{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-0001","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a1f2a1a44fe8d3205df49ed76b71ecf5ed9c31092238e93fe3338fe56cea42df","family_id":"qf-mod49-clt-finite-law-moments","seed":202611181001,"semantic_template":"clt-finite-law-moments|{law:[[-1,[1,2]],[-2,[1,2]]],process_certificate:\"infinite_iid\"}"},"fields":{"title":"Finite Marginal Moments Qualify the Declared IID CLT Premises","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, positive_finite_variance, declared_process_certificate, qualification_statement).","answer":{"kind":"expression","value":"([[-2, 1/2], [-1, 1/2]], -3/2, 1/4, Yes, infinite_iid, CLT_hypotheses_qualified)","explanation":"Compute the exact common mean and centered second moment, requiring strictly positive finite variance. This qualifies a declared marginal law; it does not verify independence or an infinite process from samples, nor prove the CLT. The exact result, retaining the requested order, is ([[-2, 1/2], [-1, 1/2]], -3/2, 1/4, Yes, infinite_iid, CLT_hypotheses_qualified)."},"skills":["Finite Marginal Moments Qualify the Declared IID CLT Premises"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r01-central-limit-theorem-finite-variance-iid-0002","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7b29b191d3625bd9c55a35f29e8415e90809aaf6e123674078aebe0466698a55","family_id":"qf-mod49-clt-finite-law-moments","seed":202611181002,"semantic_template":"clt-finite-law-moments|{law:[[-1,[1,2]],[0,[1,2]]],process_certificate:\"infinite_iid\"}"},"fields":{"title":"Finite Marginal Moments Qualify the Declared IID CLT Premises","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, positive_finite_variance, declared_process_certificate, qualification_statement).","answer":{"kind":"expression","value":"([[-1, 1/2], [0, 1/2]], -1/2, 1/4, Yes, infinite_iid, CLT_hypotheses_qualified)","explanation":"Compute the exact common mean and centered second moment, requiring strictly positive finite variance. This qualifies a declared marginal law; it does not verify independence or an infinite process from samples, nor prove the CLT. The exact result, retaining the requested order, is ([[-1, 1/2], [0, 1/2]], -1/2, 1/4, Yes, infinite_iid, CLT_hypotheses_qualified)."},"skills":["Finite Marginal Moments Qualify the Declared IID CLT Premises"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r01-central-limit-theorem-finite-variance-iid-0003","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"efb31c172de7dfcf7648b286646a6ea19fda9ef4d069f7b5bad87d3e4215d40f","family_id":"qf-mod49-clt-finite-law-moments","seed":202611181003,"semantic_template":"clt-finite-law-moments|{law:[[0,[1,2]],[1,[1,2]]],process_certificate:\"infinite_iid\"}"},"fields":{"title":"Finite Marginal Moments Qualify the Declared IID CLT Premises","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, positive_finite_variance, declared_process_certificate, qualification_statement).","answer":{"kind":"expression","value":"([[0, 1/2], [1, 1/2]], 1/2, 1/4, Yes, infinite_iid, CLT_hypotheses_qualified)","explanation":"Compute the exact common mean and centered second moment, requiring strictly positive finite variance. This qualifies a declared marginal law; it does not verify independence or an infinite process from samples, nor prove the CLT. The exact result, retaining the requested order, is ([[0, 1/2], [1, 1/2]], 1/2, 1/4, Yes, infinite_iid, CLT_hypotheses_qualified)."},"skills":["Finite Marginal Moments Qualify the Declared IID CLT Premises"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r01-central-limit-theorem-finite-variance-iid-0004","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2b0a8ced060ccd8c85c5868500e4a6227b5d5eacc568ce5f9455a7746d6980dd","family_id":"qf-mod49-clt-finite-law-moments","seed":202611181004,"semantic_template":"clt-finite-law-moments|{law:[[1,[1,2]],[2,[1,2]]],process_certificate:\"infinite_iid\"}"},"fields":{"title":"Finite Marginal Moments Qualify the Declared IID CLT Premises","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, positive_finite_variance, declared_process_certificate, qualification_statement).","answer":{"kind":"expression","value":"([[1, 1/2], [2, 1/2]], 3/2, 1/4, Yes, infinite_iid, CLT_hypotheses_qualified)","explanation":"Compute the exact common mean and centered second moment, requiring strictly positive finite variance. This qualifies a declared marginal law; it does not verify independence or an infinite process from samples, nor prove the CLT. The exact result, retaining the requested order, is ([[1, 1/2], [2, 1/2]], 3/2, 1/4, Yes, infinite_iid, CLT_hypotheses_qualified)."},"skills":["Finite Marginal Moments Qualify the Declared IID CLT Premises"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r01-central-limit-theorem-finite-variance-iid-0005","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7d24e673b964d04e272ad4826a528ad9e7a0938e26a8280f13037d3701e20cea","family_id":"qf-mod49-clt-finite-law-moments","seed":202611181005,"semantic_template":"clt-finite-law-moments|{law:[[2,[1,2]],[3,[1,2]]],process_certificate:\"infinite_iid\"}"},"fields":{"title":"Finite Marginal Moments Qualify the Declared IID CLT Premises","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, positive_finite_variance, declared_process_certificate, qualification_statement).","answer":{"kind":"expression","value":"([[2, 1/2], [3, 1/2]], 5/2, 1/4, Yes, infinite_iid, CLT_hypotheses_qualified)","explanation":"Compute the exact common mean and centered second moment, requiring strictly positive finite variance. This qualifies a declared marginal law; it does not verify independence or an infinite process from samples, nor prove the CLT. The exact result, retaining the requested order, is ([[2, 1/2], [3, 1/2]], 5/2, 1/4, Yes, infinite_iid, CLT_hypotheses_qualified)."},"skills":["Finite Marginal Moments Qualify the Declared IID CLT Premises"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r01-central-limit-theorem-finite-variance-iid-0006","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4f2e7459102c3519869f4dc9f738fcbc2818a1fc8e7f0b86b41ad6684c0097a9","family_id":"qf-mod49-clt-finite-law-moments","seed":202611181006,"semantic_template":"clt-finite-law-moments|{law:[[-1,[2,3]],[-2,[1,3]]],process_certificate:\"infinite_iid\"}"},"fields":{"title":"Finite Marginal Moments Qualify the Declared IID CLT Premises","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, positive_finite_variance, declared_process_certificate, qualification_statement).","answer":{"kind":"expression","value":"([[-2, 1/3], [-1, 2/3]], -4/3, 2/9, Yes, infinite_iid, CLT_hypotheses_qualified)","explanation":"Compute the exact common mean and centered second moment, requiring strictly positive finite variance. This qualifies a declared marginal law; it does not verify independence or an infinite process from samples, nor prove the CLT. The exact result, retaining the requested order, is ([[-2, 1/3], [-1, 2/3]], -4/3, 2/9, Yes, infinite_iid, CLT_hypotheses_qualified)."},"skills":["Finite Marginal Moments Qualify the Declared IID CLT Premises"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r01-central-limit-theorem-finite-variance-iid-0007","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b6462b106f2996a67e40de1e6dc42b93bb4b8af436baa428dffad152f0790017","family_id":"qf-mod49-clt-finite-law-moments","seed":202611181007,"semantic_template":"clt-finite-law-moments|{law:[[-1,[1,3]],[0,[2,3]]],process_certificate:\"infinite_iid\"}"},"fields":{"title":"Finite Marginal Moments Qualify the Declared IID CLT Premises","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, positive_finite_variance, declared_process_certificate, qualification_statement).","answer":{"kind":"expression","value":"([[-1, 1/3], [0, 2/3]], -1/3, 2/9, Yes, infinite_iid, CLT_hypotheses_qualified)","explanation":"Compute the exact common mean and centered second moment, requiring strictly positive finite variance. This qualifies a declared marginal law; it does not verify independence or an infinite process from samples, nor prove the CLT. The exact result, retaining the requested order, is ([[-1, 1/3], [0, 2/3]], -1/3, 2/9, Yes, infinite_iid, CLT_hypotheses_qualified)."},"skills":["Finite Marginal Moments Qualify the Declared IID CLT Premises"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r01-central-limit-theorem-finite-variance-iid-0008","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0b940de2662afcabdde9cb9707a90126d2049e0f2b96763f3db1783d16463349","family_id":"qf-mod49-clt-finite-law-moments","seed":202611181008,"semantic_template":"clt-finite-law-moments|{law:[[0,[1,3]],[1,[2,3]]],process_certificate:\"infinite_iid\"}"},"fields":{"title":"Finite Marginal Moments Qualify the Declared IID CLT Premises","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, positive_finite_variance, declared_process_certificate, qualification_statement).","answer":{"kind":"expression","value":"([[0, 1/3], [1, 2/3]], 2/3, 2/9, Yes, infinite_iid, CLT_hypotheses_qualified)","explanation":"Compute the exact common mean and centered second moment, requiring strictly positive finite variance. This qualifies a declared marginal law; it does not verify independence or an infinite process from samples, nor prove the CLT. The exact result, retaining the requested order, is ([[0, 1/3], [1, 2/3]], 2/3, 2/9, Yes, infinite_iid, CLT_hypotheses_qualified)."},"skills":["Finite Marginal Moments Qualify the Declared IID CLT Premises"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r01-central-limit-theorem-finite-variance-iid-0009","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dfb8f195affcf9a9a36f67fd907954ddd54900be514c8e620c564c9fb4c83a93","family_id":"qf-mod49-clt-finite-law-moments","seed":202611181009,"semantic_template":"clt-finite-law-moments|{law:[[1,[1,3]],[2,[2,3]]],process_certificate:\"infinite_iid\"}"},"fields":{"title":"Finite Marginal Moments Qualify the Declared IID CLT Premises","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, positive_finite_variance, declared_process_certificate, qualification_statement).","answer":{"kind":"expression","value":"([[1, 1/3], [2, 2/3]], 5/3, 2/9, Yes, infinite_iid, CLT_hypotheses_qualified)","explanation":"Compute the exact common mean and centered second moment, requiring strictly positive finite variance. This qualifies a declared marginal law; it does not verify independence or an infinite process from samples, nor prove the CLT. The exact result, retaining the requested order, is ([[1, 1/3], [2, 2/3]], 5/3, 2/9, Yes, infinite_iid, CLT_hypotheses_qualified)."},"skills":["Finite Marginal Moments Qualify the Declared IID CLT Premises"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r01-central-limit-theorem-finite-variance-iid-0010","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"37345900df18c541137a321a7da71fbb134d5509fd7831ae9ce76795403f0b88","family_id":"qf-mod49-clt-finite-law-moments","seed":202611181010,"semantic_template":"clt-finite-law-moments|{law:[[2,[1,3]],[3,[2,3]]],process_certificate:\"infinite_iid\"}"},"fields":{"title":"Finite Marginal Moments Qualify the Declared IID CLT Premises","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, positive_finite_variance, declared_process_certificate, qualification_statement).","answer":{"kind":"expression","value":"([[2, 1/3], [3, 2/3]], 8/3, 2/9, Yes, infinite_iid, CLT_hypotheses_qualified)","explanation":"Compute the exact common mean and centered second moment, requiring strictly positive finite variance. This qualifies a declared marginal law; it does not verify independence or an infinite process from samples, nor prove the CLT. The exact result, retaining the requested order, is ([[2, 1/3], [3, 2/3]], 8/3, 2/9, Yes, infinite_iid, CLT_hypotheses_qualified)."},"skills":["Finite Marginal Moments Qualify the Declared IID CLT Premises"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r01-central-limit-theorem-finite-variance-iid-0011","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"71668f5102fc28f30bedc3f8121ab683651a2003f4fa26b6797d0c6f83aea364","family_id":"qf-mod49-clt-finite-law-moments","seed":202611181011,"semantic_template":"clt-finite-law-moments|{law:[[-1,[3,4]],[-2,[1,4]]],process_certificate:\"infinite_iid\"}"},"fields":{"title":"Finite Marginal Moments Qualify the Declared IID CLT Premises","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, positive_finite_variance, declared_process_certificate, qualification_statement).","answer":{"kind":"expression","value":"([[-2, 1/4], [-1, 3/4]], -5/4, 3/16, Yes, infinite_iid, CLT_hypotheses_qualified)","explanation":"Compute the exact common mean and centered second moment, requiring strictly positive finite variance. This qualifies a declared marginal law; it does not verify independence or an infinite process from samples, nor prove the CLT. The exact result, retaining the requested order, is ([[-2, 1/4], [-1, 3/4]], -5/4, 3/16, Yes, infinite_iid, CLT_hypotheses_qualified)."},"skills":["Finite Marginal Moments Qualify the Declared IID CLT Premises"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r01-central-limit-theorem-finite-variance-iid-0012","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d69eba3b764a4136a618e0fa49accddda5b532b97d38312d90e3dce15f1fd350","family_id":"qf-mod49-clt-finite-law-moments","seed":202611181012,"semantic_template":"clt-finite-law-moments|{law:[[-1,[1,4]],[0,[3,4]]],process_certificate:\"infinite_iid\"}"},"fields":{"title":"Finite Marginal Moments Qualify the Declared IID CLT Premises","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, positive_finite_variance, declared_process_certificate, qualification_statement).","answer":{"kind":"expression","value":"([[-1, 1/4], [0, 3/4]], -1/4, 3/16, Yes, infinite_iid, CLT_hypotheses_qualified)","explanation":"Compute the exact common mean and centered second moment, requiring strictly positive finite variance. This qualifies a declared marginal law; it does not verify independence or an infinite process from samples, nor prove the CLT. The exact result, retaining the requested order, is ([[-1, 1/4], [0, 3/4]], -1/4, 3/16, Yes, infinite_iid, CLT_hypotheses_qualified)."},"skills":["Finite Marginal Moments Qualify the Declared IID CLT Premises"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r01-central-limit-theorem-finite-variance-iid-0013","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fd50cce4ef64a4030734e883ea9225ccf5bba0750f5be46c39b1f9d4a692ce6c","family_id":"qf-mod49-clt-finite-law-moments","seed":202611181013,"semantic_template":"clt-finite-law-moments|{law:[[0,[1,4]],[1,[3,4]]],process_certificate:\"infinite_iid\"}"},"fields":{"title":"Finite Marginal Moments Qualify the Declared IID CLT Premises","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, positive_finite_variance, declared_process_certificate, qualification_statement).","answer":{"kind":"expression","value":"([[0, 1/4], [1, 3/4]], 3/4, 3/16, Yes, infinite_iid, CLT_hypotheses_qualified)","explanation":"Compute the exact common mean and centered second moment, requiring strictly positive finite variance. This qualifies a declared marginal law; it does not verify independence or an infinite process from samples, nor prove the CLT. The exact result, retaining the requested order, is ([[0, 1/4], [1, 3/4]], 3/4, 3/16, Yes, infinite_iid, CLT_hypotheses_qualified)."},"skills":["Finite Marginal Moments Qualify the Declared IID CLT Premises"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r01-central-limit-theorem-finite-variance-iid-0014","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3e5daff3b26292f84f8c4ba18d474fc87276f3de4ca52ba8a691d81b21da6a94","family_id":"qf-mod49-clt-finite-law-moments","seed":202611181014,"semantic_template":"clt-finite-law-moments|{law:[[1,[1,4]],[2,[3,4]]],process_certificate:\"infinite_iid\"}"},"fields":{"title":"Finite Marginal Moments Qualify the Declared IID CLT Premises","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, positive_finite_variance, declared_process_certificate, qualification_statement).","answer":{"kind":"expression","value":"([[1, 1/4], [2, 3/4]], 7/4, 3/16, Yes, infinite_iid, CLT_hypotheses_qualified)","explanation":"Compute the exact common mean and centered second moment, requiring strictly positive finite variance. This qualifies a declared marginal law; it does not verify independence or an infinite process from samples, nor prove the CLT. The exact result, retaining the requested order, is ([[1, 1/4], [2, 3/4]], 7/4, 3/16, Yes, infinite_iid, CLT_hypotheses_qualified)."},"skills":["Finite Marginal Moments Qualify the Declared IID CLT Premises"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r01-central-limit-theorem-finite-variance-iid-0015","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"148753e4e8d0f3ad69e23fa5bff70b0f0478befab4eb92d00a4748891f71ba89","family_id":"qf-mod49-clt-finite-law-moments","seed":202611181015,"semantic_template":"clt-finite-law-moments|{law:[[2,[1,4]],[3,[3,4]]],process_certificate:\"infinite_iid\"}"},"fields":{"title":"Finite Marginal Moments Qualify the Declared IID CLT Premises","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, positive_finite_variance, declared_process_certificate, qualification_statement).","answer":{"kind":"expression","value":"([[2, 1/4], [3, 3/4]], 11/4, 3/16, Yes, infinite_iid, CLT_hypotheses_qualified)","explanation":"Compute the exact common mean and centered second moment, requiring strictly positive finite variance. This qualifies a declared marginal law; it does not verify independence or an infinite process from samples, nor prove the CLT. The exact result, retaining the requested order, is ([[2, 1/4], [3, 3/4]], 11/4, 3/16, Yes, infinite_iid, CLT_hypotheses_qualified)."},"skills":["Finite Marginal Moments Qualify the Declared IID CLT Premises"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r01-central-limit-theorem-finite-variance-iid-0016","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d74b17cf00db2dfea414de46dcc608b59b335d9c266d60880f98dca12a36b2f1","family_id":"qf-mod49-clt-finite-law-moments","seed":202611181016,"semantic_template":"clt-finite-law-moments|{law:[[-1,[3,5]],[-2,[2,5]]],process_certificate:\"infinite_iid\"}"},"fields":{"title":"Finite Marginal Moments Qualify the Declared IID CLT Premises","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, positive_finite_variance, declared_process_certificate, qualification_statement).","answer":{"kind":"expression","value":"([[-2, 2/5], [-1, 3/5]], -7/5, 6/25, Yes, infinite_iid, CLT_hypotheses_qualified)","explanation":"Compute the exact common mean and centered second moment, requiring strictly positive finite variance. This qualifies a declared marginal law; it does not verify independence or an infinite process from samples, nor prove the CLT. The exact result, retaining the requested order, is ([[-2, 2/5], [-1, 3/5]], -7/5, 6/25, Yes, infinite_iid, CLT_hypotheses_qualified)."},"skills":["Finite Marginal Moments Qualify the Declared IID CLT Premises"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r01-central-limit-theorem-finite-variance-iid-0017","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8867d388b21926110de032e6cd3cefe67f9659876f22d33269c2a1bd77de07bb","family_id":"qf-mod49-clt-finite-law-moments","seed":202611181017,"semantic_template":"clt-finite-law-moments|{law:[[-1,[2,5]],[0,[3,5]]],process_certificate:\"infinite_iid\"}"},"fields":{"title":"Finite Marginal Moments Qualify the Declared IID CLT Premises","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, positive_finite_variance, declared_process_certificate, qualification_statement).","answer":{"kind":"expression","value":"([[-1, 2/5], [0, 3/5]], -2/5, 6/25, Yes, infinite_iid, CLT_hypotheses_qualified)","explanation":"Compute the exact common mean and centered second moment, requiring strictly positive finite variance. This qualifies a declared marginal law; it does not verify independence or an infinite process from samples, nor prove the CLT. The exact result, retaining the requested order, is ([[-1, 2/5], [0, 3/5]], -2/5, 6/25, Yes, infinite_iid, CLT_hypotheses_qualified)."},"skills":["Finite Marginal Moments Qualify the Declared IID CLT Premises"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r01-central-limit-theorem-finite-variance-iid-0018","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a59ab545ccccbbf2c48bfd7b78d2e71f1875176e5720bb1c3778e0566e4510cb","family_id":"qf-mod49-clt-finite-law-moments","seed":202611181018,"semantic_template":"clt-finite-law-moments|{law:[[0,[2,5]],[1,[3,5]]],process_certificate:\"infinite_iid\"}"},"fields":{"title":"Finite Marginal Moments Qualify the Declared IID CLT Premises","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, positive_finite_variance, declared_process_certificate, qualification_statement).","answer":{"kind":"expression","value":"([[0, 2/5], [1, 3/5]], 3/5, 6/25, Yes, infinite_iid, CLT_hypotheses_qualified)","explanation":"Compute the exact common mean and centered second moment, requiring strictly positive finite variance. This qualifies a declared marginal law; it does not verify independence or an infinite process from samples, nor prove the CLT. The exact result, retaining the requested order, is ([[0, 2/5], [1, 3/5]], 3/5, 6/25, Yes, infinite_iid, CLT_hypotheses_qualified)."},"skills":["Finite Marginal Moments Qualify the Declared IID CLT Premises"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r01-central-limit-theorem-finite-variance-iid-0019","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"21e8242e08e430d644fc19cce04d7013f8721de90cd3bbaf5d37d94b918cd26a","family_id":"qf-mod49-clt-finite-law-moments","seed":202611181019,"semantic_template":"clt-finite-law-moments|{law:[[1,[2,5]],[2,[3,5]]],process_certificate:\"infinite_iid\"}"},"fields":{"title":"Finite Marginal Moments Qualify the Declared IID CLT Premises","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, positive_finite_variance, declared_process_certificate, qualification_statement).","answer":{"kind":"expression","value":"([[1, 2/5], [2, 3/5]], 8/5, 6/25, Yes, infinite_iid, CLT_hypotheses_qualified)","explanation":"Compute the exact common mean and centered second moment, requiring strictly positive finite variance. This qualifies a declared marginal law; it does not verify independence or an infinite process from samples, nor prove the CLT. The exact result, retaining the requested order, is ([[1, 2/5], [2, 3/5]], 8/5, 6/25, Yes, infinite_iid, CLT_hypotheses_qualified)."},"skills":["Finite Marginal Moments Qualify the Declared IID CLT Premises"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r01-central-limit-theorem-finite-variance-iid-0020","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"43f211ac498159581e8b14cc6279993dbec0dc946b6780cdad268f6f4ca1c485","family_id":"qf-mod49-clt-finite-law-moments","seed":202611181020,"semantic_template":"clt-finite-law-moments|{law:[[2,[2,5]],[3,[3,5]]],process_certificate:\"infinite_iid\"}"},"fields":{"title":"Finite Marginal Moments Qualify the Declared IID CLT Premises","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, positive_finite_variance, declared_process_certificate, qualification_statement).","answer":{"kind":"expression","value":"([[2, 2/5], [3, 3/5]], 13/5, 6/25, Yes, infinite_iid, CLT_hypotheses_qualified)","explanation":"Compute the exact common mean and centered second moment, requiring strictly positive finite variance. This qualifies a declared marginal law; it does not verify independence or an infinite process from samples, nor prove the CLT. The exact result, retaining the requested order, is ([[2, 2/5], [3, 3/5]], 13/5, 6/25, Yes, infinite_iid, CLT_hypotheses_qualified)."},"skills":["Finite Marginal Moments Qualify the Declared IID CLT Premises"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r01-central-limit-theorem-finite-variance-iid-0021","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a3503ad669ebe923b7f4d92921749cc1c533aaea8f33409a920043366d864082","family_id":"qf-mod49-clt-finite-law-moments","seed":202611181021,"semantic_template":"clt-finite-law-moments|{law:[[-1,[4,5]],[-2,[1,5]]],process_certificate:\"infinite_iid\"}"},"fields":{"title":"Finite Marginal Moments Qualify the Declared IID CLT Premises","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, positive_finite_variance, declared_process_certificate, qualification_statement).","answer":{"kind":"expression","value":"([[-2, 1/5], [-1, 4/5]], -6/5, 4/25, Yes, infinite_iid, CLT_hypotheses_qualified)","explanation":"Compute the exact common mean and centered second moment, requiring strictly positive finite variance. This qualifies a declared marginal law; it does not verify independence or an infinite process from samples, nor prove the CLT. The exact result, retaining the requested order, is ([[-2, 1/5], [-1, 4/5]], -6/5, 4/25, Yes, infinite_iid, CLT_hypotheses_qualified)."},"skills":["Finite Marginal Moments Qualify the Declared IID CLT Premises"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r01-central-limit-theorem-finite-variance-iid-0022","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f4aedceda6a8b28b6d8b2b2432de5d57b10d24177d0a1ab6dceed22e5c11878f","family_id":"qf-mod49-clt-finite-law-moments","seed":202611181022,"semantic_template":"clt-finite-law-moments|{law:[[-1,[1,5]],[0,[4,5]]],process_certificate:\"infinite_iid\"}"},"fields":{"title":"Finite Marginal Moments Qualify the Declared IID CLT Premises","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, positive_finite_variance, declared_process_certificate, qualification_statement).","answer":{"kind":"expression","value":"([[-1, 1/5], [0, 4/5]], -1/5, 4/25, Yes, infinite_iid, CLT_hypotheses_qualified)","explanation":"Compute the exact common mean and centered second moment, requiring strictly positive finite variance. This qualifies a declared marginal law; it does not verify independence or an infinite process from samples, nor prove the CLT. The exact result, retaining the requested order, is ([[-1, 1/5], [0, 4/5]], -1/5, 4/25, Yes, infinite_iid, CLT_hypotheses_qualified)."},"skills":["Finite Marginal Moments Qualify the Declared IID CLT Premises"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r01-central-limit-theorem-finite-variance-iid-0023","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"49fe0563f4c650ce7b2226ecb566507a590c2b0764ffeca9aa554f481f26918a","family_id":"qf-mod49-clt-finite-law-moments","seed":202611181023,"semantic_template":"clt-finite-law-moments|{law:[[0,[1,5]],[1,[4,5]]],process_certificate:\"infinite_iid\"}"},"fields":{"title":"Finite Marginal Moments Qualify the Declared IID CLT Premises","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, positive_finite_variance, declared_process_certificate, qualification_statement).","answer":{"kind":"expression","value":"([[0, 1/5], [1, 4/5]], 4/5, 4/25, Yes, infinite_iid, CLT_hypotheses_qualified)","explanation":"Compute the exact common mean and centered second moment, requiring strictly positive finite variance. This qualifies a declared marginal law; it does not verify independence or an infinite process from samples, nor prove the CLT. The exact result, retaining the requested order, is ([[0, 1/5], [1, 4/5]], 4/5, 4/25, Yes, infinite_iid, CLT_hypotheses_qualified)."},"skills":["Finite Marginal Moments Qualify the Declared IID CLT Premises"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r01-central-limit-theorem-finite-variance-iid-0024","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"73f5e8d35ed7bcecddbf2fb9cc266cc9cc7ed092cd89339fb675d9c34ddbaf0f","family_id":"qf-mod49-clt-finite-law-moments","seed":202611181024,"semantic_template":"clt-finite-law-moments|{law:[[1,[1,5]],[2,[4,5]]],process_certificate:\"infinite_iid\"}"},"fields":{"title":"Finite Marginal Moments Qualify the Declared IID CLT Premises","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, positive_finite_variance, declared_process_certificate, qualification_statement).","answer":{"kind":"expression","value":"([[1, 1/5], [2, 4/5]], 9/5, 4/25, Yes, infinite_iid, CLT_hypotheses_qualified)","explanation":"Compute the exact common mean and centered second moment, requiring strictly positive finite variance. This qualifies a declared marginal law; it does not verify independence or an infinite process from samples, nor prove the CLT. The exact result, retaining the requested order, is ([[1, 1/5], [2, 4/5]], 9/5, 4/25, Yes, infinite_iid, CLT_hypotheses_qualified)."},"skills":["Finite Marginal Moments Qualify the Declared IID CLT Premises"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r01-central-limit-theorem-finite-variance-iid-0025","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ffe903c2c749960c4f877dae44cc803ec04ff760ac0d71c7f23bd802846ee5bf","family_id":"qf-mod49-clt-finite-law-moments","seed":202611181025,"semantic_template":"clt-finite-law-moments|{law:[[2,[1,5]],[3,[4,5]]],process_certificate:\"infinite_iid\"}"},"fields":{"title":"Finite Marginal Moments Qualify the Declared IID CLT Premises","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, positive_finite_variance, declared_process_certificate, qualification_statement).","answer":{"kind":"expression","value":"([[2, 1/5], [3, 4/5]], 14/5, 4/25, Yes, infinite_iid, CLT_hypotheses_qualified)","explanation":"Compute the exact common mean and centered second moment, requiring strictly positive finite variance. This qualifies a declared marginal law; it does not verify independence or an infinite process from samples, nor prove the CLT. The exact result, retaining the requested order, is ([[2, 1/5], [3, 4/5]], 14/5, 4/25, Yes, infinite_iid, CLT_hypotheses_qualified)."},"skills":["Finite Marginal Moments Qualify the Declared IID CLT Premises"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r01-central-limit-theorem-finite-variance-iid-0026","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"29bb63275548e0419ff033d4d3f97a8a897dbfc5704d357088ad2831bd4444a4","family_id":"qf-mod49-clt-standardized-sum-moments","seed":202611181026,"semantic_template":"clt-standardized-sum-moments|{law:[[-1,[1,2]],[-2,[1,2]]],n:1,process_certificate:\"infinite_iid\"}"},"fields":{"title":"Complete Fixed-n Sum Moments Do Not Prove a Distributional 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 sample size n=1 and S_n as the sum of all n coordinates. Give (n, sum_distribution, total_probability, sum_mean, sum_variance, n_times_mean, n_times_variance, standardized_mean, standardized_variance, standardization_description, finite_scope).","answer":{"kind":"expression","value":"(1, [[-2, 1/2], [-1, 1/2]], 1, -3/2, 1/4, -3/2, 1/4, 0, 1, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), complete_joint_law_moments_do_not_prove_distributional_convergence)","explanation":"Aggregate the entire finite IID product law by sum. Retain total mass, E[S_n]=n*mu and Var(S_n)=n*sigma^2. The centered normalization Z_n=(S_n−n*mu)/(sigma*sqrt(n)) has mean0 and variance1. Those moments do not make a finite-n law normal or prove convergence in distribution. The exact result, retaining the requested order, is (1, [[-2, 1/2], [-1, 1/2]], 1, -3/2, 1/4, -3/2, 1/4, 0, 1, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), complete_joint_law_moments_do_not_prove_distributional_convergence)."},"skills":["Complete Fixed-n Sum Moments Do Not Prove a Distributional 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-r01-central-limit-theorem-finite-variance-iid-0027","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3a763d4f0de542d789f8d15cbd68ef9a4f2615899723ecede6a3bb3ff7a00fdc","family_id":"qf-mod49-clt-standardized-sum-moments","seed":202611181027,"semantic_template":"clt-standardized-sum-moments|{law:[[-1,[1,2]],[0,[1,2]]],n:1,process_certificate:\"infinite_iid\"}"},"fields":{"title":"Complete Fixed-n Sum Moments Do Not Prove a Distributional 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 sample size n=1 and S_n as the sum of all n coordinates. Give (n, sum_distribution, total_probability, sum_mean, sum_variance, n_times_mean, n_times_variance, standardized_mean, standardized_variance, standardization_description, finite_scope).","answer":{"kind":"expression","value":"(1, [[-1, 1/2], [0, 1/2]], 1, -1/2, 1/4, -1/2, 1/4, 0, 1, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), complete_joint_law_moments_do_not_prove_distributional_convergence)","explanation":"Aggregate the entire finite IID product law by sum. Retain total mass, E[S_n]=n*mu and Var(S_n)=n*sigma^2. The centered normalization Z_n=(S_n−n*mu)/(sigma*sqrt(n)) has mean0 and variance1. Those moments do not make a finite-n law normal or prove convergence in distribution. The exact result, retaining the requested order, is (1, [[-1, 1/2], [0, 1/2]], 1, -1/2, 1/4, -1/2, 1/4, 0, 1, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), complete_joint_law_moments_do_not_prove_distributional_convergence)."},"skills":["Complete Fixed-n Sum Moments Do Not Prove a Distributional 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-r01-central-limit-theorem-finite-variance-iid-0028","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"485c5fa74f752dc7afc1f035deb485ab7d1d5b60019e004c7dd9347c3bbc7035","family_id":"qf-mod49-clt-standardized-sum-moments","seed":202611181028,"semantic_template":"clt-standardized-sum-moments|{law:[[0,[1,2]],[1,[1,2]]],n:1,process_certificate:\"infinite_iid\"}"},"fields":{"title":"Complete Fixed-n Sum Moments Do Not Prove a Distributional 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 sample size n=1 and S_n as the sum of all n coordinates. Give (n, sum_distribution, total_probability, sum_mean, sum_variance, n_times_mean, n_times_variance, standardized_mean, standardized_variance, standardization_description, finite_scope).","answer":{"kind":"expression","value":"(1, [[0, 1/2], [1, 1/2]], 1, 1/2, 1/4, 1/2, 1/4, 0, 1, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), complete_joint_law_moments_do_not_prove_distributional_convergence)","explanation":"Aggregate the entire finite IID product law by sum. Retain total mass, E[S_n]=n*mu and Var(S_n)=n*sigma^2. The centered normalization Z_n=(S_n−n*mu)/(sigma*sqrt(n)) has mean0 and variance1. Those moments do not make a finite-n law normal or prove convergence in distribution. The exact result, retaining the requested order, is (1, [[0, 1/2], [1, 1/2]], 1, 1/2, 1/4, 1/2, 1/4, 0, 1, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), complete_joint_law_moments_do_not_prove_distributional_convergence)."},"skills":["Complete Fixed-n Sum Moments Do Not Prove a Distributional 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-r01-central-limit-theorem-finite-variance-iid-0029","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a1fafe2383b262da2bd8429bd60306159ca363af9180a5f642e1b16cf2fdb12f","family_id":"qf-mod49-clt-standardized-sum-moments","seed":202611181029,"semantic_template":"clt-standardized-sum-moments|{law:[[1,[1,2]],[2,[1,2]]],n:1,process_certificate:\"infinite_iid\"}"},"fields":{"title":"Complete Fixed-n Sum Moments Do Not Prove a Distributional 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 sample size n=1 and S_n as the sum of all n coordinates. Give (n, sum_distribution, total_probability, sum_mean, sum_variance, n_times_mean, n_times_variance, standardized_mean, standardized_variance, standardization_description, finite_scope).","answer":{"kind":"expression","value":"(1, [[1, 1/2], [2, 1/2]], 1, 3/2, 1/4, 3/2, 1/4, 0, 1, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), complete_joint_law_moments_do_not_prove_distributional_convergence)","explanation":"Aggregate the entire finite IID product law by sum. Retain total mass, E[S_n]=n*mu and Var(S_n)=n*sigma^2. The centered normalization Z_n=(S_n−n*mu)/(sigma*sqrt(n)) has mean0 and variance1. Those moments do not make a finite-n law normal or prove convergence in distribution. The exact result, retaining the requested order, is (1, [[1, 1/2], [2, 1/2]], 1, 3/2, 1/4, 3/2, 1/4, 0, 1, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), complete_joint_law_moments_do_not_prove_distributional_convergence)."},"skills":["Complete Fixed-n Sum Moments Do Not Prove a Distributional 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-r01-central-limit-theorem-finite-variance-iid-0030","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9f59fb288998b2a13ffae082a27f30877857bfde65abf06f461ccac69ea1579a","family_id":"qf-mod49-clt-standardized-sum-moments","seed":202611181030,"semantic_template":"clt-standardized-sum-moments|{law:[[2,[1,2]],[3,[1,2]]],n:1,process_certificate:\"infinite_iid\"}"},"fields":{"title":"Complete Fixed-n Sum Moments Do Not Prove a Distributional 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 sample size n=1 and S_n as the sum of all n coordinates. Give (n, sum_distribution, total_probability, sum_mean, sum_variance, n_times_mean, n_times_variance, standardized_mean, standardized_variance, standardization_description, finite_scope).","answer":{"kind":"expression","value":"(1, [[2, 1/2], [3, 1/2]], 1, 5/2, 1/4, 5/2, 1/4, 0, 1, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), complete_joint_law_moments_do_not_prove_distributional_convergence)","explanation":"Aggregate the entire finite IID product law by sum. Retain total mass, E[S_n]=n*mu and Var(S_n)=n*sigma^2. The centered normalization Z_n=(S_n−n*mu)/(sigma*sqrt(n)) has mean0 and variance1. Those moments do not make a finite-n law normal or prove convergence in distribution. The exact result, retaining the requested order, is (1, [[2, 1/2], [3, 1/2]], 1, 5/2, 1/4, 5/2, 1/4, 0, 1, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), complete_joint_law_moments_do_not_prove_distributional_convergence)."},"skills":["Complete Fixed-n Sum Moments Do Not Prove a Distributional 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-r01-central-limit-theorem-finite-variance-iid-0031","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6335b2041eb095f2ec142e0a6a8df2e755f6f3b9a11e82da5ce5bba0bef07a8f","family_id":"qf-mod49-clt-standardized-sum-moments","seed":202611181031,"semantic_template":"clt-standardized-sum-moments|{law:[[-1,[1,2]],[-2,[1,2]]],n:2,process_certificate:\"infinite_iid\"}"},"fields":{"title":"Complete Fixed-n Sum Moments Do Not Prove a Distributional 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 sample size n=2 and S_n as the sum of all n coordinates. Give (n, sum_distribution, total_probability, sum_mean, sum_variance, n_times_mean, n_times_variance, standardized_mean, standardized_variance, standardization_description, finite_scope).","answer":{"kind":"expression","value":"(2, [[-4, 1/4], [-3, 1/2], [-2, 1/4]], 1, -3, 1/2, -3, 1/2, 0, 1, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), complete_joint_law_moments_do_not_prove_distributional_convergence)","explanation":"Aggregate the entire finite IID product law by sum. Retain total mass, E[S_n]=n*mu and Var(S_n)=n*sigma^2. The centered normalization Z_n=(S_n−n*mu)/(sigma*sqrt(n)) has mean0 and variance1. Those moments do not make a finite-n law normal or prove convergence in distribution. The exact result, retaining the requested order, is (2, [[-4, 1/4], [-3, 1/2], [-2, 1/4]], 1, -3, 1/2, -3, 1/2, 0, 1, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), complete_joint_law_moments_do_not_prove_distributional_convergence)."},"skills":["Complete Fixed-n Sum Moments Do Not Prove a Distributional 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-r01-central-limit-theorem-finite-variance-iid-0032","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6d37232cf7c1429e8777a5ba0d38de6e7de513abcb3e772a6186ca0f0445c3ee","family_id":"qf-mod49-clt-standardized-sum-moments","seed":202611181032,"semantic_template":"clt-standardized-sum-moments|{law:[[-1,[1,2]],[0,[1,2]]],n:2,process_certificate:\"infinite_iid\"}"},"fields":{"title":"Complete Fixed-n Sum Moments Do Not Prove a Distributional 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 sample size n=2 and S_n as the sum of all n coordinates. Give (n, sum_distribution, total_probability, sum_mean, sum_variance, n_times_mean, n_times_variance, standardized_mean, standardized_variance, standardization_description, finite_scope).","answer":{"kind":"expression","value":"(2, [[-2, 1/4], [-1, 1/2], [0, 1/4]], 1, -1, 1/2, -1, 1/2, 0, 1, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), complete_joint_law_moments_do_not_prove_distributional_convergence)","explanation":"Aggregate the entire finite IID product law by sum. Retain total mass, E[S_n]=n*mu and Var(S_n)=n*sigma^2. The centered normalization Z_n=(S_n−n*mu)/(sigma*sqrt(n)) has mean0 and variance1. Those moments do not make a finite-n law normal or prove convergence in distribution. The exact result, retaining the requested order, is (2, [[-2, 1/4], [-1, 1/2], [0, 1/4]], 1, -1, 1/2, -1, 1/2, 0, 1, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), complete_joint_law_moments_do_not_prove_distributional_convergence)."},"skills":["Complete Fixed-n Sum Moments Do Not Prove a Distributional 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-r01-central-limit-theorem-finite-variance-iid-0033","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8adecfe3df466f59ab307b7c7bc764b89851d605aa8537a9069b09844c7c3919","family_id":"qf-mod49-clt-standardized-sum-moments","seed":202611181033,"semantic_template":"clt-standardized-sum-moments|{law:[[0,[1,2]],[1,[1,2]]],n:2,process_certificate:\"infinite_iid\"}"},"fields":{"title":"Complete Fixed-n Sum Moments Do Not Prove a Distributional 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 sample size n=2 and S_n as the sum of all n coordinates. Give (n, sum_distribution, total_probability, sum_mean, sum_variance, n_times_mean, n_times_variance, standardized_mean, standardized_variance, standardization_description, finite_scope).","answer":{"kind":"expression","value":"(2, [[0, 1/4], [1, 1/2], [2, 1/4]], 1, 1, 1/2, 1, 1/2, 0, 1, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), complete_joint_law_moments_do_not_prove_distributional_convergence)","explanation":"Aggregate the entire finite IID product law by sum. Retain total mass, E[S_n]=n*mu and Var(S_n)=n*sigma^2. The centered normalization Z_n=(S_n−n*mu)/(sigma*sqrt(n)) has mean0 and variance1. Those moments do not make a finite-n law normal or prove convergence in distribution. The exact result, retaining the requested order, is (2, [[0, 1/4], [1, 1/2], [2, 1/4]], 1, 1, 1/2, 1, 1/2, 0, 1, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), complete_joint_law_moments_do_not_prove_distributional_convergence)."},"skills":["Complete Fixed-n Sum Moments Do Not Prove a Distributional 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-r01-central-limit-theorem-finite-variance-iid-0034","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3e5d253825bcacfe06c230d4596d995c458b8754a189c3f330e8de767c415180","family_id":"qf-mod49-clt-standardized-sum-moments","seed":202611181034,"semantic_template":"clt-standardized-sum-moments|{law:[[1,[1,2]],[2,[1,2]]],n:2,process_certificate:\"infinite_iid\"}"},"fields":{"title":"Complete Fixed-n Sum Moments Do Not Prove a Distributional 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 sample size n=2 and S_n as the sum of all n coordinates. Give (n, sum_distribution, total_probability, sum_mean, sum_variance, n_times_mean, n_times_variance, standardized_mean, standardized_variance, standardization_description, finite_scope).","answer":{"kind":"expression","value":"(2, [[2, 1/4], [3, 1/2], [4, 1/4]], 1, 3, 1/2, 3, 1/2, 0, 1, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), complete_joint_law_moments_do_not_prove_distributional_convergence)","explanation":"Aggregate the entire finite IID product law by sum. Retain total mass, E[S_n]=n*mu and Var(S_n)=n*sigma^2. The centered normalization Z_n=(S_n−n*mu)/(sigma*sqrt(n)) has mean0 and variance1. Those moments do not make a finite-n law normal or prove convergence in distribution. The exact result, retaining the requested order, is (2, [[2, 1/4], [3, 1/2], [4, 1/4]], 1, 3, 1/2, 3, 1/2, 0, 1, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), complete_joint_law_moments_do_not_prove_distributional_convergence)."},"skills":["Complete Fixed-n Sum Moments Do Not Prove a Distributional 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-r01-central-limit-theorem-finite-variance-iid-0035","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f6b475814ef80590a36968ccc1b10cac4dd1c8a61c71437c77d84f1d1eddc36f","family_id":"qf-mod49-clt-standardized-sum-moments","seed":202611181035,"semantic_template":"clt-standardized-sum-moments|{law:[[2,[1,2]],[3,[1,2]]],n:2,process_certificate:\"infinite_iid\"}"},"fields":{"title":"Complete Fixed-n Sum Moments Do Not Prove a Distributional 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 sample size n=2 and S_n as the sum of all n coordinates. Give (n, sum_distribution, total_probability, sum_mean, sum_variance, n_times_mean, n_times_variance, standardized_mean, standardized_variance, standardization_description, finite_scope).","answer":{"kind":"expression","value":"(2, [[4, 1/4], [5, 1/2], [6, 1/4]], 1, 5, 1/2, 5, 1/2, 0, 1, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), complete_joint_law_moments_do_not_prove_distributional_convergence)","explanation":"Aggregate the entire finite IID product law by sum. Retain total mass, E[S_n]=n*mu and Var(S_n)=n*sigma^2. The centered normalization Z_n=(S_n−n*mu)/(sigma*sqrt(n)) has mean0 and variance1. Those moments do not make a finite-n law normal or prove convergence in distribution. The exact result, retaining the requested order, is (2, [[4, 1/4], [5, 1/2], [6, 1/4]], 1, 5, 1/2, 5, 1/2, 0, 1, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), complete_joint_law_moments_do_not_prove_distributional_convergence)."},"skills":["Complete Fixed-n Sum Moments Do Not Prove a Distributional 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-r01-central-limit-theorem-finite-variance-iid-0036","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a9244f0a5f3dec30c3a20aa0187d4bcd397409c25b826e90329e933e32cf204e","family_id":"qf-mod49-clt-standardized-sum-moments","seed":202611181036,"semantic_template":"clt-standardized-sum-moments|{law:[[-1,[1,2]],[-2,[1,2]]],n:3,process_certificate:\"infinite_iid\"}"},"fields":{"title":"Complete Fixed-n Sum Moments Do Not Prove a Distributional 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 sample size n=3 and S_n as the sum of all n coordinates. Give (n, sum_distribution, total_probability, sum_mean, sum_variance, n_times_mean, n_times_variance, standardized_mean, standardized_variance, standardization_description, finite_scope).","answer":{"kind":"expression","value":"(3, [[-6, 1/8], [-5, 3/8], [-4, 3/8], [-3, 1/8]], 1, -9/2, 3/4, -9/2, 3/4, 0, 1, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), complete_joint_law_moments_do_not_prove_distributional_convergence)","explanation":"Aggregate the entire finite IID product law by sum. Retain total mass, E[S_n]=n*mu and Var(S_n)=n*sigma^2. The centered normalization Z_n=(S_n−n*mu)/(sigma*sqrt(n)) has mean0 and variance1. Those moments do not make a finite-n law normal or prove convergence in distribution. The exact result, retaining the requested order, is (3, [[-6, 1/8], [-5, 3/8], [-4, 3/8], [-3, 1/8]], 1, -9/2, 3/4, -9/2, 3/4, 0, 1, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), complete_joint_law_moments_do_not_prove_distributional_convergence)."},"skills":["Complete Fixed-n Sum Moments Do Not Prove a Distributional 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-r01-central-limit-theorem-finite-variance-iid-0037","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d18eb911abc22e1e91a4dc5fec61a6efac31015641bac1fe8aea9c530e142233","family_id":"qf-mod49-clt-standardized-sum-moments","seed":202611181037,"semantic_template":"clt-standardized-sum-moments|{law:[[-1,[1,2]],[0,[1,2]]],n:3,process_certificate:\"infinite_iid\"}"},"fields":{"title":"Complete Fixed-n Sum Moments Do Not Prove a Distributional 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 sample size n=3 and S_n as the sum of all n coordinates. Give (n, sum_distribution, total_probability, sum_mean, sum_variance, n_times_mean, n_times_variance, standardized_mean, standardized_variance, standardization_description, finite_scope).","answer":{"kind":"expression","value":"(3, [[-3, 1/8], [-2, 3/8], [-1, 3/8], [0, 1/8]], 1, -3/2, 3/4, -3/2, 3/4, 0, 1, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), complete_joint_law_moments_do_not_prove_distributional_convergence)","explanation":"Aggregate the entire finite IID product law by sum. Retain total mass, E[S_n]=n*mu and Var(S_n)=n*sigma^2. The centered normalization Z_n=(S_n−n*mu)/(sigma*sqrt(n)) has mean0 and variance1. Those moments do not make a finite-n law normal or prove convergence in distribution. The exact result, retaining the requested order, is (3, [[-3, 1/8], [-2, 3/8], [-1, 3/8], [0, 1/8]], 1, -3/2, 3/4, -3/2, 3/4, 0, 1, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), complete_joint_law_moments_do_not_prove_distributional_convergence)."},"skills":["Complete Fixed-n Sum Moments Do Not Prove a Distributional 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-r01-central-limit-theorem-finite-variance-iid-0038","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2163d0bce0b2019e84dcd8dd8db1e4d56d8cbda64cf43941c6442b1235cc4467","family_id":"qf-mod49-clt-standardized-sum-moments","seed":202611181038,"semantic_template":"clt-standardized-sum-moments|{law:[[0,[1,2]],[1,[1,2]]],n:3,process_certificate:\"infinite_iid\"}"},"fields":{"title":"Complete Fixed-n Sum Moments Do Not Prove a Distributional 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 sample size n=3 and S_n as the sum of all n coordinates. Give (n, sum_distribution, total_probability, sum_mean, sum_variance, n_times_mean, n_times_variance, standardized_mean, standardized_variance, standardization_description, finite_scope).","answer":{"kind":"expression","value":"(3, [[0, 1/8], [1, 3/8], [2, 3/8], [3, 1/8]], 1, 3/2, 3/4, 3/2, 3/4, 0, 1, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), complete_joint_law_moments_do_not_prove_distributional_convergence)","explanation":"Aggregate the entire finite IID product law by sum. Retain total mass, E[S_n]=n*mu and Var(S_n)=n*sigma^2. The centered normalization Z_n=(S_n−n*mu)/(sigma*sqrt(n)) has mean0 and variance1. Those moments do not make a finite-n law normal or prove convergence in distribution. The exact result, retaining the requested order, is (3, [[0, 1/8], [1, 3/8], [2, 3/8], [3, 1/8]], 1, 3/2, 3/4, 3/2, 3/4, 0, 1, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), complete_joint_law_moments_do_not_prove_distributional_convergence)."},"skills":["Complete Fixed-n Sum Moments Do Not Prove a Distributional 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-r01-central-limit-theorem-finite-variance-iid-0039","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"632fab1a091484f6ca8ad6ca71164bffd8224e6b12ccbbd5ba8619652ec32225","family_id":"qf-mod49-clt-standardized-sum-moments","seed":202611181039,"semantic_template":"clt-standardized-sum-moments|{law:[[1,[1,2]],[2,[1,2]]],n:3,process_certificate:\"infinite_iid\"}"},"fields":{"title":"Complete Fixed-n Sum Moments Do Not Prove a Distributional 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 sample size n=3 and S_n as the sum of all n coordinates. Give (n, sum_distribution, total_probability, sum_mean, sum_variance, n_times_mean, n_times_variance, standardized_mean, standardized_variance, standardization_description, finite_scope).","answer":{"kind":"expression","value":"(3, [[3, 1/8], [4, 3/8], [5, 3/8], [6, 1/8]], 1, 9/2, 3/4, 9/2, 3/4, 0, 1, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), complete_joint_law_moments_do_not_prove_distributional_convergence)","explanation":"Aggregate the entire finite IID product law by sum. Retain total mass, E[S_n]=n*mu and Var(S_n)=n*sigma^2. The centered normalization Z_n=(S_n−n*mu)/(sigma*sqrt(n)) has mean0 and variance1. Those moments do not make a finite-n law normal or prove convergence in distribution. The exact result, retaining the requested order, is (3, [[3, 1/8], [4, 3/8], [5, 3/8], [6, 1/8]], 1, 9/2, 3/4, 9/2, 3/4, 0, 1, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), complete_joint_law_moments_do_not_prove_distributional_convergence)."},"skills":["Complete Fixed-n Sum Moments Do Not Prove a Distributional 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-r01-central-limit-theorem-finite-variance-iid-0040","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"25433dc162bb9376f1c75caa9c0e3ca5c4912c5864c1d65937c49f13da1149a0","family_id":"qf-mod49-clt-standardized-sum-moments","seed":202611181040,"semantic_template":"clt-standardized-sum-moments|{law:[[2,[1,2]],[3,[1,2]]],n:3,process_certificate:\"infinite_iid\"}"},"fields":{"title":"Complete Fixed-n Sum Moments Do Not Prove a Distributional 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 sample size n=3 and S_n as the sum of all n coordinates. Give (n, sum_distribution, total_probability, sum_mean, sum_variance, n_times_mean, n_times_variance, standardized_mean, standardized_variance, standardization_description, finite_scope).","answer":{"kind":"expression","value":"(3, [[6, 1/8], [7, 3/8], [8, 3/8], [9, 1/8]], 1, 15/2, 3/4, 15/2, 3/4, 0, 1, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), complete_joint_law_moments_do_not_prove_distributional_convergence)","explanation":"Aggregate the entire finite IID product law by sum. Retain total mass, E[S_n]=n*mu and Var(S_n)=n*sigma^2. The centered normalization Z_n=(S_n−n*mu)/(sigma*sqrt(n)) has mean0 and variance1. Those moments do not make a finite-n law normal or prove convergence in distribution. The exact result, retaining the requested order, is (3, [[6, 1/8], [7, 3/8], [8, 3/8], [9, 1/8]], 1, 15/2, 3/4, 15/2, 3/4, 0, 1, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), complete_joint_law_moments_do_not_prove_distributional_convergence)."},"skills":["Complete Fixed-n Sum Moments Do Not Prove a Distributional 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-r01-central-limit-theorem-finite-variance-iid-0041","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1c2bafda7f3b0d9e3add4190e1c694c2c4cbf583cf0c27d8c41d5d39e758bccd","family_id":"qf-mod49-clt-standardized-sum-moments","seed":202611181041,"semantic_template":"clt-standardized-sum-moments|{law:[[-1,[1,2]],[-2,[1,2]]],n:4,process_certificate:\"infinite_iid\"}"},"fields":{"title":"Complete Fixed-n Sum Moments Do Not Prove a Distributional 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 sample size n=4 and S_n as the sum of all n coordinates. Give (n, sum_distribution, total_probability, sum_mean, sum_variance, n_times_mean, n_times_variance, standardized_mean, standardized_variance, standardization_description, finite_scope).","answer":{"kind":"expression","value":"(4, [[-8, 1/16], [-7, 1/4], [-6, 3/8], [-5, 1/4], [-4, 1/16]], 1, -6, 1, -6, 1, 0, 1, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), complete_joint_law_moments_do_not_prove_distributional_convergence)","explanation":"Aggregate the entire finite IID product law by sum. Retain total mass, E[S_n]=n*mu and Var(S_n)=n*sigma^2. The centered normalization Z_n=(S_n−n*mu)/(sigma*sqrt(n)) has mean0 and variance1. Those moments do not make a finite-n law normal or prove convergence in distribution. The exact result, retaining the requested order, is (4, [[-8, 1/16], [-7, 1/4], [-6, 3/8], [-5, 1/4], [-4, 1/16]], 1, -6, 1, -6, 1, 0, 1, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), complete_joint_law_moments_do_not_prove_distributional_convergence)."},"skills":["Complete Fixed-n Sum Moments Do Not Prove a Distributional 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-r01-central-limit-theorem-finite-variance-iid-0042","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d9d6aad9f68715b00dca78f71c7353e64c4f319d418aeb67ed9f2ea6d5416ae8","family_id":"qf-mod49-clt-standardized-sum-moments","seed":202611181042,"semantic_template":"clt-standardized-sum-moments|{law:[[-1,[1,2]],[0,[1,2]]],n:4,process_certificate:\"infinite_iid\"}"},"fields":{"title":"Complete Fixed-n Sum Moments Do Not Prove a Distributional 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 sample size n=4 and S_n as the sum of all n coordinates. Give (n, sum_distribution, total_probability, sum_mean, sum_variance, n_times_mean, n_times_variance, standardized_mean, standardized_variance, standardization_description, finite_scope).","answer":{"kind":"expression","value":"(4, [[-4, 1/16], [-3, 1/4], [-2, 3/8], [-1, 1/4], [0, 1/16]], 1, -2, 1, -2, 1, 0, 1, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), complete_joint_law_moments_do_not_prove_distributional_convergence)","explanation":"Aggregate the entire finite IID product law by sum. Retain total mass, E[S_n]=n*mu and Var(S_n)=n*sigma^2. The centered normalization Z_n=(S_n−n*mu)/(sigma*sqrt(n)) has mean0 and variance1. Those moments do not make a finite-n law normal or prove convergence in distribution. The exact result, retaining the requested order, is (4, [[-4, 1/16], [-3, 1/4], [-2, 3/8], [-1, 1/4], [0, 1/16]], 1, -2, 1, -2, 1, 0, 1, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), complete_joint_law_moments_do_not_prove_distributional_convergence)."},"skills":["Complete Fixed-n Sum Moments Do Not Prove a Distributional 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-r01-central-limit-theorem-finite-variance-iid-0043","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c2002eaa4bbc3f807602ba88f6bd568ed49960701329ecac01a5605ae3cbcebc","family_id":"qf-mod49-clt-standardized-sum-moments","seed":202611181043,"semantic_template":"clt-standardized-sum-moments|{law:[[0,[1,2]],[1,[1,2]]],n:4,process_certificate:\"infinite_iid\"}"},"fields":{"title":"Complete Fixed-n Sum Moments Do Not Prove a Distributional 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 sample size n=4 and S_n as the sum of all n coordinates. Give (n, sum_distribution, total_probability, sum_mean, sum_variance, n_times_mean, n_times_variance, standardized_mean, standardized_variance, standardization_description, finite_scope).","answer":{"kind":"expression","value":"(4, [[0, 1/16], [1, 1/4], [2, 3/8], [3, 1/4], [4, 1/16]], 1, 2, 1, 2, 1, 0, 1, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), complete_joint_law_moments_do_not_prove_distributional_convergence)","explanation":"Aggregate the entire finite IID product law by sum. Retain total mass, E[S_n]=n*mu and Var(S_n)=n*sigma^2. The centered normalization Z_n=(S_n−n*mu)/(sigma*sqrt(n)) has mean0 and variance1. Those moments do not make a finite-n law normal or prove convergence in distribution. The exact result, retaining the requested order, is (4, [[0, 1/16], [1, 1/4], [2, 3/8], [3, 1/4], [4, 1/16]], 1, 2, 1, 2, 1, 0, 1, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), complete_joint_law_moments_do_not_prove_distributional_convergence)."},"skills":["Complete Fixed-n Sum Moments Do Not Prove a Distributional 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-r01-central-limit-theorem-finite-variance-iid-0044","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6d7e7979ee4dedc50e223154a0fe1e94113fc81d3e8486a0a30292d9c47a4b86","family_id":"qf-mod49-clt-standardized-sum-moments","seed":202611181044,"semantic_template":"clt-standardized-sum-moments|{law:[[1,[1,2]],[2,[1,2]]],n:4,process_certificate:\"infinite_iid\"}"},"fields":{"title":"Complete Fixed-n Sum Moments Do Not Prove a Distributional 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 sample size n=4 and S_n as the sum of all n coordinates. Give (n, sum_distribution, total_probability, sum_mean, sum_variance, n_times_mean, n_times_variance, standardized_mean, standardized_variance, standardization_description, finite_scope).","answer":{"kind":"expression","value":"(4, [[4, 1/16], [5, 1/4], [6, 3/8], [7, 1/4], [8, 1/16]], 1, 6, 1, 6, 1, 0, 1, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), complete_joint_law_moments_do_not_prove_distributional_convergence)","explanation":"Aggregate the entire finite IID product law by sum. Retain total mass, E[S_n]=n*mu and Var(S_n)=n*sigma^2. The centered normalization Z_n=(S_n−n*mu)/(sigma*sqrt(n)) has mean0 and variance1. Those moments do not make a finite-n law normal or prove convergence in distribution. The exact result, retaining the requested order, is (4, [[4, 1/16], [5, 1/4], [6, 3/8], [7, 1/4], [8, 1/16]], 1, 6, 1, 6, 1, 0, 1, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), complete_joint_law_moments_do_not_prove_distributional_convergence)."},"skills":["Complete Fixed-n Sum Moments Do Not Prove a Distributional 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-r01-central-limit-theorem-finite-variance-iid-0045","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ab91acb4d9dc927d49780dfb4d6265fdd9ad33573dc0e69521da443775348411","family_id":"qf-mod49-clt-standardized-sum-moments","seed":202611181045,"semantic_template":"clt-standardized-sum-moments|{law:[[2,[1,2]],[3,[1,2]]],n:4,process_certificate:\"infinite_iid\"}"},"fields":{"title":"Complete Fixed-n Sum Moments Do Not Prove a Distributional 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 sample size n=4 and S_n as the sum of all n coordinates. Give (n, sum_distribution, total_probability, sum_mean, sum_variance, n_times_mean, n_times_variance, standardized_mean, standardized_variance, standardization_description, finite_scope).","answer":{"kind":"expression","value":"(4, [[8, 1/16], [9, 1/4], [10, 3/8], [11, 1/4], [12, 1/16]], 1, 10, 1, 10, 1, 0, 1, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), complete_joint_law_moments_do_not_prove_distributional_convergence)","explanation":"Aggregate the entire finite IID product law by sum. Retain total mass, E[S_n]=n*mu and Var(S_n)=n*sigma^2. The centered normalization Z_n=(S_n−n*mu)/(sigma*sqrt(n)) has mean0 and variance1. Those moments do not make a finite-n law normal or prove convergence in distribution. The exact result, retaining the requested order, is (4, [[8, 1/16], [9, 1/4], [10, 3/8], [11, 1/4], [12, 1/16]], 1, 10, 1, 10, 1, 0, 1, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), complete_joint_law_moments_do_not_prove_distributional_convergence)."},"skills":["Complete Fixed-n Sum Moments Do Not Prove a Distributional 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-r01-central-limit-theorem-finite-variance-iid-0046","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6f227ce941518a7a291ce170b7e89f6b4cdfb1a5d9fcb5e5c96bd5fdc1218d87","family_id":"qf-mod49-clt-standardized-sum-moments","seed":202611181046,"semantic_template":"clt-standardized-sum-moments|{law:[[-1,[1,2]],[-2,[1,2]]],n:5,process_certificate:\"infinite_iid\"}"},"fields":{"title":"Complete Fixed-n Sum Moments Do Not Prove a Distributional 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 sample size n=5 and S_n as the sum of all n coordinates. Give (n, sum_distribution, total_probability, sum_mean, sum_variance, n_times_mean, n_times_variance, standardized_mean, standardized_variance, standardization_description, finite_scope).","answer":{"kind":"expression","value":"(5, [[-10, 1/32], [-9, 5/32], [-8, 5/16], [-7, 5/16], [-6, 5/32], [-5, 1/32]], 1, -15/2, 5/4, -15/2, 5/4, 0, 1, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), complete_joint_law_moments_do_not_prove_distributional_convergence)","explanation":"Aggregate the entire finite IID product law by sum. Retain total mass, E[S_n]=n*mu and Var(S_n)=n*sigma^2. The centered normalization Z_n=(S_n−n*mu)/(sigma*sqrt(n)) has mean0 and variance1. Those moments do not make a finite-n law normal or prove convergence in distribution. The exact result, retaining the requested order, is (5, [[-10, 1/32], [-9, 5/32], [-8, 5/16], [-7, 5/16], [-6, 5/32], [-5, 1/32]], 1, -15/2, 5/4, -15/2, 5/4, 0, 1, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), complete_joint_law_moments_do_not_prove_distributional_convergence)."},"skills":["Complete Fixed-n Sum Moments Do Not Prove a Distributional 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-r01-central-limit-theorem-finite-variance-iid-0047","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6e975b0494911614f6bfa2cd1c268ddb3b57ea7a81953e1e3ba8e3db4bfff773","family_id":"qf-mod49-clt-standardized-sum-moments","seed":202611181047,"semantic_template":"clt-standardized-sum-moments|{law:[[-1,[1,2]],[0,[1,2]]],n:5,process_certificate:\"infinite_iid\"}"},"fields":{"title":"Complete Fixed-n Sum Moments Do Not Prove a Distributional 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 sample size n=5 and S_n as the sum of all n coordinates. Give (n, sum_distribution, total_probability, sum_mean, sum_variance, n_times_mean, n_times_variance, standardized_mean, standardized_variance, standardization_description, finite_scope).","answer":{"kind":"expression","value":"(5, [[-5, 1/32], [-4, 5/32], [-3, 5/16], [-2, 5/16], [-1, 5/32], [0, 1/32]], 1, -5/2, 5/4, -5/2, 5/4, 0, 1, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), complete_joint_law_moments_do_not_prove_distributional_convergence)","explanation":"Aggregate the entire finite IID product law by sum. Retain total mass, E[S_n]=n*mu and Var(S_n)=n*sigma^2. The centered normalization Z_n=(S_n−n*mu)/(sigma*sqrt(n)) has mean0 and variance1. Those moments do not make a finite-n law normal or prove convergence in distribution. The exact result, retaining the requested order, is (5, [[-5, 1/32], [-4, 5/32], [-3, 5/16], [-2, 5/16], [-1, 5/32], [0, 1/32]], 1, -5/2, 5/4, -5/2, 5/4, 0, 1, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), complete_joint_law_moments_do_not_prove_distributional_convergence)."},"skills":["Complete Fixed-n Sum Moments Do Not Prove a Distributional 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-r01-central-limit-theorem-finite-variance-iid-0048","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ad56ee5ebceb2e9910c3d904dfdb8573ed055efe68b8773cef15146780ff8745","family_id":"qf-mod49-clt-standardized-sum-moments","seed":202611181048,"semantic_template":"clt-standardized-sum-moments|{law:[[0,[1,2]],[1,[1,2]]],n:5,process_certificate:\"infinite_iid\"}"},"fields":{"title":"Complete Fixed-n Sum Moments Do Not Prove a Distributional 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 sample size n=5 and S_n as the sum of all n coordinates. Give (n, sum_distribution, total_probability, sum_mean, sum_variance, n_times_mean, n_times_variance, standardized_mean, standardized_variance, standardization_description, finite_scope).","answer":{"kind":"expression","value":"(5, [[0, 1/32], [1, 5/32], [2, 5/16], [3, 5/16], [4, 5/32], [5, 1/32]], 1, 5/2, 5/4, 5/2, 5/4, 0, 1, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), complete_joint_law_moments_do_not_prove_distributional_convergence)","explanation":"Aggregate the entire finite IID product law by sum. Retain total mass, E[S_n]=n*mu and Var(S_n)=n*sigma^2. The centered normalization Z_n=(S_n−n*mu)/(sigma*sqrt(n)) has mean0 and variance1. Those moments do not make a finite-n law normal or prove convergence in distribution. The exact result, retaining the requested order, is (5, [[0, 1/32], [1, 5/32], [2, 5/16], [3, 5/16], [4, 5/32], [5, 1/32]], 1, 5/2, 5/4, 5/2, 5/4, 0, 1, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), complete_joint_law_moments_do_not_prove_distributional_convergence)."},"skills":["Complete Fixed-n Sum Moments Do Not Prove a Distributional 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-r01-central-limit-theorem-finite-variance-iid-0049","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"55ec7fb14d46b1f0e4f11682f28ec7977d7ce5f6540e6c79045a44f7726ed3e8","family_id":"qf-mod49-clt-standardized-sum-moments","seed":202611181049,"semantic_template":"clt-standardized-sum-moments|{law:[[1,[1,2]],[2,[1,2]]],n:5,process_certificate:\"infinite_iid\"}"},"fields":{"title":"Complete Fixed-n Sum Moments Do Not Prove a Distributional 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 sample size n=5 and S_n as the sum of all n coordinates. Give (n, sum_distribution, total_probability, sum_mean, sum_variance, n_times_mean, n_times_variance, standardized_mean, standardized_variance, standardization_description, finite_scope).","answer":{"kind":"expression","value":"(5, [[5, 1/32], [6, 5/32], [7, 5/16], [8, 5/16], [9, 5/32], [10, 1/32]], 1, 15/2, 5/4, 15/2, 5/4, 0, 1, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), complete_joint_law_moments_do_not_prove_distributional_convergence)","explanation":"Aggregate the entire finite IID product law by sum. Retain total mass, E[S_n]=n*mu and Var(S_n)=n*sigma^2. The centered normalization Z_n=(S_n−n*mu)/(sigma*sqrt(n)) has mean0 and variance1. Those moments do not make a finite-n law normal or prove convergence in distribution. The exact result, retaining the requested order, is (5, [[5, 1/32], [6, 5/32], [7, 5/16], [8, 5/16], [9, 5/32], [10, 1/32]], 1, 15/2, 5/4, 15/2, 5/4, 0, 1, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), complete_joint_law_moments_do_not_prove_distributional_convergence)."},"skills":["Complete Fixed-n Sum Moments Do Not Prove a Distributional 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-r01-central-limit-theorem-finite-variance-iid-0050","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a8762d84a4356ada169e5b5b82b89c30490960fa3cbd129d356533f97f1ada95","family_id":"qf-mod49-clt-standardized-sum-moments","seed":202611181050,"semantic_template":"clt-standardized-sum-moments|{law:[[2,[1,2]],[3,[1,2]]],n:5,process_certificate:\"infinite_iid\"}"},"fields":{"title":"Complete Fixed-n Sum Moments Do Not Prove a Distributional 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 sample size n=5 and S_n as the sum of all n coordinates. Give (n, sum_distribution, total_probability, sum_mean, sum_variance, n_times_mean, n_times_variance, standardized_mean, standardized_variance, standardization_description, finite_scope).","answer":{"kind":"expression","value":"(5, [[10, 1/32], [11, 5/32], [12, 5/16], [13, 5/16], [14, 5/32], [15, 1/32]], 1, 25/2, 5/4, 25/2, 5/4, 0, 1, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), complete_joint_law_moments_do_not_prove_distributional_convergence)","explanation":"Aggregate the entire finite IID product law by sum. Retain total mass, E[S_n]=n*mu and Var(S_n)=n*sigma^2. The centered normalization Z_n=(S_n−n*mu)/(sigma*sqrt(n)) has mean0 and variance1. Those moments do not make a finite-n law normal or prove convergence in distribution. The exact result, retaining the requested order, is (5, [[10, 1/32], [11, 5/32], [12, 5/16], [13, 5/16], [14, 5/32], [15, 1/32]], 1, 25/2, 5/4, 25/2, 5/4, 0, 1, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), complete_joint_law_moments_do_not_prove_distributional_convergence)."},"skills":["Complete Fixed-n Sum Moments Do Not Prove a Distributional 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-r01-central-limit-theorem-finite-variance-iid-0051","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"25fe9d8a6c587c7bf833e953b4d8628f704512a4306ce42921d80c6147192fe2","family_id":"qf-mod49-clt-binomial-lattice","seed":202611181051,"semantic_template":"clt-binomial-lattice|{n:2,success_probability:[1,2],threshold:0}"},"fields":{"title":"An Exact Binomial Lattice Is Not a Normal Approximation","stem":"Use Bernoulli success probability p=1/2, sample size n=2, integer threshold=0, and independent coordinates for this finite sum. Give (n, success_probability, binomial_distribution, total_probability, threshold, cdf_at_threshold, normalized, standardized_threshold_description, finite_scope).","answer":{"kind":"expression","value":"(2, 1/2, [[0, 1/4], [1, 1/2], [2, 1/4]], 1, 0, 1/4, Yes, z_threshold=(0-n*p)/sqrt(n*p*(1-p)), exact_lattice_not_normal_approximation_or_convergence_evidence)","explanation":"Compute the exact probabilities for every integer count0 through n and the inclusive CDF at the threshold. Retain the symbolic standardized threshold; neither that expression nor a finite lattice table establishes an asymptotic limit or an approximation error. The exact result, retaining the requested order, is (2, 1/2, [[0, 1/4], [1, 1/2], [2, 1/4]], 1, 0, 1/4, Yes, z_threshold=(0-n*p)/sqrt(n*p*(1-p)), exact_lattice_not_normal_approximation_or_convergence_evidence)."},"skills":["An Exact Binomial Lattice Is Not a Normal Approximation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r01-central-limit-theorem-finite-variance-iid-0052","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8f6d7a36e780b21443a41bae575999402b2516cf3e5f5150c80a21cdb4a2fee0","family_id":"qf-mod49-clt-binomial-lattice","seed":202611181052,"semantic_template":"clt-binomial-lattice|{n:2,success_probability:[1,2],threshold:1}"},"fields":{"title":"An Exact Binomial Lattice Is Not a Normal Approximation","stem":"Use Bernoulli success probability p=1/2, sample size n=2, integer threshold=1, and independent coordinates for this finite sum. Give (n, success_probability, binomial_distribution, total_probability, threshold, cdf_at_threshold, normalized, standardized_threshold_description, finite_scope).","answer":{"kind":"expression","value":"(2, 1/2, [[0, 1/4], [1, 1/2], [2, 1/4]], 1, 1, 3/4, Yes, z_threshold=(1-n*p)/sqrt(n*p*(1-p)), exact_lattice_not_normal_approximation_or_convergence_evidence)","explanation":"Compute the exact probabilities for every integer count0 through n and the inclusive CDF at the threshold. Retain the symbolic standardized threshold; neither that expression nor a finite lattice table establishes an asymptotic limit or an approximation error. The exact result, retaining the requested order, is (2, 1/2, [[0, 1/4], [1, 1/2], [2, 1/4]], 1, 1, 3/4, Yes, z_threshold=(1-n*p)/sqrt(n*p*(1-p)), exact_lattice_not_normal_approximation_or_convergence_evidence)."},"skills":["An Exact Binomial Lattice Is Not a Normal Approximation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r01-central-limit-theorem-finite-variance-iid-0053","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"80c3a121d2637aa47fba56a998a28c6997e8ec4f4b2f3e636831d5e3764cb325","family_id":"qf-mod49-clt-binomial-lattice","seed":202611181053,"semantic_template":"clt-binomial-lattice|{n:2,success_probability:[1,2],threshold:2}"},"fields":{"title":"An Exact Binomial Lattice Is Not a Normal Approximation","stem":"Use Bernoulli success probability p=1/2, sample size n=2, integer threshold=2, and independent coordinates for this finite sum. Give (n, success_probability, binomial_distribution, total_probability, threshold, cdf_at_threshold, normalized, standardized_threshold_description, finite_scope).","answer":{"kind":"expression","value":"(2, 1/2, [[0, 1/4], [1, 1/2], [2, 1/4]], 1, 2, 1, Yes, z_threshold=(2-n*p)/sqrt(n*p*(1-p)), exact_lattice_not_normal_approximation_or_convergence_evidence)","explanation":"Compute the exact probabilities for every integer count0 through n and the inclusive CDF at the threshold. Retain the symbolic standardized threshold; neither that expression nor a finite lattice table establishes an asymptotic limit or an approximation error. The exact result, retaining the requested order, is (2, 1/2, [[0, 1/4], [1, 1/2], [2, 1/4]], 1, 2, 1, Yes, z_threshold=(2-n*p)/sqrt(n*p*(1-p)), exact_lattice_not_normal_approximation_or_convergence_evidence)."},"skills":["An Exact Binomial Lattice Is Not a Normal Approximation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r01-central-limit-theorem-finite-variance-iid-0054","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4e10c1b59e2de0f76cfde794e996d2f6608934406007ac879dc019837ac24cb2","family_id":"qf-mod49-clt-binomial-lattice","seed":202611181054,"semantic_template":"clt-binomial-lattice|{n:3,success_probability:[1,2],threshold:0}"},"fields":{"title":"An Exact Binomial Lattice Is Not a Normal Approximation","stem":"Use Bernoulli success probability p=1/2, sample size n=3, integer threshold=0, and independent coordinates for this finite sum. Give (n, success_probability, binomial_distribution, total_probability, threshold, cdf_at_threshold, normalized, standardized_threshold_description, finite_scope).","answer":{"kind":"expression","value":"(3, 1/2, [[0, 1/8], [1, 3/8], [2, 3/8], [3, 1/8]], 1, 0, 1/8, Yes, z_threshold=(0-n*p)/sqrt(n*p*(1-p)), exact_lattice_not_normal_approximation_or_convergence_evidence)","explanation":"Compute the exact probabilities for every integer count0 through n and the inclusive CDF at the threshold. Retain the symbolic standardized threshold; neither that expression nor a finite lattice table establishes an asymptotic limit or an approximation error. The exact result, retaining the requested order, is (3, 1/2, [[0, 1/8], [1, 3/8], [2, 3/8], [3, 1/8]], 1, 0, 1/8, Yes, z_threshold=(0-n*p)/sqrt(n*p*(1-p)), exact_lattice_not_normal_approximation_or_convergence_evidence)."},"skills":["An Exact Binomial Lattice Is Not a Normal Approximation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r01-central-limit-theorem-finite-variance-iid-0055","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ff2b863d9d934d9a8b2e6092af2a19941ec86d449dcf7abe942c6cf56304a8e5","family_id":"qf-mod49-clt-binomial-lattice","seed":202611181055,"semantic_template":"clt-binomial-lattice|{n:3,success_probability:[1,2],threshold:1}"},"fields":{"title":"An Exact Binomial Lattice Is Not a Normal Approximation","stem":"Use Bernoulli success probability p=1/2, sample size n=3, integer threshold=1, and independent coordinates for this finite sum. Give (n, success_probability, binomial_distribution, total_probability, threshold, cdf_at_threshold, normalized, standardized_threshold_description, finite_scope).","answer":{"kind":"expression","value":"(3, 1/2, [[0, 1/8], [1, 3/8], [2, 3/8], [3, 1/8]], 1, 1, 1/2, Yes, z_threshold=(1-n*p)/sqrt(n*p*(1-p)), exact_lattice_not_normal_approximation_or_convergence_evidence)","explanation":"Compute the exact probabilities for every integer count0 through n and the inclusive CDF at the threshold. Retain the symbolic standardized threshold; neither that expression nor a finite lattice table establishes an asymptotic limit or an approximation error. The exact result, retaining the requested order, is (3, 1/2, [[0, 1/8], [1, 3/8], [2, 3/8], [3, 1/8]], 1, 1, 1/2, Yes, z_threshold=(1-n*p)/sqrt(n*p*(1-p)), exact_lattice_not_normal_approximation_or_convergence_evidence)."},"skills":["An Exact Binomial Lattice Is Not a Normal Approximation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r01-central-limit-theorem-finite-variance-iid-0056","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8e7e9e303fe7588a4da65374e893326c560796e64974521210ac90fb7dd013fe","family_id":"qf-mod49-clt-binomial-lattice","seed":202611181056,"semantic_template":"clt-binomial-lattice|{n:3,success_probability:[1,2],threshold:2}"},"fields":{"title":"An Exact Binomial Lattice Is Not a Normal Approximation","stem":"Use Bernoulli success probability p=1/2, sample size n=3, integer threshold=2, and independent coordinates for this finite sum. Give (n, success_probability, binomial_distribution, total_probability, threshold, cdf_at_threshold, normalized, standardized_threshold_description, finite_scope).","answer":{"kind":"expression","value":"(3, 1/2, [[0, 1/8], [1, 3/8], [2, 3/8], [3, 1/8]], 1, 2, 7/8, Yes, z_threshold=(2-n*p)/sqrt(n*p*(1-p)), exact_lattice_not_normal_approximation_or_convergence_evidence)","explanation":"Compute the exact probabilities for every integer count0 through n and the inclusive CDF at the threshold. Retain the symbolic standardized threshold; neither that expression nor a finite lattice table establishes an asymptotic limit or an approximation error. The exact result, retaining the requested order, is (3, 1/2, [[0, 1/8], [1, 3/8], [2, 3/8], [3, 1/8]], 1, 2, 7/8, Yes, z_threshold=(2-n*p)/sqrt(n*p*(1-p)), exact_lattice_not_normal_approximation_or_convergence_evidence)."},"skills":["An Exact Binomial Lattice Is Not a Normal Approximation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r01-central-limit-theorem-finite-variance-iid-0057","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2264cdfcb4524ab3facc71a6848b87e1926db078ab52a690e746774a326cbccb","family_id":"qf-mod49-clt-binomial-lattice","seed":202611181057,"semantic_template":"clt-binomial-lattice|{n:3,success_probability:[1,2],threshold:3}"},"fields":{"title":"An Exact Binomial Lattice Is Not a Normal Approximation","stem":"Use Bernoulli success probability p=1/2, sample size n=3, integer threshold=3, and independent coordinates for this finite sum. Give (n, success_probability, binomial_distribution, total_probability, threshold, cdf_at_threshold, normalized, standardized_threshold_description, finite_scope).","answer":{"kind":"expression","value":"(3, 1/2, [[0, 1/8], [1, 3/8], [2, 3/8], [3, 1/8]], 1, 3, 1, Yes, z_threshold=(3-n*p)/sqrt(n*p*(1-p)), exact_lattice_not_normal_approximation_or_convergence_evidence)","explanation":"Compute the exact probabilities for every integer count0 through n and the inclusive CDF at the threshold. Retain the symbolic standardized threshold; neither that expression nor a finite lattice table establishes an asymptotic limit or an approximation error. The exact result, retaining the requested order, is (3, 1/2, [[0, 1/8], [1, 3/8], [2, 3/8], [3, 1/8]], 1, 3, 1, Yes, z_threshold=(3-n*p)/sqrt(n*p*(1-p)), exact_lattice_not_normal_approximation_or_convergence_evidence)."},"skills":["An Exact Binomial Lattice Is Not a Normal Approximation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r01-central-limit-theorem-finite-variance-iid-0058","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e49b8c8120f7313115d90eb5f16ce7dd1696f5f7c428ed5ffe7eb4e5608a0434","family_id":"qf-mod49-clt-binomial-lattice","seed":202611181058,"semantic_template":"clt-binomial-lattice|{n:4,success_probability:[1,2],threshold:0}"},"fields":{"title":"An Exact Binomial Lattice Is Not a Normal Approximation","stem":"Use Bernoulli success probability p=1/2, sample size n=4, integer threshold=0, and independent coordinates for this finite sum. Give (n, success_probability, binomial_distribution, total_probability, threshold, cdf_at_threshold, normalized, standardized_threshold_description, finite_scope).","answer":{"kind":"expression","value":"(4, 1/2, [[0, 1/16], [1, 1/4], [2, 3/8], [3, 1/4], [4, 1/16]], 1, 0, 1/16, Yes, z_threshold=(0-n*p)/sqrt(n*p*(1-p)), exact_lattice_not_normal_approximation_or_convergence_evidence)","explanation":"Compute the exact probabilities for every integer count0 through n and the inclusive CDF at the threshold. Retain the symbolic standardized threshold; neither that expression nor a finite lattice table establishes an asymptotic limit or an approximation error. The exact result, retaining the requested order, is (4, 1/2, [[0, 1/16], [1, 1/4], [2, 3/8], [3, 1/4], [4, 1/16]], 1, 0, 1/16, Yes, z_threshold=(0-n*p)/sqrt(n*p*(1-p)), exact_lattice_not_normal_approximation_or_convergence_evidence)."},"skills":["An Exact Binomial Lattice Is Not a Normal Approximation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r01-central-limit-theorem-finite-variance-iid-0059","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"703c9489eff847117107d47821f55c1134b0695d11da186f9052f3619517ef80","family_id":"qf-mod49-clt-binomial-lattice","seed":202611181059,"semantic_template":"clt-binomial-lattice|{n:4,success_probability:[1,2],threshold:1}"},"fields":{"title":"An Exact Binomial Lattice Is Not a Normal Approximation","stem":"Use Bernoulli success probability p=1/2, sample size n=4, integer threshold=1, and independent coordinates for this finite sum. Give (n, success_probability, binomial_distribution, total_probability, threshold, cdf_at_threshold, normalized, standardized_threshold_description, finite_scope).","answer":{"kind":"expression","value":"(4, 1/2, [[0, 1/16], [1, 1/4], [2, 3/8], [3, 1/4], [4, 1/16]], 1, 1, 5/16, Yes, z_threshold=(1-n*p)/sqrt(n*p*(1-p)), exact_lattice_not_normal_approximation_or_convergence_evidence)","explanation":"Compute the exact probabilities for every integer count0 through n and the inclusive CDF at the threshold. Retain the symbolic standardized threshold; neither that expression nor a finite lattice table establishes an asymptotic limit or an approximation error. The exact result, retaining the requested order, is (4, 1/2, [[0, 1/16], [1, 1/4], [2, 3/8], [3, 1/4], [4, 1/16]], 1, 1, 5/16, Yes, z_threshold=(1-n*p)/sqrt(n*p*(1-p)), exact_lattice_not_normal_approximation_or_convergence_evidence)."},"skills":["An Exact Binomial Lattice Is Not a Normal Approximation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r01-central-limit-theorem-finite-variance-iid-0060","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4c6a304a3a3354f72b5048cb082bd0d0ea2d34efaf498a67b9bbefe1a3ae4fce","family_id":"qf-mod49-clt-binomial-lattice","seed":202611181060,"semantic_template":"clt-binomial-lattice|{n:4,success_probability:[1,2],threshold:2}"},"fields":{"title":"An Exact Binomial Lattice Is Not a Normal Approximation","stem":"Use Bernoulli success probability p=1/2, sample size n=4, integer threshold=2, and independent coordinates for this finite sum. Give (n, success_probability, binomial_distribution, total_probability, threshold, cdf_at_threshold, normalized, standardized_threshold_description, finite_scope).","answer":{"kind":"expression","value":"(4, 1/2, [[0, 1/16], [1, 1/4], [2, 3/8], [3, 1/4], [4, 1/16]], 1, 2, 11/16, Yes, z_threshold=(2-n*p)/sqrt(n*p*(1-p)), exact_lattice_not_normal_approximation_or_convergence_evidence)","explanation":"Compute the exact probabilities for every integer count0 through n and the inclusive CDF at the threshold. Retain the symbolic standardized threshold; neither that expression nor a finite lattice table establishes an asymptotic limit or an approximation error. The exact result, retaining the requested order, is (4, 1/2, [[0, 1/16], [1, 1/4], [2, 3/8], [3, 1/4], [4, 1/16]], 1, 2, 11/16, Yes, z_threshold=(2-n*p)/sqrt(n*p*(1-p)), exact_lattice_not_normal_approximation_or_convergence_evidence)."},"skills":["An Exact Binomial Lattice Is Not a Normal Approximation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r01-central-limit-theorem-finite-variance-iid-0061","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"aa86c361bcee67d2ab30152503768423b8f8b08bd6dcf430aa973a8c30f42125","family_id":"qf-mod49-clt-binomial-lattice","seed":202611181061,"semantic_template":"clt-binomial-lattice|{n:4,success_probability:[1,2],threshold:3}"},"fields":{"title":"An Exact Binomial Lattice Is Not a Normal Approximation","stem":"Use Bernoulli success probability p=1/2, sample size n=4, integer threshold=3, and independent coordinates for this finite sum. Give (n, success_probability, binomial_distribution, total_probability, threshold, cdf_at_threshold, normalized, standardized_threshold_description, finite_scope).","answer":{"kind":"expression","value":"(4, 1/2, [[0, 1/16], [1, 1/4], [2, 3/8], [3, 1/4], [4, 1/16]], 1, 3, 15/16, Yes, z_threshold=(3-n*p)/sqrt(n*p*(1-p)), exact_lattice_not_normal_approximation_or_convergence_evidence)","explanation":"Compute the exact probabilities for every integer count0 through n and the inclusive CDF at the threshold. Retain the symbolic standardized threshold; neither that expression nor a finite lattice table establishes an asymptotic limit or an approximation error. The exact result, retaining the requested order, is (4, 1/2, [[0, 1/16], [1, 1/4], [2, 3/8], [3, 1/4], [4, 1/16]], 1, 3, 15/16, Yes, z_threshold=(3-n*p)/sqrt(n*p*(1-p)), exact_lattice_not_normal_approximation_or_convergence_evidence)."},"skills":["An Exact Binomial Lattice Is Not a Normal Approximation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r01-central-limit-theorem-finite-variance-iid-0062","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1a3eaacc3826fad4e2791c0aa3fbd2e908a157045eda394c0dc4683e8ed9b431","family_id":"qf-mod49-clt-binomial-lattice","seed":202611181062,"semantic_template":"clt-binomial-lattice|{n:4,success_probability:[1,2],threshold:4}"},"fields":{"title":"An Exact Binomial Lattice Is Not a Normal Approximation","stem":"Use Bernoulli success probability p=1/2, sample size n=4, integer threshold=4, and independent coordinates for this finite sum. Give (n, success_probability, binomial_distribution, total_probability, threshold, cdf_at_threshold, normalized, standardized_threshold_description, finite_scope).","answer":{"kind":"expression","value":"(4, 1/2, [[0, 1/16], [1, 1/4], [2, 3/8], [3, 1/4], [4, 1/16]], 1, 4, 1, Yes, z_threshold=(4-n*p)/sqrt(n*p*(1-p)), exact_lattice_not_normal_approximation_or_convergence_evidence)","explanation":"Compute the exact probabilities for every integer count0 through n and the inclusive CDF at the threshold. Retain the symbolic standardized threshold; neither that expression nor a finite lattice table establishes an asymptotic limit or an approximation error. The exact result, retaining the requested order, is (4, 1/2, [[0, 1/16], [1, 1/4], [2, 3/8], [3, 1/4], [4, 1/16]], 1, 4, 1, Yes, z_threshold=(4-n*p)/sqrt(n*p*(1-p)), exact_lattice_not_normal_approximation_or_convergence_evidence)."},"skills":["An Exact Binomial Lattice Is Not a Normal Approximation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r01-central-limit-theorem-finite-variance-iid-0063","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b9bf75a1877ae4ba2eace9eaf85e980015b41a2d4f03b2e476b8c726dc915f0e","family_id":"qf-mod49-clt-binomial-lattice","seed":202611181063,"semantic_template":"clt-binomial-lattice|{n:5,success_probability:[1,2],threshold:0}"},"fields":{"title":"An Exact Binomial Lattice Is Not a Normal Approximation","stem":"Use Bernoulli success probability p=1/2, sample size n=5, integer threshold=0, and independent coordinates for this finite sum. Give (n, success_probability, binomial_distribution, total_probability, threshold, cdf_at_threshold, normalized, standardized_threshold_description, finite_scope).","answer":{"kind":"expression","value":"(5, 1/2, [[0, 1/32], [1, 5/32], [2, 5/16], [3, 5/16], [4, 5/32], [5, 1/32]], 1, 0, 1/32, Yes, z_threshold=(0-n*p)/sqrt(n*p*(1-p)), exact_lattice_not_normal_approximation_or_convergence_evidence)","explanation":"Compute the exact probabilities for every integer count0 through n and the inclusive CDF at the threshold. Retain the symbolic standardized threshold; neither that expression nor a finite lattice table establishes an asymptotic limit or an approximation error. The exact result, retaining the requested order, is (5, 1/2, [[0, 1/32], [1, 5/32], [2, 5/16], [3, 5/16], [4, 5/32], [5, 1/32]], 1, 0, 1/32, Yes, z_threshold=(0-n*p)/sqrt(n*p*(1-p)), exact_lattice_not_normal_approximation_or_convergence_evidence)."},"skills":["An Exact Binomial Lattice Is Not a Normal Approximation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r01-central-limit-theorem-finite-variance-iid-0064","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"24b6f02d06ccd6e04690fd5e679d461d8c70ec5543eb166925c0733361715416","family_id":"qf-mod49-clt-binomial-lattice","seed":202611181064,"semantic_template":"clt-binomial-lattice|{n:5,success_probability:[1,2],threshold:1}"},"fields":{"title":"An Exact Binomial Lattice Is Not a Normal Approximation","stem":"Use Bernoulli success probability p=1/2, sample size n=5, integer threshold=1, and independent coordinates for this finite sum. Give (n, success_probability, binomial_distribution, total_probability, threshold, cdf_at_threshold, normalized, standardized_threshold_description, finite_scope).","answer":{"kind":"expression","value":"(5, 1/2, [[0, 1/32], [1, 5/32], [2, 5/16], [3, 5/16], [4, 5/32], [5, 1/32]], 1, 1, 3/16, Yes, z_threshold=(1-n*p)/sqrt(n*p*(1-p)), exact_lattice_not_normal_approximation_or_convergence_evidence)","explanation":"Compute the exact probabilities for every integer count0 through n and the inclusive CDF at the threshold. Retain the symbolic standardized threshold; neither that expression nor a finite lattice table establishes an asymptotic limit or an approximation error. The exact result, retaining the requested order, is (5, 1/2, [[0, 1/32], [1, 5/32], [2, 5/16], [3, 5/16], [4, 5/32], [5, 1/32]], 1, 1, 3/16, Yes, z_threshold=(1-n*p)/sqrt(n*p*(1-p)), exact_lattice_not_normal_approximation_or_convergence_evidence)."},"skills":["An Exact Binomial Lattice Is Not a Normal Approximation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r01-central-limit-theorem-finite-variance-iid-0065","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c2bbc9e321b79dcabbac4f2730b1a72b7fe8fe5f0a529fda17887f3806457319","family_id":"qf-mod49-clt-binomial-lattice","seed":202611181065,"semantic_template":"clt-binomial-lattice|{n:5,success_probability:[1,2],threshold:2}"},"fields":{"title":"An Exact Binomial Lattice Is Not a Normal Approximation","stem":"Use Bernoulli success probability p=1/2, sample size n=5, integer threshold=2, and independent coordinates for this finite sum. Give (n, success_probability, binomial_distribution, total_probability, threshold, cdf_at_threshold, normalized, standardized_threshold_description, finite_scope).","answer":{"kind":"expression","value":"(5, 1/2, [[0, 1/32], [1, 5/32], [2, 5/16], [3, 5/16], [4, 5/32], [5, 1/32]], 1, 2, 1/2, Yes, z_threshold=(2-n*p)/sqrt(n*p*(1-p)), exact_lattice_not_normal_approximation_or_convergence_evidence)","explanation":"Compute the exact probabilities for every integer count0 through n and the inclusive CDF at the threshold. Retain the symbolic standardized threshold; neither that expression nor a finite lattice table establishes an asymptotic limit or an approximation error. The exact result, retaining the requested order, is (5, 1/2, [[0, 1/32], [1, 5/32], [2, 5/16], [3, 5/16], [4, 5/32], [5, 1/32]], 1, 2, 1/2, Yes, z_threshold=(2-n*p)/sqrt(n*p*(1-p)), exact_lattice_not_normal_approximation_or_convergence_evidence)."},"skills":["An Exact Binomial Lattice Is Not a Normal Approximation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r01-central-limit-theorem-finite-variance-iid-0066","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bb6f07ebf7725c6c76779dc5581a2395f275bed72e147614a21e91221f1eff0c","family_id":"qf-mod49-clt-binomial-lattice","seed":202611181066,"semantic_template":"clt-binomial-lattice|{n:5,success_probability:[1,2],threshold:3}"},"fields":{"title":"An Exact Binomial Lattice Is Not a Normal Approximation","stem":"Use Bernoulli success probability p=1/2, sample size n=5, integer threshold=3, and independent coordinates for this finite sum. Give (n, success_probability, binomial_distribution, total_probability, threshold, cdf_at_threshold, normalized, standardized_threshold_description, finite_scope).","answer":{"kind":"expression","value":"(5, 1/2, [[0, 1/32], [1, 5/32], [2, 5/16], [3, 5/16], [4, 5/32], [5, 1/32]], 1, 3, 13/16, Yes, z_threshold=(3-n*p)/sqrt(n*p*(1-p)), exact_lattice_not_normal_approximation_or_convergence_evidence)","explanation":"Compute the exact probabilities for every integer count0 through n and the inclusive CDF at the threshold. Retain the symbolic standardized threshold; neither that expression nor a finite lattice table establishes an asymptotic limit or an approximation error. The exact result, retaining the requested order, is (5, 1/2, [[0, 1/32], [1, 5/32], [2, 5/16], [3, 5/16], [4, 5/32], [5, 1/32]], 1, 3, 13/16, Yes, z_threshold=(3-n*p)/sqrt(n*p*(1-p)), exact_lattice_not_normal_approximation_or_convergence_evidence)."},"skills":["An Exact Binomial Lattice Is Not a Normal Approximation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r01-central-limit-theorem-finite-variance-iid-0067","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ca1722c357d0957c195493ecce1f12188c391b069c126d5456c0a4c52952bd17","family_id":"qf-mod49-clt-binomial-lattice","seed":202611181067,"semantic_template":"clt-binomial-lattice|{n:5,success_probability:[1,2],threshold:4}"},"fields":{"title":"An Exact Binomial Lattice Is Not a Normal Approximation","stem":"Use Bernoulli success probability p=1/2, sample size n=5, integer threshold=4, and independent coordinates for this finite sum. Give (n, success_probability, binomial_distribution, total_probability, threshold, cdf_at_threshold, normalized, standardized_threshold_description, finite_scope).","answer":{"kind":"expression","value":"(5, 1/2, [[0, 1/32], [1, 5/32], [2, 5/16], [3, 5/16], [4, 5/32], [5, 1/32]], 1, 4, 31/32, Yes, z_threshold=(4-n*p)/sqrt(n*p*(1-p)), exact_lattice_not_normal_approximation_or_convergence_evidence)","explanation":"Compute the exact probabilities for every integer count0 through n and the inclusive CDF at the threshold. Retain the symbolic standardized threshold; neither that expression nor a finite lattice table establishes an asymptotic limit or an approximation error. The exact result, retaining the requested order, is (5, 1/2, [[0, 1/32], [1, 5/32], [2, 5/16], [3, 5/16], [4, 5/32], [5, 1/32]], 1, 4, 31/32, Yes, z_threshold=(4-n*p)/sqrt(n*p*(1-p)), exact_lattice_not_normal_approximation_or_convergence_evidence)."},"skills":["An Exact Binomial Lattice Is Not a Normal Approximation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r01-central-limit-theorem-finite-variance-iid-0068","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"98375caef369606bac516abac5751979290b175bd5c2062ee22ec97706e08acf","family_id":"qf-mod49-clt-binomial-lattice","seed":202611181068,"semantic_template":"clt-binomial-lattice|{n:5,success_probability:[1,2],threshold:5}"},"fields":{"title":"An Exact Binomial Lattice Is Not a Normal Approximation","stem":"Use Bernoulli success probability p=1/2, sample size n=5, integer threshold=5, and independent coordinates for this finite sum. Give (n, success_probability, binomial_distribution, total_probability, threshold, cdf_at_threshold, normalized, standardized_threshold_description, finite_scope).","answer":{"kind":"expression","value":"(5, 1/2, [[0, 1/32], [1, 5/32], [2, 5/16], [3, 5/16], [4, 5/32], [5, 1/32]], 1, 5, 1, Yes, z_threshold=(5-n*p)/sqrt(n*p*(1-p)), exact_lattice_not_normal_approximation_or_convergence_evidence)","explanation":"Compute the exact probabilities for every integer count0 through n and the inclusive CDF at the threshold. Retain the symbolic standardized threshold; neither that expression nor a finite lattice table establishes an asymptotic limit or an approximation error. The exact result, retaining the requested order, is (5, 1/2, [[0, 1/32], [1, 5/32], [2, 5/16], [3, 5/16], [4, 5/32], [5, 1/32]], 1, 5, 1, Yes, z_threshold=(5-n*p)/sqrt(n*p*(1-p)), exact_lattice_not_normal_approximation_or_convergence_evidence)."},"skills":["An Exact Binomial Lattice Is Not a Normal Approximation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r01-central-limit-theorem-finite-variance-iid-0069","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"aac95813752c2238cba75e38a2ce96a38d6b350383c15a545b0ab0740f8fc2ef","family_id":"qf-mod49-clt-binomial-lattice","seed":202611181069,"semantic_template":"clt-binomial-lattice|{n:6,success_probability:[1,2],threshold:0}"},"fields":{"title":"An Exact Binomial Lattice Is Not a Normal Approximation","stem":"Use Bernoulli success probability p=1/2, sample size n=6, integer threshold=0, and independent coordinates for this finite sum. Give (n, success_probability, binomial_distribution, total_probability, threshold, cdf_at_threshold, normalized, standardized_threshold_description, finite_scope).","answer":{"kind":"expression","value":"(6, 1/2, [[0, 1/64], [1, 3/32], [2, 15/64], [3, 5/16], [4, 15/64], [5, 3/32], [6, 1/64]], 1, 0, 1/64, Yes, z_threshold=(0-n*p)/sqrt(n*p*(1-p)), exact_lattice_not_normal_approximation_or_convergence_evidence)","explanation":"Compute the exact probabilities for every integer count0 through n and the inclusive CDF at the threshold. Retain the symbolic standardized threshold; neither that expression nor a finite lattice table establishes an asymptotic limit or an approximation error. The exact result, retaining the requested order, is (6, 1/2, [[0, 1/64], [1, 3/32], [2, 15/64], [3, 5/16], [4, 15/64], [5, 3/32], [6, 1/64]], 1, 0, 1/64, Yes, z_threshold=(0-n*p)/sqrt(n*p*(1-p)), exact_lattice_not_normal_approximation_or_convergence_evidence)."},"skills":["An Exact Binomial Lattice Is Not a Normal Approximation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r01-central-limit-theorem-finite-variance-iid-0070","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"80e071f75989fcb389867f20630a4ea636a51f9235b0b70952c3449cbcf17252","family_id":"qf-mod49-clt-binomial-lattice","seed":202611181070,"semantic_template":"clt-binomial-lattice|{n:6,success_probability:[1,2],threshold:1}"},"fields":{"title":"An Exact Binomial Lattice Is Not a Normal Approximation","stem":"Use Bernoulli success probability p=1/2, sample size n=6, integer threshold=1, and independent coordinates for this finite sum. Give (n, success_probability, binomial_distribution, total_probability, threshold, cdf_at_threshold, normalized, standardized_threshold_description, finite_scope).","answer":{"kind":"expression","value":"(6, 1/2, [[0, 1/64], [1, 3/32], [2, 15/64], [3, 5/16], [4, 15/64], [5, 3/32], [6, 1/64]], 1, 1, 7/64, Yes, z_threshold=(1-n*p)/sqrt(n*p*(1-p)), exact_lattice_not_normal_approximation_or_convergence_evidence)","explanation":"Compute the exact probabilities for every integer count0 through n and the inclusive CDF at the threshold. Retain the symbolic standardized threshold; neither that expression nor a finite lattice table establishes an asymptotic limit or an approximation error. The exact result, retaining the requested order, is (6, 1/2, [[0, 1/64], [1, 3/32], [2, 15/64], [3, 5/16], [4, 15/64], [5, 3/32], [6, 1/64]], 1, 1, 7/64, Yes, z_threshold=(1-n*p)/sqrt(n*p*(1-p)), exact_lattice_not_normal_approximation_or_convergence_evidence)."},"skills":["An Exact Binomial Lattice Is Not a Normal Approximation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r01-central-limit-theorem-finite-variance-iid-0071","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"553491c5a59c16eb474c7be4021518d85d705863d618dec8e6b6cbf239657f05","family_id":"qf-mod49-clt-binomial-lattice","seed":202611181071,"semantic_template":"clt-binomial-lattice|{n:6,success_probability:[1,2],threshold:2}"},"fields":{"title":"An Exact Binomial Lattice Is Not a Normal Approximation","stem":"Use Bernoulli success probability p=1/2, sample size n=6, integer threshold=2, and independent coordinates for this finite sum. Give (n, success_probability, binomial_distribution, total_probability, threshold, cdf_at_threshold, normalized, standardized_threshold_description, finite_scope).","answer":{"kind":"expression","value":"(6, 1/2, [[0, 1/64], [1, 3/32], [2, 15/64], [3, 5/16], [4, 15/64], [5, 3/32], [6, 1/64]], 1, 2, 11/32, Yes, z_threshold=(2-n*p)/sqrt(n*p*(1-p)), exact_lattice_not_normal_approximation_or_convergence_evidence)","explanation":"Compute the exact probabilities for every integer count0 through n and the inclusive CDF at the threshold. Retain the symbolic standardized threshold; neither that expression nor a finite lattice table establishes an asymptotic limit or an approximation error. The exact result, retaining the requested order, is (6, 1/2, [[0, 1/64], [1, 3/32], [2, 15/64], [3, 5/16], [4, 15/64], [5, 3/32], [6, 1/64]], 1, 2, 11/32, Yes, z_threshold=(2-n*p)/sqrt(n*p*(1-p)), exact_lattice_not_normal_approximation_or_convergence_evidence)."},"skills":["An Exact Binomial Lattice Is Not a Normal Approximation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r01-central-limit-theorem-finite-variance-iid-0072","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e9932a2781c56a8fb5ee686a1310152fc1f3eb934bfbfc5131589833889ed8b3","family_id":"qf-mod49-clt-binomial-lattice","seed":202611181072,"semantic_template":"clt-binomial-lattice|{n:6,success_probability:[1,2],threshold:3}"},"fields":{"title":"An Exact Binomial Lattice Is Not a Normal Approximation","stem":"Use Bernoulli success probability p=1/2, sample size n=6, integer threshold=3, and independent coordinates for this finite sum. Give (n, success_probability, binomial_distribution, total_probability, threshold, cdf_at_threshold, normalized, standardized_threshold_description, finite_scope).","answer":{"kind":"expression","value":"(6, 1/2, [[0, 1/64], [1, 3/32], [2, 15/64], [3, 5/16], [4, 15/64], [5, 3/32], [6, 1/64]], 1, 3, 21/32, Yes, z_threshold=(3-n*p)/sqrt(n*p*(1-p)), exact_lattice_not_normal_approximation_or_convergence_evidence)","explanation":"Compute the exact probabilities for every integer count0 through n and the inclusive CDF at the threshold. Retain the symbolic standardized threshold; neither that expression nor a finite lattice table establishes an asymptotic limit or an approximation error. The exact result, retaining the requested order, is (6, 1/2, [[0, 1/64], [1, 3/32], [2, 15/64], [3, 5/16], [4, 15/64], [5, 3/32], [6, 1/64]], 1, 3, 21/32, Yes, z_threshold=(3-n*p)/sqrt(n*p*(1-p)), exact_lattice_not_normal_approximation_or_convergence_evidence)."},"skills":["An Exact Binomial Lattice Is Not a Normal Approximation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r01-central-limit-theorem-finite-variance-iid-0073","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4e99a56969254e4793ab99edc280ece46b4232404a447cac9a32b0a5b44670e8","family_id":"qf-mod49-clt-binomial-lattice","seed":202611181073,"semantic_template":"clt-binomial-lattice|{n:6,success_probability:[1,2],threshold:4}"},"fields":{"title":"An Exact Binomial Lattice Is Not a Normal Approximation","stem":"Use Bernoulli success probability p=1/2, sample size n=6, integer threshold=4, and independent coordinates for this finite sum. Give (n, success_probability, binomial_distribution, total_probability, threshold, cdf_at_threshold, normalized, standardized_threshold_description, finite_scope).","answer":{"kind":"expression","value":"(6, 1/2, [[0, 1/64], [1, 3/32], [2, 15/64], [3, 5/16], [4, 15/64], [5, 3/32], [6, 1/64]], 1, 4, 57/64, Yes, z_threshold=(4-n*p)/sqrt(n*p*(1-p)), exact_lattice_not_normal_approximation_or_convergence_evidence)","explanation":"Compute the exact probabilities for every integer count0 through n and the inclusive CDF at the threshold. Retain the symbolic standardized threshold; neither that expression nor a finite lattice table establishes an asymptotic limit or an approximation error. The exact result, retaining the requested order, is (6, 1/2, [[0, 1/64], [1, 3/32], [2, 15/64], [3, 5/16], [4, 15/64], [5, 3/32], [6, 1/64]], 1, 4, 57/64, Yes, z_threshold=(4-n*p)/sqrt(n*p*(1-p)), exact_lattice_not_normal_approximation_or_convergence_evidence)."},"skills":["An Exact Binomial Lattice Is Not a Normal Approximation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r01-central-limit-theorem-finite-variance-iid-0074","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5da96055d8d70543d16914f320dfe9a843a0c86157b48be3d411d4f1b32d5e5f","family_id":"qf-mod49-clt-binomial-lattice","seed":202611181074,"semantic_template":"clt-binomial-lattice|{n:6,success_probability:[1,2],threshold:5}"},"fields":{"title":"An Exact Binomial Lattice Is Not a Normal Approximation","stem":"Use Bernoulli success probability p=1/2, sample size n=6, integer threshold=5, and independent coordinates for this finite sum. Give (n, success_probability, binomial_distribution, total_probability, threshold, cdf_at_threshold, normalized, standardized_threshold_description, finite_scope).","answer":{"kind":"expression","value":"(6, 1/2, [[0, 1/64], [1, 3/32], [2, 15/64], [3, 5/16], [4, 15/64], [5, 3/32], [6, 1/64]], 1, 5, 63/64, Yes, z_threshold=(5-n*p)/sqrt(n*p*(1-p)), exact_lattice_not_normal_approximation_or_convergence_evidence)","explanation":"Compute the exact probabilities for every integer count0 through n and the inclusive CDF at the threshold. Retain the symbolic standardized threshold; neither that expression nor a finite lattice table establishes an asymptotic limit or an approximation error. The exact result, retaining the requested order, is (6, 1/2, [[0, 1/64], [1, 3/32], [2, 15/64], [3, 5/16], [4, 15/64], [5, 3/32], [6, 1/64]], 1, 5, 63/64, Yes, z_threshold=(5-n*p)/sqrt(n*p*(1-p)), exact_lattice_not_normal_approximation_or_convergence_evidence)."},"skills":["An Exact Binomial Lattice Is Not a Normal Approximation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r01-central-limit-theorem-finite-variance-iid-0075","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a83a0fb7fe6e024125cde02116b5cffb10d420eed6cec6f1a84b76a44edacb76","family_id":"qf-mod49-clt-binomial-lattice","seed":202611181075,"semantic_template":"clt-binomial-lattice|{n:6,success_probability:[1,2],threshold:6}"},"fields":{"title":"An Exact Binomial Lattice Is Not a Normal Approximation","stem":"Use Bernoulli success probability p=1/2, sample size n=6, integer threshold=6, and independent coordinates for this finite sum. Give (n, success_probability, binomial_distribution, total_probability, threshold, cdf_at_threshold, normalized, standardized_threshold_description, finite_scope).","answer":{"kind":"expression","value":"(6, 1/2, [[0, 1/64], [1, 3/32], [2, 15/64], [3, 5/16], [4, 15/64], [5, 3/32], [6, 1/64]], 1, 6, 1, Yes, z_threshold=(6-n*p)/sqrt(n*p*(1-p)), exact_lattice_not_normal_approximation_or_convergence_evidence)","explanation":"Compute the exact probabilities for every integer count0 through n and the inclusive CDF at the threshold. Retain the symbolic standardized threshold; neither that expression nor a finite lattice table establishes an asymptotic limit or an approximation error. The exact result, retaining the requested order, is (6, 1/2, [[0, 1/64], [1, 3/32], [2, 15/64], [3, 5/16], [4, 15/64], [5, 3/32], [6, 1/64]], 1, 6, 1, Yes, z_threshold=(6-n*p)/sqrt(n*p*(1-p)), exact_lattice_not_normal_approximation_or_convergence_evidence)."},"skills":["An Exact Binomial Lattice Is Not a Normal Approximation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r01-central-limit-theorem-finite-variance-iid-0076","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c916430c874c29c3580e37f519f9cfe7f1618f3265a6eb3fe52a743fd8351e83","family_id":"qf-mod49-clt-application-boundary","seed":202611181076,"semantic_template":"clt-application-boundary|{mode:\"symbolic_application\",model:{law:[[-1,[1,2]],[-2,[1,2]]],n:2,process_certificate:\"infinite_iid\"}}"},"fields":{"title":"Declared Theorem Application, Failed Hypotheses and Finite Evidence Have Different Roles","stem":"Use mode=symbolic_application and complete model={\"values\":[-2,-1],\"probabilities\":[[1,2],[1,2]],\"process_certificate\":\"infinite_iid\",\"n\":2}. Give (mode, declared_process_certificate, mean, variance, n, standardization_description, named_theorem_conclusion, conditional_application_qualified).","answer":{"kind":"expression","value":"(symbolic_application, infinite_iid, -3/2, 1/4, 2, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), Z_n_converges_in_distribution_to_N(0,1), Yes)","explanation":"symbolic_application invokes the named CLT conditionally on the declared infinite IID law and positive variance; it is not a proof performed here. Different marginals exclude this IID theorem only. Zero variance makes this normalization undefined. n*sigma is the wrong scaling. One finite histogram does not establish a distributional limit. Keep the five branch-specific field layouts. The exact result, retaining the requested order, is (symbolic_application, infinite_iid, -3/2, 1/4, 2, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), Z_n_converges_in_distribution_to_N(0,1), Yes)."},"skills":["Declared Theorem Application, Failed Hypotheses and Finite Evidence Have Different Roles"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r01-central-limit-theorem-finite-variance-iid-0077","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fd696a039c5f97e7ab4489eca7bfc5ef527985588d4cbb93480225fd636ed26b","family_id":"qf-mod49-clt-application-boundary","seed":202611181077,"semantic_template":"clt-application-boundary|{mode:\"non_iid\",model:{law_a:[[-1,[1,2]],[0,[1,2]]],law_b:[[-1,[1,7]],[0,[6,7]]]}}"},"fields":{"title":"Declared Theorem Application, Failed Hypotheses and Finite Evidence Have Different Roles","stem":"Use mode=non_iid and complete model={\"values_a\":[-1,0],\"probabilities_a\":[[1,2],[1,2]],\"values_b\":[-1,0],\"probabilities_b\":[[1,7],[6,7]]}. Give (mode, first_marginal_law, second_marginal_law, identical_marginals, this_IID_theorem_applicable, scope).","answer":{"kind":"expression","value":"(non_iid, [[-1, 1/2], [0, 1/2]], [[-1, 1/7], [0, 6/7]], No, No, this_IID_CLT_not_applicable_not_a_general_counterexample)","explanation":"symbolic_application invokes the named CLT conditionally on the declared infinite IID law and positive variance; it is not a proof performed here. Different marginals exclude this IID theorem only. Zero variance makes this normalization undefined. n*sigma is the wrong scaling. One finite histogram does not establish a distributional limit. Keep the five branch-specific field layouts. The exact result, retaining the requested order, is (non_iid, [[-1, 1/2], [0, 1/2]], [[-1, 1/7], [0, 6/7]], No, No, this_IID_CLT_not_applicable_not_a_general_counterexample)."},"skills":["Declared Theorem Application, Failed Hypotheses and Finite Evidence Have Different Roles"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r01-central-limit-theorem-finite-variance-iid-0078","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8ba13981c9315f142b2924706b1a5b4764d3e3fe9b8e28228f81d10e7aefa2ff","family_id":"qf-mod49-clt-application-boundary","seed":202611181078,"semantic_template":"clt-application-boundary|{mode:\"zero_variance\",model:{n:4,value:-1}}"},"fields":{"title":"Declared Theorem Application, Failed Hypotheses and Finite Evidence Have Different Roles","stem":"Use mode=zero_variance and complete model={\"value\":-1,\"n\":4}. Give (mode, constant_value, n, variance, standardization_obstruction, this_standardization_claim).","answer":{"kind":"expression","value":"(zero_variance, -1, 4, 0, sigma=0_makes_standardization_undefined, No)","explanation":"symbolic_application invokes the named CLT conditionally on the declared infinite IID law and positive variance; it is not a proof performed here. Different marginals exclude this IID theorem only. Zero variance makes this normalization undefined. n*sigma is the wrong scaling. One finite histogram does not establish a distributional limit. Keep the five branch-specific field layouts. The exact result, retaining the requested order, is (zero_variance, -1, 4, 0, sigma=0_makes_standardization_undefined, No)."},"skills":["Declared Theorem Application, Failed Hypotheses and Finite Evidence Have Different Roles"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r01-central-limit-theorem-finite-variance-iid-0079","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"461606279384502fa06d8dbcf33ceca9c81d9d457f61207be7051f8ed267a9e8","family_id":"qf-mod49-clt-application-boundary","seed":202611181079,"semantic_template":"clt-application-boundary|{mode:\"wrong_scaling\",model:{law:[[1,[1,2]],[2,[1,2]]],n:5,process_certificate:\"infinite_iid\"}}"},"fields":{"title":"Declared Theorem Application, Failed Hypotheses and Finite Evidence Have Different Roles","stem":"Use mode=wrong_scaling and complete model={\"values\":[1,2],\"probabilities\":[[1,2],[1,2]],\"process_certificate\":\"infinite_iid\",\"n\":5}. Give (mode, mean, variance, n, proposed_denominator, required_denominator, proposed_scaling_claim).","answer":{"kind":"expression","value":"(wrong_scaling, 3/2, 1/4, 5, denominator=n*sigma, required=sigma*sqrt(n), No)","explanation":"symbolic_application invokes the named CLT conditionally on the declared infinite IID law and positive variance; it is not a proof performed here. Different marginals exclude this IID theorem only. Zero variance makes this normalization undefined. n*sigma is the wrong scaling. One finite histogram does not establish a distributional limit. Keep the five branch-specific field layouts. The exact result, retaining the requested order, is (wrong_scaling, 3/2, 1/4, 5, denominator=n*sigma, required=sigma*sqrt(n), No)."},"skills":["Declared Theorem Application, Failed Hypotheses and Finite Evidence Have Different Roles"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r01-central-limit-theorem-finite-variance-iid-0080","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b60ebeb0b2bddad7c9f27ee4edba7415a8a8c637c860e4dd3c540bb2452b1ed9","family_id":"qf-mod49-clt-application-boundary","seed":202611181080,"semantic_template":"clt-application-boundary|{mode:\"finite_histogram\",model:{law:[[2,[1,2]],[3,[1,2]]],n:6,process_certificate:\"infinite_iid\"}}"},"fields":{"title":"Declared Theorem Application, Failed Hypotheses and Finite Evidence Have Different Roles","stem":"Use mode=finite_histogram and complete model={\"values\":[2,3],\"probabilities\":[[1,2],[1,2]],\"process_certificate\":\"infinite_iid\",\"n\":6}. Give (mode, mean, variance, n, evidence_scope, limit_proof_claim, boundary).","answer":{"kind":"expression","value":"(finite_histogram, 5/2, 1/4, 6, one_finite_histogram, No, finite_histogram_does_not_prove_distributional_limit)","explanation":"symbolic_application invokes the named CLT conditionally on the declared infinite IID law and positive variance; it is not a proof performed here. Different marginals exclude this IID theorem only. Zero variance makes this normalization undefined. n*sigma is the wrong scaling. One finite histogram does not establish a distributional limit. Keep the five branch-specific field layouts. The exact result, retaining the requested order, is (finite_histogram, 5/2, 1/4, 6, one_finite_histogram, No, finite_histogram_does_not_prove_distributional_limit)."},"skills":["Declared Theorem Application, Failed Hypotheses and Finite Evidence Have Different Roles"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r01-central-limit-theorem-finite-variance-iid-0081","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1bb10205536613d76432e54f91d224973f87bc09f40aca402e9dcdf1bbe8c665","family_id":"qf-mod49-clt-application-boundary","seed":202611181081,"semantic_template":"clt-application-boundary|{mode:\"symbolic_application\",model:{law:[[-1,[2,3]],[-2,[1,3]]],n:2,process_certificate:\"infinite_iid\"}}"},"fields":{"title":"Declared Theorem Application, Failed Hypotheses and Finite Evidence Have Different Roles","stem":"Use mode=symbolic_application and complete model={\"values\":[-2,-1],\"probabilities\":[[1,3],[2,3]],\"process_certificate\":\"infinite_iid\",\"n\":2}. Give (mode, declared_process_certificate, mean, variance, n, standardization_description, named_theorem_conclusion, conditional_application_qualified).","answer":{"kind":"expression","value":"(symbolic_application, infinite_iid, -4/3, 2/9, 2, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), Z_n_converges_in_distribution_to_N(0,1), Yes)","explanation":"symbolic_application invokes the named CLT conditionally on the declared infinite IID law and positive variance; it is not a proof performed here. Different marginals exclude this IID theorem only. Zero variance makes this normalization undefined. n*sigma is the wrong scaling. One finite histogram does not establish a distributional limit. Keep the five branch-specific field layouts. The exact result, retaining the requested order, is (symbolic_application, infinite_iid, -4/3, 2/9, 2, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), Z_n_converges_in_distribution_to_N(0,1), Yes)."},"skills":["Declared Theorem Application, Failed Hypotheses and Finite Evidence Have Different Roles"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r01-central-limit-theorem-finite-variance-iid-0082","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"19ad3b34c073fae4cda9fe0e8d537d2ba8b34f53f063f22cd0dbe2cd073f9f11","family_id":"qf-mod49-clt-application-boundary","seed":202611181082,"semantic_template":"clt-application-boundary|{mode:\"non_iid\",model:{law_a:[[-1,[1,3]],[0,[2,3]]],law_b:[[-1,[1,7]],[0,[6,7]]]}}"},"fields":{"title":"Declared Theorem Application, Failed Hypotheses and Finite Evidence Have Different Roles","stem":"Use mode=non_iid and complete model={\"values_a\":[-1,0],\"probabilities_a\":[[1,3],[2,3]],\"values_b\":[-1,0],\"probabilities_b\":[[1,7],[6,7]]}. Give (mode, first_marginal_law, second_marginal_law, identical_marginals, this_IID_theorem_applicable, scope).","answer":{"kind":"expression","value":"(non_iid, [[-1, 1/3], [0, 2/3]], [[-1, 1/7], [0, 6/7]], No, No, this_IID_CLT_not_applicable_not_a_general_counterexample)","explanation":"symbolic_application invokes the named CLT conditionally on the declared infinite IID law and positive variance; it is not a proof performed here. Different marginals exclude this IID theorem only. Zero variance makes this normalization undefined. n*sigma is the wrong scaling. One finite histogram does not establish a distributional limit. Keep the five branch-specific field layouts. The exact result, retaining the requested order, is (non_iid, [[-1, 1/3], [0, 2/3]], [[-1, 1/7], [0, 6/7]], No, No, this_IID_CLT_not_applicable_not_a_general_counterexample)."},"skills":["Declared Theorem Application, Failed Hypotheses and Finite Evidence Have Different Roles"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r01-central-limit-theorem-finite-variance-iid-0083","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"21b83b5ce68791419dbf7e467ac70804bfe8aeb1131bc3af63d0c326f7dad56f","family_id":"qf-mod49-clt-application-boundary","seed":202611181083,"semantic_template":"clt-application-boundary|{mode:\"zero_variance\",model:{n:4,value:-3}}"},"fields":{"title":"Declared Theorem Application, Failed Hypotheses and Finite Evidence Have Different Roles","stem":"Use mode=zero_variance and complete model={\"value\":-3,\"n\":4}. Give (mode, constant_value, n, variance, standardization_obstruction, this_standardization_claim).","answer":{"kind":"expression","value":"(zero_variance, -3, 4, 0, sigma=0_makes_standardization_undefined, No)","explanation":"symbolic_application invokes the named CLT conditionally on the declared infinite IID law and positive variance; it is not a proof performed here. Different marginals exclude this IID theorem only. Zero variance makes this normalization undefined. n*sigma is the wrong scaling. One finite histogram does not establish a distributional limit. Keep the five branch-specific field layouts. The exact result, retaining the requested order, is (zero_variance, -3, 4, 0, sigma=0_makes_standardization_undefined, No)."},"skills":["Declared Theorem Application, Failed Hypotheses and Finite Evidence Have Different Roles"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r01-central-limit-theorem-finite-variance-iid-0084","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d2744bd081a97d0c66619e5dab87bcf07299ff0c92f1e8ff28bbd663d3b3c346","family_id":"qf-mod49-clt-application-boundary","seed":202611181084,"semantic_template":"clt-application-boundary|{mode:\"wrong_scaling\",model:{law:[[1,[1,3]],[2,[2,3]]],n:5,process_certificate:\"infinite_iid\"}}"},"fields":{"title":"Declared Theorem Application, Failed Hypotheses and Finite Evidence Have Different Roles","stem":"Use mode=wrong_scaling and complete model={\"values\":[1,2],\"probabilities\":[[1,3],[2,3]],\"process_certificate\":\"infinite_iid\",\"n\":5}. Give (mode, mean, variance, n, proposed_denominator, required_denominator, proposed_scaling_claim).","answer":{"kind":"expression","value":"(wrong_scaling, 5/3, 2/9, 5, denominator=n*sigma, required=sigma*sqrt(n), No)","explanation":"symbolic_application invokes the named CLT conditionally on the declared infinite IID law and positive variance; it is not a proof performed here. Different marginals exclude this IID theorem only. Zero variance makes this normalization undefined. n*sigma is the wrong scaling. One finite histogram does not establish a distributional limit. Keep the five branch-specific field layouts. The exact result, retaining the requested order, is (wrong_scaling, 5/3, 2/9, 5, denominator=n*sigma, required=sigma*sqrt(n), No)."},"skills":["Declared Theorem Application, Failed Hypotheses and Finite Evidence Have Different Roles"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r01-central-limit-theorem-finite-variance-iid-0085","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"55e238562f1e9cfc634be59de3474f0273b0a6fb1c7445fbb1fa00d76d7d68d7","family_id":"qf-mod49-clt-application-boundary","seed":202611181085,"semantic_template":"clt-application-boundary|{mode:\"finite_histogram\",model:{law:[[2,[1,3]],[3,[2,3]]],n:6,process_certificate:\"infinite_iid\"}}"},"fields":{"title":"Declared Theorem Application, Failed Hypotheses and Finite Evidence Have Different Roles","stem":"Use mode=finite_histogram and complete model={\"values\":[2,3],\"probabilities\":[[1,3],[2,3]],\"process_certificate\":\"infinite_iid\",\"n\":6}. Give (mode, mean, variance, n, evidence_scope, limit_proof_claim, boundary).","answer":{"kind":"expression","value":"(finite_histogram, 8/3, 2/9, 6, one_finite_histogram, No, finite_histogram_does_not_prove_distributional_limit)","explanation":"symbolic_application invokes the named CLT conditionally on the declared infinite IID law and positive variance; it is not a proof performed here. Different marginals exclude this IID theorem only. Zero variance makes this normalization undefined. n*sigma is the wrong scaling. One finite histogram does not establish a distributional limit. Keep the five branch-specific field layouts. The exact result, retaining the requested order, is (finite_histogram, 8/3, 2/9, 6, one_finite_histogram, No, finite_histogram_does_not_prove_distributional_limit)."},"skills":["Declared Theorem Application, Failed Hypotheses and Finite Evidence Have Different Roles"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r01-central-limit-theorem-finite-variance-iid-0086","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c3556898f8259ea46dbc4fb208e532fad625d818b700fad59b6a46827486519b","family_id":"qf-mod49-clt-application-boundary","seed":202611181086,"semantic_template":"clt-application-boundary|{mode:\"symbolic_application\",model:{law:[[-1,[3,4]],[-2,[1,4]]],n:2,process_certificate:\"infinite_iid\"}}"},"fields":{"title":"Declared Theorem Application, Failed Hypotheses and Finite Evidence Have Different Roles","stem":"Use mode=symbolic_application and complete model={\"values\":[-2,-1],\"probabilities\":[[1,4],[3,4]],\"process_certificate\":\"infinite_iid\",\"n\":2}. Give (mode, declared_process_certificate, mean, variance, n, standardization_description, named_theorem_conclusion, conditional_application_qualified).","answer":{"kind":"expression","value":"(symbolic_application, infinite_iid, -5/4, 3/16, 2, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), Z_n_converges_in_distribution_to_N(0,1), Yes)","explanation":"symbolic_application invokes the named CLT conditionally on the declared infinite IID law and positive variance; it is not a proof performed here. Different marginals exclude this IID theorem only. Zero variance makes this normalization undefined. n*sigma is the wrong scaling. One finite histogram does not establish a distributional limit. Keep the five branch-specific field layouts. The exact result, retaining the requested order, is (symbolic_application, infinite_iid, -5/4, 3/16, 2, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), Z_n_converges_in_distribution_to_N(0,1), Yes)."},"skills":["Declared Theorem Application, Failed Hypotheses and Finite Evidence Have Different Roles"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r01-central-limit-theorem-finite-variance-iid-0087","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"325de25cb073b30fb46a1ca3a99d6570117ee17759a21f3738e2ca947729bbab","family_id":"qf-mod49-clt-application-boundary","seed":202611181087,"semantic_template":"clt-application-boundary|{mode:\"non_iid\",model:{law_a:[[-1,[1,4]],[0,[3,4]]],law_b:[[-1,[1,7]],[0,[6,7]]]}}"},"fields":{"title":"Declared Theorem Application, Failed Hypotheses and Finite Evidence Have Different Roles","stem":"Use mode=non_iid and complete model={\"values_a\":[-1,0],\"probabilities_a\":[[1,4],[3,4]],\"values_b\":[-1,0],\"probabilities_b\":[[1,7],[6,7]]}. Give (mode, first_marginal_law, second_marginal_law, identical_marginals, this_IID_theorem_applicable, scope).","answer":{"kind":"expression","value":"(non_iid, [[-1, 1/4], [0, 3/4]], [[-1, 1/7], [0, 6/7]], No, No, this_IID_CLT_not_applicable_not_a_general_counterexample)","explanation":"symbolic_application invokes the named CLT conditionally on the declared infinite IID law and positive variance; it is not a proof performed here. Different marginals exclude this IID theorem only. Zero variance makes this normalization undefined. n*sigma is the wrong scaling. One finite histogram does not establish a distributional limit. Keep the five branch-specific field layouts. The exact result, retaining the requested order, is (non_iid, [[-1, 1/4], [0, 3/4]], [[-1, 1/7], [0, 6/7]], No, No, this_IID_CLT_not_applicable_not_a_general_counterexample)."},"skills":["Declared Theorem Application, Failed Hypotheses and Finite Evidence Have Different Roles"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r01-central-limit-theorem-finite-variance-iid-0088","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cb78cb5ff785802bf1c99e219c2edb3857c810737901df1a4b73e6e34f5c8f2c","family_id":"qf-mod49-clt-application-boundary","seed":202611181088,"semantic_template":"clt-application-boundary|{mode:\"zero_variance\",model:{n:4,value:2}}"},"fields":{"title":"Declared Theorem Application, Failed Hypotheses and Finite Evidence Have Different Roles","stem":"Use mode=zero_variance and complete model={\"value\":2,\"n\":4}. Give (mode, constant_value, n, variance, standardization_obstruction, this_standardization_claim).","answer":{"kind":"expression","value":"(zero_variance, 2, 4, 0, sigma=0_makes_standardization_undefined, No)","explanation":"symbolic_application invokes the named CLT conditionally on the declared infinite IID law and positive variance; it is not a proof performed here. Different marginals exclude this IID theorem only. Zero variance makes this normalization undefined. n*sigma is the wrong scaling. One finite histogram does not establish a distributional limit. Keep the five branch-specific field layouts. The exact result, retaining the requested order, is (zero_variance, 2, 4, 0, sigma=0_makes_standardization_undefined, No)."},"skills":["Declared Theorem Application, Failed Hypotheses and Finite Evidence Have Different Roles"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r01-central-limit-theorem-finite-variance-iid-0089","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"525a1336010fe0ef9401797662a4f660c66302ba149bec21eb1fe4d268af3be8","family_id":"qf-mod49-clt-application-boundary","seed":202611181089,"semantic_template":"clt-application-boundary|{mode:\"wrong_scaling\",model:{law:[[1,[1,4]],[2,[3,4]]],n:5,process_certificate:\"infinite_iid\"}}"},"fields":{"title":"Declared Theorem Application, Failed Hypotheses and Finite Evidence Have Different Roles","stem":"Use mode=wrong_scaling and complete model={\"values\":[1,2],\"probabilities\":[[1,4],[3,4]],\"process_certificate\":\"infinite_iid\",\"n\":5}. Give (mode, mean, variance, n, proposed_denominator, required_denominator, proposed_scaling_claim).","answer":{"kind":"expression","value":"(wrong_scaling, 7/4, 3/16, 5, denominator=n*sigma, required=sigma*sqrt(n), No)","explanation":"symbolic_application invokes the named CLT conditionally on the declared infinite IID law and positive variance; it is not a proof performed here. Different marginals exclude this IID theorem only. Zero variance makes this normalization undefined. n*sigma is the wrong scaling. One finite histogram does not establish a distributional limit. Keep the five branch-specific field layouts. The exact result, retaining the requested order, is (wrong_scaling, 7/4, 3/16, 5, denominator=n*sigma, required=sigma*sqrt(n), No)."},"skills":["Declared Theorem Application, Failed Hypotheses and Finite Evidence Have Different Roles"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r01-central-limit-theorem-finite-variance-iid-0090","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"693a2ed264ebf50bbabf342d1f474f21b638d9745fbf127524d0701b3529c676","family_id":"qf-mod49-clt-application-boundary","seed":202611181090,"semantic_template":"clt-application-boundary|{mode:\"finite_histogram\",model:{law:[[2,[1,4]],[3,[3,4]]],n:6,process_certificate:\"infinite_iid\"}}"},"fields":{"title":"Declared Theorem Application, Failed Hypotheses and Finite Evidence Have Different Roles","stem":"Use mode=finite_histogram and complete model={\"values\":[2,3],\"probabilities\":[[1,4],[3,4]],\"process_certificate\":\"infinite_iid\",\"n\":6}. Give (mode, mean, variance, n, evidence_scope, limit_proof_claim, boundary).","answer":{"kind":"expression","value":"(finite_histogram, 11/4, 3/16, 6, one_finite_histogram, No, finite_histogram_does_not_prove_distributional_limit)","explanation":"symbolic_application invokes the named CLT conditionally on the declared infinite IID law and positive variance; it is not a proof performed here. Different marginals exclude this IID theorem only. Zero variance makes this normalization undefined. n*sigma is the wrong scaling. One finite histogram does not establish a distributional limit. Keep the five branch-specific field layouts. The exact result, retaining the requested order, is (finite_histogram, 11/4, 3/16, 6, one_finite_histogram, No, finite_histogram_does_not_prove_distributional_limit)."},"skills":["Declared Theorem Application, Failed Hypotheses and Finite Evidence Have Different Roles"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r01-central-limit-theorem-finite-variance-iid-0091","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"48a720308011d05a9b1686b656fe47cb88298e5b35fff4f0f77cf3bec87bfaf7","family_id":"qf-mod49-clt-application-boundary","seed":202611181091,"semantic_template":"clt-application-boundary|{mode:\"symbolic_application\",model:{law:[[-1,[3,5]],[-2,[2,5]]],n:2,process_certificate:\"infinite_iid\"}}"},"fields":{"title":"Declared Theorem Application, Failed Hypotheses and Finite Evidence Have Different Roles","stem":"Use mode=symbolic_application and complete model={\"values\":[-2,-1],\"probabilities\":[[2,5],[3,5]],\"process_certificate\":\"infinite_iid\",\"n\":2}. Give (mode, declared_process_certificate, mean, variance, n, standardization_description, named_theorem_conclusion, conditional_application_qualified).","answer":{"kind":"expression","value":"(symbolic_application, infinite_iid, -7/5, 6/25, 2, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), Z_n_converges_in_distribution_to_N(0,1), Yes)","explanation":"symbolic_application invokes the named CLT conditionally on the declared infinite IID law and positive variance; it is not a proof performed here. Different marginals exclude this IID theorem only. Zero variance makes this normalization undefined. n*sigma is the wrong scaling. One finite histogram does not establish a distributional limit. Keep the five branch-specific field layouts. The exact result, retaining the requested order, is (symbolic_application, infinite_iid, -7/5, 6/25, 2, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), Z_n_converges_in_distribution_to_N(0,1), Yes)."},"skills":["Declared Theorem Application, Failed Hypotheses and Finite Evidence Have Different Roles"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r01-central-limit-theorem-finite-variance-iid-0092","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0c7c16a336b0f984ec036e550336c18275dfcd2930acc576aaecc2142bd97eae","family_id":"qf-mod49-clt-application-boundary","seed":202611181092,"semantic_template":"clt-application-boundary|{mode:\"non_iid\",model:{law_a:[[-1,[2,5]],[0,[3,5]]],law_b:[[-1,[1,7]],[0,[6,7]]]}}"},"fields":{"title":"Declared Theorem Application, Failed Hypotheses and Finite Evidence Have Different Roles","stem":"Use mode=non_iid and complete model={\"values_a\":[-1,0],\"probabilities_a\":[[2,5],[3,5]],\"values_b\":[-1,0],\"probabilities_b\":[[1,7],[6,7]]}. Give (mode, first_marginal_law, second_marginal_law, identical_marginals, this_IID_theorem_applicable, scope).","answer":{"kind":"expression","value":"(non_iid, [[-1, 2/5], [0, 3/5]], [[-1, 1/7], [0, 6/7]], No, No, this_IID_CLT_not_applicable_not_a_general_counterexample)","explanation":"symbolic_application invokes the named CLT conditionally on the declared infinite IID law and positive variance; it is not a proof performed here. Different marginals exclude this IID theorem only. Zero variance makes this normalization undefined. n*sigma is the wrong scaling. One finite histogram does not establish a distributional limit. Keep the five branch-specific field layouts. The exact result, retaining the requested order, is (non_iid, [[-1, 2/5], [0, 3/5]], [[-1, 1/7], [0, 6/7]], No, No, this_IID_CLT_not_applicable_not_a_general_counterexample)."},"skills":["Declared Theorem Application, Failed Hypotheses and Finite Evidence Have Different Roles"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r01-central-limit-theorem-finite-variance-iid-0093","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"99456ae8dbe625e34cd5a3f4ccd813511987780b18f425700b26a7fa077b8c9e","family_id":"qf-mod49-clt-application-boundary","seed":202611181093,"semantic_template":"clt-application-boundary|{mode:\"zero_variance\",model:{n:4,value:0}}"},"fields":{"title":"Declared Theorem Application, Failed Hypotheses and Finite Evidence Have Different Roles","stem":"Use mode=zero_variance and complete model={\"value\":0,\"n\":4}. Give (mode, constant_value, n, variance, standardization_obstruction, this_standardization_claim).","answer":{"kind":"expression","value":"(zero_variance, 0, 4, 0, sigma=0_makes_standardization_undefined, No)","explanation":"symbolic_application invokes the named CLT conditionally on the declared infinite IID law and positive variance; it is not a proof performed here. Different marginals exclude this IID theorem only. Zero variance makes this normalization undefined. n*sigma is the wrong scaling. One finite histogram does not establish a distributional limit. Keep the five branch-specific field layouts. The exact result, retaining the requested order, is (zero_variance, 0, 4, 0, sigma=0_makes_standardization_undefined, No)."},"skills":["Declared Theorem Application, Failed Hypotheses and Finite Evidence Have Different Roles"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r01-central-limit-theorem-finite-variance-iid-0094","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"18d343d97c8109cf9e2f760746857ba399c5fd9eab1419217d3a6eff9dfb1cfb","family_id":"qf-mod49-clt-application-boundary","seed":202611181094,"semantic_template":"clt-application-boundary|{mode:\"wrong_scaling\",model:{law:[[1,[2,5]],[2,[3,5]]],n:5,process_certificate:\"infinite_iid\"}}"},"fields":{"title":"Declared Theorem Application, Failed Hypotheses and Finite Evidence Have Different Roles","stem":"Use mode=wrong_scaling and complete model={\"values\":[1,2],\"probabilities\":[[2,5],[3,5]],\"process_certificate\":\"infinite_iid\",\"n\":5}. Give (mode, mean, variance, n, proposed_denominator, required_denominator, proposed_scaling_claim).","answer":{"kind":"expression","value":"(wrong_scaling, 8/5, 6/25, 5, denominator=n*sigma, required=sigma*sqrt(n), No)","explanation":"symbolic_application invokes the named CLT conditionally on the declared infinite IID law and positive variance; it is not a proof performed here. Different marginals exclude this IID theorem only. Zero variance makes this normalization undefined. n*sigma is the wrong scaling. One finite histogram does not establish a distributional limit. Keep the five branch-specific field layouts. The exact result, retaining the requested order, is (wrong_scaling, 8/5, 6/25, 5, denominator=n*sigma, required=sigma*sqrt(n), No)."},"skills":["Declared Theorem Application, Failed Hypotheses and Finite Evidence Have Different Roles"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r01-central-limit-theorem-finite-variance-iid-0095","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4a0f90d1aaed6e7de097ac5b238dbc9ee0cb3abd83f7f2c7960e5d76c0068fe3","family_id":"qf-mod49-clt-application-boundary","seed":202611181095,"semantic_template":"clt-application-boundary|{mode:\"finite_histogram\",model:{law:[[2,[2,5]],[3,[3,5]]],n:6,process_certificate:\"infinite_iid\"}}"},"fields":{"title":"Declared Theorem Application, Failed Hypotheses and Finite Evidence Have Different Roles","stem":"Use mode=finite_histogram and complete model={\"values\":[2,3],\"probabilities\":[[2,5],[3,5]],\"process_certificate\":\"infinite_iid\",\"n\":6}. Give (mode, mean, variance, n, evidence_scope, limit_proof_claim, boundary).","answer":{"kind":"expression","value":"(finite_histogram, 13/5, 6/25, 6, one_finite_histogram, No, finite_histogram_does_not_prove_distributional_limit)","explanation":"symbolic_application invokes the named CLT conditionally on the declared infinite IID law and positive variance; it is not a proof performed here. Different marginals exclude this IID theorem only. Zero variance makes this normalization undefined. n*sigma is the wrong scaling. One finite histogram does not establish a distributional limit. Keep the five branch-specific field layouts. The exact result, retaining the requested order, is (finite_histogram, 13/5, 6/25, 6, one_finite_histogram, No, finite_histogram_does_not_prove_distributional_limit)."},"skills":["Declared Theorem Application, Failed Hypotheses and Finite Evidence Have Different Roles"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r01-central-limit-theorem-finite-variance-iid-0096","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d0d8420629b9805919ce839d42033f18ddb162da170746403f05124e9ccf1edc","family_id":"qf-mod49-clt-application-boundary","seed":202611181096,"semantic_template":"clt-application-boundary|{mode:\"symbolic_application\",model:{law:[[-1,[4,5]],[-2,[1,5]]],n:2,process_certificate:\"infinite_iid\"}}"},"fields":{"title":"Declared Theorem Application, Failed Hypotheses and Finite Evidence Have Different Roles","stem":"Use mode=symbolic_application and complete model={\"values\":[-2,-1],\"probabilities\":[[1,5],[4,5]],\"process_certificate\":\"infinite_iid\",\"n\":2}. Give (mode, declared_process_certificate, mean, variance, n, standardization_description, named_theorem_conclusion, conditional_application_qualified).","answer":{"kind":"expression","value":"(symbolic_application, infinite_iid, -6/5, 4/25, 2, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), Z_n_converges_in_distribution_to_N(0,1), Yes)","explanation":"symbolic_application invokes the named CLT conditionally on the declared infinite IID law and positive variance; it is not a proof performed here. Different marginals exclude this IID theorem only. Zero variance makes this normalization undefined. n*sigma is the wrong scaling. One finite histogram does not establish a distributional limit. Keep the five branch-specific field layouts. The exact result, retaining the requested order, is (symbolic_application, infinite_iid, -6/5, 4/25, 2, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), Z_n_converges_in_distribution_to_N(0,1), Yes)."},"skills":["Declared Theorem Application, Failed Hypotheses and Finite Evidence Have Different Roles"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r01-central-limit-theorem-finite-variance-iid-0097","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1fbd99de225e1a33fa031ef3c6f5ff34a87a939a728f46089fac5091008b840f","family_id":"qf-mod49-clt-application-boundary","seed":202611181097,"semantic_template":"clt-application-boundary|{mode:\"non_iid\",model:{law_a:[[-1,[1,5]],[0,[4,5]]],law_b:[[-1,[1,7]],[0,[6,7]]]}}"},"fields":{"title":"Declared Theorem Application, Failed Hypotheses and Finite Evidence Have Different Roles","stem":"Use mode=non_iid and complete model={\"values_a\":[-1,0],\"probabilities_a\":[[1,5],[4,5]],\"values_b\":[-1,0],\"probabilities_b\":[[1,7],[6,7]]}. Give (mode, first_marginal_law, second_marginal_law, identical_marginals, this_IID_theorem_applicable, scope).","answer":{"kind":"expression","value":"(non_iid, [[-1, 1/5], [0, 4/5]], [[-1, 1/7], [0, 6/7]], No, No, this_IID_CLT_not_applicable_not_a_general_counterexample)","explanation":"symbolic_application invokes the named CLT conditionally on the declared infinite IID law and positive variance; it is not a proof performed here. Different marginals exclude this IID theorem only. Zero variance makes this normalization undefined. n*sigma is the wrong scaling. One finite histogram does not establish a distributional limit. Keep the five branch-specific field layouts. The exact result, retaining the requested order, is (non_iid, [[-1, 1/5], [0, 4/5]], [[-1, 1/7], [0, 6/7]], No, No, this_IID_CLT_not_applicable_not_a_general_counterexample)."},"skills":["Declared Theorem Application, Failed Hypotheses and Finite Evidence Have Different Roles"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r01-central-limit-theorem-finite-variance-iid-0098","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"30379825074dce82e8718d14b966dc64b2f44b152c289f49a513ad2f3b30b0ec","family_id":"qf-mod49-clt-application-boundary","seed":202611181098,"semantic_template":"clt-application-boundary|{mode:\"zero_variance\",model:{n:4,value:-2}}"},"fields":{"title":"Declared Theorem Application, Failed Hypotheses and Finite Evidence Have Different Roles","stem":"Use mode=zero_variance and complete model={\"value\":-2,\"n\":4}. Give (mode, constant_value, n, variance, standardization_obstruction, this_standardization_claim).","answer":{"kind":"expression","value":"(zero_variance, -2, 4, 0, sigma=0_makes_standardization_undefined, No)","explanation":"symbolic_application invokes the named CLT conditionally on the declared infinite IID law and positive variance; it is not a proof performed here. Different marginals exclude this IID theorem only. Zero variance makes this normalization undefined. n*sigma is the wrong scaling. One finite histogram does not establish a distributional limit. Keep the five branch-specific field layouts. The exact result, retaining the requested order, is (zero_variance, -2, 4, 0, sigma=0_makes_standardization_undefined, No)."},"skills":["Declared Theorem Application, Failed Hypotheses and Finite Evidence Have Different Roles"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r01-central-limit-theorem-finite-variance-iid-0099","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"edefa987438d9b0811219e2aa2382e867a9a5908e8fb38a7d5e1a7300e61ad93","family_id":"qf-mod49-clt-application-boundary","seed":202611181099,"semantic_template":"clt-application-boundary|{mode:\"wrong_scaling\",model:{law:[[1,[1,5]],[2,[4,5]]],n:5,process_certificate:\"infinite_iid\"}}"},"fields":{"title":"Declared Theorem Application, Failed Hypotheses and Finite Evidence Have Different Roles","stem":"Use mode=wrong_scaling and complete model={\"values\":[1,2],\"probabilities\":[[1,5],[4,5]],\"process_certificate\":\"infinite_iid\",\"n\":5}. Give (mode, mean, variance, n, proposed_denominator, required_denominator, proposed_scaling_claim).","answer":{"kind":"expression","value":"(wrong_scaling, 9/5, 4/25, 5, denominator=n*sigma, required=sigma*sqrt(n), No)","explanation":"symbolic_application invokes the named CLT conditionally on the declared infinite IID law and positive variance; it is not a proof performed here. Different marginals exclude this IID theorem only. Zero variance makes this normalization undefined. n*sigma is the wrong scaling. One finite histogram does not establish a distributional limit. Keep the five branch-specific field layouts. The exact result, retaining the requested order, is (wrong_scaling, 9/5, 4/25, 5, denominator=n*sigma, required=sigma*sqrt(n), No)."},"skills":["Declared Theorem Application, Failed Hypotheses and Finite Evidence Have Different Roles"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r01-central-limit-theorem-finite-variance-iid-0100","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"663e5cf4e7e385dcf37c2992bfc9de0a8a90b0a666b6126892e9f214f35d0612","family_id":"qf-mod49-clt-application-boundary","seed":202611181100,"semantic_template":"clt-application-boundary|{mode:\"finite_histogram\",model:{law:[[2,[1,5]],[3,[4,5]]],n:6,process_certificate:\"infinite_iid\"}}"},"fields":{"title":"Declared Theorem Application, Failed Hypotheses and Finite Evidence Have Different Roles","stem":"Use mode=finite_histogram and complete model={\"values\":[2,3],\"probabilities\":[[1,5],[4,5]],\"process_certificate\":\"infinite_iid\",\"n\":6}. Give (mode, mean, variance, n, evidence_scope, limit_proof_claim, boundary).","answer":{"kind":"expression","value":"(finite_histogram, 14/5, 4/25, 6, one_finite_histogram, No, finite_histogram_does_not_prove_distributional_limit)","explanation":"symbolic_application invokes the named CLT conditionally on the declared infinite IID law and positive variance; it is not a proof performed here. Different marginals exclude this IID theorem only. Zero variance makes this normalization undefined. n*sigma is the wrong scaling. One finite histogram does not establish a distributional limit. Keep the five branch-specific field layouts. The exact result, retaining the requested order, is (finite_histogram, 14/5, 4/25, 6, one_finite_histogram, No, finite_histogram_does_not_prove_distributional_limit)."},"skills":["Declared Theorem Application, Failed Hypotheses and Finite Evidence Have Different Roles"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}