{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r03-convergence-in-distribution-0001","batch_id":"batch-20261014-advance-20261031-r03-convergence-in-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2dd75362a3580db64851271266f05f22cd54764b00fe3b75606403837b54584b","family_id":"qf-mod31-convdist-bernoulli-cdf","seed":202610313001,"semantic_template":"qf-mod31-convdist-bernoulli-cdf/{\"c\":[1,20],\"n\":1,\"p\":[1,2],\"shift\":2,\"threshold\":[-1,1]}"},"fields":{"title":"Compare Bernoulli CDF Values at an Explicit Threshold","stem":"Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=1/20, positive shift=2, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. n=1, threshold=-1/1. Give (n,p_n,threshold,F_n,F,limit_CDF_continuity_point?).","answer":{"kind":"expression","value":"(1, 31/60, -1, 0, 0, Yes)","explanation":"Compute p_n exactly, then evaluate both complete CDFs with the same threshold conventions. The final flag describes continuity of the limit CDF, not whether these two finite-n values happen to be equal. The exact result, retaining the requested order, is (1, 31/60, -1, 0, 0, Yes)."},"skills":["Compare Bernoulli CDF Values at an Explicit Threshold"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261031-r03-convergence-in-distribution-0002","batch_id":"batch-20261014-advance-20261031-r03-convergence-in-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0dfa80d64ed3e8ccda829ec0d8d8d41c4ed77d2941b446fd022f44173d3f6152","family_id":"qf-mod31-convdist-bernoulli-cdf","seed":202610313002,"semantic_template":"qf-mod31-convdist-bernoulli-cdf/{\"c\":[2,20],\"n\":2,\"p\":[1,2],\"shift\":3,\"threshold\":[1,2]}"},"fields":{"title":"Compare Bernoulli CDF Values at an Explicit Threshold","stem":"Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=2/20, positive shift=3, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. n=2, threshold=1/2. Give (n,p_n,threshold,F_n,F,limit_CDF_continuity_point?).","answer":{"kind":"expression","value":"(2, 13/25, 1/2, 12/25, 1/2, Yes)","explanation":"Compute p_n exactly, then evaluate both complete CDFs with the same threshold conventions. The final flag describes continuity of the limit CDF, not whether these two finite-n values happen to be equal. The exact result, retaining the requested order, is (2, 13/25, 1/2, 12/25, 1/2, Yes)."},"skills":["Compare Bernoulli CDF Values at an Explicit Threshold"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261031-r03-convergence-in-distribution-0003","batch_id":"batch-20261014-advance-20261031-r03-convergence-in-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"25f340b3567874c4881485902c9286d5afb10ecc5f623e57d0bf7ce185d9f265","family_id":"qf-mod31-convdist-bernoulli-cdf","seed":202610313003,"semantic_template":"qf-mod31-convdist-bernoulli-cdf/{\"c\":[3,20],\"n\":3,\"p\":[1,2],\"shift\":4,\"threshold\":[2,1]}"},"fields":{"title":"Compare Bernoulli CDF Values at an Explicit Threshold","stem":"Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=3/20, positive shift=4, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. n=3, threshold=2/1. Give (n,p_n,threshold,F_n,F,limit_CDF_continuity_point?).","answer":{"kind":"expression","value":"(3, 73/140, 2, 1, 1, Yes)","explanation":"Compute p_n exactly, then evaluate both complete CDFs with the same threshold conventions. The final flag describes continuity of the limit CDF, not whether these two finite-n values happen to be equal. The exact result, retaining the requested order, is (3, 73/140, 2, 1, 1, Yes)."},"skills":["Compare Bernoulli CDF Values at an Explicit Threshold"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261031-r03-convergence-in-distribution-0004","batch_id":"batch-20261014-advance-20261031-r03-convergence-in-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f7d9b8cfffe7cd2512b5832cc313d751ff841542e547492d14c63e11b0c7494b","family_id":"qf-mod31-convdist-bernoulli-cdf","seed":202610313004,"semantic_template":"qf-mod31-convdist-bernoulli-cdf/{\"c\":[4,20],\"n\":4,\"p\":[1,2],\"shift\":5,\"threshold\":[-1,1]}"},"fields":{"title":"Compare Bernoulli CDF Values at an Explicit Threshold","stem":"Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=4/20, positive shift=5, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. n=4, threshold=-1/1. Give (n,p_n,threshold,F_n,F,limit_CDF_continuity_point?).","answer":{"kind":"expression","value":"(4, 47/90, -1, 0, 0, Yes)","explanation":"Compute p_n exactly, then evaluate both complete CDFs with the same threshold conventions. The final flag describes continuity of the limit CDF, not whether these two finite-n values happen to be equal. The exact result, retaining the requested order, is (4, 47/90, -1, 0, 0, Yes)."},"skills":["Compare Bernoulli CDF Values at an Explicit Threshold"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261031-r03-convergence-in-distribution-0005","batch_id":"batch-20261014-advance-20261031-r03-convergence-in-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a488e18a0640f49b0f80f06704b443d5c12ccce437c69efaf061ed4ac4671dd9","family_id":"qf-mod31-convdist-bernoulli-cdf","seed":202610313005,"semantic_template":"qf-mod31-convdist-bernoulli-cdf/{\"c\":[5,20],\"n\":5,\"p\":[1,2],\"shift\":6,\"threshold\":[1,2]}"},"fields":{"title":"Compare Bernoulli CDF Values at an Explicit Threshold","stem":"Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=5/20, positive shift=6, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. n=5, threshold=1/2. Give (n,p_n,threshold,F_n,F,limit_CDF_continuity_point?).","answer":{"kind":"expression","value":"(5, 23/44, 1/2, 21/44, 1/2, Yes)","explanation":"Compute p_n exactly, then evaluate both complete CDFs with the same threshold conventions. The final flag describes continuity of the limit CDF, not whether these two finite-n values happen to be equal. The exact result, retaining the requested order, is (5, 23/44, 1/2, 21/44, 1/2, Yes)."},"skills":["Compare Bernoulli CDF Values at an Explicit Threshold"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261031-r03-convergence-in-distribution-0006","batch_id":"batch-20261014-advance-20261031-r03-convergence-in-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a47f506e29235f5fec55df39fe36a0e6923b8d796674a108a14edeb9f287b5f6","family_id":"qf-mod31-convdist-bernoulli-cdf","seed":202610313006,"semantic_template":"qf-mod31-convdist-bernoulli-cdf/{\"c\":[1,20],\"n\":6,\"p\":[1,2],\"shift\":7,\"threshold\":[2,1]}"},"fields":{"title":"Compare Bernoulli CDF Values at an Explicit Threshold","stem":"Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=1/20, positive shift=7, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. n=6, threshold=2/1. Give (n,p_n,threshold,F_n,F,limit_CDF_continuity_point?).","answer":{"kind":"expression","value":"(6, 131/260, 2, 1, 1, Yes)","explanation":"Compute p_n exactly, then evaluate both complete CDFs with the same threshold conventions. The final flag describes continuity of the limit CDF, not whether these two finite-n values happen to be equal. The exact result, retaining the requested order, is (6, 131/260, 2, 1, 1, Yes)."},"skills":["Compare Bernoulli CDF Values at an Explicit Threshold"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261031-r03-convergence-in-distribution-0007","batch_id":"batch-20261014-advance-20261031-r03-convergence-in-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bf1c69d8a1a8c94fc5b3a47e3188988e5fa9a7ba9b69e07bcd9aa9250eacce7d","family_id":"qf-mod31-convdist-bernoulli-cdf","seed":202610313007,"semantic_template":"qf-mod31-convdist-bernoulli-cdf/{\"c\":[2,20],\"n\":7,\"p\":[1,2],\"shift\":8,\"threshold\":[-1,1]}"},"fields":{"title":"Compare Bernoulli CDF Values at an Explicit Threshold","stem":"Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=2/20, positive shift=8, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. n=7, threshold=-1/1. Give (n,p_n,threshold,F_n,F,limit_CDF_continuity_point?).","answer":{"kind":"expression","value":"(7, 38/75, -1, 0, 0, Yes)","explanation":"Compute p_n exactly, then evaluate both complete CDFs with the same threshold conventions. The final flag describes continuity of the limit CDF, not whether these two finite-n values happen to be equal. The exact result, retaining the requested order, is (7, 38/75, -1, 0, 0, Yes)."},"skills":["Compare Bernoulli CDF Values at an Explicit Threshold"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261031-r03-convergence-in-distribution-0008","batch_id":"batch-20261014-advance-20261031-r03-convergence-in-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"383d01e64e821948fedd10bf6ab61bb6be56847bd4536767a1de01b67f81fa8a","family_id":"qf-mod31-convdist-bernoulli-cdf","seed":202610313008,"semantic_template":"qf-mod31-convdist-bernoulli-cdf/{\"c\":[3,20],\"n\":8,\"p\":[1,2],\"shift\":9,\"threshold\":[1,2]}"},"fields":{"title":"Compare Bernoulli CDF Values at an Explicit Threshold","stem":"Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=3/20, positive shift=9, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. n=8, threshold=1/2. Give (n,p_n,threshold,F_n,F,limit_CDF_continuity_point?).","answer":{"kind":"expression","value":"(8, 173/340, 1/2, 167/340, 1/2, Yes)","explanation":"Compute p_n exactly, then evaluate both complete CDFs with the same threshold conventions. The final flag describes continuity of the limit CDF, not whether these two finite-n values happen to be equal. The exact result, retaining the requested order, is (8, 173/340, 1/2, 167/340, 1/2, Yes)."},"skills":["Compare Bernoulli CDF Values at an Explicit Threshold"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261031-r03-convergence-in-distribution-0009","batch_id":"batch-20261014-advance-20261031-r03-convergence-in-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8b5816d70ca3b58f17a8af2d8329add6a4293d592685bf267462409e7e1b0ceb","family_id":"qf-mod31-convdist-bernoulli-cdf","seed":202610313009,"semantic_template":"qf-mod31-convdist-bernoulli-cdf/{\"c\":[4,20],\"n\":9,\"p\":[1,2],\"shift\":10,\"threshold\":[2,1]}"},"fields":{"title":"Compare Bernoulli CDF Values at an Explicit Threshold","stem":"Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=4/20, positive shift=10, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. n=9, threshold=2/1. Give (n,p_n,threshold,F_n,F,limit_CDF_continuity_point?).","answer":{"kind":"expression","value":"(9, 97/190, 2, 1, 1, Yes)","explanation":"Compute p_n exactly, then evaluate both complete CDFs with the same threshold conventions. The final flag describes continuity of the limit CDF, not whether these two finite-n values happen to be equal. The exact result, retaining the requested order, is (9, 97/190, 2, 1, 1, Yes)."},"skills":["Compare Bernoulli CDF Values at an Explicit Threshold"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261031-r03-convergence-in-distribution-0010","batch_id":"batch-20261014-advance-20261031-r03-convergence-in-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a0ef0d0062988aad37fe4e85355bb4b9c7f07884aa27eb7031c6a388e2f7c753","family_id":"qf-mod31-convdist-bernoulli-cdf","seed":202610313010,"semantic_template":"qf-mod31-convdist-bernoulli-cdf/{\"c\":[5,20],\"n\":10,\"p\":[1,2],\"shift\":11,\"threshold\":[-1,1]}"},"fields":{"title":"Compare Bernoulli CDF Values at an Explicit Threshold","stem":"Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=5/20, positive shift=11, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. n=10, threshold=-1/1. Give (n,p_n,threshold,F_n,F,limit_CDF_continuity_point?).","answer":{"kind":"expression","value":"(10, 43/84, -1, 0, 0, Yes)","explanation":"Compute p_n exactly, then evaluate both complete CDFs with the same threshold conventions. The final flag describes continuity of the limit CDF, not whether these two finite-n values happen to be equal. The exact result, retaining the requested order, is (10, 43/84, -1, 0, 0, Yes)."},"skills":["Compare Bernoulli CDF Values at an Explicit Threshold"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261031-r03-convergence-in-distribution-0011","batch_id":"batch-20261014-advance-20261031-r03-convergence-in-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3fb86fd50bab1cef9e20c6db75f0cd1755d74fe9736429ad70bcb4f29cd2af08","family_id":"qf-mod31-convdist-bernoulli-cdf","seed":202610313011,"semantic_template":"qf-mod31-convdist-bernoulli-cdf/{\"c\":[1,20],\"n\":11,\"p\":[1,2],\"shift\":12,\"threshold\":[1,2]}"},"fields":{"title":"Compare Bernoulli CDF Values at an Explicit Threshold","stem":"Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=1/20, positive shift=12, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. n=11, threshold=1/2. Give (n,p_n,threshold,F_n,F,limit_CDF_continuity_point?).","answer":{"kind":"expression","value":"(11, 231/460, 1/2, 229/460, 1/2, Yes)","explanation":"Compute p_n exactly, then evaluate both complete CDFs with the same threshold conventions. The final flag describes continuity of the limit CDF, not whether these two finite-n values happen to be equal. The exact result, retaining the requested order, is (11, 231/460, 1/2, 229/460, 1/2, Yes)."},"skills":["Compare Bernoulli CDF Values at an Explicit Threshold"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261031-r03-convergence-in-distribution-0012","batch_id":"batch-20261014-advance-20261031-r03-convergence-in-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dfd11f245df21b19f2e6fc2cac889b185f7e738ca8cc089bb11ca71d6c418d31","family_id":"qf-mod31-convdist-bernoulli-cdf","seed":202610313012,"semantic_template":"qf-mod31-convdist-bernoulli-cdf/{\"c\":[2,20],\"n\":12,\"p\":[1,2],\"shift\":13,\"threshold\":[2,1]}"},"fields":{"title":"Compare Bernoulli CDF Values at an Explicit Threshold","stem":"Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=2/20, positive shift=13, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. n=12, threshold=2/1. Give (n,p_n,threshold,F_n,F,limit_CDF_continuity_point?).","answer":{"kind":"expression","value":"(12, 63/125, 2, 1, 1, Yes)","explanation":"Compute p_n exactly, then evaluate both complete CDFs with the same threshold conventions. The final flag describes continuity of the limit CDF, not whether these two finite-n values happen to be equal. The exact result, retaining the requested order, is (12, 63/125, 2, 1, 1, Yes)."},"skills":["Compare Bernoulli CDF Values at an Explicit Threshold"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261031-r03-convergence-in-distribution-0013","batch_id":"batch-20261014-advance-20261031-r03-convergence-in-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c02808c9821156e61d567a6b3f1ee15bc6574d3262532cde79ba22bb432447f7","family_id":"qf-mod31-convdist-bernoulli-cdf","seed":202610313013,"semantic_template":"qf-mod31-convdist-bernoulli-cdf/{\"c\":[3,20],\"n\":13,\"p\":[1,2],\"shift\":14,\"threshold\":[-1,1]}"},"fields":{"title":"Compare Bernoulli CDF Values at an Explicit Threshold","stem":"Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=3/20, positive shift=14, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. n=13, threshold=-1/1. Give (n,p_n,threshold,F_n,F,limit_CDF_continuity_point?).","answer":{"kind":"expression","value":"(13, 91/180, -1, 0, 0, Yes)","explanation":"Compute p_n exactly, then evaluate both complete CDFs with the same threshold conventions. The final flag describes continuity of the limit CDF, not whether these two finite-n values happen to be equal. The exact result, retaining the requested order, is (13, 91/180, -1, 0, 0, Yes)."},"skills":["Compare Bernoulli CDF Values at an Explicit Threshold"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261031-r03-convergence-in-distribution-0014","batch_id":"batch-20261014-advance-20261031-r03-convergence-in-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c768f097c5802c801878a4e2c9caf0acd19193963da3d95ca9858ca22327c581","family_id":"qf-mod31-convdist-bernoulli-cdf","seed":202610313014,"semantic_template":"qf-mod31-convdist-bernoulli-cdf/{\"c\":[4,20],\"n\":14,\"p\":[1,2],\"shift\":15,\"threshold\":[1,2]}"},"fields":{"title":"Compare Bernoulli CDF Values at an Explicit Threshold","stem":"Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=4/20, positive shift=15, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. n=14, threshold=1/2. Give (n,p_n,threshold,F_n,F,limit_CDF_continuity_point?).","answer":{"kind":"expression","value":"(14, 147/290, 1/2, 143/290, 1/2, Yes)","explanation":"Compute p_n exactly, then evaluate both complete CDFs with the same threshold conventions. The final flag describes continuity of the limit CDF, not whether these two finite-n values happen to be equal. The exact result, retaining the requested order, is (14, 147/290, 1/2, 143/290, 1/2, Yes)."},"skills":["Compare Bernoulli CDF Values at an Explicit Threshold"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261031-r03-convergence-in-distribution-0015","batch_id":"batch-20261014-advance-20261031-r03-convergence-in-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ead06886a7a28157a9c0bf4aaa62de4c6f08ec8b54117356bc170c7397862c09","family_id":"qf-mod31-convdist-bernoulli-cdf","seed":202610313015,"semantic_template":"qf-mod31-convdist-bernoulli-cdf/{\"c\":[5,20],\"n\":15,\"p\":[1,2],\"shift\":16,\"threshold\":[2,1]}"},"fields":{"title":"Compare Bernoulli CDF Values at an Explicit Threshold","stem":"Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=5/20, positive shift=16, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. n=15, threshold=2/1. Give (n,p_n,threshold,F_n,F,limit_CDF_continuity_point?).","answer":{"kind":"expression","value":"(15, 63/124, 2, 1, 1, Yes)","explanation":"Compute p_n exactly, then evaluate both complete CDFs with the same threshold conventions. The final flag describes continuity of the limit CDF, not whether these two finite-n values happen to be equal. The exact result, retaining the requested order, is (15, 63/124, 2, 1, 1, Yes)."},"skills":["Compare Bernoulli CDF Values at an Explicit Threshold"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261031-r03-convergence-in-distribution-0016","batch_id":"batch-20261014-advance-20261031-r03-convergence-in-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b22a109c894531d7947600014c0ca08dbc5905ab2b6941f16edcb8bfabefb006","family_id":"qf-mod31-convdist-bernoulli-cdf","seed":202610313016,"semantic_template":"qf-mod31-convdist-bernoulli-cdf/{\"c\":[1,20],\"n\":16,\"p\":[1,2],\"shift\":17,\"threshold\":[-1,1]}"},"fields":{"title":"Compare Bernoulli CDF Values at an Explicit Threshold","stem":"Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=1/20, positive shift=17, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. n=16, threshold=-1/1. Give (n,p_n,threshold,F_n,F,limit_CDF_continuity_point?).","answer":{"kind":"expression","value":"(16, 331/660, -1, 0, 0, Yes)","explanation":"Compute p_n exactly, then evaluate both complete CDFs with the same threshold conventions. The final flag describes continuity of the limit CDF, not whether these two finite-n values happen to be equal. The exact result, retaining the requested order, is (16, 331/660, -1, 0, 0, Yes)."},"skills":["Compare Bernoulli CDF Values at an Explicit Threshold"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261031-r03-convergence-in-distribution-0017","batch_id":"batch-20261014-advance-20261031-r03-convergence-in-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6bec0d5a0be5a97e4a22cd74e26b320114c1a8d8ac47edfe54a5449fe3fd859a","family_id":"qf-mod31-convdist-bernoulli-cdf","seed":202610313017,"semantic_template":"qf-mod31-convdist-bernoulli-cdf/{\"c\":[2,20],\"n\":17,\"p\":[1,2],\"shift\":18,\"threshold\":[1,2]}"},"fields":{"title":"Compare Bernoulli CDF Values at an Explicit Threshold","stem":"Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=2/20, positive shift=18, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. n=17, threshold=1/2. Give (n,p_n,threshold,F_n,F,limit_CDF_continuity_point?).","answer":{"kind":"expression","value":"(17, 88/175, 1/2, 87/175, 1/2, Yes)","explanation":"Compute p_n exactly, then evaluate both complete CDFs with the same threshold conventions. The final flag describes continuity of the limit CDF, not whether these two finite-n values happen to be equal. The exact result, retaining the requested order, is (17, 88/175, 1/2, 87/175, 1/2, Yes)."},"skills":["Compare Bernoulli CDF Values at an Explicit Threshold"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261031-r03-convergence-in-distribution-0018","batch_id":"batch-20261014-advance-20261031-r03-convergence-in-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a74416380a38d509825c626297175b756bf44c11f21530819553449fe639652f","family_id":"qf-mod31-convdist-bernoulli-cdf","seed":202610313018,"semantic_template":"qf-mod31-convdist-bernoulli-cdf/{\"c\":[3,20],\"n\":18,\"p\":[1,2],\"shift\":19,\"threshold\":[2,1]}"},"fields":{"title":"Compare Bernoulli CDF Values at an Explicit Threshold","stem":"Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=3/20, positive shift=19, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. n=18, threshold=2/1. Give (n,p_n,threshold,F_n,F,limit_CDF_continuity_point?).","answer":{"kind":"expression","value":"(18, 373/740, 2, 1, 1, Yes)","explanation":"Compute p_n exactly, then evaluate both complete CDFs with the same threshold conventions. The final flag describes continuity of the limit CDF, not whether these two finite-n values happen to be equal. The exact result, retaining the requested order, is (18, 373/740, 2, 1, 1, Yes)."},"skills":["Compare Bernoulli CDF Values at an Explicit Threshold"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261031-r03-convergence-in-distribution-0019","batch_id":"batch-20261014-advance-20261031-r03-convergence-in-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f23c36dc3aad8db6dab7a8aeea442ccdca2811f7d15a9011076ffaadd1df06ba","family_id":"qf-mod31-convdist-bernoulli-cdf","seed":202610313019,"semantic_template":"qf-mod31-convdist-bernoulli-cdf/{\"c\":[4,20],\"n\":19,\"p\":[1,2],\"shift\":20,\"threshold\":[-1,1]}"},"fields":{"title":"Compare Bernoulli CDF Values at an Explicit Threshold","stem":"Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=4/20, positive shift=20, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. n=19, threshold=-1/1. Give (n,p_n,threshold,F_n,F,limit_CDF_continuity_point?).","answer":{"kind":"expression","value":"(19, 197/390, -1, 0, 0, Yes)","explanation":"Compute p_n exactly, then evaluate both complete CDFs with the same threshold conventions. The final flag describes continuity of the limit CDF, not whether these two finite-n values happen to be equal. The exact result, retaining the requested order, is (19, 197/390, -1, 0, 0, Yes)."},"skills":["Compare Bernoulli CDF Values at an Explicit Threshold"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261031-r03-convergence-in-distribution-0020","batch_id":"batch-20261014-advance-20261031-r03-convergence-in-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"384b30debb66a05c82f1635ec867ce7baf9c47a91ccca2127edeaf2937111a5a","family_id":"qf-mod31-convdist-bernoulli-cdf","seed":202610313020,"semantic_template":"qf-mod31-convdist-bernoulli-cdf/{\"c\":[5,20],\"n\":20,\"p\":[1,2],\"shift\":21,\"threshold\":[1,2]}"},"fields":{"title":"Compare Bernoulli CDF Values at an Explicit Threshold","stem":"Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=5/20, positive shift=21, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. n=20, threshold=1/2. Give (n,p_n,threshold,F_n,F,limit_CDF_continuity_point?).","answer":{"kind":"expression","value":"(20, 83/164, 1/2, 81/164, 1/2, Yes)","explanation":"Compute p_n exactly, then evaluate both complete CDFs with the same threshold conventions. The final flag describes continuity of the limit CDF, not whether these two finite-n values happen to be equal. The exact result, retaining the requested order, is (20, 83/164, 1/2, 81/164, 1/2, Yes)."},"skills":["Compare Bernoulli CDF Values at an Explicit Threshold"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261031-r03-convergence-in-distribution-0021","batch_id":"batch-20261014-advance-20261031-r03-convergence-in-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"59b3ed5385767bbc88f32b3b811a74a960594786a2eecddfb48e4bffae294fb5","family_id":"qf-mod31-convdist-bernoulli-cdf","seed":202610313021,"semantic_template":"qf-mod31-convdist-bernoulli-cdf/{\"c\":[1,20],\"n\":21,\"p\":[1,2],\"shift\":22,\"threshold\":[2,1]}"},"fields":{"title":"Compare Bernoulli CDF Values at an Explicit Threshold","stem":"Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=1/20, positive shift=22, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. n=21, threshold=2/1. Give (n,p_n,threshold,F_n,F,limit_CDF_continuity_point?).","answer":{"kind":"expression","value":"(21, 431/860, 2, 1, 1, Yes)","explanation":"Compute p_n exactly, then evaluate both complete CDFs with the same threshold conventions. The final flag describes continuity of the limit CDF, not whether these two finite-n values happen to be equal. The exact result, retaining the requested order, is (21, 431/860, 2, 1, 1, Yes)."},"skills":["Compare Bernoulli CDF Values at an Explicit Threshold"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261031-r03-convergence-in-distribution-0022","batch_id":"batch-20261014-advance-20261031-r03-convergence-in-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b78c25bc3392bad90eab6e98bb99e5ef52cbe5ecec4397acd109eda47a46864f","family_id":"qf-mod31-convdist-bernoulli-cdf","seed":202610313022,"semantic_template":"qf-mod31-convdist-bernoulli-cdf/{\"c\":[2,20],\"n\":22,\"p\":[1,2],\"shift\":23,\"threshold\":[-1,1]}"},"fields":{"title":"Compare Bernoulli CDF Values at an Explicit Threshold","stem":"Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=2/20, positive shift=23, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. n=22, threshold=-1/1. Give (n,p_n,threshold,F_n,F,limit_CDF_continuity_point?).","answer":{"kind":"expression","value":"(22, 113/225, -1, 0, 0, Yes)","explanation":"Compute p_n exactly, then evaluate both complete CDFs with the same threshold conventions. The final flag describes continuity of the limit CDF, not whether these two finite-n values happen to be equal. The exact result, retaining the requested order, is (22, 113/225, -1, 0, 0, Yes)."},"skills":["Compare Bernoulli CDF Values at an Explicit Threshold"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261031-r03-convergence-in-distribution-0023","batch_id":"batch-20261014-advance-20261031-r03-convergence-in-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a307fc6ba93b7bf079e862c9dd11e467c33ac8e522cb96a7b89da30bb4914213","family_id":"qf-mod31-convdist-bernoulli-cdf","seed":202610313023,"semantic_template":"qf-mod31-convdist-bernoulli-cdf/{\"c\":[3,20],\"n\":23,\"p\":[1,2],\"shift\":24,\"threshold\":[1,2]}"},"fields":{"title":"Compare Bernoulli CDF Values at an Explicit Threshold","stem":"Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=3/20, positive shift=24, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. n=23, threshold=1/2. Give (n,p_n,threshold,F_n,F,limit_CDF_continuity_point?).","answer":{"kind":"expression","value":"(23, 473/940, 1/2, 467/940, 1/2, Yes)","explanation":"Compute p_n exactly, then evaluate both complete CDFs with the same threshold conventions. The final flag describes continuity of the limit CDF, not whether these two finite-n values happen to be equal. The exact result, retaining the requested order, is (23, 473/940, 1/2, 467/940, 1/2, Yes)."},"skills":["Compare Bernoulli CDF Values at an Explicit Threshold"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261031-r03-convergence-in-distribution-0024","batch_id":"batch-20261014-advance-20261031-r03-convergence-in-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bf53349c0722bec025df18ae5765cc663428ff1b165a89b9cb7ecdb78317536b","family_id":"qf-mod31-convdist-bernoulli-cdf","seed":202610313024,"semantic_template":"qf-mod31-convdist-bernoulli-cdf/{\"c\":[4,20],\"n\":24,\"p\":[1,2],\"shift\":25,\"threshold\":[2,1]}"},"fields":{"title":"Compare Bernoulli CDF Values at an Explicit Threshold","stem":"Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=4/20, positive shift=25, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. n=24, threshold=2/1. Give (n,p_n,threshold,F_n,F,limit_CDF_continuity_point?).","answer":{"kind":"expression","value":"(24, 247/490, 2, 1, 1, Yes)","explanation":"Compute p_n exactly, then evaluate both complete CDFs with the same threshold conventions. The final flag describes continuity of the limit CDF, not whether these two finite-n values happen to be equal. The exact result, retaining the requested order, is (24, 247/490, 2, 1, 1, Yes)."},"skills":["Compare Bernoulli CDF Values at an Explicit Threshold"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261031-r03-convergence-in-distribution-0025","batch_id":"batch-20261014-advance-20261031-r03-convergence-in-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d578741865a698a22b5ffe10b883746f499b02b6486cde50f982ae4ea46128b8","family_id":"qf-mod31-convdist-bernoulli-cdf","seed":202610313025,"semantic_template":"qf-mod31-convdist-bernoulli-cdf/{\"c\":[5,20],\"n\":25,\"p\":[1,2],\"shift\":26,\"threshold\":[-1,1]}"},"fields":{"title":"Compare Bernoulli CDF Values at an Explicit Threshold","stem":"Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=5/20, positive shift=26, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. n=25, threshold=-1/1. Give (n,p_n,threshold,F_n,F,limit_CDF_continuity_point?).","answer":{"kind":"expression","value":"(25, 103/204, -1, 0, 0, Yes)","explanation":"Compute p_n exactly, then evaluate both complete CDFs with the same threshold conventions. The final flag describes continuity of the limit CDF, not whether these two finite-n values happen to be equal. The exact result, retaining the requested order, is (25, 103/204, -1, 0, 0, Yes)."},"skills":["Compare Bernoulli CDF Values at an Explicit Threshold"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261031-r03-convergence-in-distribution-0026","batch_id":"batch-20261014-advance-20261031-r03-convergence-in-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9db86f2dc5b602fac8b3ba24440ac3ea789dd8dbb4c95483ab68201ad4a08b1d","family_id":"qf-mod31-convdist-pointwise-gap","seed":202610313026,"semantic_template":"qf-mod31-convdist-pointwise-gap/{\"c\":[1,20],\"n\":1,\"p\":[1,2],\"shift\":2,\"threshold\":[-1,1]}"},"fields":{"title":"A Symbolic Parameter Tail Controls Every Required CDF Gap","stem":"Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=1/20, positive shift=2, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. n=1, threshold=-1/1. Give (n,threshold,absolute_gap,limit_CDF_continuity_point?,gap_tends_to_zero_at_every_limit_CDF_continuity_point).","answer":{"kind":"expression","value":"(1, -1, 0, Yes, gap_tends_to_zero_at_every_limit_CDF_continuity_point)","explanation":"Take the exact absolute CDF difference. On [0,1) it is |c|/(n+shift); outside that interval it is zero. The explicit all-n tail, not a finite table, supplies convergence at every required continuity point. A jump-point flag does not assert failure of convergence there; the definition simply does not require it. The exact result, retaining the requested order, is (1, -1, 0, Yes, gap_tends_to_zero_at_every_limit_CDF_continuity_point)."},"skills":["A Symbolic Parameter Tail Controls Every Required CDF Gap"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261031-r03-convergence-in-distribution-0027","batch_id":"batch-20261014-advance-20261031-r03-convergence-in-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0dd5985cdf6bc02b18a44cbf978b9e3e243e6226c3fe6fd79a71c86f51aabb1a","family_id":"qf-mod31-convdist-pointwise-gap","seed":202610313027,"semantic_template":"qf-mod31-convdist-pointwise-gap/{\"c\":[2,20],\"n\":2,\"p\":[1,2],\"shift\":3,\"threshold\":[1,2]}"},"fields":{"title":"A Symbolic Parameter Tail Controls Every Required CDF Gap","stem":"Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=2/20, positive shift=3, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. n=2, threshold=1/2. Give (n,threshold,absolute_gap,limit_CDF_continuity_point?,gap_tends_to_zero_at_every_limit_CDF_continuity_point).","answer":{"kind":"expression","value":"(2, 1/2, 1/50, Yes, gap_tends_to_zero_at_every_limit_CDF_continuity_point)","explanation":"Take the exact absolute CDF difference. On [0,1) it is |c|/(n+shift); outside that interval it is zero. The explicit all-n tail, not a finite table, supplies convergence at every required continuity point. A jump-point flag does not assert failure of convergence there; the definition simply does not require it. The exact result, retaining the requested order, is (2, 1/2, 1/50, Yes, gap_tends_to_zero_at_every_limit_CDF_continuity_point)."},"skills":["A Symbolic Parameter Tail Controls Every Required CDF Gap"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261031-r03-convergence-in-distribution-0028","batch_id":"batch-20261014-advance-20261031-r03-convergence-in-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f20368ecbe33fb6c3bc94b09331b2ab0ad9c9353d4503b4bde1a58b5fca8e03b","family_id":"qf-mod31-convdist-pointwise-gap","seed":202610313028,"semantic_template":"qf-mod31-convdist-pointwise-gap/{\"c\":[3,20],\"n\":3,\"p\":[1,2],\"shift\":4,\"threshold\":[2,1]}"},"fields":{"title":"A Symbolic Parameter Tail Controls Every Required CDF Gap","stem":"Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=3/20, positive shift=4, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. n=3, threshold=2/1. Give (n,threshold,absolute_gap,limit_CDF_continuity_point?,gap_tends_to_zero_at_every_limit_CDF_continuity_point).","answer":{"kind":"expression","value":"(3, 2, 0, Yes, gap_tends_to_zero_at_every_limit_CDF_continuity_point)","explanation":"Take the exact absolute CDF difference. On [0,1) it is |c|/(n+shift); outside that interval it is zero. The explicit all-n tail, not a finite table, supplies convergence at every required continuity point. A jump-point flag does not assert failure of convergence there; the definition simply does not require it. The exact result, retaining the requested order, is (3, 2, 0, Yes, gap_tends_to_zero_at_every_limit_CDF_continuity_point)."},"skills":["A Symbolic Parameter Tail Controls Every Required CDF Gap"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261031-r03-convergence-in-distribution-0029","batch_id":"batch-20261014-advance-20261031-r03-convergence-in-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4087d6866684cace0e26eed3d3ef213dcbcddf7a315f3e52308cf4fd18bc76c5","family_id":"qf-mod31-convdist-pointwise-gap","seed":202610313029,"semantic_template":"qf-mod31-convdist-pointwise-gap/{\"c\":[4,20],\"n\":4,\"p\":[1,2],\"shift\":5,\"threshold\":[-1,1]}"},"fields":{"title":"A Symbolic Parameter Tail Controls Every Required CDF Gap","stem":"Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=4/20, positive shift=5, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. n=4, threshold=-1/1. Give (n,threshold,absolute_gap,limit_CDF_continuity_point?,gap_tends_to_zero_at_every_limit_CDF_continuity_point).","answer":{"kind":"expression","value":"(4, -1, 0, Yes, gap_tends_to_zero_at_every_limit_CDF_continuity_point)","explanation":"Take the exact absolute CDF difference. On [0,1) it is |c|/(n+shift); outside that interval it is zero. The explicit all-n tail, not a finite table, supplies convergence at every required continuity point. A jump-point flag does not assert failure of convergence there; the definition simply does not require it. The exact result, retaining the requested order, is (4, -1, 0, Yes, gap_tends_to_zero_at_every_limit_CDF_continuity_point)."},"skills":["A Symbolic Parameter Tail Controls Every Required CDF Gap"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261031-r03-convergence-in-distribution-0030","batch_id":"batch-20261014-advance-20261031-r03-convergence-in-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"09104b7957c439bc3db34263d0961982b6e8cb38f2046b46579a206590bfc1ba","family_id":"qf-mod31-convdist-pointwise-gap","seed":202610313030,"semantic_template":"qf-mod31-convdist-pointwise-gap/{\"c\":[5,20],\"n\":5,\"p\":[1,2],\"shift\":6,\"threshold\":[1,2]}"},"fields":{"title":"A Symbolic Parameter Tail Controls Every Required CDF Gap","stem":"Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=5/20, positive shift=6, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. n=5, threshold=1/2. Give (n,threshold,absolute_gap,limit_CDF_continuity_point?,gap_tends_to_zero_at_every_limit_CDF_continuity_point).","answer":{"kind":"expression","value":"(5, 1/2, 1/44, Yes, gap_tends_to_zero_at_every_limit_CDF_continuity_point)","explanation":"Take the exact absolute CDF difference. On [0,1) it is |c|/(n+shift); outside that interval it is zero. The explicit all-n tail, not a finite table, supplies convergence at every required continuity point. A jump-point flag does not assert failure of convergence there; the definition simply does not require it. The exact result, retaining the requested order, is (5, 1/2, 1/44, Yes, gap_tends_to_zero_at_every_limit_CDF_continuity_point)."},"skills":["A Symbolic Parameter Tail Controls Every Required CDF Gap"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261031-r03-convergence-in-distribution-0031","batch_id":"batch-20261014-advance-20261031-r03-convergence-in-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"24d1cdd8e5e74968dca17d9d81289330f5f9453b17b5d66cbd58e669adc07227","family_id":"qf-mod31-convdist-pointwise-gap","seed":202610313031,"semantic_template":"qf-mod31-convdist-pointwise-gap/{\"c\":[1,20],\"n\":6,\"p\":[1,2],\"shift\":7,\"threshold\":[2,1]}"},"fields":{"title":"A Symbolic Parameter Tail Controls Every Required CDF Gap","stem":"Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=1/20, positive shift=7, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. n=6, threshold=2/1. Give (n,threshold,absolute_gap,limit_CDF_continuity_point?,gap_tends_to_zero_at_every_limit_CDF_continuity_point).","answer":{"kind":"expression","value":"(6, 2, 0, Yes, gap_tends_to_zero_at_every_limit_CDF_continuity_point)","explanation":"Take the exact absolute CDF difference. On [0,1) it is |c|/(n+shift); outside that interval it is zero. The explicit all-n tail, not a finite table, supplies convergence at every required continuity point. A jump-point flag does not assert failure of convergence there; the definition simply does not require it. The exact result, retaining the requested order, is (6, 2, 0, Yes, gap_tends_to_zero_at_every_limit_CDF_continuity_point)."},"skills":["A Symbolic Parameter Tail Controls Every Required CDF Gap"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261031-r03-convergence-in-distribution-0032","batch_id":"batch-20261014-advance-20261031-r03-convergence-in-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b1f1aeb31927fcda17493aa0c4ddd0f5edb611f76df3de78fda3c6855b5026f3","family_id":"qf-mod31-convdist-pointwise-gap","seed":202610313032,"semantic_template":"qf-mod31-convdist-pointwise-gap/{\"c\":[2,20],\"n\":7,\"p\":[1,2],\"shift\":8,\"threshold\":[-1,1]}"},"fields":{"title":"A Symbolic Parameter Tail Controls Every Required CDF Gap","stem":"Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=2/20, positive shift=8, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. n=7, threshold=-1/1. Give (n,threshold,absolute_gap,limit_CDF_continuity_point?,gap_tends_to_zero_at_every_limit_CDF_continuity_point).","answer":{"kind":"expression","value":"(7, -1, 0, Yes, gap_tends_to_zero_at_every_limit_CDF_continuity_point)","explanation":"Take the exact absolute CDF difference. On [0,1) it is |c|/(n+shift); outside that interval it is zero. The explicit all-n tail, not a finite table, supplies convergence at every required continuity point. A jump-point flag does not assert failure of convergence there; the definition simply does not require it. The exact result, retaining the requested order, is (7, -1, 0, Yes, gap_tends_to_zero_at_every_limit_CDF_continuity_point)."},"skills":["A Symbolic Parameter Tail Controls Every Required CDF Gap"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261031-r03-convergence-in-distribution-0033","batch_id":"batch-20261014-advance-20261031-r03-convergence-in-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"21d469db940c1c7c6258737f3c93c3300dfae0986fe2005a342c8fb2836a61ae","family_id":"qf-mod31-convdist-pointwise-gap","seed":202610313033,"semantic_template":"qf-mod31-convdist-pointwise-gap/{\"c\":[3,20],\"n\":8,\"p\":[1,2],\"shift\":9,\"threshold\":[1,2]}"},"fields":{"title":"A Symbolic Parameter Tail Controls Every Required CDF Gap","stem":"Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=3/20, positive shift=9, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. n=8, threshold=1/2. Give (n,threshold,absolute_gap,limit_CDF_continuity_point?,gap_tends_to_zero_at_every_limit_CDF_continuity_point).","answer":{"kind":"expression","value":"(8, 1/2, 3/340, Yes, gap_tends_to_zero_at_every_limit_CDF_continuity_point)","explanation":"Take the exact absolute CDF difference. On [0,1) it is |c|/(n+shift); outside that interval it is zero. The explicit all-n tail, not a finite table, supplies convergence at every required continuity point. A jump-point flag does not assert failure of convergence there; the definition simply does not require it. The exact result, retaining the requested order, is (8, 1/2, 3/340, Yes, gap_tends_to_zero_at_every_limit_CDF_continuity_point)."},"skills":["A Symbolic Parameter Tail Controls Every Required CDF Gap"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261031-r03-convergence-in-distribution-0034","batch_id":"batch-20261014-advance-20261031-r03-convergence-in-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"63a021f0ea6f8a9cd0c0702ce15e2374f7ed48b16b42dde875d69dc755d9d750","family_id":"qf-mod31-convdist-pointwise-gap","seed":202610313034,"semantic_template":"qf-mod31-convdist-pointwise-gap/{\"c\":[4,20],\"n\":9,\"p\":[1,2],\"shift\":10,\"threshold\":[2,1]}"},"fields":{"title":"A Symbolic Parameter Tail Controls Every Required CDF Gap","stem":"Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=4/20, positive shift=10, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. n=9, threshold=2/1. Give (n,threshold,absolute_gap,limit_CDF_continuity_point?,gap_tends_to_zero_at_every_limit_CDF_continuity_point).","answer":{"kind":"expression","value":"(9, 2, 0, Yes, gap_tends_to_zero_at_every_limit_CDF_continuity_point)","explanation":"Take the exact absolute CDF difference. On [0,1) it is |c|/(n+shift); outside that interval it is zero. The explicit all-n tail, not a finite table, supplies convergence at every required continuity point. A jump-point flag does not assert failure of convergence there; the definition simply does not require it. The exact result, retaining the requested order, is (9, 2, 0, Yes, gap_tends_to_zero_at_every_limit_CDF_continuity_point)."},"skills":["A Symbolic Parameter Tail Controls Every Required CDF Gap"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261031-r03-convergence-in-distribution-0035","batch_id":"batch-20261014-advance-20261031-r03-convergence-in-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4e20bc626be26374d59e7f46780baf962475c6bcf5f6abfafcb6dc46a609805e","family_id":"qf-mod31-convdist-pointwise-gap","seed":202610313035,"semantic_template":"qf-mod31-convdist-pointwise-gap/{\"c\":[5,20],\"n\":10,\"p\":[1,2],\"shift\":11,\"threshold\":[-1,1]}"},"fields":{"title":"A Symbolic Parameter Tail Controls Every Required CDF Gap","stem":"Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=5/20, positive shift=11, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. n=10, threshold=-1/1. Give (n,threshold,absolute_gap,limit_CDF_continuity_point?,gap_tends_to_zero_at_every_limit_CDF_continuity_point).","answer":{"kind":"expression","value":"(10, -1, 0, Yes, gap_tends_to_zero_at_every_limit_CDF_continuity_point)","explanation":"Take the exact absolute CDF difference. On [0,1) it is |c|/(n+shift); outside that interval it is zero. The explicit all-n tail, not a finite table, supplies convergence at every required continuity point. A jump-point flag does not assert failure of convergence there; the definition simply does not require it. The exact result, retaining the requested order, is (10, -1, 0, Yes, gap_tends_to_zero_at_every_limit_CDF_continuity_point)."},"skills":["A Symbolic Parameter Tail Controls Every Required CDF Gap"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261031-r03-convergence-in-distribution-0036","batch_id":"batch-20261014-advance-20261031-r03-convergence-in-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6b7ddec537392387c76e3388846a179adab7450fae1d430925274ee960456236","family_id":"qf-mod31-convdist-pointwise-gap","seed":202610313036,"semantic_template":"qf-mod31-convdist-pointwise-gap/{\"c\":[1,20],\"n\":11,\"p\":[1,2],\"shift\":12,\"threshold\":[1,2]}"},"fields":{"title":"A Symbolic Parameter Tail Controls Every Required CDF Gap","stem":"Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=1/20, positive shift=12, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. n=11, threshold=1/2. Give (n,threshold,absolute_gap,limit_CDF_continuity_point?,gap_tends_to_zero_at_every_limit_CDF_continuity_point).","answer":{"kind":"expression","value":"(11, 1/2, 1/460, Yes, gap_tends_to_zero_at_every_limit_CDF_continuity_point)","explanation":"Take the exact absolute CDF difference. On [0,1) it is |c|/(n+shift); outside that interval it is zero. The explicit all-n tail, not a finite table, supplies convergence at every required continuity point. A jump-point flag does not assert failure of convergence there; the definition simply does not require it. The exact result, retaining the requested order, is (11, 1/2, 1/460, Yes, gap_tends_to_zero_at_every_limit_CDF_continuity_point)."},"skills":["A Symbolic Parameter Tail Controls Every Required CDF Gap"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261031-r03-convergence-in-distribution-0037","batch_id":"batch-20261014-advance-20261031-r03-convergence-in-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"88b519c6e754bff7c50836c03290650cfb017f56a249ad8a8e4999e805c87041","family_id":"qf-mod31-convdist-pointwise-gap","seed":202610313037,"semantic_template":"qf-mod31-convdist-pointwise-gap/{\"c\":[2,20],\"n\":12,\"p\":[1,2],\"shift\":13,\"threshold\":[2,1]}"},"fields":{"title":"A Symbolic Parameter Tail Controls Every Required CDF Gap","stem":"Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=2/20, positive shift=13, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. n=12, threshold=2/1. Give (n,threshold,absolute_gap,limit_CDF_continuity_point?,gap_tends_to_zero_at_every_limit_CDF_continuity_point).","answer":{"kind":"expression","value":"(12, 2, 0, Yes, gap_tends_to_zero_at_every_limit_CDF_continuity_point)","explanation":"Take the exact absolute CDF difference. On [0,1) it is |c|/(n+shift); outside that interval it is zero. The explicit all-n tail, not a finite table, supplies convergence at every required continuity point. A jump-point flag does not assert failure of convergence there; the definition simply does not require it. The exact result, retaining the requested order, is (12, 2, 0, Yes, gap_tends_to_zero_at_every_limit_CDF_continuity_point)."},"skills":["A Symbolic Parameter Tail Controls Every Required CDF Gap"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261031-r03-convergence-in-distribution-0038","batch_id":"batch-20261014-advance-20261031-r03-convergence-in-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e7afa3150586b085fc665520e22328bb50534238f1c8898a7ba693a76e78e68a","family_id":"qf-mod31-convdist-pointwise-gap","seed":202610313038,"semantic_template":"qf-mod31-convdist-pointwise-gap/{\"c\":[3,20],\"n\":13,\"p\":[1,2],\"shift\":14,\"threshold\":[-1,1]}"},"fields":{"title":"A Symbolic Parameter Tail Controls Every Required CDF Gap","stem":"Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=3/20, positive shift=14, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. n=13, threshold=-1/1. Give (n,threshold,absolute_gap,limit_CDF_continuity_point?,gap_tends_to_zero_at_every_limit_CDF_continuity_point).","answer":{"kind":"expression","value":"(13, -1, 0, Yes, gap_tends_to_zero_at_every_limit_CDF_continuity_point)","explanation":"Take the exact absolute CDF difference. On [0,1) it is |c|/(n+shift); outside that interval it is zero. The explicit all-n tail, not a finite table, supplies convergence at every required continuity point. A jump-point flag does not assert failure of convergence there; the definition simply does not require it. The exact result, retaining the requested order, is (13, -1, 0, Yes, gap_tends_to_zero_at_every_limit_CDF_continuity_point)."},"skills":["A Symbolic Parameter Tail Controls Every Required CDF Gap"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261031-r03-convergence-in-distribution-0039","batch_id":"batch-20261014-advance-20261031-r03-convergence-in-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e4cb8142b1eb67a77cd6ecf185fc53fa50714214ac36136f84275eef13b043e8","family_id":"qf-mod31-convdist-pointwise-gap","seed":202610313039,"semantic_template":"qf-mod31-convdist-pointwise-gap/{\"c\":[4,20],\"n\":14,\"p\":[1,2],\"shift\":15,\"threshold\":[1,2]}"},"fields":{"title":"A Symbolic Parameter Tail Controls Every Required CDF Gap","stem":"Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=4/20, positive shift=15, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. n=14, threshold=1/2. Give (n,threshold,absolute_gap,limit_CDF_continuity_point?,gap_tends_to_zero_at_every_limit_CDF_continuity_point).","answer":{"kind":"expression","value":"(14, 1/2, 1/145, Yes, gap_tends_to_zero_at_every_limit_CDF_continuity_point)","explanation":"Take the exact absolute CDF difference. On [0,1) it is |c|/(n+shift); outside that interval it is zero. The explicit all-n tail, not a finite table, supplies convergence at every required continuity point. A jump-point flag does not assert failure of convergence there; the definition simply does not require it. The exact result, retaining the requested order, is (14, 1/2, 1/145, Yes, gap_tends_to_zero_at_every_limit_CDF_continuity_point)."},"skills":["A Symbolic Parameter Tail Controls Every Required CDF Gap"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261031-r03-convergence-in-distribution-0040","batch_id":"batch-20261014-advance-20261031-r03-convergence-in-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1ba69836dbacd272837ceed379a0f365170aa8ebd3a83cf0cbb5777d946d3202","family_id":"qf-mod31-convdist-pointwise-gap","seed":202610313040,"semantic_template":"qf-mod31-convdist-pointwise-gap/{\"c\":[5,20],\"n\":15,\"p\":[1,2],\"shift\":16,\"threshold\":[2,1]}"},"fields":{"title":"A Symbolic Parameter Tail Controls Every Required CDF Gap","stem":"Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=5/20, positive shift=16, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. n=15, threshold=2/1. Give (n,threshold,absolute_gap,limit_CDF_continuity_point?,gap_tends_to_zero_at_every_limit_CDF_continuity_point).","answer":{"kind":"expression","value":"(15, 2, 0, Yes, gap_tends_to_zero_at_every_limit_CDF_continuity_point)","explanation":"Take the exact absolute CDF difference. On [0,1) it is |c|/(n+shift); outside that interval it is zero. The explicit all-n tail, not a finite table, supplies convergence at every required continuity point. A jump-point flag does not assert failure of convergence there; the definition simply does not require it. The exact result, retaining the requested order, is (15, 2, 0, Yes, gap_tends_to_zero_at_every_limit_CDF_continuity_point)."},"skills":["A Symbolic Parameter Tail Controls Every Required CDF Gap"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261031-r03-convergence-in-distribution-0041","batch_id":"batch-20261014-advance-20261031-r03-convergence-in-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fdb0d6a90425de68516118b81008621d9a459fea3e3be0f58a173a3e9efb8496","family_id":"qf-mod31-convdist-pointwise-gap","seed":202610313041,"semantic_template":"qf-mod31-convdist-pointwise-gap/{\"c\":[1,20],\"n\":16,\"p\":[1,2],\"shift\":17,\"threshold\":[-1,1]}"},"fields":{"title":"A Symbolic Parameter Tail Controls Every Required CDF Gap","stem":"Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=1/20, positive shift=17, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. n=16, threshold=-1/1. Give (n,threshold,absolute_gap,limit_CDF_continuity_point?,gap_tends_to_zero_at_every_limit_CDF_continuity_point).","answer":{"kind":"expression","value":"(16, -1, 0, Yes, gap_tends_to_zero_at_every_limit_CDF_continuity_point)","explanation":"Take the exact absolute CDF difference. On [0,1) it is |c|/(n+shift); outside that interval it is zero. The explicit all-n tail, not a finite table, supplies convergence at every required continuity point. A jump-point flag does not assert failure of convergence there; the definition simply does not require it. The exact result, retaining the requested order, is (16, -1, 0, Yes, gap_tends_to_zero_at_every_limit_CDF_continuity_point)."},"skills":["A Symbolic Parameter Tail Controls Every Required CDF Gap"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261031-r03-convergence-in-distribution-0042","batch_id":"batch-20261014-advance-20261031-r03-convergence-in-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dbf95fc37241ab641ea629eb77fa9f6e4830f549c07330985da8eb1d728940a8","family_id":"qf-mod31-convdist-pointwise-gap","seed":202610313042,"semantic_template":"qf-mod31-convdist-pointwise-gap/{\"c\":[2,20],\"n\":17,\"p\":[1,2],\"shift\":18,\"threshold\":[1,2]}"},"fields":{"title":"A Symbolic Parameter Tail Controls Every Required CDF Gap","stem":"Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=2/20, positive shift=18, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. n=17, threshold=1/2. Give (n,threshold,absolute_gap,limit_CDF_continuity_point?,gap_tends_to_zero_at_every_limit_CDF_continuity_point).","answer":{"kind":"expression","value":"(17, 1/2, 1/350, Yes, gap_tends_to_zero_at_every_limit_CDF_continuity_point)","explanation":"Take the exact absolute CDF difference. On [0,1) it is |c|/(n+shift); outside that interval it is zero. The explicit all-n tail, not a finite table, supplies convergence at every required continuity point. A jump-point flag does not assert failure of convergence there; the definition simply does not require it. The exact result, retaining the requested order, is (17, 1/2, 1/350, Yes, gap_tends_to_zero_at_every_limit_CDF_continuity_point)."},"skills":["A Symbolic Parameter Tail Controls Every Required CDF Gap"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261031-r03-convergence-in-distribution-0043","batch_id":"batch-20261014-advance-20261031-r03-convergence-in-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7bee27ea566c35ec3f8f0a1faf9ad46a9d04e7873da22215dc920e2e3b504e62","family_id":"qf-mod31-convdist-pointwise-gap","seed":202610313043,"semantic_template":"qf-mod31-convdist-pointwise-gap/{\"c\":[3,20],\"n\":18,\"p\":[1,2],\"shift\":19,\"threshold\":[2,1]}"},"fields":{"title":"A Symbolic Parameter Tail Controls Every Required CDF Gap","stem":"Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=3/20, positive shift=19, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. n=18, threshold=2/1. Give (n,threshold,absolute_gap,limit_CDF_continuity_point?,gap_tends_to_zero_at_every_limit_CDF_continuity_point).","answer":{"kind":"expression","value":"(18, 2, 0, Yes, gap_tends_to_zero_at_every_limit_CDF_continuity_point)","explanation":"Take the exact absolute CDF difference. On [0,1) it is |c|/(n+shift); outside that interval it is zero. The explicit all-n tail, not a finite table, supplies convergence at every required continuity point. A jump-point flag does not assert failure of convergence there; the definition simply does not require it. The exact result, retaining the requested order, is (18, 2, 0, Yes, gap_tends_to_zero_at_every_limit_CDF_continuity_point)."},"skills":["A Symbolic Parameter Tail Controls Every Required CDF Gap"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261031-r03-convergence-in-distribution-0044","batch_id":"batch-20261014-advance-20261031-r03-convergence-in-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"084c45b1cee5854aab4af3469813c37ac73540e07b264fee4ed4025435858ee4","family_id":"qf-mod31-convdist-pointwise-gap","seed":202610313044,"semantic_template":"qf-mod31-convdist-pointwise-gap/{\"c\":[4,20],\"n\":19,\"p\":[1,2],\"shift\":20,\"threshold\":[-1,1]}"},"fields":{"title":"A Symbolic Parameter Tail Controls Every Required CDF Gap","stem":"Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=4/20, positive shift=20, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. n=19, threshold=-1/1. Give (n,threshold,absolute_gap,limit_CDF_continuity_point?,gap_tends_to_zero_at_every_limit_CDF_continuity_point).","answer":{"kind":"expression","value":"(19, -1, 0, Yes, gap_tends_to_zero_at_every_limit_CDF_continuity_point)","explanation":"Take the exact absolute CDF difference. On [0,1) it is |c|/(n+shift); outside that interval it is zero. The explicit all-n tail, not a finite table, supplies convergence at every required continuity point. A jump-point flag does not assert failure of convergence there; the definition simply does not require it. The exact result, retaining the requested order, is (19, -1, 0, Yes, gap_tends_to_zero_at_every_limit_CDF_continuity_point)."},"skills":["A Symbolic Parameter Tail Controls Every Required CDF Gap"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261031-r03-convergence-in-distribution-0045","batch_id":"batch-20261014-advance-20261031-r03-convergence-in-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"300381aebd07bc605cfa2220ae7f62f99f9af9f39366ff82fb4d93f448f9b54e","family_id":"qf-mod31-convdist-pointwise-gap","seed":202610313045,"semantic_template":"qf-mod31-convdist-pointwise-gap/{\"c\":[5,20],\"n\":20,\"p\":[1,2],\"shift\":21,\"threshold\":[1,2]}"},"fields":{"title":"A Symbolic Parameter Tail Controls Every Required CDF Gap","stem":"Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=5/20, positive shift=21, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. n=20, threshold=1/2. Give (n,threshold,absolute_gap,limit_CDF_continuity_point?,gap_tends_to_zero_at_every_limit_CDF_continuity_point).","answer":{"kind":"expression","value":"(20, 1/2, 1/164, Yes, gap_tends_to_zero_at_every_limit_CDF_continuity_point)","explanation":"Take the exact absolute CDF difference. On [0,1) it is |c|/(n+shift); outside that interval it is zero. The explicit all-n tail, not a finite table, supplies convergence at every required continuity point. A jump-point flag does not assert failure of convergence there; the definition simply does not require it. The exact result, retaining the requested order, is (20, 1/2, 1/164, Yes, gap_tends_to_zero_at_every_limit_CDF_continuity_point)."},"skills":["A Symbolic Parameter Tail Controls Every Required CDF Gap"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261031-r03-convergence-in-distribution-0046","batch_id":"batch-20261014-advance-20261031-r03-convergence-in-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6223f9491eadd6390f8dedcbe2e7f0d4f29e5ea00ad500a4a67106acc50e1627","family_id":"qf-mod31-convdist-pointwise-gap","seed":202610313046,"semantic_template":"qf-mod31-convdist-pointwise-gap/{\"c\":[1,20],\"n\":21,\"p\":[1,2],\"shift\":22,\"threshold\":[2,1]}"},"fields":{"title":"A Symbolic Parameter Tail Controls Every Required CDF Gap","stem":"Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=1/20, positive shift=22, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. n=21, threshold=2/1. Give (n,threshold,absolute_gap,limit_CDF_continuity_point?,gap_tends_to_zero_at_every_limit_CDF_continuity_point).","answer":{"kind":"expression","value":"(21, 2, 0, Yes, gap_tends_to_zero_at_every_limit_CDF_continuity_point)","explanation":"Take the exact absolute CDF difference. On [0,1) it is |c|/(n+shift); outside that interval it is zero. The explicit all-n tail, not a finite table, supplies convergence at every required continuity point. A jump-point flag does not assert failure of convergence there; the definition simply does not require it. The exact result, retaining the requested order, is (21, 2, 0, Yes, gap_tends_to_zero_at_every_limit_CDF_continuity_point)."},"skills":["A Symbolic Parameter Tail Controls Every Required CDF Gap"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261031-r03-convergence-in-distribution-0047","batch_id":"batch-20261014-advance-20261031-r03-convergence-in-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0af53433f481231503b75261b432aaa7cedd3b28fcef3ead304bbbd75703e93f","family_id":"qf-mod31-convdist-pointwise-gap","seed":202610313047,"semantic_template":"qf-mod31-convdist-pointwise-gap/{\"c\":[2,20],\"n\":22,\"p\":[1,2],\"shift\":23,\"threshold\":[-1,1]}"},"fields":{"title":"A Symbolic Parameter Tail Controls Every Required CDF Gap","stem":"Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=2/20, positive shift=23, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. n=22, threshold=-1/1. Give (n,threshold,absolute_gap,limit_CDF_continuity_point?,gap_tends_to_zero_at_every_limit_CDF_continuity_point).","answer":{"kind":"expression","value":"(22, -1, 0, Yes, gap_tends_to_zero_at_every_limit_CDF_continuity_point)","explanation":"Take the exact absolute CDF difference. On [0,1) it is |c|/(n+shift); outside that interval it is zero. The explicit all-n tail, not a finite table, supplies convergence at every required continuity point. A jump-point flag does not assert failure of convergence there; the definition simply does not require it. The exact result, retaining the requested order, is (22, -1, 0, Yes, gap_tends_to_zero_at_every_limit_CDF_continuity_point)."},"skills":["A Symbolic Parameter Tail Controls Every Required CDF Gap"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261031-r03-convergence-in-distribution-0048","batch_id":"batch-20261014-advance-20261031-r03-convergence-in-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"40c7209a9d54af4621102e7a86b1ff6dce290af66addfef39916c45aad36136e","family_id":"qf-mod31-convdist-pointwise-gap","seed":202610313048,"semantic_template":"qf-mod31-convdist-pointwise-gap/{\"c\":[3,20],\"n\":23,\"p\":[1,2],\"shift\":24,\"threshold\":[1,2]}"},"fields":{"title":"A Symbolic Parameter Tail Controls Every Required CDF Gap","stem":"Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=3/20, positive shift=24, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. n=23, threshold=1/2. Give (n,threshold,absolute_gap,limit_CDF_continuity_point?,gap_tends_to_zero_at_every_limit_CDF_continuity_point).","answer":{"kind":"expression","value":"(23, 1/2, 3/940, Yes, gap_tends_to_zero_at_every_limit_CDF_continuity_point)","explanation":"Take the exact absolute CDF difference. On [0,1) it is |c|/(n+shift); outside that interval it is zero. The explicit all-n tail, not a finite table, supplies convergence at every required continuity point. A jump-point flag does not assert failure of convergence there; the definition simply does not require it. The exact result, retaining the requested order, is (23, 1/2, 3/940, Yes, gap_tends_to_zero_at_every_limit_CDF_continuity_point)."},"skills":["A Symbolic Parameter Tail Controls Every Required CDF Gap"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261031-r03-convergence-in-distribution-0049","batch_id":"batch-20261014-advance-20261031-r03-convergence-in-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"aecf837b4b28653d537ccbcbcfaaf0010879079f55f859b0fc7103230952c81f","family_id":"qf-mod31-convdist-pointwise-gap","seed":202610313049,"semantic_template":"qf-mod31-convdist-pointwise-gap/{\"c\":[4,20],\"n\":24,\"p\":[1,2],\"shift\":25,\"threshold\":[2,1]}"},"fields":{"title":"A Symbolic Parameter Tail Controls Every Required CDF Gap","stem":"Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=4/20, positive shift=25, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. n=24, threshold=2/1. Give (n,threshold,absolute_gap,limit_CDF_continuity_point?,gap_tends_to_zero_at_every_limit_CDF_continuity_point).","answer":{"kind":"expression","value":"(24, 2, 0, Yes, gap_tends_to_zero_at_every_limit_CDF_continuity_point)","explanation":"Take the exact absolute CDF difference. On [0,1) it is |c|/(n+shift); outside that interval it is zero. The explicit all-n tail, not a finite table, supplies convergence at every required continuity point. A jump-point flag does not assert failure of convergence there; the definition simply does not require it. The exact result, retaining the requested order, is (24, 2, 0, Yes, gap_tends_to_zero_at_every_limit_CDF_continuity_point)."},"skills":["A Symbolic Parameter Tail Controls Every Required CDF Gap"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261031-r03-convergence-in-distribution-0050","batch_id":"batch-20261014-advance-20261031-r03-convergence-in-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d8a17e43842a825ad37f503aee77529fb94619d6c3a41f1bcd954eda705f49e6","family_id":"qf-mod31-convdist-pointwise-gap","seed":202610313050,"semantic_template":"qf-mod31-convdist-pointwise-gap/{\"c\":[5,20],\"n\":25,\"p\":[1,2],\"shift\":26,\"threshold\":[-1,1]}"},"fields":{"title":"A Symbolic Parameter Tail Controls Every Required CDF Gap","stem":"Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=5/20, positive shift=26, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. n=25, threshold=-1/1. Give (n,threshold,absolute_gap,limit_CDF_continuity_point?,gap_tends_to_zero_at_every_limit_CDF_continuity_point).","answer":{"kind":"expression","value":"(25, -1, 0, Yes, gap_tends_to_zero_at_every_limit_CDF_continuity_point)","explanation":"Take the exact absolute CDF difference. On [0,1) it is |c|/(n+shift); outside that interval it is zero. The explicit all-n tail, not a finite table, supplies convergence at every required continuity point. A jump-point flag does not assert failure of convergence there; the definition simply does not require it. The exact result, retaining the requested order, is (25, -1, 0, Yes, gap_tends_to_zero_at_every_limit_CDF_continuity_point)."},"skills":["A Symbolic Parameter Tail Controls Every Required CDF Gap"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261031-r03-convergence-in-distribution-0051","batch_id":"batch-20261014-advance-20261031-r03-convergence-in-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5b9e3f3be315411e9c98b096f358ede0299b2ac71b49ca500be2b4279855f439","family_id":"qf-mod31-convdist-continuity-obligation","seed":202610313051,"semantic_template":"qf-mod31-convdist-continuity-obligation/{\"p\":[1,30],\"threshold\":[-12,7]}"},"fields":{"title":"Distinguish Required Continuity Points from Limit-CDF Jumps","stem":"Use the explicit Bernoulli symbolic family with interior limit p=1/30, 0<p<1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. threshold=-12/7. Give (p,threshold,continuity_point?,definition_obligation). The obligation is CDF_convergence_required_here or limit_CDF_jump_no_requirement_from_definition.","answer":{"kind":"expression","value":"(1/30, -12/7, Yes, CDF_convergence_required_here)","explanation":"For the registered interior Bernoulli limit, compare the threshold exactly with 0 and 1. Only the continuity points carry the defining requirement. Degenerate p=0 or p=1 has a different jump set and is not extrapolated from this projection. The exact result, retaining the requested order, is (1/30, -12/7, Yes, CDF_convergence_required_here)."},"skills":["Distinguish Required Continuity Points from Limit-CDF Jumps"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261031-r03-convergence-in-distribution-0052","batch_id":"batch-20261014-advance-20261031-r03-convergence-in-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ddd24ff872f2e2114537d731095ec177cda423c4e7af8cf682b0b5a030d868fe","family_id":"qf-mod31-convdist-continuity-obligation","seed":202610313052,"semantic_template":"qf-mod31-convdist-continuity-obligation/{\"p\":[2,30],\"threshold\":[-11,7]}"},"fields":{"title":"Distinguish Required Continuity Points from Limit-CDF Jumps","stem":"Use the explicit Bernoulli symbolic family with interior limit p=2/30, 0<p<1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. threshold=-11/7. Give (p,threshold,continuity_point?,definition_obligation). The obligation is CDF_convergence_required_here or limit_CDF_jump_no_requirement_from_definition.","answer":{"kind":"expression","value":"(1/15, -11/7, Yes, CDF_convergence_required_here)","explanation":"For the registered interior Bernoulli limit, compare the threshold exactly with 0 and 1. Only the continuity points carry the defining requirement. Degenerate p=0 or p=1 has a different jump set and is not extrapolated from this projection. The exact result, retaining the requested order, is (1/15, -11/7, Yes, CDF_convergence_required_here)."},"skills":["Distinguish Required Continuity Points from Limit-CDF Jumps"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261031-r03-convergence-in-distribution-0053","batch_id":"batch-20261014-advance-20261031-r03-convergence-in-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7b2f2f080fb9d3d48e22314127368c7b56062d74ce6783ddb58bec48c6c86a84","family_id":"qf-mod31-convdist-continuity-obligation","seed":202610313053,"semantic_template":"qf-mod31-convdist-continuity-obligation/{\"p\":[3,30],\"threshold\":[-10,7]}"},"fields":{"title":"Distinguish Required Continuity Points from Limit-CDF Jumps","stem":"Use the explicit Bernoulli symbolic family with interior limit p=3/30, 0<p<1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. threshold=-10/7. Give (p,threshold,continuity_point?,definition_obligation). The obligation is CDF_convergence_required_here or limit_CDF_jump_no_requirement_from_definition.","answer":{"kind":"expression","value":"(1/10, -10/7, Yes, CDF_convergence_required_here)","explanation":"For the registered interior Bernoulli limit, compare the threshold exactly with 0 and 1. Only the continuity points carry the defining requirement. Degenerate p=0 or p=1 has a different jump set and is not extrapolated from this projection. The exact result, retaining the requested order, is (1/10, -10/7, Yes, CDF_convergence_required_here)."},"skills":["Distinguish Required Continuity Points from Limit-CDF Jumps"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261031-r03-convergence-in-distribution-0054","batch_id":"batch-20261014-advance-20261031-r03-convergence-in-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"969b3aa34fa8290471d804bd98c660a987b1d78872c9d47a9b1db7aab51a61e5","family_id":"qf-mod31-convdist-continuity-obligation","seed":202610313054,"semantic_template":"qf-mod31-convdist-continuity-obligation/{\"p\":[4,30],\"threshold\":[-9,7]}"},"fields":{"title":"Distinguish Required Continuity Points from Limit-CDF Jumps","stem":"Use the explicit Bernoulli symbolic family with interior limit p=4/30, 0<p<1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. threshold=-9/7. Give (p,threshold,continuity_point?,definition_obligation). The obligation is CDF_convergence_required_here or limit_CDF_jump_no_requirement_from_definition.","answer":{"kind":"expression","value":"(2/15, -9/7, Yes, CDF_convergence_required_here)","explanation":"For the registered interior Bernoulli limit, compare the threshold exactly with 0 and 1. Only the continuity points carry the defining requirement. Degenerate p=0 or p=1 has a different jump set and is not extrapolated from this projection. The exact result, retaining the requested order, is (2/15, -9/7, Yes, CDF_convergence_required_here)."},"skills":["Distinguish Required Continuity Points from Limit-CDF Jumps"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261031-r03-convergence-in-distribution-0055","batch_id":"batch-20261014-advance-20261031-r03-convergence-in-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0c7f923d72a6c40a15962cd633e34640bf722c3bbeb0b8203992f63b704ba1b9","family_id":"qf-mod31-convdist-continuity-obligation","seed":202610313055,"semantic_template":"qf-mod31-convdist-continuity-obligation/{\"p\":[5,30],\"threshold\":[-8,7]}"},"fields":{"title":"Distinguish Required Continuity Points from Limit-CDF Jumps","stem":"Use the explicit Bernoulli symbolic family with interior limit p=5/30, 0<p<1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. threshold=-8/7. Give (p,threshold,continuity_point?,definition_obligation). The obligation is CDF_convergence_required_here or limit_CDF_jump_no_requirement_from_definition.","answer":{"kind":"expression","value":"(1/6, -8/7, Yes, CDF_convergence_required_here)","explanation":"For the registered interior Bernoulli limit, compare the threshold exactly with 0 and 1. Only the continuity points carry the defining requirement. Degenerate p=0 or p=1 has a different jump set and is not extrapolated from this projection. The exact result, retaining the requested order, is (1/6, -8/7, Yes, CDF_convergence_required_here)."},"skills":["Distinguish Required Continuity Points from Limit-CDF Jumps"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261031-r03-convergence-in-distribution-0056","batch_id":"batch-20261014-advance-20261031-r03-convergence-in-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"12c43ad0372e293fc0a2cd2d459c9b2a60281a454762b4d61f418dc7d00541eb","family_id":"qf-mod31-convdist-continuity-obligation","seed":202610313056,"semantic_template":"qf-mod31-convdist-continuity-obligation/{\"p\":[6,30],\"threshold\":[-7,7]}"},"fields":{"title":"Distinguish Required Continuity Points from Limit-CDF Jumps","stem":"Use the explicit Bernoulli symbolic family with interior limit p=6/30, 0<p<1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. threshold=-7/7. Give (p,threshold,continuity_point?,definition_obligation). The obligation is CDF_convergence_required_here or limit_CDF_jump_no_requirement_from_definition.","answer":{"kind":"expression","value":"(1/5, -1, Yes, CDF_convergence_required_here)","explanation":"For the registered interior Bernoulli limit, compare the threshold exactly with 0 and 1. Only the continuity points carry the defining requirement. Degenerate p=0 or p=1 has a different jump set and is not extrapolated from this projection. The exact result, retaining the requested order, is (1/5, -1, Yes, CDF_convergence_required_here)."},"skills":["Distinguish Required Continuity Points from Limit-CDF Jumps"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261031-r03-convergence-in-distribution-0057","batch_id":"batch-20261014-advance-20261031-r03-convergence-in-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8b11e6c270066c3d73e037308a2fa03ec798bbb6951c10c7eb972a754148fba4","family_id":"qf-mod31-convdist-continuity-obligation","seed":202610313057,"semantic_template":"qf-mod31-convdist-continuity-obligation/{\"p\":[7,30],\"threshold\":[-6,7]}"},"fields":{"title":"Distinguish Required Continuity Points from Limit-CDF Jumps","stem":"Use the explicit Bernoulli symbolic family with interior limit p=7/30, 0<p<1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. threshold=-6/7. Give (p,threshold,continuity_point?,definition_obligation). The obligation is CDF_convergence_required_here or limit_CDF_jump_no_requirement_from_definition.","answer":{"kind":"expression","value":"(7/30, -6/7, Yes, CDF_convergence_required_here)","explanation":"For the registered interior Bernoulli limit, compare the threshold exactly with 0 and 1. Only the continuity points carry the defining requirement. Degenerate p=0 or p=1 has a different jump set and is not extrapolated from this projection. The exact result, retaining the requested order, is (7/30, -6/7, Yes, CDF_convergence_required_here)."},"skills":["Distinguish Required Continuity Points from Limit-CDF Jumps"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261031-r03-convergence-in-distribution-0058","batch_id":"batch-20261014-advance-20261031-r03-convergence-in-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0f4f97f1a189571ff872f411d0a62230a27de6466b627d12b8b13e5b265f05cf","family_id":"qf-mod31-convdist-continuity-obligation","seed":202610313058,"semantic_template":"qf-mod31-convdist-continuity-obligation/{\"p\":[8,30],\"threshold\":[-5,7]}"},"fields":{"title":"Distinguish Required Continuity Points from Limit-CDF Jumps","stem":"Use the explicit Bernoulli symbolic family with interior limit p=8/30, 0<p<1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. threshold=-5/7. Give (p,threshold,continuity_point?,definition_obligation). The obligation is CDF_convergence_required_here or limit_CDF_jump_no_requirement_from_definition.","answer":{"kind":"expression","value":"(4/15, -5/7, Yes, CDF_convergence_required_here)","explanation":"For the registered interior Bernoulli limit, compare the threshold exactly with 0 and 1. Only the continuity points carry the defining requirement. Degenerate p=0 or p=1 has a different jump set and is not extrapolated from this projection. The exact result, retaining the requested order, is (4/15, -5/7, Yes, CDF_convergence_required_here)."},"skills":["Distinguish Required Continuity Points from Limit-CDF Jumps"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261031-r03-convergence-in-distribution-0059","batch_id":"batch-20261014-advance-20261031-r03-convergence-in-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7753c3a07171b1fe8364e9085d0dedf4cc342d4245616e9f66ed156e41d3c5ab","family_id":"qf-mod31-convdist-continuity-obligation","seed":202610313059,"semantic_template":"qf-mod31-convdist-continuity-obligation/{\"p\":[9,30],\"threshold\":[-4,7]}"},"fields":{"title":"Distinguish Required Continuity Points from Limit-CDF Jumps","stem":"Use the explicit Bernoulli symbolic family with interior limit p=9/30, 0<p<1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. threshold=-4/7. Give (p,threshold,continuity_point?,definition_obligation). The obligation is CDF_convergence_required_here or limit_CDF_jump_no_requirement_from_definition.","answer":{"kind":"expression","value":"(3/10, -4/7, Yes, CDF_convergence_required_here)","explanation":"For the registered interior Bernoulli limit, compare the threshold exactly with 0 and 1. Only the continuity points carry the defining requirement. Degenerate p=0 or p=1 has a different jump set and is not extrapolated from this projection. The exact result, retaining the requested order, is (3/10, -4/7, Yes, CDF_convergence_required_here)."},"skills":["Distinguish Required Continuity Points from Limit-CDF Jumps"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261031-r03-convergence-in-distribution-0060","batch_id":"batch-20261014-advance-20261031-r03-convergence-in-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1813abb261f614abad08e6e2b6d285989acbf6244d8083e9b9ae31fb6365ffb1","family_id":"qf-mod31-convdist-continuity-obligation","seed":202610313060,"semantic_template":"qf-mod31-convdist-continuity-obligation/{\"p\":[10,30],\"threshold\":[-3,7]}"},"fields":{"title":"Distinguish Required Continuity Points from Limit-CDF Jumps","stem":"Use the explicit Bernoulli symbolic family with interior limit p=10/30, 0<p<1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. threshold=-3/7. Give (p,threshold,continuity_point?,definition_obligation). The obligation is CDF_convergence_required_here or limit_CDF_jump_no_requirement_from_definition.","answer":{"kind":"expression","value":"(1/3, -3/7, Yes, CDF_convergence_required_here)","explanation":"For the registered interior Bernoulli limit, compare the threshold exactly with 0 and 1. Only the continuity points carry the defining requirement. Degenerate p=0 or p=1 has a different jump set and is not extrapolated from this projection. The exact result, retaining the requested order, is (1/3, -3/7, Yes, CDF_convergence_required_here)."},"skills":["Distinguish Required Continuity Points from Limit-CDF Jumps"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261031-r03-convergence-in-distribution-0061","batch_id":"batch-20261014-advance-20261031-r03-convergence-in-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2003b241df12ea2fa8e56242195e56753ed9d80f244d6b1e2a52d84b44ca00a3","family_id":"qf-mod31-convdist-continuity-obligation","seed":202610313061,"semantic_template":"qf-mod31-convdist-continuity-obligation/{\"p\":[11,30],\"threshold\":[-2,7]}"},"fields":{"title":"Distinguish Required Continuity Points from Limit-CDF Jumps","stem":"Use the explicit Bernoulli symbolic family with interior limit p=11/30, 0<p<1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. threshold=-2/7. Give (p,threshold,continuity_point?,definition_obligation). The obligation is CDF_convergence_required_here or limit_CDF_jump_no_requirement_from_definition.","answer":{"kind":"expression","value":"(11/30, -2/7, Yes, CDF_convergence_required_here)","explanation":"For the registered interior Bernoulli limit, compare the threshold exactly with 0 and 1. Only the continuity points carry the defining requirement. Degenerate p=0 or p=1 has a different jump set and is not extrapolated from this projection. The exact result, retaining the requested order, is (11/30, -2/7, Yes, CDF_convergence_required_here)."},"skills":["Distinguish Required Continuity Points from Limit-CDF Jumps"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261031-r03-convergence-in-distribution-0062","batch_id":"batch-20261014-advance-20261031-r03-convergence-in-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9de4c4bc854c37105fba1bb1919ebbb68886d2f2ae949a0deace8f346d9a05df","family_id":"qf-mod31-convdist-continuity-obligation","seed":202610313062,"semantic_template":"qf-mod31-convdist-continuity-obligation/{\"p\":[12,30],\"threshold\":[-1,7]}"},"fields":{"title":"Distinguish Required Continuity Points from Limit-CDF Jumps","stem":"Use the explicit Bernoulli symbolic family with interior limit p=12/30, 0<p<1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. threshold=-1/7. Give (p,threshold,continuity_point?,definition_obligation). The obligation is CDF_convergence_required_here or limit_CDF_jump_no_requirement_from_definition.","answer":{"kind":"expression","value":"(2/5, -1/7, Yes, CDF_convergence_required_here)","explanation":"For the registered interior Bernoulli limit, compare the threshold exactly with 0 and 1. Only the continuity points carry the defining requirement. Degenerate p=0 or p=1 has a different jump set and is not extrapolated from this projection. The exact result, retaining the requested order, is (2/5, -1/7, Yes, CDF_convergence_required_here)."},"skills":["Distinguish Required Continuity Points from Limit-CDF Jumps"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261031-r03-convergence-in-distribution-0063","batch_id":"batch-20261014-advance-20261031-r03-convergence-in-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fcaca24bd43f45a60cee1f18a2d1639a76df18f71641f668ee0784c04b50392a","family_id":"qf-mod31-convdist-continuity-obligation","seed":202610313063,"semantic_template":"qf-mod31-convdist-continuity-obligation/{\"p\":[13,30],\"threshold\":[0,7]}"},"fields":{"title":"Distinguish Required Continuity Points from Limit-CDF Jumps","stem":"Use the explicit Bernoulli symbolic family with interior limit p=13/30, 0<p<1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. threshold=0/7. Give (p,threshold,continuity_point?,definition_obligation). The obligation is CDF_convergence_required_here or limit_CDF_jump_no_requirement_from_definition.","answer":{"kind":"expression","value":"(13/30, 0, No, limit_CDF_jump_no_requirement_from_definition)","explanation":"For the registered interior Bernoulli limit, compare the threshold exactly with 0 and 1. Only the continuity points carry the defining requirement. Degenerate p=0 or p=1 has a different jump set and is not extrapolated from this projection. The exact result, retaining the requested order, is (13/30, 0, No, limit_CDF_jump_no_requirement_from_definition)."},"skills":["Distinguish Required Continuity Points from Limit-CDF Jumps"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261031-r03-convergence-in-distribution-0064","batch_id":"batch-20261014-advance-20261031-r03-convergence-in-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ae6c8491204be478b3f823f8d81939c9eaa4ada1ffba056f992d357d12575b63","family_id":"qf-mod31-convdist-continuity-obligation","seed":202610313064,"semantic_template":"qf-mod31-convdist-continuity-obligation/{\"p\":[14,30],\"threshold\":[1,7]}"},"fields":{"title":"Distinguish Required Continuity Points from Limit-CDF Jumps","stem":"Use the explicit Bernoulli symbolic family with interior limit p=14/30, 0<p<1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. threshold=1/7. Give (p,threshold,continuity_point?,definition_obligation). The obligation is CDF_convergence_required_here or limit_CDF_jump_no_requirement_from_definition.","answer":{"kind":"expression","value":"(7/15, 1/7, Yes, CDF_convergence_required_here)","explanation":"For the registered interior Bernoulli limit, compare the threshold exactly with 0 and 1. Only the continuity points carry the defining requirement. Degenerate p=0 or p=1 has a different jump set and is not extrapolated from this projection. The exact result, retaining the requested order, is (7/15, 1/7, Yes, CDF_convergence_required_here)."},"skills":["Distinguish Required Continuity Points from Limit-CDF Jumps"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261031-r03-convergence-in-distribution-0065","batch_id":"batch-20261014-advance-20261031-r03-convergence-in-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fdff105ae1d5a1bf25114371d2f4f5c003f68d9ba8ab3793c228411898ad56cc","family_id":"qf-mod31-convdist-continuity-obligation","seed":202610313065,"semantic_template":"qf-mod31-convdist-continuity-obligation/{\"p\":[15,30],\"threshold\":[2,7]}"},"fields":{"title":"Distinguish Required Continuity Points from Limit-CDF Jumps","stem":"Use the explicit Bernoulli symbolic family with interior limit p=15/30, 0<p<1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. threshold=2/7. Give (p,threshold,continuity_point?,definition_obligation). The obligation is CDF_convergence_required_here or limit_CDF_jump_no_requirement_from_definition.","answer":{"kind":"expression","value":"(1/2, 2/7, Yes, CDF_convergence_required_here)","explanation":"For the registered interior Bernoulli limit, compare the threshold exactly with 0 and 1. Only the continuity points carry the defining requirement. Degenerate p=0 or p=1 has a different jump set and is not extrapolated from this projection. The exact result, retaining the requested order, is (1/2, 2/7, Yes, CDF_convergence_required_here)."},"skills":["Distinguish Required Continuity Points from Limit-CDF Jumps"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261031-r03-convergence-in-distribution-0066","batch_id":"batch-20261014-advance-20261031-r03-convergence-in-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0fac4dc0072550ffef19e5c55ae06dbb924aba463e80fda92b5317dc81f1db2f","family_id":"qf-mod31-convdist-continuity-obligation","seed":202610313066,"semantic_template":"qf-mod31-convdist-continuity-obligation/{\"p\":[16,30],\"threshold\":[3,7]}"},"fields":{"title":"Distinguish Required Continuity Points from Limit-CDF Jumps","stem":"Use the explicit Bernoulli symbolic family with interior limit p=16/30, 0<p<1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. threshold=3/7. Give (p,threshold,continuity_point?,definition_obligation). The obligation is CDF_convergence_required_here or limit_CDF_jump_no_requirement_from_definition.","answer":{"kind":"expression","value":"(8/15, 3/7, Yes, CDF_convergence_required_here)","explanation":"For the registered interior Bernoulli limit, compare the threshold exactly with 0 and 1. Only the continuity points carry the defining requirement. Degenerate p=0 or p=1 has a different jump set and is not extrapolated from this projection. The exact result, retaining the requested order, is (8/15, 3/7, Yes, CDF_convergence_required_here)."},"skills":["Distinguish Required Continuity Points from Limit-CDF Jumps"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261031-r03-convergence-in-distribution-0067","batch_id":"batch-20261014-advance-20261031-r03-convergence-in-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b245b4c80d902de08ccbf340957a757495eb50a677d3d33236894117cc2cf647","family_id":"qf-mod31-convdist-continuity-obligation","seed":202610313067,"semantic_template":"qf-mod31-convdist-continuity-obligation/{\"p\":[17,30],\"threshold\":[4,7]}"},"fields":{"title":"Distinguish Required Continuity Points from Limit-CDF Jumps","stem":"Use the explicit Bernoulli symbolic family with interior limit p=17/30, 0<p<1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. threshold=4/7. Give (p,threshold,continuity_point?,definition_obligation). The obligation is CDF_convergence_required_here or limit_CDF_jump_no_requirement_from_definition.","answer":{"kind":"expression","value":"(17/30, 4/7, Yes, CDF_convergence_required_here)","explanation":"For the registered interior Bernoulli limit, compare the threshold exactly with 0 and 1. Only the continuity points carry the defining requirement. Degenerate p=0 or p=1 has a different jump set and is not extrapolated from this projection. The exact result, retaining the requested order, is (17/30, 4/7, Yes, CDF_convergence_required_here)."},"skills":["Distinguish Required Continuity Points from Limit-CDF Jumps"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261031-r03-convergence-in-distribution-0068","batch_id":"batch-20261014-advance-20261031-r03-convergence-in-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"85d93b5bf0acdc80fdaebe6e9dcb8ad62ae2ee528abf9bb3b245f18ad54e2d60","family_id":"qf-mod31-convdist-continuity-obligation","seed":202610313068,"semantic_template":"qf-mod31-convdist-continuity-obligation/{\"p\":[18,30],\"threshold\":[5,7]}"},"fields":{"title":"Distinguish Required Continuity Points from Limit-CDF Jumps","stem":"Use the explicit Bernoulli symbolic family with interior limit p=18/30, 0<p<1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. threshold=5/7. Give (p,threshold,continuity_point?,definition_obligation). The obligation is CDF_convergence_required_here or limit_CDF_jump_no_requirement_from_definition.","answer":{"kind":"expression","value":"(3/5, 5/7, Yes, CDF_convergence_required_here)","explanation":"For the registered interior Bernoulli limit, compare the threshold exactly with 0 and 1. Only the continuity points carry the defining requirement. Degenerate p=0 or p=1 has a different jump set and is not extrapolated from this projection. The exact result, retaining the requested order, is (3/5, 5/7, Yes, CDF_convergence_required_here)."},"skills":["Distinguish Required Continuity Points from Limit-CDF Jumps"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261031-r03-convergence-in-distribution-0069","batch_id":"batch-20261014-advance-20261031-r03-convergence-in-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"31375ebea0e26f13e500db6a13885c58ce55f0ce06bb7e28da45e63fbedddf0e","family_id":"qf-mod31-convdist-continuity-obligation","seed":202610313069,"semantic_template":"qf-mod31-convdist-continuity-obligation/{\"p\":[19,30],\"threshold\":[6,7]}"},"fields":{"title":"Distinguish Required Continuity Points from Limit-CDF Jumps","stem":"Use the explicit Bernoulli symbolic family with interior limit p=19/30, 0<p<1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. threshold=6/7. Give (p,threshold,continuity_point?,definition_obligation). The obligation is CDF_convergence_required_here or limit_CDF_jump_no_requirement_from_definition.","answer":{"kind":"expression","value":"(19/30, 6/7, Yes, CDF_convergence_required_here)","explanation":"For the registered interior Bernoulli limit, compare the threshold exactly with 0 and 1. Only the continuity points carry the defining requirement. Degenerate p=0 or p=1 has a different jump set and is not extrapolated from this projection. The exact result, retaining the requested order, is (19/30, 6/7, Yes, CDF_convergence_required_here)."},"skills":["Distinguish Required Continuity Points from Limit-CDF Jumps"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261031-r03-convergence-in-distribution-0070","batch_id":"batch-20261014-advance-20261031-r03-convergence-in-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e44b9d15bce37df18153284e9339f41490dab49557d6ad8ff8f1295b6158ac66","family_id":"qf-mod31-convdist-continuity-obligation","seed":202610313070,"semantic_template":"qf-mod31-convdist-continuity-obligation/{\"p\":[20,30],\"threshold\":[7,7]}"},"fields":{"title":"Distinguish Required Continuity Points from Limit-CDF Jumps","stem":"Use the explicit Bernoulli symbolic family with interior limit p=20/30, 0<p<1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. threshold=7/7. Give (p,threshold,continuity_point?,definition_obligation). The obligation is CDF_convergence_required_here or limit_CDF_jump_no_requirement_from_definition.","answer":{"kind":"expression","value":"(2/3, 1, No, limit_CDF_jump_no_requirement_from_definition)","explanation":"For the registered interior Bernoulli limit, compare the threshold exactly with 0 and 1. Only the continuity points carry the defining requirement. Degenerate p=0 or p=1 has a different jump set and is not extrapolated from this projection. The exact result, retaining the requested order, is (2/3, 1, No, limit_CDF_jump_no_requirement_from_definition)."},"skills":["Distinguish Required Continuity Points from Limit-CDF Jumps"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261031-r03-convergence-in-distribution-0071","batch_id":"batch-20261014-advance-20261031-r03-convergence-in-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4ff7c4accc5afc4217657a011932cc2bd9398d44c0730c0771704bc6af63a710","family_id":"qf-mod31-convdist-continuity-obligation","seed":202610313071,"semantic_template":"qf-mod31-convdist-continuity-obligation/{\"p\":[21,30],\"threshold\":[8,7]}"},"fields":{"title":"Distinguish Required Continuity Points from Limit-CDF Jumps","stem":"Use the explicit Bernoulli symbolic family with interior limit p=21/30, 0<p<1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. threshold=8/7. Give (p,threshold,continuity_point?,definition_obligation). The obligation is CDF_convergence_required_here or limit_CDF_jump_no_requirement_from_definition.","answer":{"kind":"expression","value":"(7/10, 8/7, Yes, CDF_convergence_required_here)","explanation":"For the registered interior Bernoulli limit, compare the threshold exactly with 0 and 1. Only the continuity points carry the defining requirement. Degenerate p=0 or p=1 has a different jump set and is not extrapolated from this projection. The exact result, retaining the requested order, is (7/10, 8/7, Yes, CDF_convergence_required_here)."},"skills":["Distinguish Required Continuity Points from Limit-CDF Jumps"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261031-r03-convergence-in-distribution-0072","batch_id":"batch-20261014-advance-20261031-r03-convergence-in-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bb10a5ebfb9abfe40365fa5b4153a9a025c9d21e0abe041eba653aad36751cc7","family_id":"qf-mod31-convdist-continuity-obligation","seed":202610313072,"semantic_template":"qf-mod31-convdist-continuity-obligation/{\"p\":[22,30],\"threshold\":[9,7]}"},"fields":{"title":"Distinguish Required Continuity Points from Limit-CDF Jumps","stem":"Use the explicit Bernoulli symbolic family with interior limit p=22/30, 0<p<1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. threshold=9/7. Give (p,threshold,continuity_point?,definition_obligation). The obligation is CDF_convergence_required_here or limit_CDF_jump_no_requirement_from_definition.","answer":{"kind":"expression","value":"(11/15, 9/7, Yes, CDF_convergence_required_here)","explanation":"For the registered interior Bernoulli limit, compare the threshold exactly with 0 and 1. Only the continuity points carry the defining requirement. Degenerate p=0 or p=1 has a different jump set and is not extrapolated from this projection. The exact result, retaining the requested order, is (11/15, 9/7, Yes, CDF_convergence_required_here)."},"skills":["Distinguish Required Continuity Points from Limit-CDF Jumps"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261031-r03-convergence-in-distribution-0073","batch_id":"batch-20261014-advance-20261031-r03-convergence-in-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e67e132c9c1f19178d767f08ff2fcdf9ffee12764a01f6dd918917e6f99e717f","family_id":"qf-mod31-convdist-continuity-obligation","seed":202610313073,"semantic_template":"qf-mod31-convdist-continuity-obligation/{\"p\":[23,30],\"threshold\":[10,7]}"},"fields":{"title":"Distinguish Required Continuity Points from Limit-CDF Jumps","stem":"Use the explicit Bernoulli symbolic family with interior limit p=23/30, 0<p<1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. threshold=10/7. Give (p,threshold,continuity_point?,definition_obligation). The obligation is CDF_convergence_required_here or limit_CDF_jump_no_requirement_from_definition.","answer":{"kind":"expression","value":"(23/30, 10/7, Yes, CDF_convergence_required_here)","explanation":"For the registered interior Bernoulli limit, compare the threshold exactly with 0 and 1. Only the continuity points carry the defining requirement. Degenerate p=0 or p=1 has a different jump set and is not extrapolated from this projection. The exact result, retaining the requested order, is (23/30, 10/7, Yes, CDF_convergence_required_here)."},"skills":["Distinguish Required Continuity Points from Limit-CDF Jumps"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261031-r03-convergence-in-distribution-0074","batch_id":"batch-20261014-advance-20261031-r03-convergence-in-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"da9c0c99dc2e48d61e097d611be4a5d0de58151d4e22577e7bf5ad4bd66658d4","family_id":"qf-mod31-convdist-continuity-obligation","seed":202610313074,"semantic_template":"qf-mod31-convdist-continuity-obligation/{\"p\":[24,30],\"threshold\":[11,7]}"},"fields":{"title":"Distinguish Required Continuity Points from Limit-CDF Jumps","stem":"Use the explicit Bernoulli symbolic family with interior limit p=24/30, 0<p<1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. threshold=11/7. Give (p,threshold,continuity_point?,definition_obligation). The obligation is CDF_convergence_required_here or limit_CDF_jump_no_requirement_from_definition.","answer":{"kind":"expression","value":"(4/5, 11/7, Yes, CDF_convergence_required_here)","explanation":"For the registered interior Bernoulli limit, compare the threshold exactly with 0 and 1. Only the continuity points carry the defining requirement. Degenerate p=0 or p=1 has a different jump set and is not extrapolated from this projection. The exact result, retaining the requested order, is (4/5, 11/7, Yes, CDF_convergence_required_here)."},"skills":["Distinguish Required Continuity Points from Limit-CDF Jumps"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261031-r03-convergence-in-distribution-0075","batch_id":"batch-20261014-advance-20261031-r03-convergence-in-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"76fb4334b75f27ca0eeefc6b2e02a53b24984afe2b2a1c18f33ce5e4fb91e3ff","family_id":"qf-mod31-convdist-continuity-obligation","seed":202610313075,"semantic_template":"qf-mod31-convdist-continuity-obligation/{\"p\":[25,30],\"threshold\":[12,7]}"},"fields":{"title":"Distinguish Required Continuity Points from Limit-CDF Jumps","stem":"Use the explicit Bernoulli symbolic family with interior limit p=25/30, 0<p<1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. threshold=12/7. Give (p,threshold,continuity_point?,definition_obligation). The obligation is CDF_convergence_required_here or limit_CDF_jump_no_requirement_from_definition.","answer":{"kind":"expression","value":"(5/6, 12/7, Yes, CDF_convergence_required_here)","explanation":"For the registered interior Bernoulli limit, compare the threshold exactly with 0 and 1. Only the continuity points carry the defining requirement. Degenerate p=0 or p=1 has a different jump set and is not extrapolated from this projection. The exact result, retaining the requested order, is (5/6, 12/7, Yes, CDF_convergence_required_here)."},"skills":["Distinguish Required Continuity Points from Limit-CDF Jumps"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261031-r03-convergence-in-distribution-0076","batch_id":"batch-20261014-advance-20261031-r03-convergence-in-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4e10e8eee8d668256b0d58a2a68bc0130a4e9e74b01e24de918133dac7851bf0","family_id":"qf-mod31-convdist-finite-prefix","seed":202610313076,"semantic_template":"qf-mod31-convdist-finite-prefix/{\"c\":[1,20],\"p\":[1,2],\"prefix_length\":1,\"shift\":2}"},"fields":{"title":"A Fixed Prefix Does Not Supply the All-n CDF Limit Obligation","stem":"Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=1/20, positive shift=2, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. Checked prefix_length=1. Give (p,c,shift,prefix_length,sufficient?,finite_prefix_not_convergence_certificate).","answer":{"kind":"expression","value":"(1/2, 1/20, 2, 1, No, finite_prefix_not_convergence_certificate)","explanation":"The No sufficiency flag concerns a finite-prefix inspection, not the validity of the explicitly declared symbolic family. Its whole tail can supply a separate universal certificate; a long finite prefix alone cannot. The exact result, retaining the requested order, is (1/2, 1/20, 2, 1, No, finite_prefix_not_convergence_certificate)."},"skills":["A Fixed Prefix Does Not Supply the All-n CDF Limit Obligation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261031-r03-convergence-in-distribution-0077","batch_id":"batch-20261014-advance-20261031-r03-convergence-in-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7151d6894141a7e1720095a405f0e107f40b2762e439362eeaa121e7b1546c89","family_id":"qf-mod31-convdist-finite-prefix","seed":202610313077,"semantic_template":"qf-mod31-convdist-finite-prefix/{\"c\":[2,20],\"p\":[1,2],\"prefix_length\":2,\"shift\":3}"},"fields":{"title":"A Fixed Prefix Does Not Supply the All-n CDF Limit Obligation","stem":"Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=2/20, positive shift=3, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. Checked prefix_length=2. Give (p,c,shift,prefix_length,sufficient?,finite_prefix_not_convergence_certificate).","answer":{"kind":"expression","value":"(1/2, 1/10, 3, 2, No, finite_prefix_not_convergence_certificate)","explanation":"The No sufficiency flag concerns a finite-prefix inspection, not the validity of the explicitly declared symbolic family. Its whole tail can supply a separate universal certificate; a long finite prefix alone cannot. The exact result, retaining the requested order, is (1/2, 1/10, 3, 2, No, finite_prefix_not_convergence_certificate)."},"skills":["A Fixed Prefix Does Not Supply the All-n CDF Limit Obligation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261031-r03-convergence-in-distribution-0078","batch_id":"batch-20261014-advance-20261031-r03-convergence-in-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6c2ae259fefa073146234435a6b82dda91c27b9050451a13ec79f3e563410d9d","family_id":"qf-mod31-convdist-finite-prefix","seed":202610313078,"semantic_template":"qf-mod31-convdist-finite-prefix/{\"c\":[3,20],\"p\":[1,2],\"prefix_length\":3,\"shift\":4}"},"fields":{"title":"A Fixed Prefix Does Not Supply the All-n CDF Limit Obligation","stem":"Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=3/20, positive shift=4, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. Checked prefix_length=3. Give (p,c,shift,prefix_length,sufficient?,finite_prefix_not_convergence_certificate).","answer":{"kind":"expression","value":"(1/2, 3/20, 4, 3, No, finite_prefix_not_convergence_certificate)","explanation":"The No sufficiency flag concerns a finite-prefix inspection, not the validity of the explicitly declared symbolic family. Its whole tail can supply a separate universal certificate; a long finite prefix alone cannot. The exact result, retaining the requested order, is (1/2, 3/20, 4, 3, No, finite_prefix_not_convergence_certificate)."},"skills":["A Fixed Prefix Does Not Supply the All-n CDF Limit Obligation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261031-r03-convergence-in-distribution-0079","batch_id":"batch-20261014-advance-20261031-r03-convergence-in-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0d5ee17cb5782088b0a7190ebe45ac8e6992ad96fb93ce4de1655f0408aada82","family_id":"qf-mod31-convdist-finite-prefix","seed":202610313079,"semantic_template":"qf-mod31-convdist-finite-prefix/{\"c\":[4,20],\"p\":[1,2],\"prefix_length\":4,\"shift\":5}"},"fields":{"title":"A Fixed Prefix Does Not Supply the All-n CDF Limit Obligation","stem":"Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=4/20, positive shift=5, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. Checked prefix_length=4. Give (p,c,shift,prefix_length,sufficient?,finite_prefix_not_convergence_certificate).","answer":{"kind":"expression","value":"(1/2, 1/5, 5, 4, No, finite_prefix_not_convergence_certificate)","explanation":"The No sufficiency flag concerns a finite-prefix inspection, not the validity of the explicitly declared symbolic family. Its whole tail can supply a separate universal certificate; a long finite prefix alone cannot. The exact result, retaining the requested order, is (1/2, 1/5, 5, 4, No, finite_prefix_not_convergence_certificate)."},"skills":["A Fixed Prefix Does Not Supply the All-n CDF Limit Obligation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261031-r03-convergence-in-distribution-0080","batch_id":"batch-20261014-advance-20261031-r03-convergence-in-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b3a87a12b3481f8cccf3b54ddbf8cb0dbc12cb4803b9809c566b35b1bddb01d1","family_id":"qf-mod31-convdist-finite-prefix","seed":202610313080,"semantic_template":"qf-mod31-convdist-finite-prefix/{\"c\":[5,20],\"p\":[1,2],\"prefix_length\":5,\"shift\":6}"},"fields":{"title":"A Fixed Prefix Does Not Supply the All-n CDF Limit Obligation","stem":"Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=5/20, positive shift=6, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. Checked prefix_length=5. Give (p,c,shift,prefix_length,sufficient?,finite_prefix_not_convergence_certificate).","answer":{"kind":"expression","value":"(1/2, 1/4, 6, 5, No, finite_prefix_not_convergence_certificate)","explanation":"The No sufficiency flag concerns a finite-prefix inspection, not the validity of the explicitly declared symbolic family. Its whole tail can supply a separate universal certificate; a long finite prefix alone cannot. The exact result, retaining the requested order, is (1/2, 1/4, 6, 5, No, finite_prefix_not_convergence_certificate)."},"skills":["A Fixed Prefix Does Not Supply the All-n CDF Limit Obligation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261031-r03-convergence-in-distribution-0081","batch_id":"batch-20261014-advance-20261031-r03-convergence-in-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"70c79d107be8eee55bf3efab0762d32231f769d9042083c80b309112218a43c9","family_id":"qf-mod31-convdist-finite-prefix","seed":202610313081,"semantic_template":"qf-mod31-convdist-finite-prefix/{\"c\":[1,20],\"p\":[1,2],\"prefix_length\":6,\"shift\":7}"},"fields":{"title":"A Fixed Prefix Does Not Supply the All-n CDF Limit Obligation","stem":"Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=1/20, positive shift=7, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. Checked prefix_length=6. Give (p,c,shift,prefix_length,sufficient?,finite_prefix_not_convergence_certificate).","answer":{"kind":"expression","value":"(1/2, 1/20, 7, 6, No, finite_prefix_not_convergence_certificate)","explanation":"The No sufficiency flag concerns a finite-prefix inspection, not the validity of the explicitly declared symbolic family. Its whole tail can supply a separate universal certificate; a long finite prefix alone cannot. The exact result, retaining the requested order, is (1/2, 1/20, 7, 6, No, finite_prefix_not_convergence_certificate)."},"skills":["A Fixed Prefix Does Not Supply the All-n CDF Limit Obligation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261031-r03-convergence-in-distribution-0082","batch_id":"batch-20261014-advance-20261031-r03-convergence-in-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e9be78f9daa64c83e491c797c4edc46158c32539e8c5063997317197c3aa6a02","family_id":"qf-mod31-convdist-finite-prefix","seed":202610313082,"semantic_template":"qf-mod31-convdist-finite-prefix/{\"c\":[2,20],\"p\":[1,2],\"prefix_length\":7,\"shift\":8}"},"fields":{"title":"A Fixed Prefix Does Not Supply the All-n CDF Limit Obligation","stem":"Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=2/20, positive shift=8, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. Checked prefix_length=7. Give (p,c,shift,prefix_length,sufficient?,finite_prefix_not_convergence_certificate).","answer":{"kind":"expression","value":"(1/2, 1/10, 8, 7, No, finite_prefix_not_convergence_certificate)","explanation":"The No sufficiency flag concerns a finite-prefix inspection, not the validity of the explicitly declared symbolic family. Its whole tail can supply a separate universal certificate; a long finite prefix alone cannot. The exact result, retaining the requested order, is (1/2, 1/10, 8, 7, No, finite_prefix_not_convergence_certificate)."},"skills":["A Fixed Prefix Does Not Supply the All-n CDF Limit Obligation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261031-r03-convergence-in-distribution-0083","batch_id":"batch-20261014-advance-20261031-r03-convergence-in-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f4e6bca428ea252e99e36a443ead4ef01f3a9c3243abdcf56e50349f86654947","family_id":"qf-mod31-convdist-finite-prefix","seed":202610313083,"semantic_template":"qf-mod31-convdist-finite-prefix/{\"c\":[3,20],\"p\":[1,2],\"prefix_length\":8,\"shift\":9}"},"fields":{"title":"A Fixed Prefix Does Not Supply the All-n CDF Limit Obligation","stem":"Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=3/20, positive shift=9, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. Checked prefix_length=8. Give (p,c,shift,prefix_length,sufficient?,finite_prefix_not_convergence_certificate).","answer":{"kind":"expression","value":"(1/2, 3/20, 9, 8, No, finite_prefix_not_convergence_certificate)","explanation":"The No sufficiency flag concerns a finite-prefix inspection, not the validity of the explicitly declared symbolic family. Its whole tail can supply a separate universal certificate; a long finite prefix alone cannot. The exact result, retaining the requested order, is (1/2, 3/20, 9, 8, No, finite_prefix_not_convergence_certificate)."},"skills":["A Fixed Prefix Does Not Supply the All-n CDF Limit Obligation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261031-r03-convergence-in-distribution-0084","batch_id":"batch-20261014-advance-20261031-r03-convergence-in-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5932958494b547779d0e7e214c185ab830b96763e99d6cbf65afcfc770e36d7f","family_id":"qf-mod31-convdist-finite-prefix","seed":202610313084,"semantic_template":"qf-mod31-convdist-finite-prefix/{\"c\":[4,20],\"p\":[1,2],\"prefix_length\":9,\"shift\":10}"},"fields":{"title":"A Fixed Prefix Does Not Supply the All-n CDF Limit Obligation","stem":"Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=4/20, positive shift=10, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. Checked prefix_length=9. Give (p,c,shift,prefix_length,sufficient?,finite_prefix_not_convergence_certificate).","answer":{"kind":"expression","value":"(1/2, 1/5, 10, 9, No, finite_prefix_not_convergence_certificate)","explanation":"The No sufficiency flag concerns a finite-prefix inspection, not the validity of the explicitly declared symbolic family. Its whole tail can supply a separate universal certificate; a long finite prefix alone cannot. The exact result, retaining the requested order, is (1/2, 1/5, 10, 9, No, finite_prefix_not_convergence_certificate)."},"skills":["A Fixed Prefix Does Not Supply the All-n CDF Limit Obligation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261031-r03-convergence-in-distribution-0085","batch_id":"batch-20261014-advance-20261031-r03-convergence-in-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7b454d097f56920afafc4dee0867ac26358fd7605cc396da0fbc9a2d5c9e00f4","family_id":"qf-mod31-convdist-finite-prefix","seed":202610313085,"semantic_template":"qf-mod31-convdist-finite-prefix/{\"c\":[5,20],\"p\":[1,2],\"prefix_length\":10,\"shift\":11}"},"fields":{"title":"A Fixed Prefix Does Not Supply the All-n CDF Limit Obligation","stem":"Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=5/20, positive shift=11, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. Checked prefix_length=10. Give (p,c,shift,prefix_length,sufficient?,finite_prefix_not_convergence_certificate).","answer":{"kind":"expression","value":"(1/2, 1/4, 11, 10, No, finite_prefix_not_convergence_certificate)","explanation":"The No sufficiency flag concerns a finite-prefix inspection, not the validity of the explicitly declared symbolic family. Its whole tail can supply a separate universal certificate; a long finite prefix alone cannot. The exact result, retaining the requested order, is (1/2, 1/4, 11, 10, No, finite_prefix_not_convergence_certificate)."},"skills":["A Fixed Prefix Does Not Supply the All-n CDF Limit Obligation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261031-r03-convergence-in-distribution-0086","batch_id":"batch-20261014-advance-20261031-r03-convergence-in-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4711d20ff112c07cab5528e1e5cf9d651e987d3b7ddae300c61c1b542289e054","family_id":"qf-mod31-convdist-finite-prefix","seed":202610313086,"semantic_template":"qf-mod31-convdist-finite-prefix/{\"c\":[1,20],\"p\":[1,2],\"prefix_length\":11,\"shift\":12}"},"fields":{"title":"A Fixed Prefix Does Not Supply the All-n CDF Limit Obligation","stem":"Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=1/20, positive shift=12, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. Checked prefix_length=11. Give (p,c,shift,prefix_length,sufficient?,finite_prefix_not_convergence_certificate).","answer":{"kind":"expression","value":"(1/2, 1/20, 12, 11, No, finite_prefix_not_convergence_certificate)","explanation":"The No sufficiency flag concerns a finite-prefix inspection, not the validity of the explicitly declared symbolic family. Its whole tail can supply a separate universal certificate; a long finite prefix alone cannot. The exact result, retaining the requested order, is (1/2, 1/20, 12, 11, No, finite_prefix_not_convergence_certificate)."},"skills":["A Fixed Prefix Does Not Supply the All-n CDF Limit Obligation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261031-r03-convergence-in-distribution-0087","batch_id":"batch-20261014-advance-20261031-r03-convergence-in-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e47bcbf24c125571f51c88d8450ae620b16cde65fd857f5c4635a910683c4062","family_id":"qf-mod31-convdist-finite-prefix","seed":202610313087,"semantic_template":"qf-mod31-convdist-finite-prefix/{\"c\":[2,20],\"p\":[1,2],\"prefix_length\":12,\"shift\":13}"},"fields":{"title":"A Fixed Prefix Does Not Supply the All-n CDF Limit Obligation","stem":"Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=2/20, positive shift=13, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. Checked prefix_length=12. Give (p,c,shift,prefix_length,sufficient?,finite_prefix_not_convergence_certificate).","answer":{"kind":"expression","value":"(1/2, 1/10, 13, 12, No, finite_prefix_not_convergence_certificate)","explanation":"The No sufficiency flag concerns a finite-prefix inspection, not the validity of the explicitly declared symbolic family. Its whole tail can supply a separate universal certificate; a long finite prefix alone cannot. The exact result, retaining the requested order, is (1/2, 1/10, 13, 12, No, finite_prefix_not_convergence_certificate)."},"skills":["A Fixed Prefix Does Not Supply the All-n CDF Limit Obligation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261031-r03-convergence-in-distribution-0088","batch_id":"batch-20261014-advance-20261031-r03-convergence-in-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"72952a59d33e6ca4a4916e899b7c92764aa36f17cc298ebfc6a139d645e5cb90","family_id":"qf-mod31-convdist-finite-prefix","seed":202610313088,"semantic_template":"qf-mod31-convdist-finite-prefix/{\"c\":[3,20],\"p\":[1,2],\"prefix_length\":13,\"shift\":14}"},"fields":{"title":"A Fixed Prefix Does Not Supply the All-n CDF Limit Obligation","stem":"Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=3/20, positive shift=14, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. Checked prefix_length=13. Give (p,c,shift,prefix_length,sufficient?,finite_prefix_not_convergence_certificate).","answer":{"kind":"expression","value":"(1/2, 3/20, 14, 13, No, finite_prefix_not_convergence_certificate)","explanation":"The No sufficiency flag concerns a finite-prefix inspection, not the validity of the explicitly declared symbolic family. Its whole tail can supply a separate universal certificate; a long finite prefix alone cannot. The exact result, retaining the requested order, is (1/2, 3/20, 14, 13, No, finite_prefix_not_convergence_certificate)."},"skills":["A Fixed Prefix Does Not Supply the All-n CDF Limit Obligation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261031-r03-convergence-in-distribution-0089","batch_id":"batch-20261014-advance-20261031-r03-convergence-in-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"af0423ab14ef4830bca70c6335c00b1a27697cf109f239c89a54011ed5ee0740","family_id":"qf-mod31-convdist-finite-prefix","seed":202610313089,"semantic_template":"qf-mod31-convdist-finite-prefix/{\"c\":[4,20],\"p\":[1,2],\"prefix_length\":14,\"shift\":15}"},"fields":{"title":"A Fixed Prefix Does Not Supply the All-n CDF Limit Obligation","stem":"Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=4/20, positive shift=15, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. Checked prefix_length=14. Give (p,c,shift,prefix_length,sufficient?,finite_prefix_not_convergence_certificate).","answer":{"kind":"expression","value":"(1/2, 1/5, 15, 14, No, finite_prefix_not_convergence_certificate)","explanation":"The No sufficiency flag concerns a finite-prefix inspection, not the validity of the explicitly declared symbolic family. Its whole tail can supply a separate universal certificate; a long finite prefix alone cannot. The exact result, retaining the requested order, is (1/2, 1/5, 15, 14, No, finite_prefix_not_convergence_certificate)."},"skills":["A Fixed Prefix Does Not Supply the All-n CDF Limit Obligation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261031-r03-convergence-in-distribution-0090","batch_id":"batch-20261014-advance-20261031-r03-convergence-in-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"396aae051704c09b405f2342c4940c8ff24830f7110af3cc24e90b6d4962eace","family_id":"qf-mod31-convdist-finite-prefix","seed":202610313090,"semantic_template":"qf-mod31-convdist-finite-prefix/{\"c\":[5,20],\"p\":[1,2],\"prefix_length\":15,\"shift\":16}"},"fields":{"title":"A Fixed Prefix Does Not Supply the All-n CDF Limit Obligation","stem":"Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=5/20, positive shift=16, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. Checked prefix_length=15. Give (p,c,shift,prefix_length,sufficient?,finite_prefix_not_convergence_certificate).","answer":{"kind":"expression","value":"(1/2, 1/4, 16, 15, No, finite_prefix_not_convergence_certificate)","explanation":"The No sufficiency flag concerns a finite-prefix inspection, not the validity of the explicitly declared symbolic family. Its whole tail can supply a separate universal certificate; a long finite prefix alone cannot. The exact result, retaining the requested order, is (1/2, 1/4, 16, 15, No, finite_prefix_not_convergence_certificate)."},"skills":["A Fixed Prefix Does Not Supply the All-n CDF Limit Obligation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261031-r03-convergence-in-distribution-0091","batch_id":"batch-20261014-advance-20261031-r03-convergence-in-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"899f6b819be9575300388efd24262298eeb711a1a01d30b09d2cc55fb3a5a879","family_id":"qf-mod31-convdist-finite-prefix","seed":202610313091,"semantic_template":"qf-mod31-convdist-finite-prefix/{\"c\":[1,20],\"p\":[1,2],\"prefix_length\":16,\"shift\":17}"},"fields":{"title":"A Fixed Prefix Does Not Supply the All-n CDF Limit Obligation","stem":"Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=1/20, positive shift=17, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. Checked prefix_length=16. Give (p,c,shift,prefix_length,sufficient?,finite_prefix_not_convergence_certificate).","answer":{"kind":"expression","value":"(1/2, 1/20, 17, 16, No, finite_prefix_not_convergence_certificate)","explanation":"The No sufficiency flag concerns a finite-prefix inspection, not the validity of the explicitly declared symbolic family. Its whole tail can supply a separate universal certificate; a long finite prefix alone cannot. The exact result, retaining the requested order, is (1/2, 1/20, 17, 16, No, finite_prefix_not_convergence_certificate)."},"skills":["A Fixed Prefix Does Not Supply the All-n CDF Limit Obligation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261031-r03-convergence-in-distribution-0092","batch_id":"batch-20261014-advance-20261031-r03-convergence-in-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"482e56da42f3062d3b8060290a2a14b85a2564ef3ef9c6c26e2085aa667274d5","family_id":"qf-mod31-convdist-finite-prefix","seed":202610313092,"semantic_template":"qf-mod31-convdist-finite-prefix/{\"c\":[2,20],\"p\":[1,2],\"prefix_length\":17,\"shift\":18}"},"fields":{"title":"A Fixed Prefix Does Not Supply the All-n CDF Limit Obligation","stem":"Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=2/20, positive shift=18, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. Checked prefix_length=17. Give (p,c,shift,prefix_length,sufficient?,finite_prefix_not_convergence_certificate).","answer":{"kind":"expression","value":"(1/2, 1/10, 18, 17, No, finite_prefix_not_convergence_certificate)","explanation":"The No sufficiency flag concerns a finite-prefix inspection, not the validity of the explicitly declared symbolic family. Its whole tail can supply a separate universal certificate; a long finite prefix alone cannot. The exact result, retaining the requested order, is (1/2, 1/10, 18, 17, No, finite_prefix_not_convergence_certificate)."},"skills":["A Fixed Prefix Does Not Supply the All-n CDF Limit Obligation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261031-r03-convergence-in-distribution-0093","batch_id":"batch-20261014-advance-20261031-r03-convergence-in-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"54107aec8ebfa1cb5fd71049c8a72e4c54703bb638d3baf9fd27479bcdf23b81","family_id":"qf-mod31-convdist-finite-prefix","seed":202610313093,"semantic_template":"qf-mod31-convdist-finite-prefix/{\"c\":[3,20],\"p\":[1,2],\"prefix_length\":18,\"shift\":19}"},"fields":{"title":"A Fixed Prefix Does Not Supply the All-n CDF Limit Obligation","stem":"Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=3/20, positive shift=19, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. Checked prefix_length=18. Give (p,c,shift,prefix_length,sufficient?,finite_prefix_not_convergence_certificate).","answer":{"kind":"expression","value":"(1/2, 3/20, 19, 18, No, finite_prefix_not_convergence_certificate)","explanation":"The No sufficiency flag concerns a finite-prefix inspection, not the validity of the explicitly declared symbolic family. Its whole tail can supply a separate universal certificate; a long finite prefix alone cannot. The exact result, retaining the requested order, is (1/2, 3/20, 19, 18, No, finite_prefix_not_convergence_certificate)."},"skills":["A Fixed Prefix Does Not Supply the All-n CDF Limit Obligation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261031-r03-convergence-in-distribution-0094","batch_id":"batch-20261014-advance-20261031-r03-convergence-in-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"621f9a1249c24d8b179fb9ea1914e1efb221345e7f94d529335189d540bdd720","family_id":"qf-mod31-convdist-finite-prefix","seed":202610313094,"semantic_template":"qf-mod31-convdist-finite-prefix/{\"c\":[4,20],\"p\":[1,2],\"prefix_length\":19,\"shift\":20}"},"fields":{"title":"A Fixed Prefix Does Not Supply the All-n CDF Limit Obligation","stem":"Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=4/20, positive shift=20, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. Checked prefix_length=19. Give (p,c,shift,prefix_length,sufficient?,finite_prefix_not_convergence_certificate).","answer":{"kind":"expression","value":"(1/2, 1/5, 20, 19, No, finite_prefix_not_convergence_certificate)","explanation":"The No sufficiency flag concerns a finite-prefix inspection, not the validity of the explicitly declared symbolic family. Its whole tail can supply a separate universal certificate; a long finite prefix alone cannot. The exact result, retaining the requested order, is (1/2, 1/5, 20, 19, No, finite_prefix_not_convergence_certificate)."},"skills":["A Fixed Prefix Does Not Supply the All-n CDF Limit Obligation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261031-r03-convergence-in-distribution-0095","batch_id":"batch-20261014-advance-20261031-r03-convergence-in-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"34c91f24b341cc8348e7ace0af4186d0f1c8d02f32d019089cb5333194f4f9ec","family_id":"qf-mod31-convdist-finite-prefix","seed":202610313095,"semantic_template":"qf-mod31-convdist-finite-prefix/{\"c\":[5,20],\"p\":[1,2],\"prefix_length\":20,\"shift\":21}"},"fields":{"title":"A Fixed Prefix Does Not Supply the All-n CDF Limit Obligation","stem":"Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=5/20, positive shift=21, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. Checked prefix_length=20. Give (p,c,shift,prefix_length,sufficient?,finite_prefix_not_convergence_certificate).","answer":{"kind":"expression","value":"(1/2, 1/4, 21, 20, No, finite_prefix_not_convergence_certificate)","explanation":"The No sufficiency flag concerns a finite-prefix inspection, not the validity of the explicitly declared symbolic family. Its whole tail can supply a separate universal certificate; a long finite prefix alone cannot. The exact result, retaining the requested order, is (1/2, 1/4, 21, 20, No, finite_prefix_not_convergence_certificate)."},"skills":["A Fixed Prefix Does Not Supply the All-n CDF Limit Obligation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261031-r03-convergence-in-distribution-0096","batch_id":"batch-20261014-advance-20261031-r03-convergence-in-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b8c58a050ccc2eb5e389455c1b2833d7d89ede8ad117a71ef8278d872d5c2f2f","family_id":"qf-mod31-convdist-finite-prefix","seed":202610313096,"semantic_template":"qf-mod31-convdist-finite-prefix/{\"c\":[1,20],\"p\":[1,2],\"prefix_length\":21,\"shift\":22}"},"fields":{"title":"A Fixed Prefix Does Not Supply the All-n CDF Limit Obligation","stem":"Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=1/20, positive shift=22, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. Checked prefix_length=21. Give (p,c,shift,prefix_length,sufficient?,finite_prefix_not_convergence_certificate).","answer":{"kind":"expression","value":"(1/2, 1/20, 22, 21, No, finite_prefix_not_convergence_certificate)","explanation":"The No sufficiency flag concerns a finite-prefix inspection, not the validity of the explicitly declared symbolic family. Its whole tail can supply a separate universal certificate; a long finite prefix alone cannot. The exact result, retaining the requested order, is (1/2, 1/20, 22, 21, No, finite_prefix_not_convergence_certificate)."},"skills":["A Fixed Prefix Does Not Supply the All-n CDF Limit Obligation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261031-r03-convergence-in-distribution-0097","batch_id":"batch-20261014-advance-20261031-r03-convergence-in-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d856f45f6ed0acf528da0215a432099b8498896df48a5e844e685ec1f8db2718","family_id":"qf-mod31-convdist-finite-prefix","seed":202610313097,"semantic_template":"qf-mod31-convdist-finite-prefix/{\"c\":[2,20],\"p\":[1,2],\"prefix_length\":22,\"shift\":23}"},"fields":{"title":"A Fixed Prefix Does Not Supply the All-n CDF Limit Obligation","stem":"Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=2/20, positive shift=23, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. Checked prefix_length=22. Give (p,c,shift,prefix_length,sufficient?,finite_prefix_not_convergence_certificate).","answer":{"kind":"expression","value":"(1/2, 1/10, 23, 22, No, finite_prefix_not_convergence_certificate)","explanation":"The No sufficiency flag concerns a finite-prefix inspection, not the validity of the explicitly declared symbolic family. Its whole tail can supply a separate universal certificate; a long finite prefix alone cannot. The exact result, retaining the requested order, is (1/2, 1/10, 23, 22, No, finite_prefix_not_convergence_certificate)."},"skills":["A Fixed Prefix Does Not Supply the All-n CDF Limit Obligation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261031-r03-convergence-in-distribution-0098","batch_id":"batch-20261014-advance-20261031-r03-convergence-in-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"db0807ddf177c1163c6f4065ec9ef667defd2f9eeaa166479eeb4fbea9704a35","family_id":"qf-mod31-convdist-finite-prefix","seed":202610313098,"semantic_template":"qf-mod31-convdist-finite-prefix/{\"c\":[3,20],\"p\":[1,2],\"prefix_length\":23,\"shift\":24}"},"fields":{"title":"A Fixed Prefix Does Not Supply the All-n CDF Limit Obligation","stem":"Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=3/20, positive shift=24, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. Checked prefix_length=23. Give (p,c,shift,prefix_length,sufficient?,finite_prefix_not_convergence_certificate).","answer":{"kind":"expression","value":"(1/2, 3/20, 24, 23, No, finite_prefix_not_convergence_certificate)","explanation":"The No sufficiency flag concerns a finite-prefix inspection, not the validity of the explicitly declared symbolic family. Its whole tail can supply a separate universal certificate; a long finite prefix alone cannot. The exact result, retaining the requested order, is (1/2, 3/20, 24, 23, No, finite_prefix_not_convergence_certificate)."},"skills":["A Fixed Prefix Does Not Supply the All-n CDF Limit Obligation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261031-r03-convergence-in-distribution-0099","batch_id":"batch-20261014-advance-20261031-r03-convergence-in-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bdb866bc097983239eb37d13dca93fe28796b96d6d3140fc368bbb1cfe499c9d","family_id":"qf-mod31-convdist-finite-prefix","seed":202610313099,"semantic_template":"qf-mod31-convdist-finite-prefix/{\"c\":[4,20],\"p\":[1,2],\"prefix_length\":24,\"shift\":25}"},"fields":{"title":"A Fixed Prefix Does Not Supply the All-n CDF Limit Obligation","stem":"Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=4/20, positive shift=25, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. Checked prefix_length=24. Give (p,c,shift,prefix_length,sufficient?,finite_prefix_not_convergence_certificate).","answer":{"kind":"expression","value":"(1/2, 1/5, 25, 24, No, finite_prefix_not_convergence_certificate)","explanation":"The No sufficiency flag concerns a finite-prefix inspection, not the validity of the explicitly declared symbolic family. Its whole tail can supply a separate universal certificate; a long finite prefix alone cannot. The exact result, retaining the requested order, is (1/2, 1/5, 25, 24, No, finite_prefix_not_convergence_certificate)."},"skills":["A Fixed Prefix Does Not Supply the All-n CDF Limit Obligation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261031-r03-convergence-in-distribution-0100","batch_id":"batch-20261014-advance-20261031-r03-convergence-in-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0f3e3e3f44f836248fcc2b7450e4241d34b7fc0b552aedb9c799e5a912cae8f8","family_id":"qf-mod31-convdist-finite-prefix","seed":202610313100,"semantic_template":"qf-mod31-convdist-finite-prefix/{\"c\":[5,20],\"p\":[1,2],\"prefix_length\":25,\"shift\":26}"},"fields":{"title":"A Fixed Prefix Does Not Supply the All-n CDF Limit Obligation","stem":"Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=5/20, positive shift=26, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. Checked prefix_length=25. Give (p,c,shift,prefix_length,sufficient?,finite_prefix_not_convergence_certificate).","answer":{"kind":"expression","value":"(1/2, 1/4, 26, 25, No, finite_prefix_not_convergence_certificate)","explanation":"The No sufficiency flag concerns a finite-prefix inspection, not the validity of the explicitly declared symbolic family. Its whole tail can supply a separate universal certificate; a long finite prefix alone cannot. The exact result, retaining the requested order, is (1/2, 1/4, 26, 25, No, finite_prefix_not_convergence_certificate)."},"skills":["A Fixed Prefix Does Not Supply the All-n CDF Limit Obligation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}