{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r06-iid-sequence-0001","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a32749562e03bfaf5241c7230e98a8d5fd08e0a93d42631f3a6661abfe2a803f","family_id":"qf-mod29-iid-bernoulli-cylinder","seed":202610296001,"semantic_template":"qf-mod29-iid-bernoulli-cylinder/{\"p\":[1,29],\"word\":[0,0,0,0]}"},"fields":{"title":"Exact Bernoulli Cylinder Probability under a Declared Product Law","stem":"Use the explicitly declared Bernoulli product-law model with exact p=1/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. Ordered finite binary word=[0, 0, 0, 0]. Give (word,ones,zeros,p,probability,symbolic_product_law).","answer":{"kind":"expression","value":"([0, 0, 0, 0], 0, 4, 1/29, 614656/707281, symbolic_product_law)","explanation":"Count ones and zeros and calculate p^ones*(1−p)^zeros exactly. Empty products equal 1, including degenerate p=0 or p=1. The arbitrary-word formula is the declared symbolic product-law certificate; this finite calculation alone is not a new proof of infinite IID. The exact result, retaining the requested order, is ([0, 0, 0, 0], 0, 4, 1/29, 614656/707281, symbolic_product_law)."},"skills":["Exact Bernoulli Cylinder Probability under a Declared Product Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r06-iid-sequence-0002","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5fcbf4a19847cbae8897b07eb68ef20779da13e091ffcbb57f6622c2e56cdcf2","family_id":"qf-mod29-iid-bernoulli-cylinder","seed":202610296002,"semantic_template":"qf-mod29-iid-bernoulli-cylinder/{\"p\":[2,29],\"word\":[1,0,0,0]}"},"fields":{"title":"Exact Bernoulli Cylinder Probability under a Declared Product Law","stem":"Use the explicitly declared Bernoulli product-law model with exact p=2/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. Ordered finite binary word=[1, 0, 0, 0]. Give (word,ones,zeros,p,probability,symbolic_product_law).","answer":{"kind":"expression","value":"([1, 0, 0, 0], 1, 3, 2/29, 39366/707281, symbolic_product_law)","explanation":"Count ones and zeros and calculate p^ones*(1−p)^zeros exactly. Empty products equal 1, including degenerate p=0 or p=1. The arbitrary-word formula is the declared symbolic product-law certificate; this finite calculation alone is not a new proof of infinite IID. The exact result, retaining the requested order, is ([1, 0, 0, 0], 1, 3, 2/29, 39366/707281, symbolic_product_law)."},"skills":["Exact Bernoulli Cylinder Probability under a Declared Product Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r06-iid-sequence-0003","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"20c00f52808a975604f7b2dfe79bacb404c62361e5154f29114458508683ec5b","family_id":"qf-mod29-iid-bernoulli-cylinder","seed":202610296003,"semantic_template":"qf-mod29-iid-bernoulli-cylinder/{\"p\":[3,29],\"word\":[0,1,0,0]}"},"fields":{"title":"Exact Bernoulli Cylinder Probability under a Declared Product Law","stem":"Use the explicitly declared Bernoulli product-law model with exact p=3/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. Ordered finite binary word=[0, 1, 0, 0]. Give (word,ones,zeros,p,probability,symbolic_product_law).","answer":{"kind":"expression","value":"([0, 1, 0, 0], 1, 3, 3/29, 52728/707281, symbolic_product_law)","explanation":"Count ones and zeros and calculate p^ones*(1−p)^zeros exactly. Empty products equal 1, including degenerate p=0 or p=1. The arbitrary-word formula is the declared symbolic product-law certificate; this finite calculation alone is not a new proof of infinite IID. The exact result, retaining the requested order, is ([0, 1, 0, 0], 1, 3, 3/29, 52728/707281, symbolic_product_law)."},"skills":["Exact Bernoulli Cylinder Probability under a Declared Product Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r06-iid-sequence-0004","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b86a48bb4ae9fa445f33afb86e30713dad1e6fada58b9476552aed693f2f1c01","family_id":"qf-mod29-iid-bernoulli-cylinder","seed":202610296004,"semantic_template":"qf-mod29-iid-bernoulli-cylinder/{\"p\":[4,29],\"word\":[1,1,0,0]}"},"fields":{"title":"Exact Bernoulli Cylinder Probability under a Declared Product Law","stem":"Use the explicitly declared Bernoulli product-law model with exact p=4/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. Ordered finite binary word=[1, 1, 0, 0]. Give (word,ones,zeros,p,probability,symbolic_product_law).","answer":{"kind":"expression","value":"([1, 1, 0, 0], 2, 2, 4/29, 10000/707281, symbolic_product_law)","explanation":"Count ones and zeros and calculate p^ones*(1−p)^zeros exactly. Empty products equal 1, including degenerate p=0 or p=1. The arbitrary-word formula is the declared symbolic product-law certificate; this finite calculation alone is not a new proof of infinite IID. The exact result, retaining the requested order, is ([1, 1, 0, 0], 2, 2, 4/29, 10000/707281, symbolic_product_law)."},"skills":["Exact Bernoulli Cylinder Probability under a Declared Product Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r06-iid-sequence-0005","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"303a85237bec74d345855249b68424e0dae8b1ec1e63c9a30bd2cf5469e6dd26","family_id":"qf-mod29-iid-bernoulli-cylinder","seed":202610296005,"semantic_template":"qf-mod29-iid-bernoulli-cylinder/{\"p\":[5,29],\"word\":[0,0,1,0]}"},"fields":{"title":"Exact Bernoulli Cylinder Probability under a Declared Product Law","stem":"Use the explicitly declared Bernoulli product-law model with exact p=5/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. Ordered finite binary word=[0, 0, 1, 0]. Give (word,ones,zeros,p,probability,symbolic_product_law).","answer":{"kind":"expression","value":"([0, 0, 1, 0], 1, 3, 5/29, 69120/707281, symbolic_product_law)","explanation":"Count ones and zeros and calculate p^ones*(1−p)^zeros exactly. Empty products equal 1, including degenerate p=0 or p=1. The arbitrary-word formula is the declared symbolic product-law certificate; this finite calculation alone is not a new proof of infinite IID. The exact result, retaining the requested order, is ([0, 0, 1, 0], 1, 3, 5/29, 69120/707281, symbolic_product_law)."},"skills":["Exact Bernoulli Cylinder Probability under a Declared Product Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r06-iid-sequence-0006","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"74e13bbb68f129370ae0614844805fd82aba1ccc0447eb544f1490bb2273249e","family_id":"qf-mod29-iid-bernoulli-cylinder","seed":202610296006,"semantic_template":"qf-mod29-iid-bernoulli-cylinder/{\"p\":[6,29],\"word\":[1,0,1,0]}"},"fields":{"title":"Exact Bernoulli Cylinder Probability under a Declared Product Law","stem":"Use the explicitly declared Bernoulli product-law model with exact p=6/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. Ordered finite binary word=[1, 0, 1, 0]. Give (word,ones,zeros,p,probability,symbolic_product_law).","answer":{"kind":"expression","value":"([1, 0, 1, 0], 2, 2, 6/29, 19044/707281, symbolic_product_law)","explanation":"Count ones and zeros and calculate p^ones*(1−p)^zeros exactly. Empty products equal 1, including degenerate p=0 or p=1. The arbitrary-word formula is the declared symbolic product-law certificate; this finite calculation alone is not a new proof of infinite IID. The exact result, retaining the requested order, is ([1, 0, 1, 0], 2, 2, 6/29, 19044/707281, symbolic_product_law)."},"skills":["Exact Bernoulli Cylinder Probability under a Declared Product Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r06-iid-sequence-0007","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6fdcb374985083f636e5db37307d00f7a8ee1ae9287fb41222f4a890bdb1d51c","family_id":"qf-mod29-iid-bernoulli-cylinder","seed":202610296007,"semantic_template":"qf-mod29-iid-bernoulli-cylinder/{\"p\":[7,29],\"word\":[0,1,1,0]}"},"fields":{"title":"Exact Bernoulli Cylinder Probability under a Declared Product Law","stem":"Use the explicitly declared Bernoulli product-law model with exact p=7/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. Ordered finite binary word=[0, 1, 1, 0]. Give (word,ones,zeros,p,probability,symbolic_product_law).","answer":{"kind":"expression","value":"([0, 1, 1, 0], 2, 2, 7/29, 23716/707281, symbolic_product_law)","explanation":"Count ones and zeros and calculate p^ones*(1−p)^zeros exactly. Empty products equal 1, including degenerate p=0 or p=1. The arbitrary-word formula is the declared symbolic product-law certificate; this finite calculation alone is not a new proof of infinite IID. The exact result, retaining the requested order, is ([0, 1, 1, 0], 2, 2, 7/29, 23716/707281, symbolic_product_law)."},"skills":["Exact Bernoulli Cylinder Probability under a Declared Product Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r06-iid-sequence-0008","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b4594a3730f6b980c0aaf40105782138c3fe20652584623bc5b913857e73150d","family_id":"qf-mod29-iid-bernoulli-cylinder","seed":202610296008,"semantic_template":"qf-mod29-iid-bernoulli-cylinder/{\"p\":[8,29],\"word\":[1,1,1,0]}"},"fields":{"title":"Exact Bernoulli Cylinder Probability under a Declared Product Law","stem":"Use the explicitly declared Bernoulli product-law model with exact p=8/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. Ordered finite binary word=[1, 1, 1, 0]. Give (word,ones,zeros,p,probability,symbolic_product_law).","answer":{"kind":"expression","value":"([1, 1, 1, 0], 3, 1, 8/29, 10752/707281, symbolic_product_law)","explanation":"Count ones and zeros and calculate p^ones*(1−p)^zeros exactly. Empty products equal 1, including degenerate p=0 or p=1. The arbitrary-word formula is the declared symbolic product-law certificate; this finite calculation alone is not a new proof of infinite IID. The exact result, retaining the requested order, is ([1, 1, 1, 0], 3, 1, 8/29, 10752/707281, symbolic_product_law)."},"skills":["Exact Bernoulli Cylinder Probability under a Declared Product Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r06-iid-sequence-0009","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5573b78fe3c54f6b91b0a8ec8c95f2e847831074ea83eb3f7ed274d68f477133","family_id":"qf-mod29-iid-bernoulli-cylinder","seed":202610296009,"semantic_template":"qf-mod29-iid-bernoulli-cylinder/{\"p\":[9,29],\"word\":[0,0,0,1]}"},"fields":{"title":"Exact Bernoulli Cylinder Probability under a Declared Product Law","stem":"Use the explicitly declared Bernoulli product-law model with exact p=9/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. Ordered finite binary word=[0, 0, 0, 1]. Give (word,ones,zeros,p,probability,symbolic_product_law).","answer":{"kind":"expression","value":"([0, 0, 0, 1], 1, 3, 9/29, 72000/707281, symbolic_product_law)","explanation":"Count ones and zeros and calculate p^ones*(1−p)^zeros exactly. Empty products equal 1, including degenerate p=0 or p=1. The arbitrary-word formula is the declared symbolic product-law certificate; this finite calculation alone is not a new proof of infinite IID. The exact result, retaining the requested order, is ([0, 0, 0, 1], 1, 3, 9/29, 72000/707281, symbolic_product_law)."},"skills":["Exact Bernoulli Cylinder Probability under a Declared Product Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r06-iid-sequence-0010","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"95ffb800fd904cebcc4a2c53b0cec747f0e6cf33ae39207700479163eacc88e4","family_id":"qf-mod29-iid-bernoulli-cylinder","seed":202610296010,"semantic_template":"qf-mod29-iid-bernoulli-cylinder/{\"p\":[10,29],\"word\":[1,0,0,1]}"},"fields":{"title":"Exact Bernoulli Cylinder Probability under a Declared Product Law","stem":"Use the explicitly declared Bernoulli product-law model with exact p=10/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. Ordered finite binary word=[1, 0, 0, 1]. Give (word,ones,zeros,p,probability,symbolic_product_law).","answer":{"kind":"expression","value":"([1, 0, 0, 1], 2, 2, 10/29, 36100/707281, symbolic_product_law)","explanation":"Count ones and zeros and calculate p^ones*(1−p)^zeros exactly. Empty products equal 1, including degenerate p=0 or p=1. The arbitrary-word formula is the declared symbolic product-law certificate; this finite calculation alone is not a new proof of infinite IID. The exact result, retaining the requested order, is ([1, 0, 0, 1], 2, 2, 10/29, 36100/707281, symbolic_product_law)."},"skills":["Exact Bernoulli Cylinder Probability under a Declared Product Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r06-iid-sequence-0011","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"51828e8fb921644d7dfa206528330e68f8a47a9b911244dedb48da770045e026","family_id":"qf-mod29-iid-bernoulli-cylinder","seed":202610296011,"semantic_template":"qf-mod29-iid-bernoulli-cylinder/{\"p\":[11,29],\"word\":[0,1,0,1]}"},"fields":{"title":"Exact Bernoulli Cylinder Probability under a Declared Product Law","stem":"Use the explicitly declared Bernoulli product-law model with exact p=11/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. Ordered finite binary word=[0, 1, 0, 1]. Give (word,ones,zeros,p,probability,symbolic_product_law).","answer":{"kind":"expression","value":"([0, 1, 0, 1], 2, 2, 11/29, 39204/707281, symbolic_product_law)","explanation":"Count ones and zeros and calculate p^ones*(1−p)^zeros exactly. Empty products equal 1, including degenerate p=0 or p=1. The arbitrary-word formula is the declared symbolic product-law certificate; this finite calculation alone is not a new proof of infinite IID. The exact result, retaining the requested order, is ([0, 1, 0, 1], 2, 2, 11/29, 39204/707281, symbolic_product_law)."},"skills":["Exact Bernoulli Cylinder Probability under a Declared Product Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r06-iid-sequence-0012","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9aa43f5aad89ab490638df243c656367bb0631b82d0bc4e89b8b14efae07b681","family_id":"qf-mod29-iid-bernoulli-cylinder","seed":202610296012,"semantic_template":"qf-mod29-iid-bernoulli-cylinder/{\"p\":[12,29],\"word\":[1,1,0,1]}"},"fields":{"title":"Exact Bernoulli Cylinder Probability under a Declared Product Law","stem":"Use the explicitly declared Bernoulli product-law model with exact p=12/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. Ordered finite binary word=[1, 1, 0, 1]. Give (word,ones,zeros,p,probability,symbolic_product_law).","answer":{"kind":"expression","value":"([1, 1, 0, 1], 3, 1, 12/29, 29376/707281, symbolic_product_law)","explanation":"Count ones and zeros and calculate p^ones*(1−p)^zeros exactly. Empty products equal 1, including degenerate p=0 or p=1. The arbitrary-word formula is the declared symbolic product-law certificate; this finite calculation alone is not a new proof of infinite IID. The exact result, retaining the requested order, is ([1, 1, 0, 1], 3, 1, 12/29, 29376/707281, symbolic_product_law)."},"skills":["Exact Bernoulli Cylinder Probability under a Declared Product Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r06-iid-sequence-0013","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"52d29433c966b9902b522c9af187f71690da04b0dd9812e7ff6a17453b9724e5","family_id":"qf-mod29-iid-bernoulli-cylinder","seed":202610296013,"semantic_template":"qf-mod29-iid-bernoulli-cylinder/{\"p\":[13,29],\"word\":[0,0,1,1]}"},"fields":{"title":"Exact Bernoulli Cylinder Probability under a Declared Product Law","stem":"Use the explicitly declared Bernoulli product-law model with exact p=13/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. Ordered finite binary word=[0, 0, 1, 1]. Give (word,ones,zeros,p,probability,symbolic_product_law).","answer":{"kind":"expression","value":"([0, 0, 1, 1], 2, 2, 13/29, 43264/707281, symbolic_product_law)","explanation":"Count ones and zeros and calculate p^ones*(1−p)^zeros exactly. Empty products equal 1, including degenerate p=0 or p=1. The arbitrary-word formula is the declared symbolic product-law certificate; this finite calculation alone is not a new proof of infinite IID. The exact result, retaining the requested order, is ([0, 0, 1, 1], 2, 2, 13/29, 43264/707281, symbolic_product_law)."},"skills":["Exact Bernoulli Cylinder Probability under a Declared Product Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r06-iid-sequence-0014","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5c8bdd8e1bf46c0ee1181fd4e091d249fc8550207008c68e4858fb5d68a4fbe8","family_id":"qf-mod29-iid-bernoulli-cylinder","seed":202610296014,"semantic_template":"qf-mod29-iid-bernoulli-cylinder/{\"p\":[14,29],\"word\":[1,0,1,1]}"},"fields":{"title":"Exact Bernoulli Cylinder Probability under a Declared Product Law","stem":"Use the explicitly declared Bernoulli product-law model with exact p=14/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. Ordered finite binary word=[1, 0, 1, 1]. Give (word,ones,zeros,p,probability,symbolic_product_law).","answer":{"kind":"expression","value":"([1, 0, 1, 1], 3, 1, 14/29, 41160/707281, symbolic_product_law)","explanation":"Count ones and zeros and calculate p^ones*(1−p)^zeros exactly. Empty products equal 1, including degenerate p=0 or p=1. The arbitrary-word formula is the declared symbolic product-law certificate; this finite calculation alone is not a new proof of infinite IID. The exact result, retaining the requested order, is ([1, 0, 1, 1], 3, 1, 14/29, 41160/707281, symbolic_product_law)."},"skills":["Exact Bernoulli Cylinder Probability under a Declared Product Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r06-iid-sequence-0015","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a95058567d574ac9170199ecbfb6dd01d6eb01783ee02e7ed798cdc19520a5b6","family_id":"qf-mod29-iid-bernoulli-cylinder","seed":202610296015,"semantic_template":"qf-mod29-iid-bernoulli-cylinder/{\"p\":[15,29],\"word\":[0,1,1,1]}"},"fields":{"title":"Exact Bernoulli Cylinder Probability under a Declared Product Law","stem":"Use the explicitly declared Bernoulli product-law model with exact p=15/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. Ordered finite binary word=[0, 1, 1, 1]. Give (word,ones,zeros,p,probability,symbolic_product_law).","answer":{"kind":"expression","value":"([0, 1, 1, 1], 3, 1, 15/29, 47250/707281, symbolic_product_law)","explanation":"Count ones and zeros and calculate p^ones*(1−p)^zeros exactly. Empty products equal 1, including degenerate p=0 or p=1. The arbitrary-word formula is the declared symbolic product-law certificate; this finite calculation alone is not a new proof of infinite IID. The exact result, retaining the requested order, is ([0, 1, 1, 1], 3, 1, 15/29, 47250/707281, symbolic_product_law)."},"skills":["Exact Bernoulli Cylinder Probability under a Declared Product Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r06-iid-sequence-0016","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"47976ac51f948ed594a7a0a128d997cd3dba41062dd6145b6d462d6a818cbfd1","family_id":"qf-mod29-iid-bernoulli-cylinder","seed":202610296016,"semantic_template":"qf-mod29-iid-bernoulli-cylinder/{\"p\":[16,29],\"word\":[1,1,1,1]}"},"fields":{"title":"Exact Bernoulli Cylinder Probability under a Declared Product Law","stem":"Use the explicitly declared Bernoulli product-law model with exact p=16/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. Ordered finite binary word=[1, 1, 1, 1]. Give (word,ones,zeros,p,probability,symbolic_product_law).","answer":{"kind":"expression","value":"([1, 1, 1, 1], 4, 0, 16/29, 65536/707281, symbolic_product_law)","explanation":"Count ones and zeros and calculate p^ones*(1−p)^zeros exactly. Empty products equal 1, including degenerate p=0 or p=1. The arbitrary-word formula is the declared symbolic product-law certificate; this finite calculation alone is not a new proof of infinite IID. The exact result, retaining the requested order, is ([1, 1, 1, 1], 4, 0, 16/29, 65536/707281, symbolic_product_law)."},"skills":["Exact Bernoulli Cylinder Probability under a Declared Product Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r06-iid-sequence-0017","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a3085254068cd5c313dbfb768030808933f7bd79dfc2d13fde4aa6ce9ac6c84f","family_id":"qf-mod29-iid-bernoulli-cylinder","seed":202610296017,"semantic_template":"qf-mod29-iid-bernoulli-cylinder/{\"p\":[17,29],\"word\":[0,0,0,0]}"},"fields":{"title":"Exact Bernoulli Cylinder Probability under a Declared Product Law","stem":"Use the explicitly declared Bernoulli product-law model with exact p=17/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. Ordered finite binary word=[0, 0, 0, 0]. Give (word,ones,zeros,p,probability,symbolic_product_law).","answer":{"kind":"expression","value":"([0, 0, 0, 0], 0, 4, 17/29, 20736/707281, symbolic_product_law)","explanation":"Count ones and zeros and calculate p^ones*(1−p)^zeros exactly. Empty products equal 1, including degenerate p=0 or p=1. The arbitrary-word formula is the declared symbolic product-law certificate; this finite calculation alone is not a new proof of infinite IID. The exact result, retaining the requested order, is ([0, 0, 0, 0], 0, 4, 17/29, 20736/707281, symbolic_product_law)."},"skills":["Exact Bernoulli Cylinder Probability under a Declared Product Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r06-iid-sequence-0018","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2c0dbb3fe9173a04fbae372f08db4b7abaa90ca2f1fc387d914905e274c4774a","family_id":"qf-mod29-iid-bernoulli-cylinder","seed":202610296018,"semantic_template":"qf-mod29-iid-bernoulli-cylinder/{\"p\":[18,29],\"word\":[1,0,0,0]}"},"fields":{"title":"Exact Bernoulli Cylinder Probability under a Declared Product Law","stem":"Use the explicitly declared Bernoulli product-law model with exact p=18/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. Ordered finite binary word=[1, 0, 0, 0]. Give (word,ones,zeros,p,probability,symbolic_product_law).","answer":{"kind":"expression","value":"([1, 0, 0, 0], 1, 3, 18/29, 23958/707281, symbolic_product_law)","explanation":"Count ones and zeros and calculate p^ones*(1−p)^zeros exactly. Empty products equal 1, including degenerate p=0 or p=1. The arbitrary-word formula is the declared symbolic product-law certificate; this finite calculation alone is not a new proof of infinite IID. The exact result, retaining the requested order, is ([1, 0, 0, 0], 1, 3, 18/29, 23958/707281, symbolic_product_law)."},"skills":["Exact Bernoulli Cylinder Probability under a Declared Product Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r06-iid-sequence-0019","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"77115fc12da19404a2ac4db0488eba774880ef3f559eacf36928c8b0fb38bbf4","family_id":"qf-mod29-iid-bernoulli-cylinder","seed":202610296019,"semantic_template":"qf-mod29-iid-bernoulli-cylinder/{\"p\":[19,29],\"word\":[0,1,0,0]}"},"fields":{"title":"Exact Bernoulli Cylinder Probability under a Declared Product Law","stem":"Use the explicitly declared Bernoulli product-law model with exact p=19/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. Ordered finite binary word=[0, 1, 0, 0]. Give (word,ones,zeros,p,probability,symbolic_product_law).","answer":{"kind":"expression","value":"([0, 1, 0, 0], 1, 3, 19/29, 19000/707281, symbolic_product_law)","explanation":"Count ones and zeros and calculate p^ones*(1−p)^zeros exactly. Empty products equal 1, including degenerate p=0 or p=1. The arbitrary-word formula is the declared symbolic product-law certificate; this finite calculation alone is not a new proof of infinite IID. The exact result, retaining the requested order, is ([0, 1, 0, 0], 1, 3, 19/29, 19000/707281, symbolic_product_law)."},"skills":["Exact Bernoulli Cylinder Probability under a Declared Product Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r06-iid-sequence-0020","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6b765ff9c8a7290564c03405db81200b0c02e1df9668c7bee8109af4e774a519","family_id":"qf-mod29-iid-bernoulli-cylinder","seed":202610296020,"semantic_template":"qf-mod29-iid-bernoulli-cylinder/{\"p\":[20,29],\"word\":[1,1,0,0]}"},"fields":{"title":"Exact Bernoulli Cylinder Probability under a Declared Product Law","stem":"Use the explicitly declared Bernoulli product-law model with exact p=20/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. Ordered finite binary word=[1, 1, 0, 0]. Give (word,ones,zeros,p,probability,symbolic_product_law).","answer":{"kind":"expression","value":"([1, 1, 0, 0], 2, 2, 20/29, 32400/707281, symbolic_product_law)","explanation":"Count ones and zeros and calculate p^ones*(1−p)^zeros exactly. Empty products equal 1, including degenerate p=0 or p=1. The arbitrary-word formula is the declared symbolic product-law certificate; this finite calculation alone is not a new proof of infinite IID. The exact result, retaining the requested order, is ([1, 1, 0, 0], 2, 2, 20/29, 32400/707281, symbolic_product_law)."},"skills":["Exact Bernoulli Cylinder Probability under a Declared Product Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r06-iid-sequence-0021","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4a4f17c5fe6c1f6e1c6f854201aa7038187a2afc089eb3b050887fb82c0a0245","family_id":"qf-mod29-iid-bernoulli-cylinder","seed":202610296021,"semantic_template":"qf-mod29-iid-bernoulli-cylinder/{\"p\":[21,29],\"word\":[0,0,1,0]}"},"fields":{"title":"Exact Bernoulli Cylinder Probability under a Declared Product Law","stem":"Use the explicitly declared Bernoulli product-law model with exact p=21/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. Ordered finite binary word=[0, 0, 1, 0]. Give (word,ones,zeros,p,probability,symbolic_product_law).","answer":{"kind":"expression","value":"([0, 0, 1, 0], 1, 3, 21/29, 10752/707281, symbolic_product_law)","explanation":"Count ones and zeros and calculate p^ones*(1−p)^zeros exactly. Empty products equal 1, including degenerate p=0 or p=1. The arbitrary-word formula is the declared symbolic product-law certificate; this finite calculation alone is not a new proof of infinite IID. The exact result, retaining the requested order, is ([0, 0, 1, 0], 1, 3, 21/29, 10752/707281, symbolic_product_law)."},"skills":["Exact Bernoulli Cylinder Probability under a Declared Product Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r06-iid-sequence-0022","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8c745ee700d39c81013aeea8aa2543f4e242b689e0d7ce251af1cb0a932c8829","family_id":"qf-mod29-iid-bernoulli-cylinder","seed":202610296022,"semantic_template":"qf-mod29-iid-bernoulli-cylinder/{\"p\":[22,29],\"word\":[1,0,1,0]}"},"fields":{"title":"Exact Bernoulli Cylinder Probability under a Declared Product Law","stem":"Use the explicitly declared Bernoulli product-law model with exact p=22/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. Ordered finite binary word=[1, 0, 1, 0]. Give (word,ones,zeros,p,probability,symbolic_product_law).","answer":{"kind":"expression","value":"([1, 0, 1, 0], 2, 2, 22/29, 23716/707281, symbolic_product_law)","explanation":"Count ones and zeros and calculate p^ones*(1−p)^zeros exactly. Empty products equal 1, including degenerate p=0 or p=1. The arbitrary-word formula is the declared symbolic product-law certificate; this finite calculation alone is not a new proof of infinite IID. The exact result, retaining the requested order, is ([1, 0, 1, 0], 2, 2, 22/29, 23716/707281, symbolic_product_law)."},"skills":["Exact Bernoulli Cylinder Probability under a Declared Product Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r06-iid-sequence-0023","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d6b8c6f9b2c04dba4daa5d1d86a8582af7639d36b88667ee788a49a2373e977d","family_id":"qf-mod29-iid-bernoulli-cylinder","seed":202610296023,"semantic_template":"qf-mod29-iid-bernoulli-cylinder/{\"p\":[23,29],\"word\":[0,1,1,0]}"},"fields":{"title":"Exact Bernoulli Cylinder Probability under a Declared Product Law","stem":"Use the explicitly declared Bernoulli product-law model with exact p=23/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. Ordered finite binary word=[0, 1, 1, 0]. Give (word,ones,zeros,p,probability,symbolic_product_law).","answer":{"kind":"expression","value":"([0, 1, 1, 0], 2, 2, 23/29, 19044/707281, symbolic_product_law)","explanation":"Count ones and zeros and calculate p^ones*(1−p)^zeros exactly. Empty products equal 1, including degenerate p=0 or p=1. The arbitrary-word formula is the declared symbolic product-law certificate; this finite calculation alone is not a new proof of infinite IID. The exact result, retaining the requested order, is ([0, 1, 1, 0], 2, 2, 23/29, 19044/707281, symbolic_product_law)."},"skills":["Exact Bernoulli Cylinder Probability under a Declared Product Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r06-iid-sequence-0024","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3c059eb7331642e76a9f51ac072fb4c7e798558694e98eca7a812960167d7f64","family_id":"qf-mod29-iid-bernoulli-cylinder","seed":202610296024,"semantic_template":"qf-mod29-iid-bernoulli-cylinder/{\"p\":[24,29],\"word\":[1,1,1,0]}"},"fields":{"title":"Exact Bernoulli Cylinder Probability under a Declared Product Law","stem":"Use the explicitly declared Bernoulli product-law model with exact p=24/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. Ordered finite binary word=[1, 1, 1, 0]. Give (word,ones,zeros,p,probability,symbolic_product_law).","answer":{"kind":"expression","value":"([1, 1, 1, 0], 3, 1, 24/29, 69120/707281, symbolic_product_law)","explanation":"Count ones and zeros and calculate p^ones*(1−p)^zeros exactly. Empty products equal 1, including degenerate p=0 or p=1. The arbitrary-word formula is the declared symbolic product-law certificate; this finite calculation alone is not a new proof of infinite IID. The exact result, retaining the requested order, is ([1, 1, 1, 0], 3, 1, 24/29, 69120/707281, symbolic_product_law)."},"skills":["Exact Bernoulli Cylinder Probability under a Declared Product Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r06-iid-sequence-0025","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6bae48809455cd8de40bab281d1de2df8d3a6b19fae9c9fab4f9f67c84a86bd9","family_id":"qf-mod29-iid-bernoulli-cylinder","seed":202610296025,"semantic_template":"qf-mod29-iid-bernoulli-cylinder/{\"p\":[25,29],\"word\":[0,0,0,1]}"},"fields":{"title":"Exact Bernoulli Cylinder Probability under a Declared Product Law","stem":"Use the explicitly declared Bernoulli product-law model with exact p=25/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. Ordered finite binary word=[0, 0, 0, 1]. Give (word,ones,zeros,p,probability,symbolic_product_law).","answer":{"kind":"expression","value":"([0, 0, 0, 1], 1, 3, 25/29, 1600/707281, symbolic_product_law)","explanation":"Count ones and zeros and calculate p^ones*(1−p)^zeros exactly. Empty products equal 1, including degenerate p=0 or p=1. The arbitrary-word formula is the declared symbolic product-law certificate; this finite calculation alone is not a new proof of infinite IID. The exact result, retaining the requested order, is ([0, 0, 0, 1], 1, 3, 25/29, 1600/707281, symbolic_product_law)."},"skills":["Exact Bernoulli Cylinder Probability under a Declared Product Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r06-iid-sequence-0026","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4c91af9a886b0f2307272ad3c1a6a9962a90de15918ff310d1f10375409226ab","family_id":"qf-mod29-iid-identical-marginal","seed":202610296026,"semantic_template":"qf-mod29-iid-identical-marginal/{\"left_index\":0,\"p\":[1,29],\"right_index\":17}"},"fields":{"title":"Identical Distribution Means the Complete Marginal, Not Just Its Mean","stem":"Use the explicitly declared Bernoulli product-law model with exact p=1/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. left_index=0, right_index=17. Give (left_index,right_index,[0,zero_mass],[1,one_mass],identical?,full_Bernoulli_marginal_not_mean_only).","answer":{"kind":"expression","value":"(0, 17, [0, 28/29], [1, 1/29], Yes, full_Bernoulli_marginal_not_mean_only)","explanation":"Both indices use mass 1−p at 0 and p at 1 under this same declared product law. Retain both outcome masses, including zeros; do not infer identical distribution from matching expectations alone. The exact result, retaining the requested order, is (0, 17, [0, 28/29], [1, 1/29], Yes, full_Bernoulli_marginal_not_mean_only)."},"skills":["Identical Distribution Means the Complete Marginal, Not Just Its Mean"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r06-iid-sequence-0027","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"87ea1c152656a4d0c6d31f416203346bb44b5ef97d34071d3f143b8f59290721","family_id":"qf-mod29-iid-identical-marginal","seed":202610296027,"semantic_template":"qf-mod29-iid-identical-marginal/{\"left_index\":1,\"p\":[2,29],\"right_index\":18}"},"fields":{"title":"Identical Distribution Means the Complete Marginal, Not Just Its Mean","stem":"Use the explicitly declared Bernoulli product-law model with exact p=2/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. left_index=1, right_index=18. Give (left_index,right_index,[0,zero_mass],[1,one_mass],identical?,full_Bernoulli_marginal_not_mean_only).","answer":{"kind":"expression","value":"(1, 18, [0, 27/29], [1, 2/29], Yes, full_Bernoulli_marginal_not_mean_only)","explanation":"Both indices use mass 1−p at 0 and p at 1 under this same declared product law. Retain both outcome masses, including zeros; do not infer identical distribution from matching expectations alone. The exact result, retaining the requested order, is (1, 18, [0, 27/29], [1, 2/29], Yes, full_Bernoulli_marginal_not_mean_only)."},"skills":["Identical Distribution Means the Complete Marginal, Not Just Its Mean"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r06-iid-sequence-0028","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0132a5a6219acde233e8dc689871a499fc283bb5eb7df9617ceca57c91ca513f","family_id":"qf-mod29-iid-identical-marginal","seed":202610296028,"semantic_template":"qf-mod29-iid-identical-marginal/{\"left_index\":2,\"p\":[3,29],\"right_index\":19}"},"fields":{"title":"Identical Distribution Means the Complete Marginal, Not Just Its Mean","stem":"Use the explicitly declared Bernoulli product-law model with exact p=3/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. left_index=2, right_index=19. Give (left_index,right_index,[0,zero_mass],[1,one_mass],identical?,full_Bernoulli_marginal_not_mean_only).","answer":{"kind":"expression","value":"(2, 19, [0, 26/29], [1, 3/29], Yes, full_Bernoulli_marginal_not_mean_only)","explanation":"Both indices use mass 1−p at 0 and p at 1 under this same declared product law. Retain both outcome masses, including zeros; do not infer identical distribution from matching expectations alone. The exact result, retaining the requested order, is (2, 19, [0, 26/29], [1, 3/29], Yes, full_Bernoulli_marginal_not_mean_only)."},"skills":["Identical Distribution Means the Complete Marginal, Not Just Its Mean"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r06-iid-sequence-0029","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"788051005cfe339f74d6aaea116257eef6dbf57d44e2e886b84cb282676e3b7d","family_id":"qf-mod29-iid-identical-marginal","seed":202610296029,"semantic_template":"qf-mod29-iid-identical-marginal/{\"left_index\":3,\"p\":[4,29],\"right_index\":20}"},"fields":{"title":"Identical Distribution Means the Complete Marginal, Not Just Its Mean","stem":"Use the explicitly declared Bernoulli product-law model with exact p=4/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. left_index=3, right_index=20. Give (left_index,right_index,[0,zero_mass],[1,one_mass],identical?,full_Bernoulli_marginal_not_mean_only).","answer":{"kind":"expression","value":"(3, 20, [0, 25/29], [1, 4/29], Yes, full_Bernoulli_marginal_not_mean_only)","explanation":"Both indices use mass 1−p at 0 and p at 1 under this same declared product law. Retain both outcome masses, including zeros; do not infer identical distribution from matching expectations alone. The exact result, retaining the requested order, is (3, 20, [0, 25/29], [1, 4/29], Yes, full_Bernoulli_marginal_not_mean_only)."},"skills":["Identical Distribution Means the Complete Marginal, Not Just Its Mean"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r06-iid-sequence-0030","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4497b44f8cb6940bf206b7097db537a5912f7022246b94f490c3097ff7703270","family_id":"qf-mod29-iid-identical-marginal","seed":202610296030,"semantic_template":"qf-mod29-iid-identical-marginal/{\"left_index\":4,\"p\":[5,29],\"right_index\":21}"},"fields":{"title":"Identical Distribution Means the Complete Marginal, Not Just Its Mean","stem":"Use the explicitly declared Bernoulli product-law model with exact p=5/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. left_index=4, right_index=21. Give (left_index,right_index,[0,zero_mass],[1,one_mass],identical?,full_Bernoulli_marginal_not_mean_only).","answer":{"kind":"expression","value":"(4, 21, [0, 24/29], [1, 5/29], Yes, full_Bernoulli_marginal_not_mean_only)","explanation":"Both indices use mass 1−p at 0 and p at 1 under this same declared product law. Retain both outcome masses, including zeros; do not infer identical distribution from matching expectations alone. The exact result, retaining the requested order, is (4, 21, [0, 24/29], [1, 5/29], Yes, full_Bernoulli_marginal_not_mean_only)."},"skills":["Identical Distribution Means the Complete Marginal, Not Just Its Mean"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r06-iid-sequence-0031","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4ca991a2e4a448943c76056e7000904c387dfd263ba845b5e35fba078608d93f","family_id":"qf-mod29-iid-identical-marginal","seed":202610296031,"semantic_template":"qf-mod29-iid-identical-marginal/{\"left_index\":5,\"p\":[6,29],\"right_index\":22}"},"fields":{"title":"Identical Distribution Means the Complete Marginal, Not Just Its Mean","stem":"Use the explicitly declared Bernoulli product-law model with exact p=6/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. left_index=5, right_index=22. Give (left_index,right_index,[0,zero_mass],[1,one_mass],identical?,full_Bernoulli_marginal_not_mean_only).","answer":{"kind":"expression","value":"(5, 22, [0, 23/29], [1, 6/29], Yes, full_Bernoulli_marginal_not_mean_only)","explanation":"Both indices use mass 1−p at 0 and p at 1 under this same declared product law. Retain both outcome masses, including zeros; do not infer identical distribution from matching expectations alone. The exact result, retaining the requested order, is (5, 22, [0, 23/29], [1, 6/29], Yes, full_Bernoulli_marginal_not_mean_only)."},"skills":["Identical Distribution Means the Complete Marginal, Not Just Its Mean"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r06-iid-sequence-0032","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cd264e796edd34e67c9c7a8a939cf757401ec307f04955adc8553b1fd4b205db","family_id":"qf-mod29-iid-identical-marginal","seed":202610296032,"semantic_template":"qf-mod29-iid-identical-marginal/{\"left_index\":6,\"p\":[7,29],\"right_index\":23}"},"fields":{"title":"Identical Distribution Means the Complete Marginal, Not Just Its Mean","stem":"Use the explicitly declared Bernoulli product-law model with exact p=7/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. left_index=6, right_index=23. Give (left_index,right_index,[0,zero_mass],[1,one_mass],identical?,full_Bernoulli_marginal_not_mean_only).","answer":{"kind":"expression","value":"(6, 23, [0, 22/29], [1, 7/29], Yes, full_Bernoulli_marginal_not_mean_only)","explanation":"Both indices use mass 1−p at 0 and p at 1 under this same declared product law. Retain both outcome masses, including zeros; do not infer identical distribution from matching expectations alone. The exact result, retaining the requested order, is (6, 23, [0, 22/29], [1, 7/29], Yes, full_Bernoulli_marginal_not_mean_only)."},"skills":["Identical Distribution Means the Complete Marginal, Not Just Its Mean"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r06-iid-sequence-0033","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b18ffc592e710151e629d9eb2e6c19fb782dfc16d9f4ef4fc71f4a1d7561c148","family_id":"qf-mod29-iid-identical-marginal","seed":202610296033,"semantic_template":"qf-mod29-iid-identical-marginal/{\"left_index\":7,\"p\":[8,29],\"right_index\":24}"},"fields":{"title":"Identical Distribution Means the Complete Marginal, Not Just Its Mean","stem":"Use the explicitly declared Bernoulli product-law model with exact p=8/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. left_index=7, right_index=24. Give (left_index,right_index,[0,zero_mass],[1,one_mass],identical?,full_Bernoulli_marginal_not_mean_only).","answer":{"kind":"expression","value":"(7, 24, [0, 21/29], [1, 8/29], Yes, full_Bernoulli_marginal_not_mean_only)","explanation":"Both indices use mass 1−p at 0 and p at 1 under this same declared product law. Retain both outcome masses, including zeros; do not infer identical distribution from matching expectations alone. The exact result, retaining the requested order, is (7, 24, [0, 21/29], [1, 8/29], Yes, full_Bernoulli_marginal_not_mean_only)."},"skills":["Identical Distribution Means the Complete Marginal, Not Just Its Mean"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r06-iid-sequence-0034","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4d415473f75428e013b80df5e31b3d02e1b12510152215cd1ebd038bebea8fa8","family_id":"qf-mod29-iid-identical-marginal","seed":202610296034,"semantic_template":"qf-mod29-iid-identical-marginal/{\"left_index\":8,\"p\":[9,29],\"right_index\":25}"},"fields":{"title":"Identical Distribution Means the Complete Marginal, Not Just Its Mean","stem":"Use the explicitly declared Bernoulli product-law model with exact p=9/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. left_index=8, right_index=25. Give (left_index,right_index,[0,zero_mass],[1,one_mass],identical?,full_Bernoulli_marginal_not_mean_only).","answer":{"kind":"expression","value":"(8, 25, [0, 20/29], [1, 9/29], Yes, full_Bernoulli_marginal_not_mean_only)","explanation":"Both indices use mass 1−p at 0 and p at 1 under this same declared product law. Retain both outcome masses, including zeros; do not infer identical distribution from matching expectations alone. The exact result, retaining the requested order, is (8, 25, [0, 20/29], [1, 9/29], Yes, full_Bernoulli_marginal_not_mean_only)."},"skills":["Identical Distribution Means the Complete Marginal, Not Just Its Mean"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r06-iid-sequence-0035","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8820329715dedd51c2ece7d932de9a30781667e279afbfc85fb475d2dce7ff4c","family_id":"qf-mod29-iid-identical-marginal","seed":202610296035,"semantic_template":"qf-mod29-iid-identical-marginal/{\"left_index\":9,\"p\":[10,29],\"right_index\":26}"},"fields":{"title":"Identical Distribution Means the Complete Marginal, Not Just Its Mean","stem":"Use the explicitly declared Bernoulli product-law model with exact p=10/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. left_index=9, right_index=26. Give (left_index,right_index,[0,zero_mass],[1,one_mass],identical?,full_Bernoulli_marginal_not_mean_only).","answer":{"kind":"expression","value":"(9, 26, [0, 19/29], [1, 10/29], Yes, full_Bernoulli_marginal_not_mean_only)","explanation":"Both indices use mass 1−p at 0 and p at 1 under this same declared product law. Retain both outcome masses, including zeros; do not infer identical distribution from matching expectations alone. The exact result, retaining the requested order, is (9, 26, [0, 19/29], [1, 10/29], Yes, full_Bernoulli_marginal_not_mean_only)."},"skills":["Identical Distribution Means the Complete Marginal, Not Just Its Mean"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r06-iid-sequence-0036","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"70ebaaee020831aee8d0e307be153415c1e9409e63d60d2cb9882deb96343ae8","family_id":"qf-mod29-iid-identical-marginal","seed":202610296036,"semantic_template":"qf-mod29-iid-identical-marginal/{\"left_index\":10,\"p\":[11,29],\"right_index\":27}"},"fields":{"title":"Identical Distribution Means the Complete Marginal, Not Just Its Mean","stem":"Use the explicitly declared Bernoulli product-law model with exact p=11/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. left_index=10, right_index=27. Give (left_index,right_index,[0,zero_mass],[1,one_mass],identical?,full_Bernoulli_marginal_not_mean_only).","answer":{"kind":"expression","value":"(10, 27, [0, 18/29], [1, 11/29], Yes, full_Bernoulli_marginal_not_mean_only)","explanation":"Both indices use mass 1−p at 0 and p at 1 under this same declared product law. Retain both outcome masses, including zeros; do not infer identical distribution from matching expectations alone. The exact result, retaining the requested order, is (10, 27, [0, 18/29], [1, 11/29], Yes, full_Bernoulli_marginal_not_mean_only)."},"skills":["Identical Distribution Means the Complete Marginal, Not Just Its Mean"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r06-iid-sequence-0037","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3cac80615d4781abade9db2ac4e7361a17d6bcc5c1ecc106408371a143f8d33d","family_id":"qf-mod29-iid-identical-marginal","seed":202610296037,"semantic_template":"qf-mod29-iid-identical-marginal/{\"left_index\":11,\"p\":[12,29],\"right_index\":28}"},"fields":{"title":"Identical Distribution Means the Complete Marginal, Not Just Its Mean","stem":"Use the explicitly declared Bernoulli product-law model with exact p=12/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. left_index=11, right_index=28. Give (left_index,right_index,[0,zero_mass],[1,one_mass],identical?,full_Bernoulli_marginal_not_mean_only).","answer":{"kind":"expression","value":"(11, 28, [0, 17/29], [1, 12/29], Yes, full_Bernoulli_marginal_not_mean_only)","explanation":"Both indices use mass 1−p at 0 and p at 1 under this same declared product law. Retain both outcome masses, including zeros; do not infer identical distribution from matching expectations alone. The exact result, retaining the requested order, is (11, 28, [0, 17/29], [1, 12/29], Yes, full_Bernoulli_marginal_not_mean_only)."},"skills":["Identical Distribution Means the Complete Marginal, Not Just Its Mean"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r06-iid-sequence-0038","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f74baccdc226e85f9df7519cb4364971b7dbb1425af6a46c970645fe6ea4a724","family_id":"qf-mod29-iid-identical-marginal","seed":202610296038,"semantic_template":"qf-mod29-iid-identical-marginal/{\"left_index\":12,\"p\":[13,29],\"right_index\":29}"},"fields":{"title":"Identical Distribution Means the Complete Marginal, Not Just Its Mean","stem":"Use the explicitly declared Bernoulli product-law model with exact p=13/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. left_index=12, right_index=29. Give (left_index,right_index,[0,zero_mass],[1,one_mass],identical?,full_Bernoulli_marginal_not_mean_only).","answer":{"kind":"expression","value":"(12, 29, [0, 16/29], [1, 13/29], Yes, full_Bernoulli_marginal_not_mean_only)","explanation":"Both indices use mass 1−p at 0 and p at 1 under this same declared product law. Retain both outcome masses, including zeros; do not infer identical distribution from matching expectations alone. The exact result, retaining the requested order, is (12, 29, [0, 16/29], [1, 13/29], Yes, full_Bernoulli_marginal_not_mean_only)."},"skills":["Identical Distribution Means the Complete Marginal, Not Just Its Mean"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r06-iid-sequence-0039","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bc46a4d0bd955293b4064544e1315dfc652fd08181234e8d160d7a91ef600102","family_id":"qf-mod29-iid-identical-marginal","seed":202610296039,"semantic_template":"qf-mod29-iid-identical-marginal/{\"left_index\":13,\"p\":[14,29],\"right_index\":30}"},"fields":{"title":"Identical Distribution Means the Complete Marginal, Not Just Its Mean","stem":"Use the explicitly declared Bernoulli product-law model with exact p=14/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. left_index=13, right_index=30. Give (left_index,right_index,[0,zero_mass],[1,one_mass],identical?,full_Bernoulli_marginal_not_mean_only).","answer":{"kind":"expression","value":"(13, 30, [0, 15/29], [1, 14/29], Yes, full_Bernoulli_marginal_not_mean_only)","explanation":"Both indices use mass 1−p at 0 and p at 1 under this same declared product law. Retain both outcome masses, including zeros; do not infer identical distribution from matching expectations alone. The exact result, retaining the requested order, is (13, 30, [0, 15/29], [1, 14/29], Yes, full_Bernoulli_marginal_not_mean_only)."},"skills":["Identical Distribution Means the Complete Marginal, Not Just Its Mean"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r06-iid-sequence-0040","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2329ccdb24cdab8e6a43e7cf36c9ae247e075c9c5b9601e329bbfd9ccf32444d","family_id":"qf-mod29-iid-identical-marginal","seed":202610296040,"semantic_template":"qf-mod29-iid-identical-marginal/{\"left_index\":14,\"p\":[15,29],\"right_index\":31}"},"fields":{"title":"Identical Distribution Means the Complete Marginal, Not Just Its Mean","stem":"Use the explicitly declared Bernoulli product-law model with exact p=15/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. left_index=14, right_index=31. Give (left_index,right_index,[0,zero_mass],[1,one_mass],identical?,full_Bernoulli_marginal_not_mean_only).","answer":{"kind":"expression","value":"(14, 31, [0, 14/29], [1, 15/29], Yes, full_Bernoulli_marginal_not_mean_only)","explanation":"Both indices use mass 1−p at 0 and p at 1 under this same declared product law. Retain both outcome masses, including zeros; do not infer identical distribution from matching expectations alone. The exact result, retaining the requested order, is (14, 31, [0, 14/29], [1, 15/29], Yes, full_Bernoulli_marginal_not_mean_only)."},"skills":["Identical Distribution Means the Complete Marginal, Not Just Its Mean"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r06-iid-sequence-0041","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b68b452eab09cc14cb760d12543e762ae117e977f2f4571659a834c019cbd6f9","family_id":"qf-mod29-iid-identical-marginal","seed":202610296041,"semantic_template":"qf-mod29-iid-identical-marginal/{\"left_index\":15,\"p\":[16,29],\"right_index\":32}"},"fields":{"title":"Identical Distribution Means the Complete Marginal, Not Just Its Mean","stem":"Use the explicitly declared Bernoulli product-law model with exact p=16/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. left_index=15, right_index=32. Give (left_index,right_index,[0,zero_mass],[1,one_mass],identical?,full_Bernoulli_marginal_not_mean_only).","answer":{"kind":"expression","value":"(15, 32, [0, 13/29], [1, 16/29], Yes, full_Bernoulli_marginal_not_mean_only)","explanation":"Both indices use mass 1−p at 0 and p at 1 under this same declared product law. Retain both outcome masses, including zeros; do not infer identical distribution from matching expectations alone. The exact result, retaining the requested order, is (15, 32, [0, 13/29], [1, 16/29], Yes, full_Bernoulli_marginal_not_mean_only)."},"skills":["Identical Distribution Means the Complete Marginal, Not Just Its Mean"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r06-iid-sequence-0042","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b1cc6ba451ae995dc044b9eed40f38b5a69e87da846120081c44fba1e38fefe1","family_id":"qf-mod29-iid-identical-marginal","seed":202610296042,"semantic_template":"qf-mod29-iid-identical-marginal/{\"left_index\":16,\"p\":[17,29],\"right_index\":33}"},"fields":{"title":"Identical Distribution Means the Complete Marginal, Not Just Its Mean","stem":"Use the explicitly declared Bernoulli product-law model with exact p=17/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. left_index=16, right_index=33. Give (left_index,right_index,[0,zero_mass],[1,one_mass],identical?,full_Bernoulli_marginal_not_mean_only).","answer":{"kind":"expression","value":"(16, 33, [0, 12/29], [1, 17/29], Yes, full_Bernoulli_marginal_not_mean_only)","explanation":"Both indices use mass 1−p at 0 and p at 1 under this same declared product law. Retain both outcome masses, including zeros; do not infer identical distribution from matching expectations alone. The exact result, retaining the requested order, is (16, 33, [0, 12/29], [1, 17/29], Yes, full_Bernoulli_marginal_not_mean_only)."},"skills":["Identical Distribution Means the Complete Marginal, Not Just Its Mean"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r06-iid-sequence-0043","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"23e42021ba807a2f2f20e49a9b01d336849e83d491971d30fbe829443029efe7","family_id":"qf-mod29-iid-identical-marginal","seed":202610296043,"semantic_template":"qf-mod29-iid-identical-marginal/{\"left_index\":17,\"p\":[18,29],\"right_index\":34}"},"fields":{"title":"Identical Distribution Means the Complete Marginal, Not Just Its Mean","stem":"Use the explicitly declared Bernoulli product-law model with exact p=18/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. left_index=17, right_index=34. Give (left_index,right_index,[0,zero_mass],[1,one_mass],identical?,full_Bernoulli_marginal_not_mean_only).","answer":{"kind":"expression","value":"(17, 34, [0, 11/29], [1, 18/29], Yes, full_Bernoulli_marginal_not_mean_only)","explanation":"Both indices use mass 1−p at 0 and p at 1 under this same declared product law. Retain both outcome masses, including zeros; do not infer identical distribution from matching expectations alone. The exact result, retaining the requested order, is (17, 34, [0, 11/29], [1, 18/29], Yes, full_Bernoulli_marginal_not_mean_only)."},"skills":["Identical Distribution Means the Complete Marginal, Not Just Its Mean"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r06-iid-sequence-0044","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"eb6b6f82bc848136cc0e2dc7ea92186ac54dff8a1a01e1b84437b911a4447b4c","family_id":"qf-mod29-iid-identical-marginal","seed":202610296044,"semantic_template":"qf-mod29-iid-identical-marginal/{\"left_index\":18,\"p\":[19,29],\"right_index\":35}"},"fields":{"title":"Identical Distribution Means the Complete Marginal, Not Just Its Mean","stem":"Use the explicitly declared Bernoulli product-law model with exact p=19/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. left_index=18, right_index=35. Give (left_index,right_index,[0,zero_mass],[1,one_mass],identical?,full_Bernoulli_marginal_not_mean_only).","answer":{"kind":"expression","value":"(18, 35, [0, 10/29], [1, 19/29], Yes, full_Bernoulli_marginal_not_mean_only)","explanation":"Both indices use mass 1−p at 0 and p at 1 under this same declared product law. Retain both outcome masses, including zeros; do not infer identical distribution from matching expectations alone. The exact result, retaining the requested order, is (18, 35, [0, 10/29], [1, 19/29], Yes, full_Bernoulli_marginal_not_mean_only)."},"skills":["Identical Distribution Means the Complete Marginal, Not Just Its Mean"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r06-iid-sequence-0045","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fc1fbb0b20d586c330eed571ad84ad3d575a3cae1be4e2d1ef21551b2809a395","family_id":"qf-mod29-iid-identical-marginal","seed":202610296045,"semantic_template":"qf-mod29-iid-identical-marginal/{\"left_index\":19,\"p\":[20,29],\"right_index\":36}"},"fields":{"title":"Identical Distribution Means the Complete Marginal, Not Just Its Mean","stem":"Use the explicitly declared Bernoulli product-law model with exact p=20/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. left_index=19, right_index=36. Give (left_index,right_index,[0,zero_mass],[1,one_mass],identical?,full_Bernoulli_marginal_not_mean_only).","answer":{"kind":"expression","value":"(19, 36, [0, 9/29], [1, 20/29], Yes, full_Bernoulli_marginal_not_mean_only)","explanation":"Both indices use mass 1−p at 0 and p at 1 under this same declared product law. Retain both outcome masses, including zeros; do not infer identical distribution from matching expectations alone. The exact result, retaining the requested order, is (19, 36, [0, 9/29], [1, 20/29], Yes, full_Bernoulli_marginal_not_mean_only)."},"skills":["Identical Distribution Means the Complete Marginal, Not Just Its Mean"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r06-iid-sequence-0046","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0e25cb4c8a056ff80456b1c261f1047c5ea1504c26354b6a000b30d0df080f45","family_id":"qf-mod29-iid-identical-marginal","seed":202610296046,"semantic_template":"qf-mod29-iid-identical-marginal/{\"left_index\":20,\"p\":[21,29],\"right_index\":37}"},"fields":{"title":"Identical Distribution Means the Complete Marginal, Not Just Its Mean","stem":"Use the explicitly declared Bernoulli product-law model with exact p=21/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. left_index=20, right_index=37. Give (left_index,right_index,[0,zero_mass],[1,one_mass],identical?,full_Bernoulli_marginal_not_mean_only).","answer":{"kind":"expression","value":"(20, 37, [0, 8/29], [1, 21/29], Yes, full_Bernoulli_marginal_not_mean_only)","explanation":"Both indices use mass 1−p at 0 and p at 1 under this same declared product law. Retain both outcome masses, including zeros; do not infer identical distribution from matching expectations alone. The exact result, retaining the requested order, is (20, 37, [0, 8/29], [1, 21/29], Yes, full_Bernoulli_marginal_not_mean_only)."},"skills":["Identical Distribution Means the Complete Marginal, Not Just Its Mean"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r06-iid-sequence-0047","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7add0f4972fa6f06e37fb73355a7c4b3c84880b002a4e3bff90653af3320a2c5","family_id":"qf-mod29-iid-identical-marginal","seed":202610296047,"semantic_template":"qf-mod29-iid-identical-marginal/{\"left_index\":21,\"p\":[22,29],\"right_index\":38}"},"fields":{"title":"Identical Distribution Means the Complete Marginal, Not Just Its Mean","stem":"Use the explicitly declared Bernoulli product-law model with exact p=22/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. left_index=21, right_index=38. Give (left_index,right_index,[0,zero_mass],[1,one_mass],identical?,full_Bernoulli_marginal_not_mean_only).","answer":{"kind":"expression","value":"(21, 38, [0, 7/29], [1, 22/29], Yes, full_Bernoulli_marginal_not_mean_only)","explanation":"Both indices use mass 1−p at 0 and p at 1 under this same declared product law. Retain both outcome masses, including zeros; do not infer identical distribution from matching expectations alone. The exact result, retaining the requested order, is (21, 38, [0, 7/29], [1, 22/29], Yes, full_Bernoulli_marginal_not_mean_only)."},"skills":["Identical Distribution Means the Complete Marginal, Not Just Its Mean"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r06-iid-sequence-0048","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a53118dc6b4b4a7315447deed8788b5cddf99992e8c0be252836032f1fd6d8c3","family_id":"qf-mod29-iid-identical-marginal","seed":202610296048,"semantic_template":"qf-mod29-iid-identical-marginal/{\"left_index\":22,\"p\":[23,29],\"right_index\":39}"},"fields":{"title":"Identical Distribution Means the Complete Marginal, Not Just Its Mean","stem":"Use the explicitly declared Bernoulli product-law model with exact p=23/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. left_index=22, right_index=39. Give (left_index,right_index,[0,zero_mass],[1,one_mass],identical?,full_Bernoulli_marginal_not_mean_only).","answer":{"kind":"expression","value":"(22, 39, [0, 6/29], [1, 23/29], Yes, full_Bernoulli_marginal_not_mean_only)","explanation":"Both indices use mass 1−p at 0 and p at 1 under this same declared product law. Retain both outcome masses, including zeros; do not infer identical distribution from matching expectations alone. The exact result, retaining the requested order, is (22, 39, [0, 6/29], [1, 23/29], Yes, full_Bernoulli_marginal_not_mean_only)."},"skills":["Identical Distribution Means the Complete Marginal, Not Just Its Mean"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r06-iid-sequence-0049","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"51265e1a23625b4837b0e748f795778f8e7ebac714aebcbe6c43cf03cb5c0346","family_id":"qf-mod29-iid-identical-marginal","seed":202610296049,"semantic_template":"qf-mod29-iid-identical-marginal/{\"left_index\":23,\"p\":[24,29],\"right_index\":40}"},"fields":{"title":"Identical Distribution Means the Complete Marginal, Not Just Its Mean","stem":"Use the explicitly declared Bernoulli product-law model with exact p=24/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. left_index=23, right_index=40. Give (left_index,right_index,[0,zero_mass],[1,one_mass],identical?,full_Bernoulli_marginal_not_mean_only).","answer":{"kind":"expression","value":"(23, 40, [0, 5/29], [1, 24/29], Yes, full_Bernoulli_marginal_not_mean_only)","explanation":"Both indices use mass 1−p at 0 and p at 1 under this same declared product law. Retain both outcome masses, including zeros; do not infer identical distribution from matching expectations alone. The exact result, retaining the requested order, is (23, 40, [0, 5/29], [1, 24/29], Yes, full_Bernoulli_marginal_not_mean_only)."},"skills":["Identical Distribution Means the Complete Marginal, Not Just Its Mean"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r06-iid-sequence-0050","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ae31c10084aa85a6f6d1e8d3a7a6a7c519823f9dad763918b6c4ca7a4d035eaf","family_id":"qf-mod29-iid-identical-marginal","seed":202610296050,"semantic_template":"qf-mod29-iid-identical-marginal/{\"left_index\":24,\"p\":[25,29],\"right_index\":41}"},"fields":{"title":"Identical Distribution Means the Complete Marginal, Not Just Its Mean","stem":"Use the explicitly declared Bernoulli product-law model with exact p=25/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. left_index=24, right_index=41. Give (left_index,right_index,[0,zero_mass],[1,one_mass],identical?,full_Bernoulli_marginal_not_mean_only).","answer":{"kind":"expression","value":"(24, 41, [0, 4/29], [1, 25/29], Yes, full_Bernoulli_marginal_not_mean_only)","explanation":"Both indices use mass 1−p at 0 and p at 1 under this same declared product law. Retain both outcome masses, including zeros; do not infer identical distribution from matching expectations alone. The exact result, retaining the requested order, is (24, 41, [0, 4/29], [1, 25/29], Yes, full_Bernoulli_marginal_not_mean_only)."},"skills":["Identical Distribution Means the Complete Marginal, Not Just Its Mean"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r06-iid-sequence-0051","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"22877e387172a1cab83a689911e4aa429d48524f6cfe22debe8aaafa94594f6b","family_id":"qf-mod29-iid-finite-factorization","seed":202610296051,"semantic_template":"qf-mod29-iid-finite-factorization/{\"p\":[1,29],\"word\":[0,0,0,0]}"},"fields":{"title":"Multiply Every Ordered Atom Factor in the Finite Word","stem":"Use the explicitly declared Bernoulli product-law model with exact p=1/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. Ordered binary word=[0, 0, 0, 0]. Give (word,atom_factors,product,pass?).","answer":{"kind":"expression","value":"([0, 0, 0, 0], [28/29, 28/29, 28/29, 28/29], 614656/707281, Yes)","explanation":"Assign factor p to each 1 and 1−p to each 0 in original word order, then multiply exactly. The factorization comes from the same product-law premise; it is not a test of an unknown dependent law. The exact result, retaining the requested order, is ([0, 0, 0, 0], [28/29, 28/29, 28/29, 28/29], 614656/707281, Yes)."},"skills":["Multiply Every Ordered Atom Factor in the Finite Word"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r06-iid-sequence-0052","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e06eca07455c01f3a42fa3cef36873d47baae5657f4153b65da1d52b8c1ea252","family_id":"qf-mod29-iid-finite-factorization","seed":202610296052,"semantic_template":"qf-mod29-iid-finite-factorization/{\"p\":[2,29],\"word\":[1,0,0,0]}"},"fields":{"title":"Multiply Every Ordered Atom Factor in the Finite Word","stem":"Use the explicitly declared Bernoulli product-law model with exact p=2/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. Ordered binary word=[1, 0, 0, 0]. Give (word,atom_factors,product,pass?).","answer":{"kind":"expression","value":"([1, 0, 0, 0], [2/29, 27/29, 27/29, 27/29], 39366/707281, Yes)","explanation":"Assign factor p to each 1 and 1−p to each 0 in original word order, then multiply exactly. The factorization comes from the same product-law premise; it is not a test of an unknown dependent law. The exact result, retaining the requested order, is ([1, 0, 0, 0], [2/29, 27/29, 27/29, 27/29], 39366/707281, Yes)."},"skills":["Multiply Every Ordered Atom Factor in the Finite Word"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r06-iid-sequence-0053","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1c72da04b72b5474b67a66419d5280a7c68db789f76bbdc071c994f57479e499","family_id":"qf-mod29-iid-finite-factorization","seed":202610296053,"semantic_template":"qf-mod29-iid-finite-factorization/{\"p\":[3,29],\"word\":[0,1,0,0]}"},"fields":{"title":"Multiply Every Ordered Atom Factor in the Finite Word","stem":"Use the explicitly declared Bernoulli product-law model with exact p=3/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. Ordered binary word=[0, 1, 0, 0]. Give (word,atom_factors,product,pass?).","answer":{"kind":"expression","value":"([0, 1, 0, 0], [26/29, 3/29, 26/29, 26/29], 52728/707281, Yes)","explanation":"Assign factor p to each 1 and 1−p to each 0 in original word order, then multiply exactly. The factorization comes from the same product-law premise; it is not a test of an unknown dependent law. The exact result, retaining the requested order, is ([0, 1, 0, 0], [26/29, 3/29, 26/29, 26/29], 52728/707281, Yes)."},"skills":["Multiply Every Ordered Atom Factor in the Finite Word"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r06-iid-sequence-0054","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cc327f20ef1cfa112dc6efa68f2a4c2930e5e04c4ff8308d1f1880fff4f4c6a8","family_id":"qf-mod29-iid-finite-factorization","seed":202610296054,"semantic_template":"qf-mod29-iid-finite-factorization/{\"p\":[4,29],\"word\":[1,1,0,0]}"},"fields":{"title":"Multiply Every Ordered Atom Factor in the Finite Word","stem":"Use the explicitly declared Bernoulli product-law model with exact p=4/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. Ordered binary word=[1, 1, 0, 0]. Give (word,atom_factors,product,pass?).","answer":{"kind":"expression","value":"([1, 1, 0, 0], [4/29, 4/29, 25/29, 25/29], 10000/707281, Yes)","explanation":"Assign factor p to each 1 and 1−p to each 0 in original word order, then multiply exactly. The factorization comes from the same product-law premise; it is not a test of an unknown dependent law. The exact result, retaining the requested order, is ([1, 1, 0, 0], [4/29, 4/29, 25/29, 25/29], 10000/707281, Yes)."},"skills":["Multiply Every Ordered Atom Factor in the Finite Word"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r06-iid-sequence-0055","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"340e5e12931855e0b2b85162e5f412371c6e328b0030fdfef1403e1094c3712e","family_id":"qf-mod29-iid-finite-factorization","seed":202610296055,"semantic_template":"qf-mod29-iid-finite-factorization/{\"p\":[5,29],\"word\":[0,0,1,0]}"},"fields":{"title":"Multiply Every Ordered Atom Factor in the Finite Word","stem":"Use the explicitly declared Bernoulli product-law model with exact p=5/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. Ordered binary word=[0, 0, 1, 0]. Give (word,atom_factors,product,pass?).","answer":{"kind":"expression","value":"([0, 0, 1, 0], [24/29, 24/29, 5/29, 24/29], 69120/707281, Yes)","explanation":"Assign factor p to each 1 and 1−p to each 0 in original word order, then multiply exactly. The factorization comes from the same product-law premise; it is not a test of an unknown dependent law. The exact result, retaining the requested order, is ([0, 0, 1, 0], [24/29, 24/29, 5/29, 24/29], 69120/707281, Yes)."},"skills":["Multiply Every Ordered Atom Factor in the Finite Word"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r06-iid-sequence-0056","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9d233666218450be963c0e7b9f78a0c10f08975fdfd7ad890e522bedcec2cc37","family_id":"qf-mod29-iid-finite-factorization","seed":202610296056,"semantic_template":"qf-mod29-iid-finite-factorization/{\"p\":[6,29],\"word\":[1,0,1,0]}"},"fields":{"title":"Multiply Every Ordered Atom Factor in the Finite Word","stem":"Use the explicitly declared Bernoulli product-law model with exact p=6/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. Ordered binary word=[1, 0, 1, 0]. Give (word,atom_factors,product,pass?).","answer":{"kind":"expression","value":"([1, 0, 1, 0], [6/29, 23/29, 6/29, 23/29], 19044/707281, Yes)","explanation":"Assign factor p to each 1 and 1−p to each 0 in original word order, then multiply exactly. The factorization comes from the same product-law premise; it is not a test of an unknown dependent law. The exact result, retaining the requested order, is ([1, 0, 1, 0], [6/29, 23/29, 6/29, 23/29], 19044/707281, Yes)."},"skills":["Multiply Every Ordered Atom Factor in the Finite Word"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r06-iid-sequence-0057","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bf7be0ca4463cc0e783a4b8ca2013709f3c530638a3e8039e71c2a7aaba86e5c","family_id":"qf-mod29-iid-finite-factorization","seed":202610296057,"semantic_template":"qf-mod29-iid-finite-factorization/{\"p\":[7,29],\"word\":[0,1,1,0]}"},"fields":{"title":"Multiply Every Ordered Atom Factor in the Finite Word","stem":"Use the explicitly declared Bernoulli product-law model with exact p=7/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. Ordered binary word=[0, 1, 1, 0]. Give (word,atom_factors,product,pass?).","answer":{"kind":"expression","value":"([0, 1, 1, 0], [22/29, 7/29, 7/29, 22/29], 23716/707281, Yes)","explanation":"Assign factor p to each 1 and 1−p to each 0 in original word order, then multiply exactly. The factorization comes from the same product-law premise; it is not a test of an unknown dependent law. The exact result, retaining the requested order, is ([0, 1, 1, 0], [22/29, 7/29, 7/29, 22/29], 23716/707281, Yes)."},"skills":["Multiply Every Ordered Atom Factor in the Finite Word"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r06-iid-sequence-0058","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"489aa78382dd45fdd891e87e065495788e778fa059bd0ef075bd7044d41f347a","family_id":"qf-mod29-iid-finite-factorization","seed":202610296058,"semantic_template":"qf-mod29-iid-finite-factorization/{\"p\":[8,29],\"word\":[1,1,1,0]}"},"fields":{"title":"Multiply Every Ordered Atom Factor in the Finite Word","stem":"Use the explicitly declared Bernoulli product-law model with exact p=8/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. Ordered binary word=[1, 1, 1, 0]. Give (word,atom_factors,product,pass?).","answer":{"kind":"expression","value":"([1, 1, 1, 0], [8/29, 8/29, 8/29, 21/29], 10752/707281, Yes)","explanation":"Assign factor p to each 1 and 1−p to each 0 in original word order, then multiply exactly. The factorization comes from the same product-law premise; it is not a test of an unknown dependent law. The exact result, retaining the requested order, is ([1, 1, 1, 0], [8/29, 8/29, 8/29, 21/29], 10752/707281, Yes)."},"skills":["Multiply Every Ordered Atom Factor in the Finite Word"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r06-iid-sequence-0059","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9d144a7a4316be8bd9c5e5d66bd061838c85dbd507d2049899cf66b6fbc6a57d","family_id":"qf-mod29-iid-finite-factorization","seed":202610296059,"semantic_template":"qf-mod29-iid-finite-factorization/{\"p\":[9,29],\"word\":[0,0,0,1]}"},"fields":{"title":"Multiply Every Ordered Atom Factor in the Finite Word","stem":"Use the explicitly declared Bernoulli product-law model with exact p=9/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. Ordered binary word=[0, 0, 0, 1]. Give (word,atom_factors,product,pass?).","answer":{"kind":"expression","value":"([0, 0, 0, 1], [20/29, 20/29, 20/29, 9/29], 72000/707281, Yes)","explanation":"Assign factor p to each 1 and 1−p to each 0 in original word order, then multiply exactly. The factorization comes from the same product-law premise; it is not a test of an unknown dependent law. The exact result, retaining the requested order, is ([0, 0, 0, 1], [20/29, 20/29, 20/29, 9/29], 72000/707281, Yes)."},"skills":["Multiply Every Ordered Atom Factor in the Finite Word"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r06-iid-sequence-0060","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6e53d0a832daa1a4fd68feac01b56b90a425be5ca975d19e0c2a9639d96a3b44","family_id":"qf-mod29-iid-finite-factorization","seed":202610296060,"semantic_template":"qf-mod29-iid-finite-factorization/{\"p\":[10,29],\"word\":[1,0,0,1]}"},"fields":{"title":"Multiply Every Ordered Atom Factor in the Finite Word","stem":"Use the explicitly declared Bernoulli product-law model with exact p=10/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. Ordered binary word=[1, 0, 0, 1]. Give (word,atom_factors,product,pass?).","answer":{"kind":"expression","value":"([1, 0, 0, 1], [10/29, 19/29, 19/29, 10/29], 36100/707281, Yes)","explanation":"Assign factor p to each 1 and 1−p to each 0 in original word order, then multiply exactly. The factorization comes from the same product-law premise; it is not a test of an unknown dependent law. The exact result, retaining the requested order, is ([1, 0, 0, 1], [10/29, 19/29, 19/29, 10/29], 36100/707281, Yes)."},"skills":["Multiply Every Ordered Atom Factor in the Finite Word"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r06-iid-sequence-0061","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"767c3172d7e4c6a1b6e5d49c6a092fa568fd810cf1fc411e0b5cddd801ed82fb","family_id":"qf-mod29-iid-finite-factorization","seed":202610296061,"semantic_template":"qf-mod29-iid-finite-factorization/{\"p\":[11,29],\"word\":[0,1,0,1]}"},"fields":{"title":"Multiply Every Ordered Atom Factor in the Finite Word","stem":"Use the explicitly declared Bernoulli product-law model with exact p=11/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. Ordered binary word=[0, 1, 0, 1]. Give (word,atom_factors,product,pass?).","answer":{"kind":"expression","value":"([0, 1, 0, 1], [18/29, 11/29, 18/29, 11/29], 39204/707281, Yes)","explanation":"Assign factor p to each 1 and 1−p to each 0 in original word order, then multiply exactly. The factorization comes from the same product-law premise; it is not a test of an unknown dependent law. The exact result, retaining the requested order, is ([0, 1, 0, 1], [18/29, 11/29, 18/29, 11/29], 39204/707281, Yes)."},"skills":["Multiply Every Ordered Atom Factor in the Finite Word"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r06-iid-sequence-0062","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d753dd57432618885147b10f8b73c63029ddf3ae21a1ecee86631be9620b99cf","family_id":"qf-mod29-iid-finite-factorization","seed":202610296062,"semantic_template":"qf-mod29-iid-finite-factorization/{\"p\":[12,29],\"word\":[1,1,0,1]}"},"fields":{"title":"Multiply Every Ordered Atom Factor in the Finite Word","stem":"Use the explicitly declared Bernoulli product-law model with exact p=12/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. Ordered binary word=[1, 1, 0, 1]. Give (word,atom_factors,product,pass?).","answer":{"kind":"expression","value":"([1, 1, 0, 1], [12/29, 12/29, 17/29, 12/29], 29376/707281, Yes)","explanation":"Assign factor p to each 1 and 1−p to each 0 in original word order, then multiply exactly. The factorization comes from the same product-law premise; it is not a test of an unknown dependent law. The exact result, retaining the requested order, is ([1, 1, 0, 1], [12/29, 12/29, 17/29, 12/29], 29376/707281, Yes)."},"skills":["Multiply Every Ordered Atom Factor in the Finite Word"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r06-iid-sequence-0063","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ed25f28f5dfa9d557992945bb29a6d3423cf43ba8650835dd5c668ab675a3a39","family_id":"qf-mod29-iid-finite-factorization","seed":202610296063,"semantic_template":"qf-mod29-iid-finite-factorization/{\"p\":[13,29],\"word\":[0,0,1,1]}"},"fields":{"title":"Multiply Every Ordered Atom Factor in the Finite Word","stem":"Use the explicitly declared Bernoulli product-law model with exact p=13/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. Ordered binary word=[0, 0, 1, 1]. Give (word,atom_factors,product,pass?).","answer":{"kind":"expression","value":"([0, 0, 1, 1], [16/29, 16/29, 13/29, 13/29], 43264/707281, Yes)","explanation":"Assign factor p to each 1 and 1−p to each 0 in original word order, then multiply exactly. The factorization comes from the same product-law premise; it is not a test of an unknown dependent law. The exact result, retaining the requested order, is ([0, 0, 1, 1], [16/29, 16/29, 13/29, 13/29], 43264/707281, Yes)."},"skills":["Multiply Every Ordered Atom Factor in the Finite Word"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r06-iid-sequence-0064","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9264b3709bcc7d60b2ad4b9094a7d96ee3b15d3c3f0092dfe436d1265f74882d","family_id":"qf-mod29-iid-finite-factorization","seed":202610296064,"semantic_template":"qf-mod29-iid-finite-factorization/{\"p\":[14,29],\"word\":[1,0,1,1]}"},"fields":{"title":"Multiply Every Ordered Atom Factor in the Finite Word","stem":"Use the explicitly declared Bernoulli product-law model with exact p=14/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. Ordered binary word=[1, 0, 1, 1]. Give (word,atom_factors,product,pass?).","answer":{"kind":"expression","value":"([1, 0, 1, 1], [14/29, 15/29, 14/29, 14/29], 41160/707281, Yes)","explanation":"Assign factor p to each 1 and 1−p to each 0 in original word order, then multiply exactly. The factorization comes from the same product-law premise; it is not a test of an unknown dependent law. The exact result, retaining the requested order, is ([1, 0, 1, 1], [14/29, 15/29, 14/29, 14/29], 41160/707281, Yes)."},"skills":["Multiply Every Ordered Atom Factor in the Finite Word"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r06-iid-sequence-0065","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"58dce8329c6523388b2b0acd242c2d9462f937de73d2c03e1ca93349fa2e44cd","family_id":"qf-mod29-iid-finite-factorization","seed":202610296065,"semantic_template":"qf-mod29-iid-finite-factorization/{\"p\":[15,29],\"word\":[0,1,1,1]}"},"fields":{"title":"Multiply Every Ordered Atom Factor in the Finite Word","stem":"Use the explicitly declared Bernoulli product-law model with exact p=15/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. Ordered binary word=[0, 1, 1, 1]. Give (word,atom_factors,product,pass?).","answer":{"kind":"expression","value":"([0, 1, 1, 1], [14/29, 15/29, 15/29, 15/29], 47250/707281, Yes)","explanation":"Assign factor p to each 1 and 1−p to each 0 in original word order, then multiply exactly. The factorization comes from the same product-law premise; it is not a test of an unknown dependent law. The exact result, retaining the requested order, is ([0, 1, 1, 1], [14/29, 15/29, 15/29, 15/29], 47250/707281, Yes)."},"skills":["Multiply Every Ordered Atom Factor in the Finite Word"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r06-iid-sequence-0066","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d36284082095d0028ee48ab01c82d83440fe78f7c774c58a0489d3e89701abbc","family_id":"qf-mod29-iid-finite-factorization","seed":202610296066,"semantic_template":"qf-mod29-iid-finite-factorization/{\"p\":[16,29],\"word\":[1,1,1,1]}"},"fields":{"title":"Multiply Every Ordered Atom Factor in the Finite Word","stem":"Use the explicitly declared Bernoulli product-law model with exact p=16/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. Ordered binary word=[1, 1, 1, 1]. Give (word,atom_factors,product,pass?).","answer":{"kind":"expression","value":"([1, 1, 1, 1], [16/29, 16/29, 16/29, 16/29], 65536/707281, Yes)","explanation":"Assign factor p to each 1 and 1−p to each 0 in original word order, then multiply exactly. The factorization comes from the same product-law premise; it is not a test of an unknown dependent law. The exact result, retaining the requested order, is ([1, 1, 1, 1], [16/29, 16/29, 16/29, 16/29], 65536/707281, Yes)."},"skills":["Multiply Every Ordered Atom Factor in the Finite Word"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r06-iid-sequence-0067","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c2b066438fb15cee8c0908f9743a5313db35a47990c90b096ddb7b233c5ddefb","family_id":"qf-mod29-iid-finite-factorization","seed":202610296067,"semantic_template":"qf-mod29-iid-finite-factorization/{\"p\":[17,29],\"word\":[0,0,0,0]}"},"fields":{"title":"Multiply Every Ordered Atom Factor in the Finite Word","stem":"Use the explicitly declared Bernoulli product-law model with exact p=17/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. Ordered binary word=[0, 0, 0, 0]. Give (word,atom_factors,product,pass?).","answer":{"kind":"expression","value":"([0, 0, 0, 0], [12/29, 12/29, 12/29, 12/29], 20736/707281, Yes)","explanation":"Assign factor p to each 1 and 1−p to each 0 in original word order, then multiply exactly. The factorization comes from the same product-law premise; it is not a test of an unknown dependent law. The exact result, retaining the requested order, is ([0, 0, 0, 0], [12/29, 12/29, 12/29, 12/29], 20736/707281, Yes)."},"skills":["Multiply Every Ordered Atom Factor in the Finite Word"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r06-iid-sequence-0068","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f0ad8d1c28c4564501c963881d81263f7f19c64dbf90c884054c900126a3c85f","family_id":"qf-mod29-iid-finite-factorization","seed":202610296068,"semantic_template":"qf-mod29-iid-finite-factorization/{\"p\":[18,29],\"word\":[1,0,0,0]}"},"fields":{"title":"Multiply Every Ordered Atom Factor in the Finite Word","stem":"Use the explicitly declared Bernoulli product-law model with exact p=18/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. Ordered binary word=[1, 0, 0, 0]. Give (word,atom_factors,product,pass?).","answer":{"kind":"expression","value":"([1, 0, 0, 0], [18/29, 11/29, 11/29, 11/29], 23958/707281, Yes)","explanation":"Assign factor p to each 1 and 1−p to each 0 in original word order, then multiply exactly. The factorization comes from the same product-law premise; it is not a test of an unknown dependent law. The exact result, retaining the requested order, is ([1, 0, 0, 0], [18/29, 11/29, 11/29, 11/29], 23958/707281, Yes)."},"skills":["Multiply Every Ordered Atom Factor in the Finite Word"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r06-iid-sequence-0069","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"73bb8fda78dcaf7d230beca5121d3754f06bdffad48546daf58ecee7371eabbd","family_id":"qf-mod29-iid-finite-factorization","seed":202610296069,"semantic_template":"qf-mod29-iid-finite-factorization/{\"p\":[19,29],\"word\":[0,1,0,0]}"},"fields":{"title":"Multiply Every Ordered Atom Factor in the Finite Word","stem":"Use the explicitly declared Bernoulli product-law model with exact p=19/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. Ordered binary word=[0, 1, 0, 0]. Give (word,atom_factors,product,pass?).","answer":{"kind":"expression","value":"([0, 1, 0, 0], [10/29, 19/29, 10/29, 10/29], 19000/707281, Yes)","explanation":"Assign factor p to each 1 and 1−p to each 0 in original word order, then multiply exactly. The factorization comes from the same product-law premise; it is not a test of an unknown dependent law. The exact result, retaining the requested order, is ([0, 1, 0, 0], [10/29, 19/29, 10/29, 10/29], 19000/707281, Yes)."},"skills":["Multiply Every Ordered Atom Factor in the Finite Word"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r06-iid-sequence-0070","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9197f91646daf7efa92195c1b5a60623fc61f5e30c187774abe29de2b04b2262","family_id":"qf-mod29-iid-finite-factorization","seed":202610296070,"semantic_template":"qf-mod29-iid-finite-factorization/{\"p\":[20,29],\"word\":[1,1,0,0]}"},"fields":{"title":"Multiply Every Ordered Atom Factor in the Finite Word","stem":"Use the explicitly declared Bernoulli product-law model with exact p=20/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. Ordered binary word=[1, 1, 0, 0]. Give (word,atom_factors,product,pass?).","answer":{"kind":"expression","value":"([1, 1, 0, 0], [20/29, 20/29, 9/29, 9/29], 32400/707281, Yes)","explanation":"Assign factor p to each 1 and 1−p to each 0 in original word order, then multiply exactly. The factorization comes from the same product-law premise; it is not a test of an unknown dependent law. The exact result, retaining the requested order, is ([1, 1, 0, 0], [20/29, 20/29, 9/29, 9/29], 32400/707281, Yes)."},"skills":["Multiply Every Ordered Atom Factor in the Finite Word"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r06-iid-sequence-0071","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"17628c09346cf7d79626225c9bd666fed82bd3f65b70ce5e14c2b63768432aab","family_id":"qf-mod29-iid-finite-factorization","seed":202610296071,"semantic_template":"qf-mod29-iid-finite-factorization/{\"p\":[21,29],\"word\":[0,0,1,0]}"},"fields":{"title":"Multiply Every Ordered Atom Factor in the Finite Word","stem":"Use the explicitly declared Bernoulli product-law model with exact p=21/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. Ordered binary word=[0, 0, 1, 0]. Give (word,atom_factors,product,pass?).","answer":{"kind":"expression","value":"([0, 0, 1, 0], [8/29, 8/29, 21/29, 8/29], 10752/707281, Yes)","explanation":"Assign factor p to each 1 and 1−p to each 0 in original word order, then multiply exactly. The factorization comes from the same product-law premise; it is not a test of an unknown dependent law. The exact result, retaining the requested order, is ([0, 0, 1, 0], [8/29, 8/29, 21/29, 8/29], 10752/707281, Yes)."},"skills":["Multiply Every Ordered Atom Factor in the Finite Word"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r06-iid-sequence-0072","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4fb5248a455abbcdfe233066aa1f0c0cab38ad7e8a06d29e792d9d46dbc450ae","family_id":"qf-mod29-iid-finite-factorization","seed":202610296072,"semantic_template":"qf-mod29-iid-finite-factorization/{\"p\":[22,29],\"word\":[1,0,1,0]}"},"fields":{"title":"Multiply Every Ordered Atom Factor in the Finite Word","stem":"Use the explicitly declared Bernoulli product-law model with exact p=22/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. Ordered binary word=[1, 0, 1, 0]. Give (word,atom_factors,product,pass?).","answer":{"kind":"expression","value":"([1, 0, 1, 0], [22/29, 7/29, 22/29, 7/29], 23716/707281, Yes)","explanation":"Assign factor p to each 1 and 1−p to each 0 in original word order, then multiply exactly. The factorization comes from the same product-law premise; it is not a test of an unknown dependent law. The exact result, retaining the requested order, is ([1, 0, 1, 0], [22/29, 7/29, 22/29, 7/29], 23716/707281, Yes)."},"skills":["Multiply Every Ordered Atom Factor in the Finite Word"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r06-iid-sequence-0073","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5a234dcffcdb6ccd2b6e7769dc7d6bcd270b9cae5d22024cd1b4204f42794f08","family_id":"qf-mod29-iid-finite-factorization","seed":202610296073,"semantic_template":"qf-mod29-iid-finite-factorization/{\"p\":[23,29],\"word\":[0,1,1,0]}"},"fields":{"title":"Multiply Every Ordered Atom Factor in the Finite Word","stem":"Use the explicitly declared Bernoulli product-law model with exact p=23/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. Ordered binary word=[0, 1, 1, 0]. Give (word,atom_factors,product,pass?).","answer":{"kind":"expression","value":"([0, 1, 1, 0], [6/29, 23/29, 23/29, 6/29], 19044/707281, Yes)","explanation":"Assign factor p to each 1 and 1−p to each 0 in original word order, then multiply exactly. The factorization comes from the same product-law premise; it is not a test of an unknown dependent law. The exact result, retaining the requested order, is ([0, 1, 1, 0], [6/29, 23/29, 23/29, 6/29], 19044/707281, Yes)."},"skills":["Multiply Every Ordered Atom Factor in the Finite Word"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r06-iid-sequence-0074","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"55f9016a957009c75d3ee1c4a9d6d081005ec1083bf6edf99f82c4bb81c0231d","family_id":"qf-mod29-iid-finite-factorization","seed":202610296074,"semantic_template":"qf-mod29-iid-finite-factorization/{\"p\":[24,29],\"word\":[1,1,1,0]}"},"fields":{"title":"Multiply Every Ordered Atom Factor in the Finite Word","stem":"Use the explicitly declared Bernoulli product-law model with exact p=24/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. Ordered binary word=[1, 1, 1, 0]. Give (word,atom_factors,product,pass?).","answer":{"kind":"expression","value":"([1, 1, 1, 0], [24/29, 24/29, 24/29, 5/29], 69120/707281, Yes)","explanation":"Assign factor p to each 1 and 1−p to each 0 in original word order, then multiply exactly. The factorization comes from the same product-law premise; it is not a test of an unknown dependent law. The exact result, retaining the requested order, is ([1, 1, 1, 0], [24/29, 24/29, 24/29, 5/29], 69120/707281, Yes)."},"skills":["Multiply Every Ordered Atom Factor in the Finite Word"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r06-iid-sequence-0075","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"aa5d7db6e47aad8d6ccf83e577d3dc68c22b2260da9f5eec7e6016f3b9f21a7b","family_id":"qf-mod29-iid-finite-factorization","seed":202610296075,"semantic_template":"qf-mod29-iid-finite-factorization/{\"p\":[25,29],\"word\":[0,0,0,1]}"},"fields":{"title":"Multiply Every Ordered Atom Factor in the Finite Word","stem":"Use the explicitly declared Bernoulli product-law model with exact p=25/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. Ordered binary word=[0, 0, 0, 1]. Give (word,atom_factors,product,pass?).","answer":{"kind":"expression","value":"([0, 0, 0, 1], [4/29, 4/29, 4/29, 25/29], 1600/707281, Yes)","explanation":"Assign factor p to each 1 and 1−p to each 0 in original word order, then multiply exactly. The factorization comes from the same product-law premise; it is not a test of an unknown dependent law. The exact result, retaining the requested order, is ([0, 0, 0, 1], [4/29, 4/29, 4/29, 25/29], 1600/707281, Yes)."},"skills":["Multiply Every Ordered Atom Factor in the Finite Word"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r06-iid-sequence-0076","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2f990197892bc252d5772a00fe14f41afad0e4d286d8b0696ebb5c5a72b3b7ae","family_id":"qf-mod29-iid-evidence-mode","seed":202610296076,"semantic_template":"qf-mod29-iid-evidence-mode/{\"checked_prefix\":1,\"evidence_mode\":\"finite_prefix_only\",\"p\":[1,29]}"},"fields":{"title":"A Longer Fixed Prefix Still Does Not Certify an Infinite IID Sequence","stem":"Use the explicitly declared Bernoulli product-law model with exact p=1/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. evidence_mode=finite_prefix_only, checked_prefix=1. Give (p,evidence_mode,checked_prefix,sufficient?,boundary). symbolic_product_law yields infinite_IID_certificate; finite_prefix_only yields finite_prefix_only_not_infinite_IID.","answer":{"kind":"expression","value":"(1/29, finite_prefix_only, 1, No, finite_prefix_only_not_infinite_IID)","explanation":"Judge the evidence mode, not how large the checked prefix looks. The finite-prefix No rejects sufficiency of that evidence, not the existence of the declared Bernoulli product model. The exact result, retaining the requested order, is (1/29, finite_prefix_only, 1, No, finite_prefix_only_not_infinite_IID)."},"skills":["A Longer Fixed Prefix Still Does Not Certify an Infinite IID Sequence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r06-iid-sequence-0077","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"929fa4f49421d0558d8114661b12be0869bfe35c0349ff628a4fa7000c4d7e84","family_id":"qf-mod29-iid-evidence-mode","seed":202610296077,"semantic_template":"qf-mod29-iid-evidence-mode/{\"checked_prefix\":2,\"evidence_mode\":\"symbolic_product_law\",\"p\":[2,29]}"},"fields":{"title":"A Longer Fixed Prefix Still Does Not Certify an Infinite IID Sequence","stem":"Use the explicitly declared Bernoulli product-law model with exact p=2/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. evidence_mode=symbolic_product_law, checked_prefix=2. Give (p,evidence_mode,checked_prefix,sufficient?,boundary). symbolic_product_law yields infinite_IID_certificate; finite_prefix_only yields finite_prefix_only_not_infinite_IID.","answer":{"kind":"expression","value":"(2/29, symbolic_product_law, 2, Yes, infinite_IID_certificate)","explanation":"Judge the evidence mode, not how large the checked prefix looks. The finite-prefix No rejects sufficiency of that evidence, not the existence of the declared Bernoulli product model. The exact result, retaining the requested order, is (2/29, symbolic_product_law, 2, Yes, infinite_IID_certificate)."},"skills":["A Longer Fixed Prefix Still Does Not Certify an Infinite IID Sequence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r06-iid-sequence-0078","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b0af61830acf327992918396eb20a518af2565a3cbbedf74a597fe472e59ea80","family_id":"qf-mod29-iid-evidence-mode","seed":202610296078,"semantic_template":"qf-mod29-iid-evidence-mode/{\"checked_prefix\":3,\"evidence_mode\":\"symbolic_product_law\",\"p\":[3,29]}"},"fields":{"title":"A Longer Fixed Prefix Still Does Not Certify an Infinite IID Sequence","stem":"Use the explicitly declared Bernoulli product-law model with exact p=3/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. evidence_mode=symbolic_product_law, checked_prefix=3. Give (p,evidence_mode,checked_prefix,sufficient?,boundary). symbolic_product_law yields infinite_IID_certificate; finite_prefix_only yields finite_prefix_only_not_infinite_IID.","answer":{"kind":"expression","value":"(3/29, symbolic_product_law, 3, Yes, infinite_IID_certificate)","explanation":"Judge the evidence mode, not how large the checked prefix looks. The finite-prefix No rejects sufficiency of that evidence, not the existence of the declared Bernoulli product model. The exact result, retaining the requested order, is (3/29, symbolic_product_law, 3, Yes, infinite_IID_certificate)."},"skills":["A Longer Fixed Prefix Still Does Not Certify an Infinite IID Sequence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r06-iid-sequence-0079","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"89be71b807dd122e44eddda9dccef957630249339a78f8a5293cddaa54da3082","family_id":"qf-mod29-iid-evidence-mode","seed":202610296079,"semantic_template":"qf-mod29-iid-evidence-mode/{\"checked_prefix\":4,\"evidence_mode\":\"symbolic_product_law\",\"p\":[4,29]}"},"fields":{"title":"A Longer Fixed Prefix Still Does Not Certify an Infinite IID Sequence","stem":"Use the explicitly declared Bernoulli product-law model with exact p=4/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. evidence_mode=symbolic_product_law, checked_prefix=4. Give (p,evidence_mode,checked_prefix,sufficient?,boundary). symbolic_product_law yields infinite_IID_certificate; finite_prefix_only yields finite_prefix_only_not_infinite_IID.","answer":{"kind":"expression","value":"(4/29, symbolic_product_law, 4, Yes, infinite_IID_certificate)","explanation":"Judge the evidence mode, not how large the checked prefix looks. The finite-prefix No rejects sufficiency of that evidence, not the existence of the declared Bernoulli product model. The exact result, retaining the requested order, is (4/29, symbolic_product_law, 4, Yes, infinite_IID_certificate)."},"skills":["A Longer Fixed Prefix Still Does Not Certify an Infinite IID Sequence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r06-iid-sequence-0080","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f74f5183812298493979495567c6ba773fa8f5209b39ef2ab2b331673761514f","family_id":"qf-mod29-iid-evidence-mode","seed":202610296080,"semantic_template":"qf-mod29-iid-evidence-mode/{\"checked_prefix\":5,\"evidence_mode\":\"finite_prefix_only\",\"p\":[5,29]}"},"fields":{"title":"A Longer Fixed Prefix Still Does Not Certify an Infinite IID Sequence","stem":"Use the explicitly declared Bernoulli product-law model with exact p=5/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. evidence_mode=finite_prefix_only, checked_prefix=5. Give (p,evidence_mode,checked_prefix,sufficient?,boundary). symbolic_product_law yields infinite_IID_certificate; finite_prefix_only yields finite_prefix_only_not_infinite_IID.","answer":{"kind":"expression","value":"(5/29, finite_prefix_only, 5, No, finite_prefix_only_not_infinite_IID)","explanation":"Judge the evidence mode, not how large the checked prefix looks. The finite-prefix No rejects sufficiency of that evidence, not the existence of the declared Bernoulli product model. The exact result, retaining the requested order, is (5/29, finite_prefix_only, 5, No, finite_prefix_only_not_infinite_IID)."},"skills":["A Longer Fixed Prefix Still Does Not Certify an Infinite IID Sequence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r06-iid-sequence-0081","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"49c0e3301016e47f1aec64fb98d438e579a5d69d4cfee5ee262eefb7382b6c10","family_id":"qf-mod29-iid-evidence-mode","seed":202610296081,"semantic_template":"qf-mod29-iid-evidence-mode/{\"checked_prefix\":6,\"evidence_mode\":\"symbolic_product_law\",\"p\":[6,29]}"},"fields":{"title":"A Longer Fixed Prefix Still Does Not Certify an Infinite IID Sequence","stem":"Use the explicitly declared Bernoulli product-law model with exact p=6/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. evidence_mode=symbolic_product_law, checked_prefix=6. Give (p,evidence_mode,checked_prefix,sufficient?,boundary). symbolic_product_law yields infinite_IID_certificate; finite_prefix_only yields finite_prefix_only_not_infinite_IID.","answer":{"kind":"expression","value":"(6/29, symbolic_product_law, 6, Yes, infinite_IID_certificate)","explanation":"Judge the evidence mode, not how large the checked prefix looks. The finite-prefix No rejects sufficiency of that evidence, not the existence of the declared Bernoulli product model. The exact result, retaining the requested order, is (6/29, symbolic_product_law, 6, Yes, infinite_IID_certificate)."},"skills":["A Longer Fixed Prefix Still Does Not Certify an Infinite IID Sequence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r06-iid-sequence-0082","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"18368c162eaa32809faf0483353a279aa920cae652277a2d2a0f34b17af7ddee","family_id":"qf-mod29-iid-evidence-mode","seed":202610296082,"semantic_template":"qf-mod29-iid-evidence-mode/{\"checked_prefix\":7,\"evidence_mode\":\"symbolic_product_law\",\"p\":[7,29]}"},"fields":{"title":"A Longer Fixed Prefix Still Does Not Certify an Infinite IID Sequence","stem":"Use the explicitly declared Bernoulli product-law model with exact p=7/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. evidence_mode=symbolic_product_law, checked_prefix=7. Give (p,evidence_mode,checked_prefix,sufficient?,boundary). symbolic_product_law yields infinite_IID_certificate; finite_prefix_only yields finite_prefix_only_not_infinite_IID.","answer":{"kind":"expression","value":"(7/29, symbolic_product_law, 7, Yes, infinite_IID_certificate)","explanation":"Judge the evidence mode, not how large the checked prefix looks. The finite-prefix No rejects sufficiency of that evidence, not the existence of the declared Bernoulli product model. The exact result, retaining the requested order, is (7/29, symbolic_product_law, 7, Yes, infinite_IID_certificate)."},"skills":["A Longer Fixed Prefix Still Does Not Certify an Infinite IID Sequence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r06-iid-sequence-0083","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b18588b33e7fa2c4effd57bbe715b3bba65f03d1bbc0e2e8c01ce3bfc813a859","family_id":"qf-mod29-iid-evidence-mode","seed":202610296083,"semantic_template":"qf-mod29-iid-evidence-mode/{\"checked_prefix\":8,\"evidence_mode\":\"symbolic_product_law\",\"p\":[8,29]}"},"fields":{"title":"A Longer Fixed Prefix Still Does Not Certify an Infinite IID Sequence","stem":"Use the explicitly declared Bernoulli product-law model with exact p=8/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. evidence_mode=symbolic_product_law, checked_prefix=8. Give (p,evidence_mode,checked_prefix,sufficient?,boundary). symbolic_product_law yields infinite_IID_certificate; finite_prefix_only yields finite_prefix_only_not_infinite_IID.","answer":{"kind":"expression","value":"(8/29, symbolic_product_law, 8, Yes, infinite_IID_certificate)","explanation":"Judge the evidence mode, not how large the checked prefix looks. The finite-prefix No rejects sufficiency of that evidence, not the existence of the declared Bernoulli product model. The exact result, retaining the requested order, is (8/29, symbolic_product_law, 8, Yes, infinite_IID_certificate)."},"skills":["A Longer Fixed Prefix Still Does Not Certify an Infinite IID Sequence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r06-iid-sequence-0084","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3166ca441a283037bd33411cc3e16b1da9f8586efce25192a83089af2408b0ef","family_id":"qf-mod29-iid-evidence-mode","seed":202610296084,"semantic_template":"qf-mod29-iid-evidence-mode/{\"checked_prefix\":9,\"evidence_mode\":\"finite_prefix_only\",\"p\":[9,29]}"},"fields":{"title":"A Longer Fixed Prefix Still Does Not Certify an Infinite IID Sequence","stem":"Use the explicitly declared Bernoulli product-law model with exact p=9/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. evidence_mode=finite_prefix_only, checked_prefix=9. Give (p,evidence_mode,checked_prefix,sufficient?,boundary). symbolic_product_law yields infinite_IID_certificate; finite_prefix_only yields finite_prefix_only_not_infinite_IID.","answer":{"kind":"expression","value":"(9/29, finite_prefix_only, 9, No, finite_prefix_only_not_infinite_IID)","explanation":"Judge the evidence mode, not how large the checked prefix looks. The finite-prefix No rejects sufficiency of that evidence, not the existence of the declared Bernoulli product model. The exact result, retaining the requested order, is (9/29, finite_prefix_only, 9, No, finite_prefix_only_not_infinite_IID)."},"skills":["A Longer Fixed Prefix Still Does Not Certify an Infinite IID Sequence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r06-iid-sequence-0085","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cf280cf694db640c0d74dd89b226f16f7ba44c69daa66c4ccaeb755db7de29c1","family_id":"qf-mod29-iid-evidence-mode","seed":202610296085,"semantic_template":"qf-mod29-iid-evidence-mode/{\"checked_prefix\":10,\"evidence_mode\":\"symbolic_product_law\",\"p\":[10,29]}"},"fields":{"title":"A Longer Fixed Prefix Still Does Not Certify an Infinite IID Sequence","stem":"Use the explicitly declared Bernoulli product-law model with exact p=10/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. evidence_mode=symbolic_product_law, checked_prefix=10. Give (p,evidence_mode,checked_prefix,sufficient?,boundary). symbolic_product_law yields infinite_IID_certificate; finite_prefix_only yields finite_prefix_only_not_infinite_IID.","answer":{"kind":"expression","value":"(10/29, symbolic_product_law, 10, Yes, infinite_IID_certificate)","explanation":"Judge the evidence mode, not how large the checked prefix looks. The finite-prefix No rejects sufficiency of that evidence, not the existence of the declared Bernoulli product model. The exact result, retaining the requested order, is (10/29, symbolic_product_law, 10, Yes, infinite_IID_certificate)."},"skills":["A Longer Fixed Prefix Still Does Not Certify an Infinite IID Sequence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r06-iid-sequence-0086","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fbf2ebda2a4ab8fe8e5fe2127c4102ec8e09a9a3add56d72afa6de339e977d6c","family_id":"qf-mod29-iid-evidence-mode","seed":202610296086,"semantic_template":"qf-mod29-iid-evidence-mode/{\"checked_prefix\":11,\"evidence_mode\":\"symbolic_product_law\",\"p\":[11,29]}"},"fields":{"title":"A Longer Fixed Prefix Still Does Not Certify an Infinite IID Sequence","stem":"Use the explicitly declared Bernoulli product-law model with exact p=11/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. evidence_mode=symbolic_product_law, checked_prefix=11. Give (p,evidence_mode,checked_prefix,sufficient?,boundary). symbolic_product_law yields infinite_IID_certificate; finite_prefix_only yields finite_prefix_only_not_infinite_IID.","answer":{"kind":"expression","value":"(11/29, symbolic_product_law, 11, Yes, infinite_IID_certificate)","explanation":"Judge the evidence mode, not how large the checked prefix looks. The finite-prefix No rejects sufficiency of that evidence, not the existence of the declared Bernoulli product model. The exact result, retaining the requested order, is (11/29, symbolic_product_law, 11, Yes, infinite_IID_certificate)."},"skills":["A Longer Fixed Prefix Still Does Not Certify an Infinite IID Sequence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r06-iid-sequence-0087","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2f6afa738e6badd825d0fccb9e235c8d7fc57a1c37e0c25435c6d7dfb1827afc","family_id":"qf-mod29-iid-evidence-mode","seed":202610296087,"semantic_template":"qf-mod29-iid-evidence-mode/{\"checked_prefix\":12,\"evidence_mode\":\"symbolic_product_law\",\"p\":[12,29]}"},"fields":{"title":"A Longer Fixed Prefix Still Does Not Certify an Infinite IID Sequence","stem":"Use the explicitly declared Bernoulli product-law model with exact p=12/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. evidence_mode=symbolic_product_law, checked_prefix=12. Give (p,evidence_mode,checked_prefix,sufficient?,boundary). symbolic_product_law yields infinite_IID_certificate; finite_prefix_only yields finite_prefix_only_not_infinite_IID.","answer":{"kind":"expression","value":"(12/29, symbolic_product_law, 12, Yes, infinite_IID_certificate)","explanation":"Judge the evidence mode, not how large the checked prefix looks. The finite-prefix No rejects sufficiency of that evidence, not the existence of the declared Bernoulli product model. The exact result, retaining the requested order, is (12/29, symbolic_product_law, 12, Yes, infinite_IID_certificate)."},"skills":["A Longer Fixed Prefix Still Does Not Certify an Infinite IID Sequence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r06-iid-sequence-0088","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"427361d0485236d8ecd234b2a48949ab3025351ea20b1e8b38b73957db84073c","family_id":"qf-mod29-iid-evidence-mode","seed":202610296088,"semantic_template":"qf-mod29-iid-evidence-mode/{\"checked_prefix\":13,\"evidence_mode\":\"finite_prefix_only\",\"p\":[13,29]}"},"fields":{"title":"A Longer Fixed Prefix Still Does Not Certify an Infinite IID Sequence","stem":"Use the explicitly declared Bernoulli product-law model with exact p=13/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. evidence_mode=finite_prefix_only, checked_prefix=13. Give (p,evidence_mode,checked_prefix,sufficient?,boundary). symbolic_product_law yields infinite_IID_certificate; finite_prefix_only yields finite_prefix_only_not_infinite_IID.","answer":{"kind":"expression","value":"(13/29, finite_prefix_only, 13, No, finite_prefix_only_not_infinite_IID)","explanation":"Judge the evidence mode, not how large the checked prefix looks. The finite-prefix No rejects sufficiency of that evidence, not the existence of the declared Bernoulli product model. The exact result, retaining the requested order, is (13/29, finite_prefix_only, 13, No, finite_prefix_only_not_infinite_IID)."},"skills":["A Longer Fixed Prefix Still Does Not Certify an Infinite IID Sequence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r06-iid-sequence-0089","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ecd0e4e578b2142b1148c6e394761211463c6cd9366b7c1696452bbd38af5e9c","family_id":"qf-mod29-iid-evidence-mode","seed":202610296089,"semantic_template":"qf-mod29-iid-evidence-mode/{\"checked_prefix\":14,\"evidence_mode\":\"symbolic_product_law\",\"p\":[14,29]}"},"fields":{"title":"A Longer Fixed Prefix Still Does Not Certify an Infinite IID Sequence","stem":"Use the explicitly declared Bernoulli product-law model with exact p=14/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. evidence_mode=symbolic_product_law, checked_prefix=14. Give (p,evidence_mode,checked_prefix,sufficient?,boundary). symbolic_product_law yields infinite_IID_certificate; finite_prefix_only yields finite_prefix_only_not_infinite_IID.","answer":{"kind":"expression","value":"(14/29, symbolic_product_law, 14, Yes, infinite_IID_certificate)","explanation":"Judge the evidence mode, not how large the checked prefix looks. The finite-prefix No rejects sufficiency of that evidence, not the existence of the declared Bernoulli product model. The exact result, retaining the requested order, is (14/29, symbolic_product_law, 14, Yes, infinite_IID_certificate)."},"skills":["A Longer Fixed Prefix Still Does Not Certify an Infinite IID Sequence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r06-iid-sequence-0090","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b4200e8e8d85393d464accada802b797fa9461da8cd401db6c9439a6edd7f391","family_id":"qf-mod29-iid-evidence-mode","seed":202610296090,"semantic_template":"qf-mod29-iid-evidence-mode/{\"checked_prefix\":15,\"evidence_mode\":\"symbolic_product_law\",\"p\":[15,29]}"},"fields":{"title":"A Longer Fixed Prefix Still Does Not Certify an Infinite IID Sequence","stem":"Use the explicitly declared Bernoulli product-law model with exact p=15/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. evidence_mode=symbolic_product_law, checked_prefix=15. Give (p,evidence_mode,checked_prefix,sufficient?,boundary). symbolic_product_law yields infinite_IID_certificate; finite_prefix_only yields finite_prefix_only_not_infinite_IID.","answer":{"kind":"expression","value":"(15/29, symbolic_product_law, 15, Yes, infinite_IID_certificate)","explanation":"Judge the evidence mode, not how large the checked prefix looks. The finite-prefix No rejects sufficiency of that evidence, not the existence of the declared Bernoulli product model. The exact result, retaining the requested order, is (15/29, symbolic_product_law, 15, Yes, infinite_IID_certificate)."},"skills":["A Longer Fixed Prefix Still Does Not Certify an Infinite IID Sequence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r06-iid-sequence-0091","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"64df5d992d565405bf4841c4bc03a0f9c69b54242f830baad3533a7e59ac64f6","family_id":"qf-mod29-iid-evidence-mode","seed":202610296091,"semantic_template":"qf-mod29-iid-evidence-mode/{\"checked_prefix\":16,\"evidence_mode\":\"symbolic_product_law\",\"p\":[16,29]}"},"fields":{"title":"A Longer Fixed Prefix Still Does Not Certify an Infinite IID Sequence","stem":"Use the explicitly declared Bernoulli product-law model with exact p=16/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. evidence_mode=symbolic_product_law, checked_prefix=16. Give (p,evidence_mode,checked_prefix,sufficient?,boundary). symbolic_product_law yields infinite_IID_certificate; finite_prefix_only yields finite_prefix_only_not_infinite_IID.","answer":{"kind":"expression","value":"(16/29, symbolic_product_law, 16, Yes, infinite_IID_certificate)","explanation":"Judge the evidence mode, not how large the checked prefix looks. The finite-prefix No rejects sufficiency of that evidence, not the existence of the declared Bernoulli product model. The exact result, retaining the requested order, is (16/29, symbolic_product_law, 16, Yes, infinite_IID_certificate)."},"skills":["A Longer Fixed Prefix Still Does Not Certify an Infinite IID Sequence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r06-iid-sequence-0092","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"af6846e315e40555f80bf4f879cc5c8479fb56c2f9d3bcbc7489b4e416bc6371","family_id":"qf-mod29-iid-evidence-mode","seed":202610296092,"semantic_template":"qf-mod29-iid-evidence-mode/{\"checked_prefix\":17,\"evidence_mode\":\"finite_prefix_only\",\"p\":[17,29]}"},"fields":{"title":"A Longer Fixed Prefix Still Does Not Certify an Infinite IID Sequence","stem":"Use the explicitly declared Bernoulli product-law model with exact p=17/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. evidence_mode=finite_prefix_only, checked_prefix=17. Give (p,evidence_mode,checked_prefix,sufficient?,boundary). symbolic_product_law yields infinite_IID_certificate; finite_prefix_only yields finite_prefix_only_not_infinite_IID.","answer":{"kind":"expression","value":"(17/29, finite_prefix_only, 17, No, finite_prefix_only_not_infinite_IID)","explanation":"Judge the evidence mode, not how large the checked prefix looks. The finite-prefix No rejects sufficiency of that evidence, not the existence of the declared Bernoulli product model. The exact result, retaining the requested order, is (17/29, finite_prefix_only, 17, No, finite_prefix_only_not_infinite_IID)."},"skills":["A Longer Fixed Prefix Still Does Not Certify an Infinite IID Sequence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r06-iid-sequence-0093","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b06647bdd93b8550a6cb8e521527c293cbb3d1462adef2a35625cc6c43d4d455","family_id":"qf-mod29-iid-evidence-mode","seed":202610296093,"semantic_template":"qf-mod29-iid-evidence-mode/{\"checked_prefix\":18,\"evidence_mode\":\"symbolic_product_law\",\"p\":[18,29]}"},"fields":{"title":"A Longer Fixed Prefix Still Does Not Certify an Infinite IID Sequence","stem":"Use the explicitly declared Bernoulli product-law model with exact p=18/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. evidence_mode=symbolic_product_law, checked_prefix=18. Give (p,evidence_mode,checked_prefix,sufficient?,boundary). symbolic_product_law yields infinite_IID_certificate; finite_prefix_only yields finite_prefix_only_not_infinite_IID.","answer":{"kind":"expression","value":"(18/29, symbolic_product_law, 18, Yes, infinite_IID_certificate)","explanation":"Judge the evidence mode, not how large the checked prefix looks. The finite-prefix No rejects sufficiency of that evidence, not the existence of the declared Bernoulli product model. The exact result, retaining the requested order, is (18/29, symbolic_product_law, 18, Yes, infinite_IID_certificate)."},"skills":["A Longer Fixed Prefix Still Does Not Certify an Infinite IID Sequence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r06-iid-sequence-0094","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c6e39f44219ad7bf649563acf3ab0eeac904e0262f7259e834eb845bb3cfb290","family_id":"qf-mod29-iid-evidence-mode","seed":202610296094,"semantic_template":"qf-mod29-iid-evidence-mode/{\"checked_prefix\":19,\"evidence_mode\":\"symbolic_product_law\",\"p\":[19,29]}"},"fields":{"title":"A Longer Fixed Prefix Still Does Not Certify an Infinite IID Sequence","stem":"Use the explicitly declared Bernoulli product-law model with exact p=19/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. evidence_mode=symbolic_product_law, checked_prefix=19. Give (p,evidence_mode,checked_prefix,sufficient?,boundary). symbolic_product_law yields infinite_IID_certificate; finite_prefix_only yields finite_prefix_only_not_infinite_IID.","answer":{"kind":"expression","value":"(19/29, symbolic_product_law, 19, Yes, infinite_IID_certificate)","explanation":"Judge the evidence mode, not how large the checked prefix looks. The finite-prefix No rejects sufficiency of that evidence, not the existence of the declared Bernoulli product model. The exact result, retaining the requested order, is (19/29, symbolic_product_law, 19, Yes, infinite_IID_certificate)."},"skills":["A Longer Fixed Prefix Still Does Not Certify an Infinite IID Sequence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r06-iid-sequence-0095","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"044926a3350d2981919e56e45f7524c195349886637c0f504c323cbbb33ae386","family_id":"qf-mod29-iid-evidence-mode","seed":202610296095,"semantic_template":"qf-mod29-iid-evidence-mode/{\"checked_prefix\":20,\"evidence_mode\":\"symbolic_product_law\",\"p\":[20,29]}"},"fields":{"title":"A Longer Fixed Prefix Still Does Not Certify an Infinite IID Sequence","stem":"Use the explicitly declared Bernoulli product-law model with exact p=20/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. evidence_mode=symbolic_product_law, checked_prefix=20. Give (p,evidence_mode,checked_prefix,sufficient?,boundary). symbolic_product_law yields infinite_IID_certificate; finite_prefix_only yields finite_prefix_only_not_infinite_IID.","answer":{"kind":"expression","value":"(20/29, symbolic_product_law, 20, Yes, infinite_IID_certificate)","explanation":"Judge the evidence mode, not how large the checked prefix looks. The finite-prefix No rejects sufficiency of that evidence, not the existence of the declared Bernoulli product model. The exact result, retaining the requested order, is (20/29, symbolic_product_law, 20, Yes, infinite_IID_certificate)."},"skills":["A Longer Fixed Prefix Still Does Not Certify an Infinite IID Sequence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r06-iid-sequence-0096","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"29df13fe09ee2f7128e5afcc65211c08e940797444a750136a79b205cf360f28","family_id":"qf-mod29-iid-evidence-mode","seed":202610296096,"semantic_template":"qf-mod29-iid-evidence-mode/{\"checked_prefix\":21,\"evidence_mode\":\"finite_prefix_only\",\"p\":[21,29]}"},"fields":{"title":"A Longer Fixed Prefix Still Does Not Certify an Infinite IID Sequence","stem":"Use the explicitly declared Bernoulli product-law model with exact p=21/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. evidence_mode=finite_prefix_only, checked_prefix=21. Give (p,evidence_mode,checked_prefix,sufficient?,boundary). symbolic_product_law yields infinite_IID_certificate; finite_prefix_only yields finite_prefix_only_not_infinite_IID.","answer":{"kind":"expression","value":"(21/29, finite_prefix_only, 21, No, finite_prefix_only_not_infinite_IID)","explanation":"Judge the evidence mode, not how large the checked prefix looks. The finite-prefix No rejects sufficiency of that evidence, not the existence of the declared Bernoulli product model. The exact result, retaining the requested order, is (21/29, finite_prefix_only, 21, No, finite_prefix_only_not_infinite_IID)."},"skills":["A Longer Fixed Prefix Still Does Not Certify an Infinite IID Sequence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r06-iid-sequence-0097","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"910cab6d9f8723841e08e604a4d40006a88163999a292ae77e496d42392bba09","family_id":"qf-mod29-iid-evidence-mode","seed":202610296097,"semantic_template":"qf-mod29-iid-evidence-mode/{\"checked_prefix\":22,\"evidence_mode\":\"symbolic_product_law\",\"p\":[22,29]}"},"fields":{"title":"A Longer Fixed Prefix Still Does Not Certify an Infinite IID Sequence","stem":"Use the explicitly declared Bernoulli product-law model with exact p=22/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. evidence_mode=symbolic_product_law, checked_prefix=22. Give (p,evidence_mode,checked_prefix,sufficient?,boundary). symbolic_product_law yields infinite_IID_certificate; finite_prefix_only yields finite_prefix_only_not_infinite_IID.","answer":{"kind":"expression","value":"(22/29, symbolic_product_law, 22, Yes, infinite_IID_certificate)","explanation":"Judge the evidence mode, not how large the checked prefix looks. The finite-prefix No rejects sufficiency of that evidence, not the existence of the declared Bernoulli product model. The exact result, retaining the requested order, is (22/29, symbolic_product_law, 22, Yes, infinite_IID_certificate)."},"skills":["A Longer Fixed Prefix Still Does Not Certify an Infinite IID Sequence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r06-iid-sequence-0098","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7571d8a4a4c67ecb4d2d844e00d90cb35678129649f955531500c9dfc6140ad5","family_id":"qf-mod29-iid-evidence-mode","seed":202610296098,"semantic_template":"qf-mod29-iid-evidence-mode/{\"checked_prefix\":23,\"evidence_mode\":\"symbolic_product_law\",\"p\":[23,29]}"},"fields":{"title":"A Longer Fixed Prefix Still Does Not Certify an Infinite IID Sequence","stem":"Use the explicitly declared Bernoulli product-law model with exact p=23/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. evidence_mode=symbolic_product_law, checked_prefix=23. Give (p,evidence_mode,checked_prefix,sufficient?,boundary). symbolic_product_law yields infinite_IID_certificate; finite_prefix_only yields finite_prefix_only_not_infinite_IID.","answer":{"kind":"expression","value":"(23/29, symbolic_product_law, 23, Yes, infinite_IID_certificate)","explanation":"Judge the evidence mode, not how large the checked prefix looks. The finite-prefix No rejects sufficiency of that evidence, not the existence of the declared Bernoulli product model. The exact result, retaining the requested order, is (23/29, symbolic_product_law, 23, Yes, infinite_IID_certificate)."},"skills":["A Longer Fixed Prefix Still Does Not Certify an Infinite IID Sequence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r06-iid-sequence-0099","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2d184b98028e47a051cba93ca0796ae4ad944b2baa6af4d142120bc2c7a4968d","family_id":"qf-mod29-iid-evidence-mode","seed":202610296099,"semantic_template":"qf-mod29-iid-evidence-mode/{\"checked_prefix\":24,\"evidence_mode\":\"symbolic_product_law\",\"p\":[24,29]}"},"fields":{"title":"A Longer Fixed Prefix Still Does Not Certify an Infinite IID Sequence","stem":"Use the explicitly declared Bernoulli product-law model with exact p=24/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. evidence_mode=symbolic_product_law, checked_prefix=24. Give (p,evidence_mode,checked_prefix,sufficient?,boundary). symbolic_product_law yields infinite_IID_certificate; finite_prefix_only yields finite_prefix_only_not_infinite_IID.","answer":{"kind":"expression","value":"(24/29, symbolic_product_law, 24, Yes, infinite_IID_certificate)","explanation":"Judge the evidence mode, not how large the checked prefix looks. The finite-prefix No rejects sufficiency of that evidence, not the existence of the declared Bernoulli product model. The exact result, retaining the requested order, is (24/29, symbolic_product_law, 24, Yes, infinite_IID_certificate)."},"skills":["A Longer Fixed Prefix Still Does Not Certify an Infinite IID Sequence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r06-iid-sequence-0100","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b3bde272e33b29834e6b3f22159ed38affdfd0e7d3e9b03ae044401d49e762d6","family_id":"qf-mod29-iid-evidence-mode","seed":202610296100,"semantic_template":"qf-mod29-iid-evidence-mode/{\"checked_prefix\":25,\"evidence_mode\":\"finite_prefix_only\",\"p\":[25,29]}"},"fields":{"title":"A Longer Fixed Prefix Still Does Not Certify an Infinite IID Sequence","stem":"Use the explicitly declared Bernoulli product-law model with exact p=25/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. evidence_mode=finite_prefix_only, checked_prefix=25. Give (p,evidence_mode,checked_prefix,sufficient?,boundary). symbolic_product_law yields infinite_IID_certificate; finite_prefix_only yields finite_prefix_only_not_infinite_IID.","answer":{"kind":"expression","value":"(25/29, finite_prefix_only, 25, No, finite_prefix_only_not_infinite_IID)","explanation":"Judge the evidence mode, not how large the checked prefix looks. The finite-prefix No rejects sufficiency of that evidence, not the existence of the declared Bernoulli product model. The exact result, retaining the requested order, is (25/29, finite_prefix_only, 25, No, finite_prefix_only_not_infinite_IID)."},"skills":["A Longer Fixed Prefix Still Does Not Certify an Infinite IID Sequence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}