-n Deviation Event Is Separate from an Almost-Sure 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-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-0054","batch_id":"batch-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"398b93ee3639824afdb1a9635de4a62a529ef278c474d33442797221c5965bfd","family_id":"qf-mod49-slln-fixed-n-deviation","seed":202611182054,"semantic_template":"slln-fixed-n-deviation|{epsilon:[1,3],law:[[1,[1,2]],[2,[1,2]]],n:1,process_certificate:\"infinite_iid\"}"},"fields":{"title":"The Complete Fixed-n Deviation Event Is Separate from an Almost-Sure Limit","stem":"Use the aligned finite law values=[1, 2], probabilities=[1/2, 1/2], with declared process certificate=infinite_iid. The token is an explicit infinite IID premise, not a process construction or proof by this finite calculation. Use n=1 and epsilon=1/3>0; the event is |sample_mean−mu|≥epsilon. Give (n, common_mean, epsilon, deviating_joint_word_count, deviation_probability, event_scope, almost_sure_limit_proof_claim, boundary).","answer":{"kind":"expression","value":"(1, 3/2, 1/3, 2, 1, fixed_n_deviation_event, No, not_an_almost_sure_limit_certificate)","explanation":"Count every qualifying joint word and separately sum its exact product probability. Zero-mass words still enter the combinatorial count. A fixed-n deviation event, even when fully enumerated, is not an almost-sure convergence certificate. The exact result, retaining the requested order, is (1, 3/2, 1/3, 2, 1, fixed_n_deviation_event, No, not_an_almost_sure_limit_certificate)."},"skills":["The Complete Fixed-n Deviation Event Is Separate from an Almost-Sure 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-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-0055","batch_id":"batch-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e9ffd3f641a871054af546317a47394bb7d6b4e6e58ef9969224c2e7c232e8a6","family_id":"qf-mod49-slln-fixed-n-deviation","seed":202611182055,"semantic_template":"slln-fixed-n-deviation|{epsilon:[1,2],law:[[2,[1,2]],[3,[1,2]]],n:1,process_certificate:\"infinite_iid\"}"},"fields":{"title":"The Complete Fixed-n Deviation Event Is Separate from an Almost-Sure Limit","stem":"Use the aligned finite law values=[2, 3], probabilities=[1/2, 1/2], with declared process certificate=infinite_iid. The token is an explicit infinite IID premise, not a process construction or proof by this finite calculation. Use n=1 and epsilon=2/4>0; the event is |sample_mean−mu|≥epsilon. Give (n, common_mean, epsilon, deviating_joint_word_count, deviation_probability, event_scope, almost_sure_limit_proof_claim, boundary).","answer":{"kind":"expression","value":"(1, 5/2, 1/2, 2, 1, fixed_n_deviation_event, No, not_an_almost_sure_limit_certificate)","explanation":"Count every qualifying joint word and separately sum its exact product probability. Zero-mass words still enter the combinatorial count. A fixed-n deviation event, even when fully enumerated, is not an almost-sure convergence certificate. The exact result, retaining the requested order, is (1, 5/2, 1/2, 2, 1, fixed_n_deviation_event, No, not_an_almost_sure_limit_certificate)."},"skills":["The Complete Fixed-n Deviation Event Is Separate from an Almost-Sure 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-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-0056","batch_id":"batch-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0232d3e0aa1f5ffc00d574c23a526f231a1d85ef8a3d9a24fd3996442712cafd","family_id":"qf-mod49-slln-fixed-n-deviation","seed":202611182056,"semantic_template":"slln-fixed-n-deviation|{epsilon:[3,5],law:[[-1,[1,2]],[-2,[1,2]]],n:2,process_certificate:\"infinite_iid\"}"},"fields":{"title":"The Complete Fixed-n Deviation Event Is Separate from an Almost-Sure Limit","stem":"Use the aligned finite law values=[-2, -1], probabilities=[1/2, 1/2], with declared process certificate=infinite_iid. The token is an explicit infinite IID premise, not a process construction or proof by this finite calculation. Use n=2 and epsilon=3/5>0; the event is |sample_mean−mu|≥epsilon. Give (n, common_mean, epsilon, deviating_joint_word_count, deviation_probability, event_scope, almost_sure_limit_proof_claim, boundary).","answer":{"kind":"expression","value":"(2, -3/2, 3/5, 0, 0, fixed_n_deviation_event, No, not_an_almost_sure_limit_certificate)","explanation":"Count every qualifying joint word and separately sum its exact product probability. Zero-mass words still enter the combinatorial count. A fixed-n deviation event, even when fully enumerated, is not an almost-sure convergence certificate. The exact result, retaining the requested order, is (2, -3/2, 3/5, 0, 0, fixed_n_deviation_event, No, not_an_almost_sure_limit_certificate)."},"skills":["The Complete Fixed-n Deviation Event Is Separate from an Almost-Sure 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-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-0057","batch_id":"batch-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e34088897b4cb2d7825a11787f874ef19bb8ce5c745b180099483fae87d091e8","family_id":"qf-mod49-slln-fixed-n-deviation","seed":202611182057,"semantic_template":"slln-fixed-n-deviation|{epsilon:[1,3],law:[[-1,[1,2]],[0,[1,2]]],n:2,process_certificate:\"infinite_iid\"}"},"fields":{"title":"The Complete Fixed-n Deviation Event Is Separate from an Almost-Sure Limit","stem":"Use the aligned finite law values=[-1, 0], probabilities=[1/2, 1/2], with declared process certificate=infinite_iid. The token is an explicit infinite IID premise, not a process construction or proof by this finite calculation. Use n=2 and epsilon=1/3>0; the event is |sample_mean−mu|≥epsilon. Give (n, common_mean, epsilon, deviating_joint_word_count, deviation_probability, event_scope, almost_sure_limit_proof_claim, boundary).","answer":{"kind":"expression","value":"(2, -1/2, 1/3, 2, 1/2, fixed_n_deviation_event, No, not_an_almost_sure_limit_certificate)","explanation":"Count every qualifying joint word and separately sum its exact product probability. Zero-mass words still enter the combinatorial count. A fixed-n deviation event, even when fully enumerated, is not an almost-sure convergence certificate. The exact result, retaining the requested order, is (2, -1/2, 1/3, 2, 1/2, fixed_n_deviation_event, No, not_an_almost_sure_limit_certificate)."},"skills":["The Complete Fixed-n Deviation Event Is Separate from an Almost-Sure 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-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-0058","batch_id":"batch-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a2c73adc778d1c0a98bdac8d9d90bed9040a77bf60bd7edc77ebfb7b476deb79","family_id":"qf-mod49-slln-fixed-n-deviation","seed":202611182058,"semantic_template":"slln-fixed-n-deviation|{epsilon:[1,2],law:[[0,[1,2]],[1,[1,2]]],n:2,process_certificate:\"infinite_iid\"}"},"fields":{"title":"The Complete Fixed-n Deviation Event Is Separate from an Almost-Sure Limit","stem":"Use the aligned finite law values=[0, 1], probabilities=[1/2, 1/2], with declared process certificate=infinite_iid. The token is an explicit infinite IID premise, not a process construction or proof by this finite calculation. Use n=2 and epsilon=2/4>0; the event is |sample_mean−mu|≥epsilon. Give (n, common_mean, epsilon, deviating_joint_word_count, deviation_probability, event_scope, almost_sure_limit_proof_claim, boundary).","answer":{"kind":"expression","value":"(2, 1/2, 1/2, 2, 1/2, fixed_n_deviation_event, No, not_an_almost_sure_limit_certificate)","explanation":"Count every qualifying joint word and separately sum its exact product probability. Zero-mass words still enter the combinatorial count. A fixed-n deviation event, even when fully enumerated, is not an almost-sure convergence certificate. The exact result, retaining the requested order, is (2, 1/2, 1/2, 2, 1/2, fixed_n_deviation_event, No, not_an_almost_sure_limit_certificate)."},"skills":["The Complete Fixed-n Deviation Event Is Separate from an Almost-Sure 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-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-0059","batch_id":"batch-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d057c2c37f29e94254fc442e05209007cf7eb0139bf4b693054cea8241292252","family_id":"qf-mod49-slln-fixed-n-deviation","seed":202611182059,"semantic_template":"slln-fixed-n-deviation|{epsilon:[3,5],law:[[1,[1,2]],[2,[1,2]]],n:2,process_certificate:\"infinite_iid\"}"},"fields":{"title":"The Complete Fixed-n Deviation Event Is Separate from an Almost-Sure Limit","stem":"Use the aligned finite law values=[1, 2], probabilities=[1/2, 1/2], with declared process certificate=infinite_iid. The token is an explicit infinite IID premise, not a process construction or proof by this finite calculation. Use n=2 and epsilon=3/5>0; the event is |sample_mean−mu|≥epsilon. Give (n, common_mean, epsilon, deviating_joint_word_count, deviation_probability, event_scope, almost_sure_limit_proof_claim, boundary).","answer":{"kind":"expression","value":"(2, 3/2, 3/5, 0, 0, fixed_n_deviation_event, No, not_an_almost_sure_limit_certificate)","explanation":"Count every qualifying joint word and separately sum its exact product probability. Zero-mass words still enter the combinatorial count. A fixed-n deviation event, even when fully enumerated, is not an almost-sure convergence certificate. The exact result, retaining the requested order, is (2, 3/2, 3/5, 0, 0, fixed_n_deviation_event, No, not_an_almost_sure_limit_certificate)."},"skills":["The Complete Fixed-n Deviation Event Is Separate from an Almost-Sure 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-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-0060","batch_id":"batch-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d3c42601313c5b5aa4e942397a4a02d6cbca5e535f152be0004b60d0e9da8583","family_id":"qf-mod49-slln-fixed-n-deviation","seed":202611182060,"semantic_template":"slln-fixed-n-deviation|{epsilon:[1,3],law:[[2,[1,2]],[3,[1,2]]],n:2,process_certificate:\"infinite_iid\"}"},"fields":{"title":"The Complete Fixed-n Deviation Event Is Separate from an Almost-Sure Limit","stem":"Use the aligned finite law values=[2, 3], probabilities=[1/2, 1/2], with declared process certificate=infinite_iid. The token is an explicit infinite IID premise, not a process construction or proof by this finite calculation. Use n=2 and epsilon=1/3>0; the event is |sample_mean−mu|≥epsilon. Give (n, common_mean, epsilon, deviating_joint_word_count, deviation_probability, event_scope, almost_sure_limit_proof_claim, boundary).","answer":{"kind":"expression","value":"(2, 5/2, 1/3, 2, 1/2, fixed_n_deviation_event, No, not_an_almost_sure_limit_certificate)","explanation":"Count every qualifying joint word and separately sum its exact product probability. Zero-mass words still enter the combinatorial count. A fixed-n deviation event, even when fully enumerated, is not an almost-sure convergence certificate. The exact result, retaining the requested order, is (2, 5/2, 1/3, 2, 1/2, fixed_n_deviation_event, No, not_an_almost_sure_limit_certificate)."},"skills":["The Complete Fixed-n Deviation Event Is Separate from an Almost-Sure 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-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-0061","batch_id":"batch-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b5def7c9e95851b697f0bd7d8e81d520b5747e300396241d64c3b93db524ad89","family_id":"qf-mod49-slln-fixed-n-deviation","seed":202611182061,"semantic_template":"slln-fixed-n-deviation|{epsilon:[1,2],law:[[-1,[1,2]],[-2,[1,2]]],n:3,process_certificate:\"infinite_iid\"}"},"fields":{"title":"The Complete Fixed-n Deviation Event Is Separate from an Almost-Sure Limit","stem":"Use the aligned finite law values=[-2, -1], probabilities=[1/2, 1/2], with declared process certificate=infinite_iid. The token is an explicit infinite IID premise, not a process construction or proof by this finite calculation. Use n=3 and epsilon=2/4>0; the event is |sample_mean−mu|≥epsilon. Give (n, common_mean, epsilon, deviating_joint_word_count, deviation_probability, event_scope, almost_sure_limit_proof_claim, boundary).","answer":{"kind":"expression","value":"(3, -3/2, 1/2, 2, 1/4, fixed_n_deviation_event, No, not_an_almost_sure_limit_certificate)","explanation":"Count every qualifying joint word and separately sum its exact product probability. Zero-mass words still enter the combinatorial count. A fixed-n deviation event, even when fully enumerated, is not an almost-sure convergence certificate. The exact result, retaining the requested order, is (3, -3/2, 1/2, 2, 1/4, fixed_n_deviation_event, No, not_an_almost_sure_limit_certificate)."},"skills":["The Complete Fixed-n Deviation Event Is Separate from an Almost-Sure 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-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-0062","batch_id":"batch-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"042d139864abe75dec5af6e9f09888cc4902b354ed228311f8696863b2c76524","family_id":"qf-mod49-slln-fixed-n-deviation","seed":202611182062,"semantic_template":"slln-fixed-n-deviation|{epsilon:[3,5],law:[[-1,[1,2]],[0,[1,2]]],n:3,process_certificate:\"infinite_iid\"}"},"fields":{"title":"The Complete Fixed-n Deviation Event Is Separate from an Almost-Sure Limit","stem":"Use the aligned finite law values=[-1, 0], probabilities=[1/2, 1/2], with declared process certificate=infinite_iid. The token is an explicit infinite IID premise, not a process construction or proof by this finite calculation. Use n=3 and epsilon=3/5>0; the event is |sample_mean−mu|≥epsilon. Give (n, common_mean, epsilon, deviating_joint_word_count, deviation_probability, event_scope, almost_sure_limit_proof_claim, boundary).","answer":{"kind":"expression","value":"(3, -1/2, 3/5, 0, 0, fixed_n_deviation_event, No, not_an_almost_sure_limit_certificate)","explanation":"Count every qualifying joint word and separately sum its exact product probability. Zero-mass words still enter the combinatorial count. A fixed-n deviation event, even when fully enumerated, is not an almost-sure convergence certificate. The exact result, retaining the requested order, is (3, -1/2, 3/5, 0, 0, fixed_n_deviation_event, No, not_an_almost_sure_limit_certificate)."},"skills":["The Complete Fixed-n Deviation Event Is Separate from an Almost-Sure 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-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-0063","batch_id":"batch-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"554869bee2a82ee0986e8539161e2768ba9ba0979db28de9fd6f35a73fe97e18","family_id":"qf-mod49-slln-fixed-n-deviation","seed":202611182063,"semantic_template":"slln-fixed-n-deviation|{epsilon:[1,3],law:[[0,[1,2]],[1,[1,2]]],n:3,process_certificate:\"infinite_iid\"}"},"fields":{"title":"The Complete Fixed-n Deviation Event Is Separate from an Almost-Sure Limit","stem":"Use the aligned finite law values=[0, 1], probabilities=[1/2, 1/2], with declared process certificate=infinite_iid. The token is an explicit infinite IID premise, not a process construction or proof by this finite calculation. Use n=3 and epsilon=1/3>0; the event is |sample_mean−mu|≥epsilon. Give (n, common_mean, epsilon, deviating_joint_word_count, deviation_probability, event_scope, almost_sure_limit_proof_claim, boundary).","answer":{"kind":"expression","value":"(3, 1/2, 1/3, 2, 1/4, fixed_n_deviation_event, No, not_an_almost_sure_limit_certificate)","explanation":"Count every qualifying joint word and separately sum its exact product probability. Zero-mass words still enter the combinatorial count. A fixed-n deviation event, even when fully enumerated, is not an almost-sure convergence certificate. The exact result, retaining the requested order, is (3, 1/2, 1/3, 2, 1/4, fixed_n_deviation_event, No, not_an_almost_sure_limit_certificate)."},"skills":["The Complete Fixed-n Deviation Event Is Separate from an Almost-Sure 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-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-0064","batch_id":"batch-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c90144573e3cade5f27dfdcaeadfe4508e38dc44501b56658881992afc91f06c","family_id":"qf-mod49-slln-fixed-n-deviation","seed":202611182064,"semantic_template":"slln-fixed-n-deviation|{epsilon:[1,2],law:[[1,[1,2]],[2,[1,2]]],n:3,process_certificate:\"infinite_iid\"}"},"fields":{"title":"The Complete Fixed-n Deviation Event Is Separate from an Almost-Sure Limit","stem":"Use the aligned finite law values=[1, 2], probabilities=[1/2, 1/2], with declared process certificate=infinite_iid. The token is an explicit infinite IID premise, not a process construction or proof by this finite calculation. Use n=3 and epsilon=2/4>0; the event is |sample_mean−mu|≥epsilon. Give (n, common_mean, epsilon, deviating_joint_word_count, deviation_probability, event_scope, almost_sure_limit_proof_claim, boundary).","answer":{"kind":"expression","value":"(3, 3/2, 1/2, 2, 1/4, fixed_n_deviation_event, No, not_an_almost_sure_limit_certificate)","explanation":"Count every qualifying joint word and separately sum its exact product probability. Zero-mass words still enter the combinatorial count. A fixed-n deviation event, even when fully enumerated, is not an almost-sure convergence certificate. The exact result, retaining the requested order, is (3, 3/2, 1/2, 2, 1/4, fixed_n_deviation_event, No, not_an_almost_sure_limit_certificate)."},"skills":["The Complete Fixed-n Deviation Event Is Separate from an Almost-Sure 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-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-0065","batch_id":"batch-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"93e4964f1fa2119ddd2663b7ef0e256f121f613b55dea6a701948b4cff8ad96a","family_id":"qf-mod49-slln-fixed-n-deviation","seed":202611182065,"semantic_template":"slln-fixed-n-deviation|{epsilon:[3,5],law:[[2,[1,2]],[3,[1,2]]],n:3,process_certificate:\"infinite_iid\"}"},"fields":{"title":"The Complete Fixed-n Deviation Event Is Separate from an Almost-Sure Limit","stem":"Use the aligned finite law values=[2, 3], probabilities=[1/2, 1/2], with declared process certificate=infinite_iid. The token is an explicit infinite IID premise, not a process construction or proof by this finite calculation. Use n=3 and epsilon=3/5>0; the event is |sample_mean−mu|≥epsilon. Give (n, common_mean, epsilon, deviating_joint_word_count, deviation_probability, event_scope, almost_sure_limit_proof_claim, boundary).","answer":{"kind":"expression","value":"(3, 5/2, 3/5, 0, 0, fixed_n_deviation_event, No, not_an_almost_sure_limit_certificate)","explanation":"Count every qualifying joint word and separately sum its exact product probability. Zero-mass words still enter the combinatorial count. A fixed-n deviation event, even when fully enumerated, is not an almost-sure convergence certificate. The exact result, retaining the requested order, is (3, 5/2, 3/5, 0, 0, fixed_n_deviation_event, No, not_an_almost_sure_limit_certificate)."},"skills":["The Complete Fixed-n Deviation Event Is Separate from an Almost-Sure 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-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-0066","batch_id":"batch-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bcb6ae9e0bb37cb6a1ba72792cdc076f1763608b02ebd9b01246d86843ffad7a","family_id":"qf-mod49-slln-fixed-n-deviation","seed":202611182066,"semantic_template":"slln-fixed-n-deviation|{epsilon:[1,3],law:[[-1,[1,2]],[-2,[1,2]]],n:4,process_certificate:\"infinite_iid\"}"},"fields":{"title":"The Complete Fixed-n Deviation Event Is Separate from an Almost-Sure Limit","stem":"Use the aligned finite law values=[-2, -1], probabilities=[1/2, 1/2], with declared process certificate=infinite_iid. The token is an explicit infinite IID premise, not a process construction or proof by this finite calculation. Use n=4 and epsilon=1/3>0; the event is |sample_mean−mu|≥epsilon. Give (n, common_mean, epsilon, deviating_joint_word_count, deviation_probability, event_scope, almost_sure_limit_proof_claim, boundary).","answer":{"kind":"expression","value":"(4, -3/2, 1/3, 2, 1/8, fixed_n_deviation_event, No, not_an_almost_sure_limit_certificate)","explanation":"Count every qualifying joint word and separately sum its exact product probability. Zero-mass words still enter the combinatorial count. A fixed-n deviation event, even when fully enumerated, is not an almost-sure convergence certificate. The exact result, retaining the requested order, is (4, -3/2, 1/3, 2, 1/8, fixed_n_deviation_event, No, not_an_almost_sure_limit_certificate)."},"skills":["The Complete Fixed-n Deviation Event Is Separate from an Almost-Sure 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-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-0067","batch_id":"batch-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fe81c76b7ede8f4fb36e8891fa3213b3bce75d35b7836f2f5c44ef683ded923e","family_id":"qf-mod49-slln-fixed-n-deviation","seed":202611182067,"semantic_template":"slln-fixed-n-deviation|{epsilon:[1,2],law:[[-1,[1,2]],[0,[1,2]]],n:4,process_certificate:\"infinite_iid\"}"},"fields":{"title":"The Complete Fixed-n Deviation Event Is Separate from an Almost-Sure Limit","stem":"Use the aligned finite law values=[-1, 0], probabilities=[1/2, 1/2], with declared process certificate=infinite_iid. The token is an explicit infinite IID premise, not a process construction or proof by this finite calculation. Use n=4 and epsilon=2/4>0; the event is |sample_mean−mu|≥epsilon. Give (n, common_mean, epsilon, deviating_joint_word_count, deviation_probability, event_scope, almost_sure_limit_proof_claim, boundary).","answer":{"kind":"expression","value":"(4, -1/2, 1/2, 2, 1/8, fixed_n_deviation_event, No, not_an_almost_sure_limit_certificate)","explanation":"Count every qualifying joint word and separately sum its exact product probability. Zero-mass words still enter the combinatorial count. A fixed-n deviation event, even when fully enumerated, is not an almost-sure convergence certificate. The exact result, retaining the requested order, is (4, -1/2, 1/2, 2, 1/8, fixed_n_deviation_event, No, not_an_almost_sure_limit_certificate)."},"skills":["The Complete Fixed-n Deviation Event Is Separate from an Almost-Sure 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-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-0068","batch_id":"batch-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c68d1556ff34335b4776ac65c295b99d2f4073be3a35b62668f41bb67ce903b4","family_id":"qf-mod49-slln-fixed-n-deviation","seed":202611182068,"semantic_template":"slln-fixed-n-deviation|{epsilon:[3,5],law:[[0,[1,2]],[1,[1,2]]],n:4,process_certificate:\"infinite_iid\"}"},"fields":{"title":"The Complete Fixed-n Deviation Event Is Separate from an Almost-Sure Limit","stem":"Use the aligned finite law values=[0, 1], probabilities=[1/2, 1/2], with declared process certificate=infinite_iid. The token is an explicit infinite IID premise, not a process construction or proof by this finite calculation. Use n=4 and epsilon=3/5>0; the event is |sample_mean−mu|≥epsilon. Give (n, common_mean, epsilon, deviating_joint_word_count, deviation_probability, event_scope, almost_sure_limit_proof_claim, boundary).","answer":{"kind":"expression","value":"(4, 1/2, 3/5, 0, 0, fixed_n_deviation_event, No, not_an_almost_sure_limit_certificate)","explanation":"Count every qualifying joint word and separately sum its exact product probability. Zero-mass words still enter the combinatorial count. A fixed-n deviation event, even when fully enumerated, is not an almost-sure convergence certificate. The exact result, retaining the requested order, is (4, 1/2, 3/5, 0, 0, fixed_n_deviation_event, No, not_an_almost_sure_limit_certificate)."},"skills":["The Complete Fixed-n Deviation Event Is Separate from an Almost-Sure 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-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-0069","batch_id":"batch-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f96a957ede28f8fe5a8635bf245f00c71c590d39ab0e7456abd6cfa9da3dd25b","family_id":"qf-mod49-slln-fixed-n-deviation","seed":202611182069,"semantic_template":"slln-fixed-n-deviation|{epsilon:[1,3],law:[[1,[1,2]],[2,[1,2]]],n:4,process_certificate:\"infinite_iid\"}"},"fields":{"title":"The Complete Fixed-n Deviation Event Is Separate from an Almost-Sure Limit","stem":"Use the aligned finite law values=[1, 2], probabilities=[1/2, 1/2], with declared process certificate=infinite_iid. The token is an explicit infinite IID premise, not a process construction or proof by this finite calculation. Use n=4 and epsilon=1/3>0; the event is |sample_mean−mu|≥epsilon. Give (n, common_mean, epsilon, deviating_joint_word_count, deviation_probability, event_scope, almost_sure_limit_proof_claim, boundary).","answer":{"kind":"expression","value":"(4, 3/2, 1/3, 2, 1/8, fixed_n_deviation_event, No, not_an_almost_sure_limit_certificate)","explanation":"Count every qualifying joint word and separately sum its exact product probability. Zero-mass words still enter the combinatorial count. A fixed-n deviation event, even when fully enumerated, is not an almost-sure convergence certificate. The exact result, retaining the requested order, is (4, 3/2, 1/3, 2, 1/8, fixed_n_deviation_event, No, not_an_almost_sure_limit_certificate)."},"skills":["The Complete Fixed-n Deviation Event Is Separate from an Almost-Sure 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-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-0070","batch_id":"batch-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a5ba2147717fe1d762f17741b3e8838bece523e81fb7691f713b3697f1a4c14e","family_id":"qf-mod49-slln-fixed-n-deviation","seed":202611182070,"semantic_template":"slln-fixed-n-deviation|{epsilon:[1,2],law:[[2,[1,2]],[3,[1,2]]],n:4,process_certificate:\"infinite_iid\"}"},"fields":{"title":"The Complete Fixed-n Deviation Event Is Separate from an Almost-Sure Limit","stem":"Use the aligned finite law values=[2, 3], probabilities=[1/2, 1/2], with declared process certificate=infinite_iid. The token is an explicit infinite IID premise, not a process construction or proof by this finite calculation. Use n=4 and epsilon=2/4>0; the event is |sample_mean−mu|≥epsilon. Give (n, common_mean, epsilon, deviating_joint_word_count, deviation_probability, event_scope, almost_sure_limit_proof_claim, boundary).","answer":{"kind":"expression","value":"(4, 5/2, 1/2, 2, 1/8, fixed_n_deviation_event, No, not_an_almost_sure_limit_certificate)","explanation":"Count every qualifying joint word and separately sum its exact product probability. Zero-mass words still enter the combinatorial count. A fixed-n deviation event, even when fully enumerated, is not an almost-sure convergence certificate. The exact result, retaining the requested order, is (4, 5/2, 1/2, 2, 1/8, fixed_n_deviation_event, No, not_an_almost_sure_limit_certificate)."},"skills":["The Complete Fixed-n Deviation Event Is Separate from an Almost-Sure 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-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-0071","batch_id":"batch-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"66d7c621343480cea8146e5116b440c4fffaab75e4e3eb14cfcc5f970e5504f2","family_id":"qf-mod49-slln-fixed-n-deviation","seed":202611182071,"semantic_template":"slln-fixed-n-deviation|{epsilon:[3,5],law:[[-1,[1,2]],[-2,[1,2]]],n:5,process_certificate:\"infinite_iid\"}"},"fields":{"title":"The Complete Fixed-n Deviation Event Is Separate from an Almost-Sure Limit","stem":"Use the aligned finite law values=[-2, -1], probabilities=[1/2, 1/2], with declared process certificate=infinite_iid. The token is an explicit infinite IID premise, not a process construction or proof by this finite calculation. Use n=5 and epsilon=3/5>0; the event is |sample_mean−mu|≥epsilon. Give (n, common_mean, epsilon, deviating_joint_word_count, deviation_probability, event_scope, almost_sure_limit_proof_claim, boundary).","answer":{"kind":"expression","value":"(5, -3/2, 3/5, 0, 0, fixed_n_deviation_event, No, not_an_almost_sure_limit_certificate)","explanation":"Count every qualifying joint word and separately sum its exact product probability. Zero-mass words still enter the combinatorial count. A fixed-n deviation event, even when fully enumerated, is not an almost-sure convergence certificate. The exact result, retaining the requested order, is (5, -3/2, 3/5, 0, 0, fixed_n_deviation_event, No, not_an_almost_sure_limit_certificate)."},"skills":["The Complete Fixed-n Deviation Event Is Separate from an Almost-Sure 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-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-0072","batch_id":"batch-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c3851c6726c0b296fdc025b8682eb2ad772716c4cce3576573105b425ad2215c","family_id":"qf-mod49-slln-fixed-n-deviation","seed":202611182072,"semantic_template":"slln-fixed-n-deviation|{epsilon:[1,3],law:[[-1,[1,2]],[0,[1,2]]],n:5,process_certificate:\"infinite_iid\"}"},"fields":{"title":"The Complete Fixed-n Deviation Event Is Separate from an Almost-Sure Limit","stem":"Use the aligned finite law values=[-1, 0], probabilities=[1/2, 1/2], with declared process certificate=infinite_iid. The token is an explicit infinite IID premise, not a process construction or proof by this finite calculation. Use n=5 and epsilon=1/3>0; the event is |sample_mean−mu|≥epsilon. Give (n, common_mean, epsilon, deviating_joint_word_count, deviation_probability, event_scope, almost_sure_limit_proof_claim, boundary).","answer":{"kind":"expression","value":"(5, -1/2, 1/3, 2, 1/16, fixed_n_deviation_event, No, not_an_almost_sure_limit_certificate)","explanation":"Count every qualifying joint word and separately sum its exact product probability. Zero-mass words still enter the combinatorial count. A fixed-n deviation event, even when fully enumerated, is not an almost-sure convergence certificate. The exact result, retaining the requested order, is (5, -1/2, 1/3, 2, 1/16, fixed_n_deviation_event, No, not_an_almost_sure_limit_certificate)."},"skills":["The Complete Fixed-n Deviation Event Is Separate from an Almost-Sure 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-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-0073","batch_id":"batch-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7194d5356178c1a0dcdcd85d093f788ed75db5ef1ca9ca19304ae60162819b29","family_id":"qf-mod49-slln-fixed-n-deviation","seed":202611182073,"semantic_template":"slln-fixed-n-deviation|{epsilon:[1,2],law:[[0,[1,2]],[1,[1,2]]],n:5,process_certificate:\"infinite_iid\"}"},"fields":{"title":"The Complete Fixed-n Deviation Event Is Separate from an Almost-Sure Limit","stem":"Use the aligned finite law values=[0, 1], probabilities=[1/2, 1/2], with declared process certificate=infinite_iid. The token is an explicit infinite IID premise, not a process construction or proof by this finite calculation. Use n=5 and epsilon=2/4>0; the event is |sample_mean−mu|≥epsilon. Give (n, common_mean, epsilon, deviating_joint_word_count, deviation_probability, event_scope, almost_sure_limit_proof_claim, boundary).","answer":{"kind":"expression","value":"(5, 1/2, 1/2, 2, 1/16, fixed_n_deviation_event, No, not_an_almost_sure_limit_certificate)","explanation":"Count every qualifying joint word and separately sum its exact product probability. Zero-mass words still enter the combinatorial count. A fixed-n deviation event, even when fully enumerated, is not an almost-sure convergence certificate. The exact result, retaining the requested order, is (5, 1/2, 1/2, 2, 1/16, fixed_n_deviation_event, No, not_an_almost_sure_limit_certificate)."},"skills":["The Complete Fixed-n Deviation Event Is Separate from an Almost-Sure 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-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-0074","batch_id":"batch-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"afb1e2fa643996e9cd55f307d751bda338fc6515818f332be909460ece6074bf","family_id":"qf-mod49-slln-fixed-n-deviation","seed":202611182074,"semantic_template":"slln-fixed-n-deviation|{epsilon:[3,5],law:[[1,[1,2]],[2,[1,2]]],n:5,process_certificate:\"infinite_iid\"}"},"fields":{"title":"The Complete Fixed-n Deviation Event Is Separate from an Almost-Sure Limit","stem":"Use the aligned finite law values=[1, 2], probabilities=[1/2, 1/2], with declared process certificate=infinite_iid. The token is an explicit infinite IID premise, not a process construction or proof by this finite calculation. Use n=5 and epsilon=3/5>0; the event is |sample_mean−mu|≥epsilon. Give (n, common_mean, epsilon, deviating_joint_word_count, deviation_probability, event_scope, almost_sure_limit_proof_claim, boundary).","answer":{"kind":"expression","value":"(5, 3/2, 3/5, 0, 0, fixed_n_deviation_event, No, not_an_almost_sure_limit_certificate)","explanation":"Count every qualifying joint word and separately sum its exact product probability. Zero-mass words still enter the combinatorial count. A fixed-n deviation event, even when fully enumerated, is not an almost-sure convergence certificate. The exact result, retaining the requested order, is (5, 3/2, 3/5, 0, 0, fixed_n_deviation_event, No, not_an_almost_sure_limit_certificate)."},"skills":["The Complete Fixed-n Deviation Event Is Separate from an Almost-Sure 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-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-0075","batch_id":"batch-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e1a1c808f14530e27b3ae817a791834b689d8edc79f0006a4b8d5fbfeb00767b","family_id":"qf-mod49-slln-fixed-n-deviation","seed":202611182075,"semantic_template":"slln-fixed-n-deviation|{epsilon:[1,3],law:[[2,[1,2]],[3,[1,2]]],n:5,process_certificate:\"infinite_iid\"}"},"fields":{"title":"The Complete Fixed-n Deviation Event Is Separate from an Almost-Sure Limit","stem":"Use the aligned finite law values=[2, 3], probabilities=[1/2, 1/2], with declared process certificate=infinite_iid. The token is an explicit infinite IID premise, not a process construction or proof by this finite calculation. Use n=5 and epsilon=1/3>0; the event is |sample_mean−mu|≥epsilon. Give (n, common_mean, epsilon, deviating_joint_word_count, deviation_probability, event_scope, almost_sure_limit_proof_claim, boundary).","answer":{"kind":"expression","value":"(5, 5/2, 1/3, 2, 1/16, fixed_n_deviation_event, No, not_an_almost_sure_limit_certificate)","explanation":"Count every qualifying joint word and separately sum its exact product probability. Zero-mass words still enter the combinatorial count. A fixed-n deviation event, even when fully enumerated, is not an almost-sure convergence certificate. The exact result, retaining the requested order, is (5, 5/2, 1/3, 2, 1/16, fixed_n_deviation_event, No, not_an_almost_sure_limit_certificate)."},"skills":["The Complete Fixed-n Deviation Event Is Separate from an Almost-Sure 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-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-0076","batch_id":"batch-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d82890ee53a8284a7bb7cd7b02d7e799c7af841e10afc765e93be7e5ee290f1e","family_id":"qf-mod49-slln-scope-boundary","seed":202611182076,"semantic_template":"slln-scope-boundary|{mode:\"nonfinite_support\",model:{moment_status:\"integrable\",support_kind:\"countably_infinite\"}}"},"fields":{"title":"Outside-Scope Inputs Are Not General Strong-Law Counterexamples","stem":"Use mode=nonfinite_support, complete model={\"support_kind\":\"countably_infinite\",\"moment_status\":\"integrable\"}. Give (mode, support_kind, moment_status, this_finite_valued_theorem_applicable, scope).","answer":{"kind":"expression","value":"(nonfinite_support, countably_infinite, integrable, No, outside_finite_valued_scope_not_a_counterexample_to_general_SLLN)","explanation":"Different marginals or nonfinite support do not meet this finite-valued IID theorem scope; this is not a claim that every general SLLN fails. A finite prefix does not establish an infinite process or limit. A proposed target different from the common expectation is not the theorem target. The exact result, retaining the requested order, is (nonfinite_support, countably_infinite, integrable, No, outside_finite_valued_scope_not_a_counterexample_to_general_SLLN)."},"skills":["Outside-Scope Inputs Are Not General Strong-Law Counterexamples"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-0077","batch_id":"batch-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"31c61491667934545ee043dc0cb7edfe60911e3787b86f542c113f9129cbf249","family_id":"qf-mod49-slln-scope-boundary","seed":202611182077,"semantic_template":"slln-scope-boundary|{mode:\"nonfinite_support\",model:{moment_status:\"unknown\",support_kind:\"countably_infinite\"}}"},"fields":{"title":"Outside-Scope Inputs Are Not General Strong-Law Counterexamples","stem":"Use mode=nonfinite_support, complete model={\"support_kind\":\"countably_infinite\",\"moment_status\":\"unknown\"}. Give (mode, support_kind, moment_status, this_finite_valued_theorem_applicable, scope).","answer":{"kind":"expression","value":"(nonfinite_support, countably_infinite, unknown, No, outside_finite_valued_scope_not_a_counterexample_to_general_SLLN)","explanation":"Different marginals or nonfinite support do not meet this finite-valued IID theorem scope; this is not a claim that every general SLLN fails. A finite prefix does not establish an infinite process or limit. A proposed target different from the common expectation is not the theorem target. The exact result, retaining the requested order, is (nonfinite_support, countably_infinite, unknown, No, outside_finite_valued_scope_not_a_counterexample_to_general_SLLN)."},"skills":["Outside-Scope Inputs Are Not General Strong-Law Counterexamples"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-0078","batch_id":"batch-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e003c384728802c65e006e2060321f97a38306a4a4e35023e142220dc39f8966","family_id":"qf-mod49-slln-scope-boundary","seed":202611182078,"semantic_template":"slln-scope-boundary|{mode:\"nonfinite_support\",model:{moment_status:\"integrable\",support_kind:\"continuous\"}}"},"fields":{"title":"Outside-Scope Inputs Are Not General Strong-Law Counterexamples","stem":"Use mode=nonfinite_support, complete model={\"support_kind\":\"continuous\",\"moment_status\":\"integrable\"}. Give (mode, support_kind, moment_status, this_finite_valued_theorem_applicable, scope).","answer":{"kind":"expression","value":"(nonfinite_support, continuous, integrable, No, outside_finite_valued_scope_not_a_counterexample_to_general_SLLN)","explanation":"Different marginals or nonfinite support do not meet this finite-valued IID theorem scope; this is not a claim that every general SLLN fails. A finite prefix does not establish an infinite process or limit. A proposed target different from the common expectation is not the theorem target. The exact result, retaining the requested order, is (nonfinite_support, continuous, integrable, No, outside_finite_valued_scope_not_a_counterexample_to_general_SLLN)."},"skills":["Outside-Scope Inputs Are Not General Strong-Law Counterexamples"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-0079","batch_id":"batch-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1daa0dd29f5d51089ed3cd7e8b80fbec75492c77bb8fae29ea315fdede207ebe","family_id":"qf-mod49-slln-scope-boundary","seed":202611182079,"semantic_template":"slln-scope-boundary|{mode:\"nonfinite_support\",model:{moment_status:\"unknown\",support_kind:\"continuous\"}}"},"fields":{"title":"Outside-Scope Inputs Are Not General Strong-Law Counterexamples","stem":"Use mode=nonfinite_support, complete model={\"support_kind\":\"continuous\",\"moment_status\":\"unknown\"}. Give (mode, support_kind, moment_status, this_finite_valued_theorem_applicable, scope).","answer":{"kind":"expression","value":"(nonfinite_support, continuous, unknown, No, outside_finite_valued_scope_not_a_counterexample_to_general_SLLN)","explanation":"Different marginals or nonfinite support do not meet this finite-valued IID theorem scope; this is not a claim that every general SLLN fails. A finite prefix does not establish an infinite process or limit. A proposed target different from the common expectation is not the theorem target. The exact result, retaining the requested order, is (nonfinite_support, continuous, unknown, No, outside_finite_valued_scope_not_a_counterexample_to_general_SLLN)."},"skills":["Outside-Scope Inputs Are Not General Strong-Law Counterexamples"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-0080","batch_id":"batch-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2df278b606e28d10e53d29886d505b5a843f4e952af11997f61114533d4afe7d","family_id":"qf-mod49-slln-scope-boundary","seed":202611182080,"semantic_template":"slln-scope-boundary|{mode:\"non_iid\",model:{law_a:[[2,[1,2]],[3,[1,2]]],law_b:[[2,[1,7]],[3,[6,7]]]}}"},"fields":{"title":"Outside-Scope Inputs Are Not General Strong-Law Counterexamples","stem":"Use mode=non_iid, complete model={\"values_a\":[2,3],\"probabilities_a\":[[1,2],[1,2]],\"values_b\":[2,3],\"probabilities_b\":[[1,7],[6,7]]}. Give (mode, identical_marginals, this_finite_valued_IID_theorem_applicable, scope).","answer":{"kind":"expression","value":"(non_iid, No, No, finite_valued_IID_SLLN_not_applicable_not_general_failure)","explanation":"Different marginals or nonfinite support do not meet this finite-valued IID theorem scope; this is not a claim that every general SLLN fails. A finite prefix does not establish an infinite process or limit. A proposed target different from the common expectation is not the theorem target. The exact result, retaining the requested order, is (non_iid, No, No, finite_valued_IID_SLLN_not_applicable_not_general_failure)."},"skills":["Outside-Scope Inputs Are Not General Strong-Law Counterexamples"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-0081","batch_id":"batch-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"76030d1177dcd8e8c2fa2c1554a1b0dd704691abd41e38a5719c5656285a3f87","family_id":"qf-mod49-slln-scope-boundary","seed":202611182081,"semantic_template":"slln-scope-boundary|{mode:\"finite_prefix\",model:{law:[[-1,[2,3]],[-2,[1,3]]],observations:[-2,-1,-1]}}"},"fields":{"title":"Outside-Scope Inputs Are Not General Strong-Law Counterexamples","stem":"Use mode=finite_prefix, complete model={\"values\":[-2,-1],\"probabilities\":[[1,3],[2,3]],\"observations\":[-2,-1,-1]}. Give (mode, observations, common_mean, infinite_process_or_limit_proof_claim, boundary).","answer":{"kind":"expression","value":"(finite_prefix, [-2, -1, -1], -4/3, No, finite_prefix_cannot_establish_infinite_process_or_a.s._limit)","explanation":"Different marginals or nonfinite support do not meet this finite-valued IID theorem scope; this is not a claim that every general SLLN fails. A finite prefix does not establish an infinite process or limit. A proposed target different from the common expectation is not the theorem target. The exact result, retaining the requested order, is (finite_prefix, [-2, -1, -1], -4/3, No, finite_prefix_cannot_establish_infinite_process_or_a.s._limit)."},"skills":["Outside-Scope Inputs Are Not General Strong-Law Counterexamples"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-0082","batch_id":"batch-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fe3a322ebfc355ca0ab7120783e3cfeaa3cb45ff157e76b0db39af921bc99497","family_id":"qf-mod49-slln-scope-boundary","seed":202611182082,"semantic_template":"slln-scope-boundary|{mode:\"wrong_target\",model:{law:[[-1,[1,3]],[0,[2,3]]],target:[8,1]}}"},"fields":{"title":"Outside-Scope Inputs Are Not General Strong-Law Counterexamples","stem":"Use mode=wrong_target, complete model={\"values\":[-1,0],\"probabilities\":[[1,3],[2,3]],\"target\":8}. Give (mode, common_mean, proposed_target, target_equals_common_mean, proposed_target_claim, boundary).","answer":{"kind":"expression","value":"(wrong_target, -1/3, 8, No, No, SLLN_target_is_common_expectation)","explanation":"Different marginals or nonfinite support do not meet this finite-valued IID theorem scope; this is not a claim that every general SLLN fails. A finite prefix does not establish an infinite process or limit. A proposed target different from the common expectation is not the theorem target. The exact result, retaining the requested order, is (wrong_target, -1/3, 8, No, No, SLLN_target_is_common_expectation)."},"skills":["Outside-Scope Inputs Are Not General Strong-Law Counterexamples"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-0083","batch_id":"batch-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c210a586dbe1823d423823c46af4176d280b6ec41636c62b79fdb89bd60930a5","family_id":"qf-mod49-slln-scope-boundary","seed":202611182083,"semantic_template":"slln-scope-boundary|{mode:\"non_iid\",model:{law_a:[[0,[1,3]],[1,[2,3]]],law_b:[[0,[1,7]],[1,[6,7]]]}}"},"fields":{"title":"Outside-Scope Inputs Are Not General Strong-Law Counterexamples","stem":"Use mode=non_iid, complete model={\"values_a\":[0,1],\"probabilities_a\":[[1,3],[2,3]],\"values_b\":[0,1],\"probabilities_b\":[[1,7],[6,7]]}. Give (mode, identical_marginals, this_finite_valued_IID_theorem_applicable, scope).","answer":{"kind":"expression","value":"(non_iid, No, No, finite_valued_IID_SLLN_not_applicable_not_general_failure)","explanation":"Different marginals or nonfinite support do not meet this finite-valued IID theorem scope; this is not a claim that every general SLLN fails. A finite prefix does not establish an infinite process or limit. A proposed target different from the common expectation is not the theorem target. The exact result, retaining the requested order, is (non_iid, No, No, finite_valued_IID_SLLN_not_applicable_not_general_failure)."},"skills":["Outside-Scope Inputs Are Not General Strong-Law Counterexamples"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-0084","batch_id":"batch-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e0526e1f47a09ada09fba524244d6e5bd576a085074f17c164ca35006ff2eae3","family_id":"qf-mod49-slln-scope-boundary","seed":202611182084,"semantic_template":"slln-scope-boundary|{mode:\"finite_prefix\",model:{law:[[1,[1,3]],[2,[2,3]]],observations:[1,2,1]}}"},"fields":{"title":"Outside-Scope Inputs Are Not General Strong-Law Counterexamples","stem":"Use mode=finite_prefix, complete model={\"values\":[1,2],\"probabilities\":[[1,3],[2,3]],\"observations\":[1,2,1]}. Give (mode, observations, common_mean, infinite_process_or_limit_proof_claim, boundary).","answer":{"kind":"expression","value":"(finite_prefix, [1, 2, 1], 5/3, No, finite_prefix_cannot_establish_infinite_process_or_a.s._limit)","explanation":"Different marginals or nonfinite support do not meet this finite-valued IID theorem scope; this is not a claim that every general SLLN fails. A finite prefix does not establish an infinite process or limit. A proposed target different from the common expectation is not the theorem target. The exact result, retaining the requested order, is (finite_prefix, [1, 2, 1], 5/3, No, finite_prefix_cannot_establish_infinite_process_or_a.s._limit)."},"skills":["Outside-Scope Inputs Are Not General Strong-Law Counterexamples"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-0085","batch_id":"batch-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"15dac1e23bef1bddd73ae85f644b988357e0d9b701fe4f1d750a02462e8ec5fd","family_id":"qf-mod49-slln-scope-boundary","seed":202611182085,"semantic_template":"slln-scope-boundary|{mode:\"wrong_target\",model:{law:[[2,[1,3]],[3,[2,3]]],target:[11,1]}}"},"fields":{"title":"Outside-Scope Inputs Are Not General Strong-Law Counterexamples","stem":"Use mode=wrong_target, complete model={\"values\":[2,3],\"probabilities\":[[1,3],[2,3]],\"target\":11}. Give (mode, common_mean, proposed_target, target_equals_common_mean, proposed_target_claim, boundary).","answer":{"kind":"expression","value":"(wrong_target, 8/3, 11, No, No, SLLN_target_is_common_expectation)","explanation":"Different marginals or nonfinite support do not meet this finite-valued IID theorem scope; this is not a claim that every general SLLN fails. A finite prefix does not establish an infinite process or limit. A proposed target different from the common expectation is not the theorem target. The exact result, retaining the requested order, is (wrong_target, 8/3, 11, No, No, SLLN_target_is_common_expectation)."},"skills":["Outside-Scope Inputs Are Not General Strong-Law Counterexamples"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-0086","batch_id":"batch-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f9da71a77b9781a3619dcf23f7e58e83d004fa6c092e1b45d4cca8480b18ccdf","family_id":"qf-mod49-slln-scope-boundary","seed":202611182086,"semantic_template":"slln-scope-boundary|{mode:\"non_iid\",model:{law_a:[[-1,[3,4]],[-2,[1,4]]],law_b:[[-1,[6,7]],[-2,[1,7]]]}}"},"fields":{"title":"Outside-Scope Inputs Are Not General Strong-Law Counterexamples","stem":"Use mode=non_iid, complete model={\"values_a\":[-2,-1],\"probabilities_a\":[[1,4],[3,4]],\"values_b\":[-2,-1],\"probabilities_b\":[[1,7],[6,7]]}. Give (mode, identical_marginals, this_finite_valued_IID_theorem_applicable, scope).","answer":{"kind":"expression","value":"(non_iid, No, No, finite_valued_IID_SLLN_not_applicable_not_general_failure)","explanation":"Different marginals or nonfinite support do not meet this finite-valued IID theorem scope; this is not a claim that every general SLLN fails. A finite prefix does not establish an infinite process or limit. A proposed target different from the common expectation is not the theorem target. The exact result, retaining the requested order, is (non_iid, No, No, finite_valued_IID_SLLN_not_applicable_not_general_failure)."},"skills":["Outside-Scope Inputs Are Not General Strong-Law Counterexamples"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-0087","batch_id":"batch-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3fc5dc6023055620dc7345bf4fd1a44a16651884a2b88409288e95d37bd64874","family_id":"qf-mod49-slln-scope-boundary","seed":202611182087,"semantic_template":"slln-scope-boundary|{mode:\"finite_prefix\",model:{law:[[-1,[1,4]],[0,[3,4]]],observations:[-1,0,0]}}"},"fields":{"title":"Outside-Scope Inputs Are Not General Strong-Law Counterexamples","stem":"Use mode=finite_prefix, complete model={\"values\":[-1,0],\"probabilities\":[[1,4],[3,4]],\"observations\":[-1,0,0]}. Give (mode, observations, common_mean, infinite_process_or_limit_proof_claim, boundary).","answer":{"kind":"expression","value":"(finite_prefix, [-1, 0, 0], -1/4, No, finite_prefix_cannot_establish_infinite_process_or_a.s._limit)","explanation":"Different marginals or nonfinite support do not meet this finite-valued IID theorem scope; this is not a claim that every general SLLN fails. A finite prefix does not establish an infinite process or limit. A proposed target different from the common expectation is not the theorem target. The exact result, retaining the requested order, is (finite_prefix, [-1, 0, 0], -1/4, No, finite_prefix_cannot_establish_infinite_process_or_a.s._limit)."},"skills":["Outside-Scope Inputs Are Not General Strong-Law Counterexamples"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-0088","batch_id":"batch-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c4c9e7640bb2660596b38d000a2308ec74ae8961379d6384d62554b8fa909259","family_id":"qf-mod49-slln-scope-boundary","seed":202611182088,"semantic_template":"slln-scope-boundary|{mode:\"wrong_target\",model:{law:[[0,[1,4]],[1,[3,4]]],target:[14,1]}}"},"fields":{"title":"Outside-Scope Inputs Are Not General Strong-Law Counterexamples","stem":"Use mode=wrong_target, complete model={\"values\":[0,1],\"probabilities\":[[1,4],[3,4]],\"target\":14}. Give (mode, common_mean, proposed_target, target_equals_common_mean, proposed_target_claim, boundary).","answer":{"kind":"expression","value":"(wrong_target, 3/4, 14, No, No, SLLN_target_is_common_expectation)","explanation":"Different marginals or nonfinite support do not meet this finite-valued IID theorem scope; this is not a claim that every general SLLN fails. A finite prefix does not establish an infinite process or limit. A proposed target different from the common expectation is not the theorem target. The exact result, retaining the requested order, is (wrong_target, 3/4, 14, No, No, SLLN_target_is_common_expectation)."},"skills":["Outside-Scope Inputs Are Not General Strong-Law Counterexamples"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-0089","batch_id":"batch-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8cb707b4084e004fef99eb088e94d82853a47e18b9a8a0794ffe1ba9d30ae0b5","family_id":"qf-mod49-slln-scope-boundary","seed":202611182089,"semantic_template":"slln-scope-boundary|{mode:\"non_iid\",model:{law_a:[[1,[1,4]],[2,[3,4]]],law_b:[[1,[1,7]],[2,[6,7]]]}}"},"fields":{"title":"Outside-Scope Inputs Are Not General Strong-Law Counterexamples","stem":"Use mode=non_iid, complete model={\"values_a\":[1,2],\"probabilities_a\":[[1,4],[3,4]],\"values_b\":[1,2],\"probabilities_b\":[[1,7],[6,7]]}. Give (mode, identical_marginals, this_finite_valued_IID_theorem_applicable, scope).","answer":{"kind":"expression","value":"(non_iid, No, No, finite_valued_IID_SLLN_not_applicable_not_general_failure)","explanation":"Different marginals or nonfinite support do not meet this finite-valued IID theorem scope; this is not a claim that every general SLLN fails. A finite prefix does not establish an infinite process or limit. A proposed target different from the common expectation is not the theorem target. The exact result, retaining the requested order, is (non_iid, No, No, finite_valued_IID_SLLN_not_applicable_not_general_failure)."},"skills":["Outside-Scope Inputs Are Not General Strong-Law Counterexamples"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-0090","batch_id":"batch-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c82b6c8d27902659e1177b175bf09df0dd9630590d0699f18539c9bf02b1e705","family_id":"qf-mod49-slln-scope-boundary","seed":202611182090,"semantic_template":"slln-scope-boundary|{mode:\"finite_prefix\",model:{law:[[2,[1,4]],[3,[3,4]]],observations:[2,3,2]}}"},"fields":{"title":"Outside-Scope Inputs Are Not General Strong-Law Counterexamples","stem":"Use mode=finite_prefix, complete model={\"values\":[2,3],\"probabilities\":[[1,4],[3,4]],\"observations\":[2,3,2]}. Give (mode, observations, common_mean, infinite_process_or_limit_proof_claim, boundary).","answer":{"kind":"expression","value":"(finite_prefix, [2, 3, 2], 11/4, No, finite_prefix_cannot_establish_infinite_process_or_a.s._limit)","explanation":"Different marginals or nonfinite support do not meet this finite-valued IID theorem scope; this is not a claim that every general SLLN fails. A finite prefix does not establish an infinite process or limit. A proposed target different from the common expectation is not the theorem target. The exact result, retaining the requested order, is (finite_prefix, [2, 3, 2], 11/4, No, finite_prefix_cannot_establish_infinite_process_or_a.s._limit)."},"skills":["Outside-Scope Inputs Are Not General Strong-Law Counterexamples"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-0091","batch_id":"batch-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dfcc259e2d3f8d361e67af54b3e329b35a1a117963313901ecdb4ea999cc0fc7","family_id":"qf-mod49-slln-scope-boundary","seed":202611182091,"semantic_template":"slln-scope-boundary|{mode:\"wrong_target\",model:{law:[[-1,[3,5]],[-2,[2,5]]],target:[17,1]}}"},"fields":{"title":"Outside-Scope Inputs Are Not General Strong-Law Counterexamples","stem":"Use mode=wrong_target, complete model={\"values\":[-2,-1],\"probabilities\":[[2,5],[3,5]],\"target\":17}. Give (mode, common_mean, proposed_target, target_equals_common_mean, proposed_target_claim, boundary).","answer":{"kind":"expression","value":"(wrong_target, -7/5, 17, No, No, SLLN_target_is_common_expectation)","explanation":"Different marginals or nonfinite support do not meet this finite-valued IID theorem scope; this is not a claim that every general SLLN fails. A finite prefix does not establish an infinite process or limit. A proposed target different from the common expectation is not the theorem target. The exact result, retaining the requested order, is (wrong_target, -7/5, 17, No, No, SLLN_target_is_common_expectation)."},"skills":["Outside-Scope Inputs Are Not General Strong-Law Counterexamples"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-0092","batch_id":"batch-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2d5b58e10572dc8970075ba1b31871a85f38e5e965908c66f5dee28f505002b6","family_id":"qf-mod49-slln-scope-boundary","seed":202611182092,"semantic_template":"slln-scope-boundary|{mode:\"non_iid\",model:{law_a:[[-1,[2,5]],[0,[3,5]]],law_b:[[-1,[1,7]],[0,[6,7]]]}}"},"fields":{"title":"Outside-Scope Inputs Are Not General Strong-Law Counterexamples","stem":"Use mode=non_iid, complete model={\"values_a\":[-1,0],\"probabilities_a\":[[2,5],[3,5]],\"values_b\":[-1,0],\"probabilities_b\":[[1,7],[6,7]]}. Give (mode, identical_marginals, this_finite_valued_IID_theorem_applicable, scope).","answer":{"kind":"expression","value":"(non_iid, No, No, finite_valued_IID_SLLN_not_applicable_not_general_failure)","explanation":"Different marginals or nonfinite support do not meet this finite-valued IID theorem scope; this is not a claim that every general SLLN fails. A finite prefix does not establish an infinite process or limit. A proposed target different from the common expectation is not the theorem target. The exact result, retaining the requested order, is (non_iid, No, No, finite_valued_IID_SLLN_not_applicable_not_general_failure)."},"skills":["Outside-Scope Inputs Are Not General Strong-Law Counterexamples"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-0093","batch_id":"batch-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"def272b63aec531e836f13c0bd8a0a07c218f332021d93aafcac702a2178b461","family_id":"qf-mod49-slln-scope-boundary","seed":202611182093,"semantic_template":"slln-scope-boundary|{mode:\"finite_prefix\",model:{law:[[0,[2,5]],[1,[3,5]]],observations:[0,1,1]}}"},"fields":{"title":"Outside-Scope Inputs Are Not General Strong-Law Counterexamples","stem":"Use mode=finite_prefix, complete model={\"values\":[0,1],\"probabilities\":[[2,5],[3,5]],\"observations\":[0,1,1]}. Give (mode, observations, common_mean, infinite_process_or_limit_proof_claim, boundary).","answer":{"kind":"expression","value":"(finite_prefix, [0, 1, 1], 3/5, No, finite_prefix_cannot_establish_infinite_process_or_a.s._limit)","explanation":"Different marginals or nonfinite support do not meet this finite-valued IID theorem scope; this is not a claim that every general SLLN fails. A finite prefix does not establish an infinite process or limit. A proposed target different from the common expectation is not the theorem target. The exact result, retaining the requested order, is (finite_prefix, [0, 1, 1], 3/5, No, finite_prefix_cannot_establish_infinite_process_or_a.s._limit)."},"skills":["Outside-Scope Inputs Are Not General Strong-Law Counterexamples"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-0094","batch_id":"batch-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d65eb61077ffd10ad0209112ec6dd376cce7873b77e44e906601a3e3378c41ba","family_id":"qf-mod49-slln-scope-boundary","seed":202611182094,"semantic_template":"slln-scope-boundary|{mode:\"wrong_target\",model:{law:[[1,[2,5]],[2,[3,5]]],target:[20,1]}}"},"fields":{"title":"Outside-Scope Inputs Are Not General Strong-Law Counterexamples","stem":"Use mode=wrong_target, complete model={\"values\":[1,2],\"probabilities\":[[2,5],[3,5]],\"target\":20}. Give (mode, common_mean, proposed_target, target_equals_common_mean, proposed_target_claim, boundary).","answer":{"kind":"expression","value":"(wrong_target, 8/5, 20, No, No, SLLN_target_is_common_expectation)","explanation":"Different marginals or nonfinite support do not meet this finite-valued IID theorem scope; this is not a claim that every general SLLN fails. A finite prefix does not establish an infinite process or limit. A proposed target different from the common expectation is not the theorem target. The exact result, retaining the requested order, is (wrong_target, 8/5, 20, No, No, SLLN_target_is_common_expectation)."},"skills":["Outside-Scope Inputs Are Not General Strong-Law Counterexamples"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-0095","batch_id":"batch-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0e745399fde9428b6cd70da46fd6f8d2f0f17a582af0ef76368a1bc4f84333db","family_id":"qf-mod49-slln-scope-boundary","seed":202611182095,"semantic_template":"slln-scope-boundary|{mode:\"non_iid\",model:{law_a:[[2,[2,5]],[3,[3,5]]],law_b:[[2,[1,7]],[3,[6,7]]]}}"},"fields":{"title":"Outside-Scope Inputs Are Not General Strong-Law Counterexamples","stem":"Use mode=non_iid, complete model={\"values_a\":[2,3],\"probabilities_a\":[[2,5],[3,5]],\"values_b\":[2,3],\"probabilities_b\":[[1,7],[6,7]]}. Give (mode, identical_marginals, this_finite_valued_IID_theorem_applicable, scope).","answer":{"kind":"expression","value":"(non_iid, No, No, finite_valued_IID_SLLN_not_applicable_not_general_failure)","explanation":"Different marginals or nonfinite support do not meet this finite-valued IID theorem scope; this is not a claim that every general SLLN fails. A finite prefix does not establish an infinite process or limit. A proposed target different from the common expectation is not the theorem target. The exact result, retaining the requested order, is (non_iid, No, No, finite_valued_IID_SLLN_not_applicable_not_general_failure)."},"skills":["Outside-Scope Inputs Are Not General Strong-Law Counterexamples"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-0096","batch_id":"batch-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ad39894e07e409a442fe8a297843ad45d1d6229e59de05d4a1aad9a6a25c2446","family_id":"qf-mod49-slln-scope-boundary","seed":202611182096,"semantic_template":"slln-scope-boundary|{mode:\"finite_prefix\",model:{law:[[-1,[4,5]],[-2,[1,5]]],observations:[-2,-1,-2]}}"},"fields":{"title":"Outside-Scope Inputs Are Not General Strong-Law Counterexamples","stem":"Use mode=finite_prefix, complete model={\"values\":[-2,-1],\"probabilities\":[[1,5],[4,5]],\"observations\":[-2,-1,-2]}. Give (mode, observations, common_mean, infinite_process_or_limit_proof_claim, boundary).","answer":{"kind":"expression","value":"(finite_prefix, [-2, -1, -2], -6/5, No, finite_prefix_cannot_establish_infinite_process_or_a.s._limit)","explanation":"Different marginals or nonfinite support do not meet this finite-valued IID theorem scope; this is not a claim that every general SLLN fails. A finite prefix does not establish an infinite process or limit. A proposed target different from the common expectation is not the theorem target. The exact result, retaining the requested order, is (finite_prefix, [-2, -1, -2], -6/5, No, finite_prefix_cannot_establish_infinite_process_or_a.s._limit)."},"skills":["Outside-Scope Inputs Are Not General Strong-Law Counterexamples"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-0097","batch_id":"batch-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"503f4042ab2288a3d51abd577c45da34ed3229a2e8d662f117656f54789fdccd","family_id":"qf-mod49-slln-scope-boundary","seed":202611182097,"semantic_template":"slln-scope-boundary|{mode:\"wrong_target\",model:{law:[[-1,[1,5]],[0,[4,5]]],target:[23,1]}}"},"fields":{"title":"Outside-Scope Inputs Are Not General Strong-Law Counterexamples","stem":"Use mode=wrong_target, complete model={\"values\":[-1,0],\"probabilities\":[[1,5],[4,5]],\"target\":23}. Give (mode, common_mean, proposed_target, target_equals_common_mean, proposed_target_claim, boundary).","answer":{"kind":"expression","value":"(wrong_target, -1/5, 23, No, No, SLLN_target_is_common_expectation)","explanation":"Different marginals or nonfinite support do not meet this finite-valued IID theorem scope; this is not a claim that every general SLLN fails. A finite prefix does not establish an infinite process or limit. A proposed target different from the common expectation is not the theorem target. The exact result, retaining the requested order, is (wrong_target, -1/5, 23, No, No, SLLN_target_is_common_expectation)."},"skills":["Outside-Scope Inputs Are Not General Strong-Law Counterexamples"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-0098","batch_id":"batch-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8cee5fcff6f3d13d9032750580828499f2cbb405e2534ac13b9a4ced3cc77f2d","family_id":"qf-mod49-slln-scope-boundary","seed":202611182098,"semantic_template":"slln-scope-boundary|{mode:\"non_iid\",model:{law_a:[[0,[1,5]],[1,[4,5]]],law_b:[[0,[1,7]],[1,[6,7]]]}}"},"fields":{"title":"Outside-Scope Inputs Are Not General Strong-Law Counterexamples","stem":"Use mode=non_iid, complete model={\"values_a\":[0,1],\"probabilities_a\":[[1,5],[4,5]],\"values_b\":[0,1],\"probabilities_b\":[[1,7],[6,7]]}. Give (mode, identical_marginals, this_finite_valued_IID_theorem_applicable, scope).","answer":{"kind":"expression","value":"(non_iid, No, No, finite_valued_IID_SLLN_not_applicable_not_general_failure)","explanation":"Different marginals or nonfinite support do not meet this finite-valued IID theorem scope; this is not a claim that every general SLLN fails. A finite prefix does not establish an infinite process or limit. A proposed target different from the common expectation is not the theorem target. The exact result, retaining the requested order, is (non_iid, No, No, finite_valued_IID_SLLN_not_applicable_not_general_failure)."},"skills":["Outside-Scope Inputs Are Not General Strong-Law Counterexamples"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-0099","batch_id":"batch-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2e4ae5f01996ac3c641db4ceece0853af0d0a34086b23bb6041ffee221b6755c","family_id":"qf-mod49-slln-scope-boundary","seed":202611182099,"semantic_template":"slln-scope-boundary|{mode:\"finite_prefix\",model:{law:[[1,[1,5]],[2,[4,5]]],observations:[1,2,2]}}"},"fields":{"title":"Outside-Scope Inputs Are Not General Strong-Law Counterexamples","stem":"Use mode=finite_prefix, complete model={\"values\":[1,2],\"probabilities\":[[1,5],[4,5]],\"observations\":[1,2,2]}. Give (mode, observations, common_mean, infinite_process_or_limit_proof_claim, boundary).","answer":{"kind":"expression","value":"(finite_prefix, [1, 2, 2], 9/5, No, finite_prefix_cannot_establish_infinite_process_or_a.s._limit)","explanation":"Different marginals or nonfinite support do not meet this finite-valued IID theorem scope; this is not a claim that every general SLLN fails. A finite prefix does not establish an infinite process or limit. A proposed target different from the common expectation is not the theorem target. The exact result, retaining the requested order, is (finite_prefix, [1, 2, 2], 9/5, No, finite_prefix_cannot_establish_infinite_process_or_a.s._limit)."},"skills":["Outside-Scope Inputs Are Not General Strong-Law Counterexamples"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-0100","batch_id":"batch-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bfbe4359bd12511a6cf2354223d6ffbdadcb9e62789975e21652354f71896927","family_id":"qf-mod49-slln-scope-boundary","seed":202611182100,"semantic_template":"slln-scope-boundary|{mode:\"wrong_target\",model:{law:[[2,[1,5]],[3,[4,5]]],target:[26,1]}}"},"fields":{"title":"Outside-Scope Inputs Are Not General Strong-Law Counterexamples","stem":"Use mode=wrong_target, complete model={\"values\":[2,3],\"probabilities\":[[1,5],[4,5]],\"target\":26}. Give (mode, common_mean, proposed_target, target_equals_common_mean, proposed_target_claim, boundary).","answer":{"kind":"expression","value":"(wrong_target, 14/5, 26, No, No, SLLN_target_is_common_expectation)","explanation":"Different marginals or nonfinite support do not meet this finite-valued IID theorem scope; this is not a claim that every general SLLN fails. A finite prefix does not establish an infinite process or limit. A proposed target different from the common expectation is not the theorem target. The exact result, retaining the requested order, is (wrong_target, 14/5, 26, No, No, SLLN_target_is_common_expectation)."},"skills":["Outside-Scope Inputs Are Not General Strong-Law Counterexamples"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}