{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r07-convergence-in-probability-0001","batch_id":"batch-20261014-advance-20261030-r07-convergence-in-probability-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"69abf5f81af43b21b6e265ceccafa497f17e2895eb6acf941d514cd0a2f9f3e2","family_id":"qf-mod30-convprob-deviation-probability","seed":202610307001,"semantic_template":"qf-mod30-convprob-deviation-probability/{\"n\":1,\"numerator\":1,\"shift\":2}"},"fields":{"title":"Evaluate the Deviation Profile Only at a Valid Probability Index","stem":"Use the declared rational deviation-probability tail p_n=1/(n+2) on nonnegative integer indices where this value belongs to[0,1], starting no earlier than max(0,numerator-shift). The common probability space and infinite-process interpretation remain declared premises, not constructions inferred from finitely many numbers. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Evaluate index n=1. Give (numerator,shift,n,deviation_probability).","answer":{"kind":"expression","value":"(1, 2, 1, 1/3)","explanation":"The supplied index must lie in the declared probability domain. Evaluate the exact rational value without clipping it to1, renormalizing it or silently shifting the index. One value does not establish convergence for every epsilon. The exact result, retaining the requested order, is (1, 2, 1, 1/3)."},"skills":["Evaluate the Deviation Profile Only at a Valid Probability Index"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r07-convergence-in-probability-0002","batch_id":"batch-20261014-advance-20261030-r07-convergence-in-probability-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7a3056094e8b794292020b5f70a24857f70b040dd05662bbf1c42a77614b5467","family_id":"qf-mod30-convprob-deviation-probability","seed":202610307002,"semantic_template":"qf-mod30-convprob-deviation-probability/{\"n\":2,\"numerator\":2,\"shift\":3}"},"fields":{"title":"Evaluate the Deviation Profile Only at a Valid Probability Index","stem":"Use the declared rational deviation-probability tail p_n=2/(n+3) on nonnegative integer indices where this value belongs to[0,1], starting no earlier than max(0,numerator-shift). The common probability space and infinite-process interpretation remain declared premises, not constructions inferred from finitely many numbers. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Evaluate index n=2. Give (numerator,shift,n,deviation_probability).","answer":{"kind":"expression","value":"(2, 3, 2, 2/5)","explanation":"The supplied index must lie in the declared probability domain. Evaluate the exact rational value without clipping it to1, renormalizing it or silently shifting the index. One value does not establish convergence for every epsilon. The exact result, retaining the requested order, is (2, 3, 2, 2/5)."},"skills":["Evaluate the Deviation Profile Only at a Valid Probability Index"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r07-convergence-in-probability-0003","batch_id":"batch-20261014-advance-20261030-r07-convergence-in-probability-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"093e4f00a89eb386217f0bd55735696a7dcb321eb60df05b60171d707f3716a4","family_id":"qf-mod30-convprob-deviation-probability","seed":202610307003,"semantic_template":"qf-mod30-convprob-deviation-probability/{\"n\":3,\"numerator\":3,\"shift\":4}"},"fields":{"title":"Evaluate the Deviation Profile Only at a Valid Probability Index","stem":"Use the declared rational deviation-probability tail p_n=3/(n+4) on nonnegative integer indices where this value belongs to[0,1], starting no earlier than max(0,numerator-shift). The common probability space and infinite-process interpretation remain declared premises, not constructions inferred from finitely many numbers. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Evaluate index n=3. Give (numerator,shift,n,deviation_probability).","answer":{"kind":"expression","value":"(3, 4, 3, 3/7)","explanation":"The supplied index must lie in the declared probability domain. Evaluate the exact rational value without clipping it to1, renormalizing it or silently shifting the index. One value does not establish convergence for every epsilon. The exact result, retaining the requested order, is (3, 4, 3, 3/7)."},"skills":["Evaluate the Deviation Profile Only at a Valid Probability Index"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r07-convergence-in-probability-0004","batch_id":"batch-20261014-advance-20261030-r07-convergence-in-probability-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"257c790a3a22b34d2664c08e9b1794e12b5ffe529634020526900f92790250fe","family_id":"qf-mod30-convprob-deviation-probability","seed":202610307004,"semantic_template":"qf-mod30-convprob-deviation-probability/{\"n\":4,\"numerator\":4,\"shift\":5}"},"fields":{"title":"Evaluate the Deviation Profile Only at a Valid Probability Index","stem":"Use the declared rational deviation-probability tail p_n=4/(n+5) on nonnegative integer indices where this value belongs to[0,1], starting no earlier than max(0,numerator-shift). The common probability space and infinite-process interpretation remain declared premises, not constructions inferred from finitely many numbers. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Evaluate index n=4. Give (numerator,shift,n,deviation_probability).","answer":{"kind":"expression","value":"(4, 5, 4, 4/9)","explanation":"The supplied index must lie in the declared probability domain. Evaluate the exact rational value without clipping it to1, renormalizing it or silently shifting the index. One value does not establish convergence for every epsilon. The exact result, retaining the requested order, is (4, 5, 4, 4/9)."},"skills":["Evaluate the Deviation Profile Only at a Valid Probability Index"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r07-convergence-in-probability-0005","batch_id":"batch-20261014-advance-20261030-r07-convergence-in-probability-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"26bcd63524fa2661add4ec65d6c4a6d6fe8723ceb55e9c15f951787b62e1b305","family_id":"qf-mod30-convprob-deviation-probability","seed":202610307005,"semantic_template":"qf-mod30-convprob-deviation-probability/{\"n\":5,\"numerator\":5,\"shift\":6}"},"fields":{"title":"Evaluate the Deviation Profile Only at a Valid Probability Index","stem":"Use the declared rational deviation-probability tail p_n=5/(n+6) on nonnegative integer indices where this value belongs to[0,1], starting no earlier than max(0,numerator-shift). The common probability space and infinite-process interpretation remain declared premises, not constructions inferred from finitely many numbers. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Evaluate index n=5. Give (numerator,shift,n,deviation_probability).","answer":{"kind":"expression","value":"(5, 6, 5, 5/11)","explanation":"The supplied index must lie in the declared probability domain. Evaluate the exact rational value without clipping it to1, renormalizing it or silently shifting the index. One value does not establish convergence for every epsilon. The exact result, retaining the requested order, is (5, 6, 5, 5/11)."},"skills":["Evaluate the Deviation Profile Only at a Valid Probability Index"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r07-convergence-in-probability-0006","batch_id":"batch-20261014-advance-20261030-r07-convergence-in-probability-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"957c4d7c33cf9621521b91e52ec1aaf9473e33dd54ffc3c5e7896d29a4fb846b","family_id":"qf-mod30-convprob-deviation-probability","seed":202610307006,"semantic_template":"qf-mod30-convprob-deviation-probability/{\"n\":6,\"numerator\":1,\"shift\":7}"},"fields":{"title":"Evaluate the Deviation Profile Only at a Valid Probability Index","stem":"Use the declared rational deviation-probability tail p_n=1/(n+7) on nonnegative integer indices where this value belongs to[0,1], starting no earlier than max(0,numerator-shift). The common probability space and infinite-process interpretation remain declared premises, not constructions inferred from finitely many numbers. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Evaluate index n=6. Give (numerator,shift,n,deviation_probability).","answer":{"kind":"expression","value":"(1, 7, 6, 1/13)","explanation":"The supplied index must lie in the declared probability domain. Evaluate the exact rational value without clipping it to1, renormalizing it or silently shifting the index. One value does not establish convergence for every epsilon. The exact result, retaining the requested order, is (1, 7, 6, 1/13)."},"skills":["Evaluate the Deviation Profile Only at a Valid Probability Index"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r07-convergence-in-probability-0007","batch_id":"batch-20261014-advance-20261030-r07-convergence-in-probability-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9e239f437917438747262cf83b59978c4de001a280ca81be8300390c5fe8b34c","family_id":"qf-mod30-convprob-deviation-probability","seed":202610307007,"semantic_template":"qf-mod30-convprob-deviation-probability/{\"n\":7,\"numerator\":2,\"shift\":8}"},"fields":{"title":"Evaluate the Deviation Profile Only at a Valid Probability Index","stem":"Use the declared rational deviation-probability tail p_n=2/(n+8) on nonnegative integer indices where this value belongs to[0,1], starting no earlier than max(0,numerator-shift). The common probability space and infinite-process interpretation remain declared premises, not constructions inferred from finitely many numbers. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Evaluate index n=7. Give (numerator,shift,n,deviation_probability).","answer":{"kind":"expression","value":"(2, 8, 7, 2/15)","explanation":"The supplied index must lie in the declared probability domain. Evaluate the exact rational value without clipping it to1, renormalizing it or silently shifting the index. One value does not establish convergence for every epsilon. The exact result, retaining the requested order, is (2, 8, 7, 2/15)."},"skills":["Evaluate the Deviation Profile Only at a Valid Probability Index"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r07-convergence-in-probability-0008","batch_id":"batch-20261014-advance-20261030-r07-convergence-in-probability-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d3442e89b8d95d366303ac0ac0e16023ea576a47239d5482f1fb9032e06b3a37","family_id":"qf-mod30-convprob-deviation-probability","seed":202610307008,"semantic_template":"qf-mod30-convprob-deviation-probability/{\"n\":8,\"numerator\":3,\"shift\":2}"},"fields":{"title":"Evaluate the Deviation Profile Only at a Valid Probability Index","stem":"Use the declared rational deviation-probability tail p_n=3/(n+2) on nonnegative integer indices where this value belongs to[0,1], starting no earlier than max(0,numerator-shift). The common probability space and infinite-process interpretation remain declared premises, not constructions inferred from finitely many numbers. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Evaluate index n=8. Give (numerator,shift,n,deviation_probability).","answer":{"kind":"expression","value":"(3, 2, 8, 3/10)","explanation":"The supplied index must lie in the declared probability domain. Evaluate the exact rational value without clipping it to1, renormalizing it or silently shifting the index. One value does not establish convergence for every epsilon. The exact result, retaining the requested order, is (3, 2, 8, 3/10)."},"skills":["Evaluate the Deviation Profile Only at a Valid Probability Index"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r07-convergence-in-probability-0009","batch_id":"batch-20261014-advance-20261030-r07-convergence-in-probability-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"20b6bf728d7d0d0810ea09456b6365a254812c5ee670b7d0540c3de15ddac3b9","family_id":"qf-mod30-convprob-deviation-probability","seed":202610307009,"semantic_template":"qf-mod30-convprob-deviation-probability/{\"n\":9,\"numerator\":4,\"shift\":3}"},"fields":{"title":"Evaluate the Deviation Profile Only at a Valid Probability Index","stem":"Use the declared rational deviation-probability tail p_n=4/(n+3) on nonnegative integer indices where this value belongs to[0,1], starting no earlier than max(0,numerator-shift). The common probability space and infinite-process interpretation remain declared premises, not constructions inferred from finitely many numbers. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Evaluate index n=9. Give (numerator,shift,n,deviation_probability).","answer":{"kind":"expression","value":"(4, 3, 9, 1/3)","explanation":"The supplied index must lie in the declared probability domain. Evaluate the exact rational value without clipping it to1, renormalizing it or silently shifting the index. One value does not establish convergence for every epsilon. The exact result, retaining the requested order, is (4, 3, 9, 1/3)."},"skills":["Evaluate the Deviation Profile Only at a Valid Probability Index"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r07-convergence-in-probability-0010","batch_id":"batch-20261014-advance-20261030-r07-convergence-in-probability-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ee51c842586c9cf0be8c4bc18861e16cb64ed7342a4999706f93c6e2e757c296","family_id":"qf-mod30-convprob-deviation-probability","seed":202610307010,"semantic_template":"qf-mod30-convprob-deviation-probability/{\"n\":10,\"numerator\":5,\"shift\":4}"},"fields":{"title":"Evaluate the Deviation Profile Only at a Valid Probability Index","stem":"Use the declared rational deviation-probability tail p_n=5/(n+4) on nonnegative integer indices where this value belongs to[0,1], starting no earlier than max(0,numerator-shift). The common probability space and infinite-process interpretation remain declared premises, not constructions inferred from finitely many numbers. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Evaluate index n=10. Give (numerator,shift,n,deviation_probability).","answer":{"kind":"expression","value":"(5, 4, 10, 5/14)","explanation":"The supplied index must lie in the declared probability domain. Evaluate the exact rational value without clipping it to1, renormalizing it or silently shifting the index. One value does not establish convergence for every epsilon. The exact result, retaining the requested order, is (5, 4, 10, 5/14)."},"skills":["Evaluate the Deviation Profile Only at a Valid Probability Index"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r07-convergence-in-probability-0011","batch_id":"batch-20261014-advance-20261030-r07-convergence-in-probability-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"33e435d2c128d86a78e17178c8aa067ac8bf557bf170911896da8ec7b1abcb30","family_id":"qf-mod30-convprob-deviation-probability","seed":202610307011,"semantic_template":"qf-mod30-convprob-deviation-probability/{\"n\":11,\"numerator\":1,\"shift\":5}"},"fields":{"title":"Evaluate the Deviation Profile Only at a Valid Probability Index","stem":"Use the declared rational deviation-probability tail p_n=1/(n+5) on nonnegative integer indices where this value belongs to[0,1], starting no earlier than max(0,numerator-shift). The common probability space and infinite-process interpretation remain declared premises, not constructions inferred from finitely many numbers. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Evaluate index n=11. Give (numerator,shift,n,deviation_probability).","answer":{"kind":"expression","value":"(1, 5, 11, 1/16)","explanation":"The supplied index must lie in the declared probability domain. Evaluate the exact rational value without clipping it to1, renormalizing it or silently shifting the index. One value does not establish convergence for every epsilon. The exact result, retaining the requested order, is (1, 5, 11, 1/16)."},"skills":["Evaluate the Deviation Profile Only at a Valid Probability Index"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r07-convergence-in-probability-0012","batch_id":"batch-20261014-advance-20261030-r07-convergence-in-probability-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9b4da8183649ad982877262b1e7ea020ef2f94de482cc94ff626fa54a1db6f4b","family_id":"qf-mod30-convprob-deviation-probability","seed":202610307012,"semantic_template":"qf-mod30-convprob-deviation-probability/{\"n\":12,\"numerator\":2,\"shift\":6}"},"fields":{"title":"Evaluate the Deviation Profile Only at a Valid Probability Index","stem":"Use the declared rational deviation-probability tail p_n=2/(n+6) on nonnegative integer indices where this value belongs to[0,1], starting no earlier than max(0,numerator-shift). The common probability space and infinite-process interpretation remain declared premises, not constructions inferred from finitely many numbers. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Evaluate index n=12. Give (numerator,shift,n,deviation_probability).","answer":{"kind":"expression","value":"(2, 6, 12, 1/9)","explanation":"The supplied index must lie in the declared probability domain. Evaluate the exact rational value without clipping it to1, renormalizing it or silently shifting the index. One value does not establish convergence for every epsilon. The exact result, retaining the requested order, is (2, 6, 12, 1/9)."},"skills":["Evaluate the Deviation Profile Only at a Valid Probability Index"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r07-convergence-in-probability-0013","batch_id":"batch-20261014-advance-20261030-r07-convergence-in-probability-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"faed14f5fe2230baaadb9bafc2e60874ad2bdaff21eac4d1d081f184344d1416","family_id":"qf-mod30-convprob-deviation-probability","seed":202610307013,"semantic_template":"qf-mod30-convprob-deviation-probability/{\"n\":13,\"numerator\":3,\"shift\":7}"},"fields":{"title":"Evaluate the Deviation Profile Only at a Valid Probability Index","stem":"Use the declared rational deviation-probability tail p_n=3/(n+7) on nonnegative integer indices where this value belongs to[0,1], starting no earlier than max(0,numerator-shift). The common probability space and infinite-process interpretation remain declared premises, not constructions inferred from finitely many numbers. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Evaluate index n=13. Give (numerator,shift,n,deviation_probability).","answer":{"kind":"expression","value":"(3, 7, 13, 3/20)","explanation":"The supplied index must lie in the declared probability domain. Evaluate the exact rational value without clipping it to1, renormalizing it or silently shifting the index. One value does not establish convergence for every epsilon. The exact result, retaining the requested order, is (3, 7, 13, 3/20)."},"skills":["Evaluate the Deviation Profile Only at a Valid Probability Index"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r07-convergence-in-probability-0014","batch_id":"batch-20261014-advance-20261030-r07-convergence-in-probability-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"060304176890018434afcb6a9ec72a6a61bc32797e4c28808abea8f1a80c0a91","family_id":"qf-mod30-convprob-deviation-probability","seed":202610307014,"semantic_template":"qf-mod30-convprob-deviation-probability/{\"n\":14,\"numerator\":4,\"shift\":8}"},"fields":{"title":"Evaluate the Deviation Profile Only at a Valid Probability Index","stem":"Use the declared rational deviation-probability tail p_n=4/(n+8) on nonnegative integer indices where this value belongs to[0,1], starting no earlier than max(0,numerator-shift). The common probability space and infinite-process interpretation remain declared premises, not constructions inferred from finitely many numbers. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Evaluate index n=14. Give (numerator,shift,n,deviation_probability).","answer":{"kind":"expression","value":"(4, 8, 14, 2/11)","explanation":"The supplied index must lie in the declared probability domain. Evaluate the exact rational value without clipping it to1, renormalizing it or silently shifting the index. One value does not establish convergence for every epsilon. The exact result, retaining the requested order, is (4, 8, 14, 2/11)."},"skills":["Evaluate the Deviation Profile Only at a Valid Probability Index"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r07-convergence-in-probability-0015","batch_id":"batch-20261014-advance-20261030-r07-convergence-in-probability-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"40d9f3e9679676fb61d85738b9b50859694a0604b209107f1091ab3212f4172d","family_id":"qf-mod30-convprob-deviation-probability","seed":202610307015,"semantic_template":"qf-mod30-convprob-deviation-probability/{\"n\":15,\"numerator\":5,\"shift\":2}"},"fields":{"title":"Evaluate the Deviation Profile Only at a Valid Probability Index","stem":"Use the declared rational deviation-probability tail p_n=5/(n+2) on nonnegative integer indices where this value belongs to[0,1], starting no earlier than max(0,numerator-shift). The common probability space and infinite-process interpretation remain declared premises, not constructions inferred from finitely many numbers. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Evaluate index n=15. Give (numerator,shift,n,deviation_probability).","answer":{"kind":"expression","value":"(5, 2, 15, 5/17)","explanation":"The supplied index must lie in the declared probability domain. Evaluate the exact rational value without clipping it to1, renormalizing it or silently shifting the index. One value does not establish convergence for every epsilon. The exact result, retaining the requested order, is (5, 2, 15, 5/17)."},"skills":["Evaluate the Deviation Profile Only at a Valid Probability Index"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r07-convergence-in-probability-0016","batch_id":"batch-20261014-advance-20261030-r07-convergence-in-probability-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0ecd89e616b4d5fb82d955cdcb00b96fb1d0058d0981f1d941a669235cf84400","family_id":"qf-mod30-convprob-deviation-probability","seed":202610307016,"semantic_template":"qf-mod30-convprob-deviation-probability/{\"n\":16,\"numerator\":1,\"shift\":3}"},"fields":{"title":"Evaluate the Deviation Profile Only at a Valid Probability Index","stem":"Use the declared rational deviation-probability tail p_n=1/(n+3) on nonnegative integer indices where this value belongs to[0,1], starting no earlier than max(0,numerator-shift). The common probability space and infinite-process interpretation remain declared premises, not constructions inferred from finitely many numbers. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Evaluate index n=16. Give (numerator,shift,n,deviation_probability).","answer":{"kind":"expression","value":"(1, 3, 16, 1/19)","explanation":"The supplied index must lie in the declared probability domain. Evaluate the exact rational value without clipping it to1, renormalizing it or silently shifting the index. One value does not establish convergence for every epsilon. The exact result, retaining the requested order, is (1, 3, 16, 1/19)."},"skills":["Evaluate the Deviation Profile Only at a Valid Probability Index"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r07-convergence-in-probability-0017","batch_id":"batch-20261014-advance-20261030-r07-convergence-in-probability-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b5d6543c6a682baf172ca4a276d39e607d8504b0da73db5256aa6b6a05363381","family_id":"qf-mod30-convprob-deviation-probability","seed":202610307017,"semantic_template":"qf-mod30-convprob-deviation-probability/{\"n\":17,\"numerator\":2,\"shift\":4}"},"fields":{"title":"Evaluate the Deviation Profile Only at a Valid Probability Index","stem":"Use the declared rational deviation-probability tail p_n=2/(n+4) on nonnegative integer indices where this value belongs to[0,1], starting no earlier than max(0,numerator-shift). The common probability space and infinite-process interpretation remain declared premises, not constructions inferred from finitely many numbers. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Evaluate index n=17. Give (numerator,shift,n,deviation_probability).","answer":{"kind":"expression","value":"(2, 4, 17, 2/21)","explanation":"The supplied index must lie in the declared probability domain. Evaluate the exact rational value without clipping it to1, renormalizing it or silently shifting the index. One value does not establish convergence for every epsilon. The exact result, retaining the requested order, is (2, 4, 17, 2/21)."},"skills":["Evaluate the Deviation Profile Only at a Valid Probability Index"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r07-convergence-in-probability-0018","batch_id":"batch-20261014-advance-20261030-r07-convergence-in-probability-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6dadd7cb3a87168a5020cca7022d903d50007db183e0b26ba9ae8c5fe00e8bb4","family_id":"qf-mod30-convprob-deviation-probability","seed":202610307018,"semantic_template":"qf-mod30-convprob-deviation-probability/{\"n\":18,\"numerator\":3,\"shift\":5}"},"fields":{"title":"Evaluate the Deviation Profile Only at a Valid Probability Index","stem":"Use the declared rational deviation-probability tail p_n=3/(n+5) on nonnegative integer indices where this value belongs to[0,1], starting no earlier than max(0,numerator-shift). The common probability space and infinite-process interpretation remain declared premises, not constructions inferred from finitely many numbers. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Evaluate index n=18. Give (numerator,shift,n,deviation_probability).","answer":{"kind":"expression","value":"(3, 5, 18, 3/23)","explanation":"The supplied index must lie in the declared probability domain. Evaluate the exact rational value without clipping it to1, renormalizing it or silently shifting the index. One value does not establish convergence for every epsilon. The exact result, retaining the requested order, is (3, 5, 18, 3/23)."},"skills":["Evaluate the Deviation Profile Only at a Valid Probability Index"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r07-convergence-in-probability-0019","batch_id":"batch-20261014-advance-20261030-r07-convergence-in-probability-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1fd0d9f0cd316ca187e38dbb9bc7de8539be890332cd49fa3a58188bc5f90f5c","family_id":"qf-mod30-convprob-deviation-probability","seed":202610307019,"semantic_template":"qf-mod30-convprob-deviation-probability/{\"n\":19,\"numerator\":4,\"shift\":6}"},"fields":{"title":"Evaluate the Deviation Profile Only at a Valid Probability Index","stem":"Use the declared rational deviation-probability tail p_n=4/(n+6) on nonnegative integer indices where this value belongs to[0,1], starting no earlier than max(0,numerator-shift). The common probability space and infinite-process interpretation remain declared premises, not constructions inferred from finitely many numbers. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Evaluate index n=19. Give (numerator,shift,n,deviation_probability).","answer":{"kind":"expression","value":"(4, 6, 19, 4/25)","explanation":"The supplied index must lie in the declared probability domain. Evaluate the exact rational value without clipping it to1, renormalizing it or silently shifting the index. One value does not establish convergence for every epsilon. The exact result, retaining the requested order, is (4, 6, 19, 4/25)."},"skills":["Evaluate the Deviation Profile Only at a Valid Probability Index"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r07-convergence-in-probability-0020","batch_id":"batch-20261014-advance-20261030-r07-convergence-in-probability-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0305fe6348dd9a72d3880e448522a6911fcaa5ec6170e00c69a40f5ab7ff9b7f","family_id":"qf-mod30-convprob-deviation-probability","seed":202610307020,"semantic_template":"qf-mod30-convprob-deviation-probability/{\"n\":20,\"numerator\":5,\"shift\":7}"},"fields":{"title":"Evaluate the Deviation Profile Only at a Valid Probability Index","stem":"Use the declared rational deviation-probability tail p_n=5/(n+7) on nonnegative integer indices where this value belongs to[0,1], starting no earlier than max(0,numerator-shift). The common probability space and infinite-process interpretation remain declared premises, not constructions inferred from finitely many numbers. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Evaluate index n=20. Give (numerator,shift,n,deviation_probability).","answer":{"kind":"expression","value":"(5, 7, 20, 5/27)","explanation":"The supplied index must lie in the declared probability domain. Evaluate the exact rational value without clipping it to1, renormalizing it or silently shifting the index. One value does not establish convergence for every epsilon. The exact result, retaining the requested order, is (5, 7, 20, 5/27)."},"skills":["Evaluate the Deviation Profile Only at a Valid Probability Index"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r07-convergence-in-probability-0021","batch_id":"batch-20261014-advance-20261030-r07-convergence-in-probability-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4c16caa2e4790579e7ca9df410c3cb9e47dfc02ff86dfc5a94a7194ef3f89329","family_id":"qf-mod30-convprob-deviation-probability","seed":202610307021,"semantic_template":"qf-mod30-convprob-deviation-probability/{\"n\":21,\"numerator\":1,\"shift\":8}"},"fields":{"title":"Evaluate the Deviation Profile Only at a Valid Probability Index","stem":"Use the declared rational deviation-probability tail p_n=1/(n+8) on nonnegative integer indices where this value belongs to[0,1], starting no earlier than max(0,numerator-shift). The common probability space and infinite-process interpretation remain declared premises, not constructions inferred from finitely many numbers. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Evaluate index n=21. Give (numerator,shift,n,deviation_probability).","answer":{"kind":"expression","value":"(1, 8, 21, 1/29)","explanation":"The supplied index must lie in the declared probability domain. Evaluate the exact rational value without clipping it to1, renormalizing it or silently shifting the index. One value does not establish convergence for every epsilon. The exact result, retaining the requested order, is (1, 8, 21, 1/29)."},"skills":["Evaluate the Deviation Profile Only at a Valid Probability Index"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r07-convergence-in-probability-0022","batch_id":"batch-20261014-advance-20261030-r07-convergence-in-probability-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f88baf64c66724c67a5beee29b39754241c945beaf0ca7a4fb72df5b22ab5104","family_id":"qf-mod30-convprob-deviation-probability","seed":202610307022,"semantic_template":"qf-mod30-convprob-deviation-probability/{\"n\":22,\"numerator\":2,\"shift\":2}"},"fields":{"title":"Evaluate the Deviation Profile Only at a Valid Probability Index","stem":"Use the declared rational deviation-probability tail p_n=2/(n+2) on nonnegative integer indices where this value belongs to[0,1], starting no earlier than max(0,numerator-shift). The common probability space and infinite-process interpretation remain declared premises, not constructions inferred from finitely many numbers. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Evaluate index n=22. Give (numerator,shift,n,deviation_probability).","answer":{"kind":"expression","value":"(2, 2, 22, 1/12)","explanation":"The supplied index must lie in the declared probability domain. Evaluate the exact rational value without clipping it to1, renormalizing it or silently shifting the index. One value does not establish convergence for every epsilon. The exact result, retaining the requested order, is (2, 2, 22, 1/12)."},"skills":["Evaluate the Deviation Profile Only at a Valid Probability Index"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r07-convergence-in-probability-0023","batch_id":"batch-20261014-advance-20261030-r07-convergence-in-probability-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4905707f3edfef4fb8c860639222d253e9426cdc6b84b71c748e14f55d44f156","family_id":"qf-mod30-convprob-deviation-probability","seed":202610307023,"semantic_template":"qf-mod30-convprob-deviation-probability/{\"n\":23,\"numerator\":3,\"shift\":3}"},"fields":{"title":"Evaluate the Deviation Profile Only at a Valid Probability Index","stem":"Use the declared rational deviation-probability tail p_n=3/(n+3) on nonnegative integer indices where this value belongs to[0,1], starting no earlier than max(0,numerator-shift). The common probability space and infinite-process interpretation remain declared premises, not constructions inferred from finitely many numbers. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Evaluate index n=23. Give (numerator,shift,n,deviation_probability).","answer":{"kind":"expression","value":"(3, 3, 23, 3/26)","explanation":"The supplied index must lie in the declared probability domain. Evaluate the exact rational value without clipping it to1, renormalizing it or silently shifting the index. One value does not establish convergence for every epsilon. The exact result, retaining the requested order, is (3, 3, 23, 3/26)."},"skills":["Evaluate the Deviation Profile Only at a Valid Probability Index"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r07-convergence-in-probability-0024","batch_id":"batch-20261014-advance-20261030-r07-convergence-in-probability-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"45176f49a87f28ccf5971aff7ef20d76616a305dd732ce59adbb19d12cd358bf","family_id":"qf-mod30-convprob-deviation-probability","seed":202610307024,"semantic_template":"qf-mod30-convprob-deviation-probability/{\"n\":24,\"numerator\":4,\"shift\":4}"},"fields":{"title":"Evaluate the Deviation Profile Only at a Valid Probability Index","stem":"Use the declared rational deviation-probability tail p_n=4/(n+4) on nonnegative integer indices where this value belongs to[0,1], starting no earlier than max(0,numerator-shift). The common probability space and infinite-process interpretation remain declared premises, not constructions inferred from finitely many numbers. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Evaluate index n=24. Give (numerator,shift,n,deviation_probability).","answer":{"kind":"expression","value":"(4, 4, 24, 1/7)","explanation":"The supplied index must lie in the declared probability domain. Evaluate the exact rational value without clipping it to1, renormalizing it or silently shifting the index. One value does not establish convergence for every epsilon. The exact result, retaining the requested order, is (4, 4, 24, 1/7)."},"skills":["Evaluate the Deviation Profile Only at a Valid Probability Index"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r07-convergence-in-probability-0025","batch_id":"batch-20261014-advance-20261030-r07-convergence-in-probability-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bde2dcc23d916c4ee169fda2958d1af3a123a3518e94d256277c058837a146ad","family_id":"qf-mod30-convprob-deviation-probability","seed":202610307025,"semantic_template":"qf-mod30-convprob-deviation-probability/{\"n\":25,\"numerator\":5,\"shift\":5}"},"fields":{"title":"Evaluate the Deviation Profile Only at a Valid Probability Index","stem":"Use the declared rational deviation-probability tail p_n=5/(n+5) on nonnegative integer indices where this value belongs to[0,1], starting no earlier than max(0,numerator-shift). The common probability space and infinite-process interpretation remain declared premises, not constructions inferred from finitely many numbers. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Evaluate index n=25. Give (numerator,shift,n,deviation_probability).","answer":{"kind":"expression","value":"(5, 5, 25, 1/6)","explanation":"The supplied index must lie in the declared probability domain. Evaluate the exact rational value without clipping it to1, renormalizing it or silently shifting the index. One value does not establish convergence for every epsilon. The exact result, retaining the requested order, is (5, 5, 25, 1/6)."},"skills":["Evaluate the Deviation Profile Only at a Valid Probability Index"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r07-convergence-in-probability-0026","batch_id":"batch-20261014-advance-20261030-r07-convergence-in-probability-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"af12ef7a0094653dd95eadeae278d66fd0cf6351a2c3ac134271751c18874072","family_id":"qf-mod30-convprob-epsilon-witness","seed":202610307026,"semantic_template":"qf-mod30-convprob-epsilon-witness/{\"delta\":[1,10],\"numerator\":1,\"shift\":2}"},"fields":{"title":"A Strict Rational Tail Cutoff Is Different from a Finite Prefix","stem":"Use the declared rational deviation-probability tail p_n=1/(n+2) on nonnegative integer indices where this value belongs to[0,1], starting no earlier than max(0,numerator-shift). The common probability space and infinite-process interpretation remain declared premises, not constructions inferred from finitely many numbers. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Use positive target delta=1/10. Give (numerator,shift,delta,minimal_nonnegative_N,bound_at_N,strict_tail?,forall_n_ge_N_tail_bound).","answer":{"kind":"expression","value":"(1, 2, 1/10, 9, 1/11, Yes, forall_n_ge_N_tail_bound)","explanation":"Solve the strict inequality c/(N+s)