{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r02-convex-function-0001","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f268f4f1ceb62e30e03db6fdd8431e6864fd289c64dc9b1dd4a2a88878350a78","family_id":"qf-mod25-convexfn-convex-domain-qualification","seed":202610252001,"semantic_template":"qf-mod25-convexfn-convex-domain-qualification/{\"interval\":[-2,3]}"},"fields":{"title":"Qualify a Declared Nonempty Interval before Using Jensen Semantics","stem":"The interval record is [lower,upper]=[-2, 3]. Give (interval,lower≤upper?,qualified convex interval?,all segment points remain in the interval). A reversed record is unqualified for this nonempty-interval protocol; its No flag does not classify the abstract empty set as nonconvex. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.","answer":{"kind":"expression","value":"([-2, 3], Yes, Yes, all segment points remain in the interval)","explanation":"For an ordered interval, arbitrary endpoint pairs and weights in [0,1] stay between its bounds. A singleton is allowed here. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-2, 3], Yes, Yes, all segment points remain in the interval)."},"skills":["Qualify a Declared Nonempty Interval before Using Jensen Semantics"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r02-convex-function-0002","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f88bb0cba3b5e10dc77309bc961322313d1aef6ef6d631c9fc281707918a7db0","family_id":"qf-mod25-convexfn-convex-domain-qualification","seed":202610252002,"semantic_template":"qf-mod25-convexfn-convex-domain-qualification/{\"interval\":[-3,4]}"},"fields":{"title":"Qualify a Declared Nonempty Interval before Using Jensen Semantics","stem":"The interval record is [lower,upper]=[-3, 4]. Give (interval,lower≤upper?,qualified convex interval?,all segment points remain in the interval). A reversed record is unqualified for this nonempty-interval protocol; its No flag does not classify the abstract empty set as nonconvex. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.","answer":{"kind":"expression","value":"([-3, 4], Yes, Yes, all segment points remain in the interval)","explanation":"For an ordered interval, arbitrary endpoint pairs and weights in [0,1] stay between its bounds. A singleton is allowed here. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-3, 4], Yes, Yes, all segment points remain in the interval)."},"skills":["Qualify a Declared Nonempty Interval before Using Jensen Semantics"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r02-convex-function-0003","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"021da7d844f1b6f8d3306c1fc136f4c2a36fbf7c346110bc2ef164b0bde24bf6","family_id":"qf-mod25-convexfn-convex-domain-qualification","seed":202610252003,"semantic_template":"qf-mod25-convexfn-convex-domain-qualification/{\"interval\":[-4,5]}"},"fields":{"title":"Qualify a Declared Nonempty Interval before Using Jensen Semantics","stem":"The interval record is [lower,upper]=[-4, 5]. Give (interval,lower≤upper?,qualified convex interval?,all segment points remain in the interval). A reversed record is unqualified for this nonempty-interval protocol; its No flag does not classify the abstract empty set as nonconvex. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.","answer":{"kind":"expression","value":"([-4, 5], Yes, Yes, all segment points remain in the interval)","explanation":"For an ordered interval, arbitrary endpoint pairs and weights in [0,1] stay between its bounds. A singleton is allowed here. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-4, 5], Yes, Yes, all segment points remain in the interval)."},"skills":["Qualify a Declared Nonempty Interval before Using Jensen Semantics"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r02-convex-function-0004","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f8f333926d6994dac5ab3009a567fdafb5c03fd30078dac4374cc4fef6f759a5","family_id":"qf-mod25-convexfn-convex-domain-qualification","seed":202610252004,"semantic_template":"qf-mod25-convexfn-convex-domain-qualification/{\"interval\":[-5,6]}"},"fields":{"title":"Qualify a Declared Nonempty Interval before Using Jensen Semantics","stem":"The interval record is [lower,upper]=[-5, 6]. Give (interval,lower≤upper?,qualified convex interval?,all segment points remain in the interval). A reversed record is unqualified for this nonempty-interval protocol; its No flag does not classify the abstract empty set as nonconvex. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.","answer":{"kind":"expression","value":"([-5, 6], Yes, Yes, all segment points remain in the interval)","explanation":"For an ordered interval, arbitrary endpoint pairs and weights in [0,1] stay between its bounds. A singleton is allowed here. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-5, 6], Yes, Yes, all segment points remain in the interval)."},"skills":["Qualify a Declared Nonempty Interval before Using Jensen Semantics"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r02-convex-function-0005","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"38ed36ed916e3bda365dd7e460da833447ec8105b64cecd25e0185ae9e63a6e6","family_id":"qf-mod25-convexfn-convex-domain-qualification","seed":202610252005,"semantic_template":"qf-mod25-convexfn-convex-domain-qualification/{\"interval\":[-6,7]}"},"fields":{"title":"Qualify a Declared Nonempty Interval before Using Jensen Semantics","stem":"The interval record is [lower,upper]=[-6, 7]. Give (interval,lower≤upper?,qualified convex interval?,all segment points remain in the interval). A reversed record is unqualified for this nonempty-interval protocol; its No flag does not classify the abstract empty set as nonconvex. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.","answer":{"kind":"expression","value":"([-6, 7], Yes, Yes, all segment points remain in the interval)","explanation":"For an ordered interval, arbitrary endpoint pairs and weights in [0,1] stay between its bounds. A singleton is allowed here. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-6, 7], Yes, Yes, all segment points remain in the interval)."},"skills":["Qualify a Declared Nonempty Interval before Using Jensen Semantics"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r02-convex-function-0006","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"02feb38ca3590f004966d0ecd8275b76b6c6ad4ebad0152f3a3bafee096b3675","family_id":"qf-mod25-convexfn-convex-domain-qualification","seed":202610252006,"semantic_template":"qf-mod25-convexfn-convex-domain-qualification/{\"interval\":[-7,8]}"},"fields":{"title":"Qualify a Declared Nonempty Interval before Using Jensen Semantics","stem":"The interval record is [lower,upper]=[-7, 8]. Give (interval,lower≤upper?,qualified convex interval?,all segment points remain in the interval). A reversed record is unqualified for this nonempty-interval protocol; its No flag does not classify the abstract empty set as nonconvex. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.","answer":{"kind":"expression","value":"([-7, 8], Yes, Yes, all segment points remain in the interval)","explanation":"For an ordered interval, arbitrary endpoint pairs and weights in [0,1] stay between its bounds. A singleton is allowed here. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-7, 8], Yes, Yes, all segment points remain in the interval)."},"skills":["Qualify a Declared Nonempty Interval before Using Jensen Semantics"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r02-convex-function-0007","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"99edff73fa397a101f744336b6661bbcd50999555b0ca7f1338580e4ad3fc193","family_id":"qf-mod25-convexfn-convex-domain-qualification","seed":202610252007,"semantic_template":"qf-mod25-convexfn-convex-domain-qualification/{\"interval\":[-8,9]}"},"fields":{"title":"Qualify a Declared Nonempty Interval before Using Jensen Semantics","stem":"The interval record is [lower,upper]=[-8, 9]. Give (interval,lower≤upper?,qualified convex interval?,all segment points remain in the interval). A reversed record is unqualified for this nonempty-interval protocol; its No flag does not classify the abstract empty set as nonconvex. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.","answer":{"kind":"expression","value":"([-8, 9], Yes, Yes, all segment points remain in the interval)","explanation":"For an ordered interval, arbitrary endpoint pairs and weights in [0,1] stay between its bounds. A singleton is allowed here. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-8, 9], Yes, Yes, all segment points remain in the interval)."},"skills":["Qualify a Declared Nonempty Interval before Using Jensen Semantics"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r02-convex-function-0008","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"94f2dedf5e494118478a993cf7360fb95ae0d1b85807f73b28f895bde0cff711","family_id":"qf-mod25-convexfn-convex-domain-qualification","seed":202610252008,"semantic_template":"qf-mod25-convexfn-convex-domain-qualification/{\"interval\":[-9,10]}"},"fields":{"title":"Qualify a Declared Nonempty Interval before Using Jensen Semantics","stem":"The interval record is [lower,upper]=[-9, 10]. Give (interval,lower≤upper?,qualified convex interval?,all segment points remain in the interval). A reversed record is unqualified for this nonempty-interval protocol; its No flag does not classify the abstract empty set as nonconvex. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.","answer":{"kind":"expression","value":"([-9, 10], Yes, Yes, all segment points remain in the interval)","explanation":"For an ordered interval, arbitrary endpoint pairs and weights in [0,1] stay between its bounds. A singleton is allowed here. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-9, 10], Yes, Yes, all segment points remain in the interval)."},"skills":["Qualify a Declared Nonempty Interval before Using Jensen Semantics"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r02-convex-function-0009","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9d37b34220a2bca0cd3e82e1989c21d59b3674baccd5ac5a33f8427bf8553fba","family_id":"qf-mod25-convexfn-convex-domain-qualification","seed":202610252009,"semantic_template":"qf-mod25-convexfn-convex-domain-qualification/{\"interval\":[-10,11]}"},"fields":{"title":"Qualify a Declared Nonempty Interval before Using Jensen Semantics","stem":"The interval record is [lower,upper]=[-10, 11]. Give (interval,lower≤upper?,qualified convex interval?,all segment points remain in the interval). A reversed record is unqualified for this nonempty-interval protocol; its No flag does not classify the abstract empty set as nonconvex. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.","answer":{"kind":"expression","value":"([-10, 11], Yes, Yes, all segment points remain in the interval)","explanation":"For an ordered interval, arbitrary endpoint pairs and weights in [0,1] stay between its bounds. A singleton is allowed here. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-10, 11], Yes, Yes, all segment points remain in the interval)."},"skills":["Qualify a Declared Nonempty Interval before Using Jensen Semantics"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r02-convex-function-0010","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2f1fd826371b9027986038c4d42c91425f33628fd8fbf9741cea507826ce1523","family_id":"qf-mod25-convexfn-convex-domain-qualification","seed":202610252010,"semantic_template":"qf-mod25-convexfn-convex-domain-qualification/{\"interval\":[-11,12]}"},"fields":{"title":"Qualify a Declared Nonempty Interval before Using Jensen Semantics","stem":"The interval record is [lower,upper]=[-11, 12]. Give (interval,lower≤upper?,qualified convex interval?,all segment points remain in the interval). A reversed record is unqualified for this nonempty-interval protocol; its No flag does not classify the abstract empty set as nonconvex. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.","answer":{"kind":"expression","value":"([-11, 12], Yes, Yes, all segment points remain in the interval)","explanation":"For an ordered interval, arbitrary endpoint pairs and weights in [0,1] stay between its bounds. A singleton is allowed here. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-11, 12], Yes, Yes, all segment points remain in the interval)."},"skills":["Qualify a Declared Nonempty Interval before Using Jensen Semantics"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r02-convex-function-0011","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"444037f3cde9e1c323ca73e19d3e07008bc95253021922199c58421b1a416b66","family_id":"qf-mod25-convexfn-convex-domain-qualification","seed":202610252011,"semantic_template":"qf-mod25-convexfn-convex-domain-qualification/{\"interval\":[-12,13]}"},"fields":{"title":"Qualify a Declared Nonempty Interval before Using Jensen Semantics","stem":"The interval record is [lower,upper]=[-12, 13]. Give (interval,lower≤upper?,qualified convex interval?,all segment points remain in the interval). A reversed record is unqualified for this nonempty-interval protocol; its No flag does not classify the abstract empty set as nonconvex. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.","answer":{"kind":"expression","value":"([-12, 13], Yes, Yes, all segment points remain in the interval)","explanation":"For an ordered interval, arbitrary endpoint pairs and weights in [0,1] stay between its bounds. A singleton is allowed here. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-12, 13], Yes, Yes, all segment points remain in the interval)."},"skills":["Qualify a Declared Nonempty Interval before Using Jensen Semantics"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r02-convex-function-0012","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"72f127ca96c9d9f051198df8db4eee29dd55747116157e15b0cc9ffa052b5da6","family_id":"qf-mod25-convexfn-convex-domain-qualification","seed":202610252012,"semantic_template":"qf-mod25-convexfn-convex-domain-qualification/{\"interval\":[-13,14]}"},"fields":{"title":"Qualify a Declared Nonempty Interval before Using Jensen Semantics","stem":"The interval record is [lower,upper]=[-13, 14]. Give (interval,lower≤upper?,qualified convex interval?,all segment points remain in the interval). A reversed record is unqualified for this nonempty-interval protocol; its No flag does not classify the abstract empty set as nonconvex. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.","answer":{"kind":"expression","value":"([-13, 14], Yes, Yes, all segment points remain in the interval)","explanation":"For an ordered interval, arbitrary endpoint pairs and weights in [0,1] stay between its bounds. A singleton is allowed here. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-13, 14], Yes, Yes, all segment points remain in the interval)."},"skills":["Qualify a Declared Nonempty Interval before Using Jensen Semantics"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r02-convex-function-0013","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"832a2c45de5f499c955a77e05cde2e32442330b0dd9bc6093fa959b70950edf0","family_id":"qf-mod25-convexfn-convex-domain-qualification","seed":202610252013,"semantic_template":"qf-mod25-convexfn-convex-domain-qualification/{\"interval\":[-14,15]}"},"fields":{"title":"Qualify a Declared Nonempty Interval before Using Jensen Semantics","stem":"The interval record is [lower,upper]=[-14, 15]. Give (interval,lower≤upper?,qualified convex interval?,all segment points remain in the interval). A reversed record is unqualified for this nonempty-interval protocol; its No flag does not classify the abstract empty set as nonconvex. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.","answer":{"kind":"expression","value":"([-14, 15], Yes, Yes, all segment points remain in the interval)","explanation":"For an ordered interval, arbitrary endpoint pairs and weights in [0,1] stay between its bounds. A singleton is allowed here. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-14, 15], Yes, Yes, all segment points remain in the interval)."},"skills":["Qualify a Declared Nonempty Interval before Using Jensen Semantics"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r02-convex-function-0014","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d47c3fecd3915e7e226c1ea323680499cabdfcf3adfd0b7477bcbbbd555f25b9","family_id":"qf-mod25-convexfn-convex-domain-qualification","seed":202610252014,"semantic_template":"qf-mod25-convexfn-convex-domain-qualification/{\"interval\":[-15,16]}"},"fields":{"title":"Qualify a Declared Nonempty Interval before Using Jensen Semantics","stem":"The interval record is [lower,upper]=[-15, 16]. Give (interval,lower≤upper?,qualified convex interval?,all segment points remain in the interval). A reversed record is unqualified for this nonempty-interval protocol; its No flag does not classify the abstract empty set as nonconvex. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.","answer":{"kind":"expression","value":"([-15, 16], Yes, Yes, all segment points remain in the interval)","explanation":"For an ordered interval, arbitrary endpoint pairs and weights in [0,1] stay between its bounds. A singleton is allowed here. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-15, 16], Yes, Yes, all segment points remain in the interval)."},"skills":["Qualify a Declared Nonempty Interval before Using Jensen Semantics"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r02-convex-function-0015","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ce308ee2afd7f6c00a03f914acad6219524dd8078ed276d89f59a4bf5143fee1","family_id":"qf-mod25-convexfn-convex-domain-qualification","seed":202610252015,"semantic_template":"qf-mod25-convexfn-convex-domain-qualification/{\"interval\":[-16,17]}"},"fields":{"title":"Qualify a Declared Nonempty Interval before Using Jensen Semantics","stem":"The interval record is [lower,upper]=[-16, 17]. Give (interval,lower≤upper?,qualified convex interval?,all segment points remain in the interval). A reversed record is unqualified for this nonempty-interval protocol; its No flag does not classify the abstract empty set as nonconvex. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.","answer":{"kind":"expression","value":"([-16, 17], Yes, Yes, all segment points remain in the interval)","explanation":"For an ordered interval, arbitrary endpoint pairs and weights in [0,1] stay between its bounds. A singleton is allowed here. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-16, 17], Yes, Yes, all segment points remain in the interval)."},"skills":["Qualify a Declared Nonempty Interval before Using Jensen Semantics"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r02-convex-function-0016","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5eee1316d53b94c138d2d55c4224db342c5a4972b70e412d6c2644dd52f07d7f","family_id":"qf-mod25-convexfn-convex-domain-qualification","seed":202610252016,"semantic_template":"qf-mod25-convexfn-convex-domain-qualification/{\"interval\":[-17,18]}"},"fields":{"title":"Qualify a Declared Nonempty Interval before Using Jensen Semantics","stem":"The interval record is [lower,upper]=[-17, 18]. Give (interval,lower≤upper?,qualified convex interval?,all segment points remain in the interval). A reversed record is unqualified for this nonempty-interval protocol; its No flag does not classify the abstract empty set as nonconvex. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.","answer":{"kind":"expression","value":"([-17, 18], Yes, Yes, all segment points remain in the interval)","explanation":"For an ordered interval, arbitrary endpoint pairs and weights in [0,1] stay between its bounds. A singleton is allowed here. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-17, 18], Yes, Yes, all segment points remain in the interval)."},"skills":["Qualify a Declared Nonempty Interval before Using Jensen Semantics"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r02-convex-function-0017","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a9fdb9fbb290a2191087db8ff3a7f66f238a7c805fed74b9c94323f8b76a350e","family_id":"qf-mod25-convexfn-convex-domain-qualification","seed":202610252017,"semantic_template":"qf-mod25-convexfn-convex-domain-qualification/{\"interval\":[-18,19]}"},"fields":{"title":"Qualify a Declared Nonempty Interval before Using Jensen Semantics","stem":"The interval record is [lower,upper]=[-18, 19]. Give (interval,lower≤upper?,qualified convex interval?,all segment points remain in the interval). A reversed record is unqualified for this nonempty-interval protocol; its No flag does not classify the abstract empty set as nonconvex. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.","answer":{"kind":"expression","value":"([-18, 19], Yes, Yes, all segment points remain in the interval)","explanation":"For an ordered interval, arbitrary endpoint pairs and weights in [0,1] stay between its bounds. A singleton is allowed here. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-18, 19], Yes, Yes, all segment points remain in the interval)."},"skills":["Qualify a Declared Nonempty Interval before Using Jensen Semantics"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r02-convex-function-0018","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ba501278c63a745271d64a0b4a75bf35b926c4ff629c131cc2ebfc9a630726f5","family_id":"qf-mod25-convexfn-convex-domain-qualification","seed":202610252018,"semantic_template":"qf-mod25-convexfn-convex-domain-qualification/{\"interval\":[-19,20]}"},"fields":{"title":"Qualify a Declared Nonempty Interval before Using Jensen Semantics","stem":"The interval record is [lower,upper]=[-19, 20]. Give (interval,lower≤upper?,qualified convex interval?,all segment points remain in the interval). A reversed record is unqualified for this nonempty-interval protocol; its No flag does not classify the abstract empty set as nonconvex. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.","answer":{"kind":"expression","value":"([-19, 20], Yes, Yes, all segment points remain in the interval)","explanation":"For an ordered interval, arbitrary endpoint pairs and weights in [0,1] stay between its bounds. A singleton is allowed here. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-19, 20], Yes, Yes, all segment points remain in the interval)."},"skills":["Qualify a Declared Nonempty Interval before Using Jensen Semantics"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r02-convex-function-0019","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ada84be99a455436352cede43587f8833d1dc9d495f87fa7f7121a5a319b5e8a","family_id":"qf-mod25-convexfn-convex-domain-qualification","seed":202610252019,"semantic_template":"qf-mod25-convexfn-convex-domain-qualification/{\"interval\":[-20,21]}"},"fields":{"title":"Qualify a Declared Nonempty Interval before Using Jensen Semantics","stem":"The interval record is [lower,upper]=[-20, 21]. Give (interval,lower≤upper?,qualified convex interval?,all segment points remain in the interval). A reversed record is unqualified for this nonempty-interval protocol; its No flag does not classify the abstract empty set as nonconvex. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.","answer":{"kind":"expression","value":"([-20, 21], Yes, Yes, all segment points remain in the interval)","explanation":"For an ordered interval, arbitrary endpoint pairs and weights in [0,1] stay between its bounds. A singleton is allowed here. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-20, 21], Yes, Yes, all segment points remain in the interval)."},"skills":["Qualify a Declared Nonempty Interval before Using Jensen Semantics"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r02-convex-function-0020","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c024ed74222c39e810095bf7cd584bd4a5f60ec703a042bdb6c3f3b839ce7e69","family_id":"qf-mod25-convexfn-convex-domain-qualification","seed":202610252020,"semantic_template":"qf-mod25-convexfn-convex-domain-qualification/{\"interval\":[-21,22]}"},"fields":{"title":"Qualify a Declared Nonempty Interval before Using Jensen Semantics","stem":"The interval record is [lower,upper]=[-21, 22]. Give (interval,lower≤upper?,qualified convex interval?,all segment points remain in the interval). A reversed record is unqualified for this nonempty-interval protocol; its No flag does not classify the abstract empty set as nonconvex. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.","answer":{"kind":"expression","value":"([-21, 22], Yes, Yes, all segment points remain in the interval)","explanation":"For an ordered interval, arbitrary endpoint pairs and weights in [0,1] stay between its bounds. A singleton is allowed here. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-21, 22], Yes, Yes, all segment points remain in the interval)."},"skills":["Qualify a Declared Nonempty Interval before Using Jensen Semantics"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r02-convex-function-0021","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fb24e75f876ebbc5c67c9e1be3f1f24c72f958f0eb47c0a55c28e2af2dcd74cf","family_id":"qf-mod25-convexfn-convex-domain-qualification","seed":202610252021,"semantic_template":"qf-mod25-convexfn-convex-domain-qualification/{\"interval\":[-22,23]}"},"fields":{"title":"Qualify a Declared Nonempty Interval before Using Jensen Semantics","stem":"The interval record is [lower,upper]=[-22, 23]. Give (interval,lower≤upper?,qualified convex interval?,all segment points remain in the interval). A reversed record is unqualified for this nonempty-interval protocol; its No flag does not classify the abstract empty set as nonconvex. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.","answer":{"kind":"expression","value":"([-22, 23], Yes, Yes, all segment points remain in the interval)","explanation":"For an ordered interval, arbitrary endpoint pairs and weights in [0,1] stay between its bounds. A singleton is allowed here. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-22, 23], Yes, Yes, all segment points remain in the interval)."},"skills":["Qualify a Declared Nonempty Interval before Using Jensen Semantics"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r02-convex-function-0022","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4015cb5d9e22a5c2a77822c324ae814ba9a67c0a887b8c5278aa92710cf8a723","family_id":"qf-mod25-convexfn-convex-domain-qualification","seed":202610252022,"semantic_template":"qf-mod25-convexfn-convex-domain-qualification/{\"interval\":[-23,24]}"},"fields":{"title":"Qualify a Declared Nonempty Interval before Using Jensen Semantics","stem":"The interval record is [lower,upper]=[-23, 24]. Give (interval,lower≤upper?,qualified convex interval?,all segment points remain in the interval). A reversed record is unqualified for this nonempty-interval protocol; its No flag does not classify the abstract empty set as nonconvex. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.","answer":{"kind":"expression","value":"([-23, 24], Yes, Yes, all segment points remain in the interval)","explanation":"For an ordered interval, arbitrary endpoint pairs and weights in [0,1] stay between its bounds. A singleton is allowed here. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-23, 24], Yes, Yes, all segment points remain in the interval)."},"skills":["Qualify a Declared Nonempty Interval before Using Jensen Semantics"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r02-convex-function-0023","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b2c095a66e43cbb41e4c578c1a9009881f13113a2eb1886b6f8be4db3575e7de","family_id":"qf-mod25-convexfn-convex-domain-qualification","seed":202610252023,"semantic_template":"qf-mod25-convexfn-convex-domain-qualification/{\"interval\":[-24,25]}"},"fields":{"title":"Qualify a Declared Nonempty Interval before Using Jensen Semantics","stem":"The interval record is [lower,upper]=[-24, 25]. Give (interval,lower≤upper?,qualified convex interval?,all segment points remain in the interval). A reversed record is unqualified for this nonempty-interval protocol; its No flag does not classify the abstract empty set as nonconvex. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.","answer":{"kind":"expression","value":"([-24, 25], Yes, Yes, all segment points remain in the interval)","explanation":"For an ordered interval, arbitrary endpoint pairs and weights in [0,1] stay between its bounds. A singleton is allowed here. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-24, 25], Yes, Yes, all segment points remain in the interval)."},"skills":["Qualify a Declared Nonempty Interval before Using Jensen Semantics"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r02-convex-function-0024","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1ad31556f7c7c9e25207d79412757a8734f4cf4bcf5c4ff1260a4d3631aedd96","family_id":"qf-mod25-convexfn-convex-domain-qualification","seed":202610252024,"semantic_template":"qf-mod25-convexfn-convex-domain-qualification/{\"interval\":[-25,26]}"},"fields":{"title":"Qualify a Declared Nonempty Interval before Using Jensen Semantics","stem":"The interval record is [lower,upper]=[-25, 26]. Give (interval,lower≤upper?,qualified convex interval?,all segment points remain in the interval). A reversed record is unqualified for this nonempty-interval protocol; its No flag does not classify the abstract empty set as nonconvex. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.","answer":{"kind":"expression","value":"([-25, 26], Yes, Yes, all segment points remain in the interval)","explanation":"For an ordered interval, arbitrary endpoint pairs and weights in [0,1] stay between its bounds. A singleton is allowed here. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-25, 26], Yes, Yes, all segment points remain in the interval)."},"skills":["Qualify a Declared Nonempty Interval before Using Jensen Semantics"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r02-convex-function-0025","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9122c0e1c4db925b342925894428575fab1bf111fdb6a3a98a65f6842e387999","family_id":"qf-mod25-convexfn-convex-domain-qualification","seed":202610252025,"semantic_template":"qf-mod25-convexfn-convex-domain-qualification/{\"interval\":[-26,27]}"},"fields":{"title":"Qualify a Declared Nonempty Interval before Using Jensen Semantics","stem":"The interval record is [lower,upper]=[-26, 27]. Give (interval,lower≤upper?,qualified convex interval?,all segment points remain in the interval). A reversed record is unqualified for this nonempty-interval protocol; its No flag does not classify the abstract empty set as nonconvex. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.","answer":{"kind":"expression","value":"([-26, 27], Yes, Yes, all segment points remain in the interval)","explanation":"For an ordered interval, arbitrary endpoint pairs and weights in [0,1] stay between its bounds. A singleton is allowed here. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-26, 27], Yes, Yes, all segment points remain in the interval)."},"skills":["Qualify a Declared Nonempty Interval before Using Jensen Semantics"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r02-convex-function-0026","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"81e2dd1bdeaa78e87b5688b6283e252a058c6fa262e4993f1ae2733099b011ac","family_id":"qf-mod25-convexfn-affine-universal-certificate","seed":202610252026,"semantic_template":"qf-mod25-convexfn-affine-universal-certificate/{\"coefficients\":[-12,-2],\"interval\":[-2,3]}"},"fields":{"title":"An Affine Jensen Identity on a Qualified Interval","stem":"Use f(t)=a*t+b with coefficients [a,b]=[-12, -2] and interval=[-2, 3]. Give (coefficients,interval,ordered nonempty domain?,f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y),qualified convex certificate?). A singleton is allowed; a reversed interval record is unqualified. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.","answer":{"kind":"expression","value":"([-12, -2], [-2, 3], Yes, f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y), Yes)","explanation":"Distributivity proves the equality for every pair of domain points and every permitted weight. No finite sampling is used. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-12, -2], [-2, 3], Yes, f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y), Yes)."},"skills":["An Affine Jensen Identity on a Qualified Interval"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r02-convex-function-0027","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ae27585ccc3cb974d8979d6a35c1e0a21f2052464eaff87570aa3d607648a8d7","family_id":"qf-mod25-convexfn-affine-universal-certificate","seed":202610252027,"semantic_template":"qf-mod25-convexfn-affine-universal-certificate/{\"coefficients\":[-11,-1],\"interval\":[-3,4]}"},"fields":{"title":"An Affine Jensen Identity on a Qualified Interval","stem":"Use f(t)=a*t+b with coefficients [a,b]=[-11, -1] and interval=[-3, 4]. Give (coefficients,interval,ordered nonempty domain?,f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y),qualified convex certificate?). A singleton is allowed; a reversed interval record is unqualified. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.","answer":{"kind":"expression","value":"([-11, -1], [-3, 4], Yes, f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y), Yes)","explanation":"Distributivity proves the equality for every pair of domain points and every permitted weight. No finite sampling is used. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-11, -1], [-3, 4], Yes, f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y), Yes)."},"skills":["An Affine Jensen Identity on a Qualified Interval"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r02-convex-function-0028","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e0e5e281fbadba2645a962827bbdac5ad6facecfcc9e66a41ee581668266bdb4","family_id":"qf-mod25-convexfn-affine-universal-certificate","seed":202610252028,"semantic_template":"qf-mod25-convexfn-affine-universal-certificate/{\"coefficients\":[-10,0],\"interval\":[-4,5]}"},"fields":{"title":"An Affine Jensen Identity on a Qualified Interval","stem":"Use f(t)=a*t+b with coefficients [a,b]=[-10, 0] and interval=[-4, 5]. Give (coefficients,interval,ordered nonempty domain?,f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y),qualified convex certificate?). A singleton is allowed; a reversed interval record is unqualified. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.","answer":{"kind":"expression","value":"([-10, 0], [-4, 5], Yes, f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y), Yes)","explanation":"Distributivity proves the equality for every pair of domain points and every permitted weight. No finite sampling is used. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-10, 0], [-4, 5], Yes, f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y), Yes)."},"skills":["An Affine Jensen Identity on a Qualified Interval"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r02-convex-function-0029","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"875212b9d4c422bc3d1d1e138cb204db978b24e30c80f1702716726091a8f65c","family_id":"qf-mod25-convexfn-affine-universal-certificate","seed":202610252029,"semantic_template":"qf-mod25-convexfn-affine-universal-certificate/{\"coefficients\":[-9,1],\"interval\":[-5,6]}"},"fields":{"title":"An Affine Jensen Identity on a Qualified Interval","stem":"Use f(t)=a*t+b with coefficients [a,b]=[-9, 1] and interval=[-5, 6]. Give (coefficients,interval,ordered nonempty domain?,f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y),qualified convex certificate?). A singleton is allowed; a reversed interval record is unqualified. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.","answer":{"kind":"expression","value":"([-9, 1], [-5, 6], Yes, f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y), Yes)","explanation":"Distributivity proves the equality for every pair of domain points and every permitted weight. No finite sampling is used. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-9, 1], [-5, 6], Yes, f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y), Yes)."},"skills":["An Affine Jensen Identity on a Qualified Interval"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r02-convex-function-0030","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1cc1243561e4fef6ac8f7baa0a24765e5750545015bc49781da1f53636c7b12c","family_id":"qf-mod25-convexfn-affine-universal-certificate","seed":202610252030,"semantic_template":"qf-mod25-convexfn-affine-universal-certificate/{\"coefficients\":[-8,2],\"interval\":[-6,7]}"},"fields":{"title":"An Affine Jensen Identity on a Qualified Interval","stem":"Use f(t)=a*t+b with coefficients [a,b]=[-8, 2] and interval=[-6, 7]. Give (coefficients,interval,ordered nonempty domain?,f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y),qualified convex certificate?). A singleton is allowed; a reversed interval record is unqualified. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.","answer":{"kind":"expression","value":"([-8, 2], [-6, 7], Yes, f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y), Yes)","explanation":"Distributivity proves the equality for every pair of domain points and every permitted weight. No finite sampling is used. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-8, 2], [-6, 7], Yes, f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y), Yes)."},"skills":["An Affine Jensen Identity on a Qualified Interval"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r02-convex-function-0031","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e3b76cc6f2f8ecbc2b51664be915b4c606414146c1ca47567d87ae7517454b17","family_id":"qf-mod25-convexfn-affine-universal-certificate","seed":202610252031,"semantic_template":"qf-mod25-convexfn-affine-universal-certificate/{\"coefficients\":[-7,-2],\"interval\":[-7,8]}"},"fields":{"title":"An Affine Jensen Identity on a Qualified Interval","stem":"Use f(t)=a*t+b with coefficients [a,b]=[-7, -2] and interval=[-7, 8]. Give (coefficients,interval,ordered nonempty domain?,f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y),qualified convex certificate?). A singleton is allowed; a reversed interval record is unqualified. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.","answer":{"kind":"expression","value":"([-7, -2], [-7, 8], Yes, f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y), Yes)","explanation":"Distributivity proves the equality for every pair of domain points and every permitted weight. No finite sampling is used. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-7, -2], [-7, 8], Yes, f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y), Yes)."},"skills":["An Affine Jensen Identity on a Qualified Interval"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r02-convex-function-0032","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e0a9de26d098b6df2d0cb3b818dfd7525b7a0dfef23dc1e7227956678ece0345","family_id":"qf-mod25-convexfn-affine-universal-certificate","seed":202610252032,"semantic_template":"qf-mod25-convexfn-affine-universal-certificate/{\"coefficients\":[-6,-1],\"interval\":[-8,9]}"},"fields":{"title":"An Affine Jensen Identity on a Qualified Interval","stem":"Use f(t)=a*t+b with coefficients [a,b]=[-6, -1] and interval=[-8, 9]. Give (coefficients,interval,ordered nonempty domain?,f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y),qualified convex certificate?). A singleton is allowed; a reversed interval record is unqualified. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.","answer":{"kind":"expression","value":"([-6, -1], [-8, 9], Yes, f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y), Yes)","explanation":"Distributivity proves the equality for every pair of domain points and every permitted weight. No finite sampling is used. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-6, -1], [-8, 9], Yes, f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y), Yes)."},"skills":["An Affine Jensen Identity on a Qualified Interval"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r02-convex-function-0033","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"45f55c69dea19fe69034396544e6dd28ce02e355da69a3dca79e778c701f3229","family_id":"qf-mod25-convexfn-affine-universal-certificate","seed":202610252033,"semantic_template":"qf-mod25-convexfn-affine-universal-certificate/{\"coefficients\":[-5,0],\"interval\":[-9,10]}"},"fields":{"title":"An Affine Jensen Identity on a Qualified Interval","stem":"Use f(t)=a*t+b with coefficients [a,b]=[-5, 0] and interval=[-9, 10]. Give (coefficients,interval,ordered nonempty domain?,f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y),qualified convex certificate?). A singleton is allowed; a reversed interval record is unqualified. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.","answer":{"kind":"expression","value":"([-5, 0], [-9, 10], Yes, f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y), Yes)","explanation":"Distributivity proves the equality for every pair of domain points and every permitted weight. No finite sampling is used. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-5, 0], [-9, 10], Yes, f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y), Yes)."},"skills":["An Affine Jensen Identity on a Qualified Interval"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r02-convex-function-0034","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b91b29ddd5827414afe6288257a887a4ff9774a029744657249c0a6ba8fa7bc6","family_id":"qf-mod25-convexfn-affine-universal-certificate","seed":202610252034,"semantic_template":"qf-mod25-convexfn-affine-universal-certificate/{\"coefficients\":[-4,1],\"interval\":[-10,11]}"},"fields":{"title":"An Affine Jensen Identity on a Qualified Interval","stem":"Use f(t)=a*t+b with coefficients [a,b]=[-4, 1] and interval=[-10, 11]. Give (coefficients,interval,ordered nonempty domain?,f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y),qualified convex certificate?). A singleton is allowed; a reversed interval record is unqualified. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.","answer":{"kind":"expression","value":"([-4, 1], [-10, 11], Yes, f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y), Yes)","explanation":"Distributivity proves the equality for every pair of domain points and every permitted weight. No finite sampling is used. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-4, 1], [-10, 11], Yes, f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y), Yes)."},"skills":["An Affine Jensen Identity on a Qualified Interval"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r02-convex-function-0035","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bef4ae68113912433de3f87b5af8c672997655d09c6810b7d8cfd7c6ff4cda38","family_id":"qf-mod25-convexfn-affine-universal-certificate","seed":202610252035,"semantic_template":"qf-mod25-convexfn-affine-universal-certificate/{\"coefficients\":[-3,2],\"interval\":[-11,12]}"},"fields":{"title":"An Affine Jensen Identity on a Qualified Interval","stem":"Use f(t)=a*t+b with coefficients [a,b]=[-3, 2] and interval=[-11, 12]. Give (coefficients,interval,ordered nonempty domain?,f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y),qualified convex certificate?). A singleton is allowed; a reversed interval record is unqualified. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.","answer":{"kind":"expression","value":"([-3, 2], [-11, 12], Yes, f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y), Yes)","explanation":"Distributivity proves the equality for every pair of domain points and every permitted weight. No finite sampling is used. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-3, 2], [-11, 12], Yes, f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y), Yes)."},"skills":["An Affine Jensen Identity on a Qualified Interval"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r02-convex-function-0036","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"768183b4e852849a96abccd475f7b1da57278634c6724f5e3c05472fb94ee692","family_id":"qf-mod25-convexfn-affine-universal-certificate","seed":202610252036,"semantic_template":"qf-mod25-convexfn-affine-universal-certificate/{\"coefficients\":[-2,-2],\"interval\":[-12,13]}"},"fields":{"title":"An Affine Jensen Identity on a Qualified Interval","stem":"Use f(t)=a*t+b with coefficients [a,b]=[-2, -2] and interval=[-12, 13]. Give (coefficients,interval,ordered nonempty domain?,f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y),qualified convex certificate?). A singleton is allowed; a reversed interval record is unqualified. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.","answer":{"kind":"expression","value":"([-2, -2], [-12, 13], Yes, f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y), Yes)","explanation":"Distributivity proves the equality for every pair of domain points and every permitted weight. No finite sampling is used. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-2, -2], [-12, 13], Yes, f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y), Yes)."},"skills":["An Affine Jensen Identity on a Qualified Interval"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r02-convex-function-0037","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"29424866b2c290a2d0bd2d62c6c860e8411000bce5f16655232fb82a3b03e9b7","family_id":"qf-mod25-convexfn-affine-universal-certificate","seed":202610252037,"semantic_template":"qf-mod25-convexfn-affine-universal-certificate/{\"coefficients\":[-1,-1],\"interval\":[-13,14]}"},"fields":{"title":"An Affine Jensen Identity on a Qualified Interval","stem":"Use f(t)=a*t+b with coefficients [a,b]=[-1, -1] and interval=[-13, 14]. Give (coefficients,interval,ordered nonempty domain?,f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y),qualified convex certificate?). A singleton is allowed; a reversed interval record is unqualified. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.","answer":{"kind":"expression","value":"([-1, -1], [-13, 14], Yes, f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y), Yes)","explanation":"Distributivity proves the equality for every pair of domain points and every permitted weight. No finite sampling is used. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-1, -1], [-13, 14], Yes, f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y), Yes)."},"skills":["An Affine Jensen Identity on a Qualified Interval"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r02-convex-function-0038","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a0c56206ff320f21c733e568c50ae4f2eec8db338eade9b0922e48df240fb95f","family_id":"qf-mod25-convexfn-affine-universal-certificate","seed":202610252038,"semantic_template":"qf-mod25-convexfn-affine-universal-certificate/{\"coefficients\":[0,0],\"interval\":[-14,15]}"},"fields":{"title":"An Affine Jensen Identity on a Qualified Interval","stem":"Use f(t)=a*t+b with coefficients [a,b]=[0, 0] and interval=[-14, 15]. Give (coefficients,interval,ordered nonempty domain?,f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y),qualified convex certificate?). A singleton is allowed; a reversed interval record is unqualified. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.","answer":{"kind":"expression","value":"([0, 0], [-14, 15], Yes, f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y), Yes)","explanation":"Distributivity proves the equality for every pair of domain points and every permitted weight. No finite sampling is used. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([0, 0], [-14, 15], Yes, f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y), Yes)."},"skills":["An Affine Jensen Identity on a Qualified Interval"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r02-convex-function-0039","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d80cd539b52c9d6671e5c43207ecadbb9ad45f5f482b536e5a5ff6b1c626aa84","family_id":"qf-mod25-convexfn-affine-universal-certificate","seed":202610252039,"semantic_template":"qf-mod25-convexfn-affine-universal-certificate/{\"coefficients\":[1,1],\"interval\":[-15,16]}"},"fields":{"title":"An Affine Jensen Identity on a Qualified Interval","stem":"Use f(t)=a*t+b with coefficients [a,b]=[1, 1] and interval=[-15, 16]. Give (coefficients,interval,ordered nonempty domain?,f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y),qualified convex certificate?). A singleton is allowed; a reversed interval record is unqualified. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.","answer":{"kind":"expression","value":"([1, 1], [-15, 16], Yes, f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y), Yes)","explanation":"Distributivity proves the equality for every pair of domain points and every permitted weight. No finite sampling is used. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([1, 1], [-15, 16], Yes, f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y), Yes)."},"skills":["An Affine Jensen Identity on a Qualified Interval"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r02-convex-function-0040","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"50e314ffb518e43ff389cfb420df36454a0d23312c66ae8435e84528d2993831","family_id":"qf-mod25-convexfn-affine-universal-certificate","seed":202610252040,"semantic_template":"qf-mod25-convexfn-affine-universal-certificate/{\"coefficients\":[2,2],\"interval\":[-16,17]}"},"fields":{"title":"An Affine Jensen Identity on a Qualified Interval","stem":"Use f(t)=a*t+b with coefficients [a,b]=[2, 2] and interval=[-16, 17]. Give (coefficients,interval,ordered nonempty domain?,f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y),qualified convex certificate?). A singleton is allowed; a reversed interval record is unqualified. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.","answer":{"kind":"expression","value":"([2, 2], [-16, 17], Yes, f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y), Yes)","explanation":"Distributivity proves the equality for every pair of domain points and every permitted weight. No finite sampling is used. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([2, 2], [-16, 17], Yes, f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y), Yes)."},"skills":["An Affine Jensen Identity on a Qualified Interval"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r02-convex-function-0041","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5af8cc596457d7a78e9680d5a7c2b2b4a6c3ec3eaf1ca2012c47cf7b071ef549","family_id":"qf-mod25-convexfn-affine-universal-certificate","seed":202610252041,"semantic_template":"qf-mod25-convexfn-affine-universal-certificate/{\"coefficients\":[3,-2],\"interval\":[-17,18]}"},"fields":{"title":"An Affine Jensen Identity on a Qualified Interval","stem":"Use f(t)=a*t+b with coefficients [a,b]=[3, -2] and interval=[-17, 18]. Give (coefficients,interval,ordered nonempty domain?,f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y),qualified convex certificate?). A singleton is allowed; a reversed interval record is unqualified. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.","answer":{"kind":"expression","value":"([3, -2], [-17, 18], Yes, f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y), Yes)","explanation":"Distributivity proves the equality for every pair of domain points and every permitted weight. No finite sampling is used. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([3, -2], [-17, 18], Yes, f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y), Yes)."},"skills":["An Affine Jensen Identity on a Qualified Interval"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r02-convex-function-0042","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"905bd48e9257ff9aba6efe179084c2252cb17f683a910a00cc91557eb2b3839e","family_id":"qf-mod25-convexfn-affine-universal-certificate","seed":202610252042,"semantic_template":"qf-mod25-convexfn-affine-universal-certificate/{\"coefficients\":[4,-1],\"interval\":[-18,19]}"},"fields":{"title":"An Affine Jensen Identity on a Qualified Interval","stem":"Use f(t)=a*t+b with coefficients [a,b]=[4, -1] and interval=[-18, 19]. Give (coefficients,interval,ordered nonempty domain?,f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y),qualified convex certificate?). A singleton is allowed; a reversed interval record is unqualified. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.","answer":{"kind":"expression","value":"([4, -1], [-18, 19], Yes, f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y), Yes)","explanation":"Distributivity proves the equality for every pair of domain points and every permitted weight. No finite sampling is used. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([4, -1], [-18, 19], Yes, f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y), Yes)."},"skills":["An Affine Jensen Identity on a Qualified Interval"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r02-convex-function-0043","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"669c73f43476680d31caa63d6874e449b0f9380fd582887058ac3844db90a468","family_id":"qf-mod25-convexfn-affine-universal-certificate","seed":202610252043,"semantic_template":"qf-mod25-convexfn-affine-universal-certificate/{\"coefficients\":[5,0],\"interval\":[-19,20]}"},"fields":{"title":"An Affine Jensen Identity on a Qualified Interval","stem":"Use f(t)=a*t+b with coefficients [a,b]=[5, 0] and interval=[-19, 20]. Give (coefficients,interval,ordered nonempty domain?,f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y),qualified convex certificate?). A singleton is allowed; a reversed interval record is unqualified. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.","answer":{"kind":"expression","value":"([5, 0], [-19, 20], Yes, f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y), Yes)","explanation":"Distributivity proves the equality for every pair of domain points and every permitted weight. No finite sampling is used. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([5, 0], [-19, 20], Yes, f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y), Yes)."},"skills":["An Affine Jensen Identity on a Qualified Interval"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r02-convex-function-0044","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"768ae6e8417d7be981e5474cdd29a71d320744284c076a46db8d64ba30e41d1a","family_id":"qf-mod25-convexfn-affine-universal-certificate","seed":202610252044,"semantic_template":"qf-mod25-convexfn-affine-universal-certificate/{\"coefficients\":[6,1],\"interval\":[-20,21]}"},"fields":{"title":"An Affine Jensen Identity on a Qualified Interval","stem":"Use f(t)=a*t+b with coefficients [a,b]=[6, 1] and interval=[-20, 21]. Give (coefficients,interval,ordered nonempty domain?,f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y),qualified convex certificate?). A singleton is allowed; a reversed interval record is unqualified. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.","answer":{"kind":"expression","value":"([6, 1], [-20, 21], Yes, f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y), Yes)","explanation":"Distributivity proves the equality for every pair of domain points and every permitted weight. No finite sampling is used. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([6, 1], [-20, 21], Yes, f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y), Yes)."},"skills":["An Affine Jensen Identity on a Qualified Interval"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r02-convex-function-0045","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ffecf89aa645274c487a062d1f787fc00ddb5b17f5b7940967190f4dcf3f9a79","family_id":"qf-mod25-convexfn-affine-universal-certificate","seed":202610252045,"semantic_template":"qf-mod25-convexfn-affine-universal-certificate/{\"coefficients\":[7,2],\"interval\":[-21,22]}"},"fields":{"title":"An Affine Jensen Identity on a Qualified Interval","stem":"Use f(t)=a*t+b with coefficients [a,b]=[7, 2] and interval=[-21, 22]. Give (coefficients,interval,ordered nonempty domain?,f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y),qualified convex certificate?). A singleton is allowed; a reversed interval record is unqualified. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.","answer":{"kind":"expression","value":"([7, 2], [-21, 22], Yes, f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y), Yes)","explanation":"Distributivity proves the equality for every pair of domain points and every permitted weight. No finite sampling is used. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([7, 2], [-21, 22], Yes, f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y), Yes)."},"skills":["An Affine Jensen Identity on a Qualified Interval"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r02-convex-function-0046","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0ead8ebb0daf872b2238a018304442e22a2459e110eb91f90ebbea7ec26407fb","family_id":"qf-mod25-convexfn-affine-universal-certificate","seed":202610252046,"semantic_template":"qf-mod25-convexfn-affine-universal-certificate/{\"coefficients\":[8,-2],\"interval\":[-22,23]}"},"fields":{"title":"An Affine Jensen Identity on a Qualified Interval","stem":"Use f(t)=a*t+b with coefficients [a,b]=[8, -2] and interval=[-22, 23]. Give (coefficients,interval,ordered nonempty domain?,f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y),qualified convex certificate?). A singleton is allowed; a reversed interval record is unqualified. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.","answer":{"kind":"expression","value":"([8, -2], [-22, 23], Yes, f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y), Yes)","explanation":"Distributivity proves the equality for every pair of domain points and every permitted weight. No finite sampling is used. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([8, -2], [-22, 23], Yes, f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y), Yes)."},"skills":["An Affine Jensen Identity on a Qualified Interval"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r02-convex-function-0047","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5e5c2c6eefb6679d79e9f0841c997752564a4c691100d1d53d1f5f22021e54c1","family_id":"qf-mod25-convexfn-affine-universal-certificate","seed":202610252047,"semantic_template":"qf-mod25-convexfn-affine-universal-certificate/{\"coefficients\":[9,-1],\"interval\":[-23,24]}"},"fields":{"title":"An Affine Jensen Identity on a Qualified Interval","stem":"Use f(t)=a*t+b with coefficients [a,b]=[9, -1] and interval=[-23, 24]. Give (coefficients,interval,ordered nonempty domain?,f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y),qualified convex certificate?). A singleton is allowed; a reversed interval record is unqualified. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.","answer":{"kind":"expression","value":"([9, -1], [-23, 24], Yes, f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y), Yes)","explanation":"Distributivity proves the equality for every pair of domain points and every permitted weight. No finite sampling is used. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([9, -1], [-23, 24], Yes, f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y), Yes)."},"skills":["An Affine Jensen Identity on a Qualified Interval"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r02-convex-function-0048","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2d6265cad05095563f80edccf7a23b1d69e24932660f4b72b11f95ad995eb1df","family_id":"qf-mod25-convexfn-affine-universal-certificate","seed":202610252048,"semantic_template":"qf-mod25-convexfn-affine-universal-certificate/{\"coefficients\":[10,0],\"interval\":[-24,25]}"},"fields":{"title":"An Affine Jensen Identity on a Qualified Interval","stem":"Use f(t)=a*t+b with coefficients [a,b]=[10, 0] and interval=[-24, 25]. Give (coefficients,interval,ordered nonempty domain?,f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y),qualified convex certificate?). A singleton is allowed; a reversed interval record is unqualified. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.","answer":{"kind":"expression","value":"([10, 0], [-24, 25], Yes, f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y), Yes)","explanation":"Distributivity proves the equality for every pair of domain points and every permitted weight. No finite sampling is used. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([10, 0], [-24, 25], Yes, f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y), Yes)."},"skills":["An Affine Jensen Identity on a Qualified Interval"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r02-convex-function-0049","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fd737a1196dd16833933050ea06f57bb163417bcaf085cd75c6d8ceab766e332","family_id":"qf-mod25-convexfn-affine-universal-certificate","seed":202610252049,"semantic_template":"qf-mod25-convexfn-affine-universal-certificate/{\"coefficients\":[11,1],\"interval\":[-25,26]}"},"fields":{"title":"An Affine Jensen Identity on a Qualified Interval","stem":"Use f(t)=a*t+b with coefficients [a,b]=[11, 1] and interval=[-25, 26]. Give (coefficients,interval,ordered nonempty domain?,f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y),qualified convex certificate?). A singleton is allowed; a reversed interval record is unqualified. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.","answer":{"kind":"expression","value":"([11, 1], [-25, 26], Yes, f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y), Yes)","explanation":"Distributivity proves the equality for every pair of domain points and every permitted weight. No finite sampling is used. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([11, 1], [-25, 26], Yes, f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y), Yes)."},"skills":["An Affine Jensen Identity on a Qualified Interval"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r02-convex-function-0050","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"42a0e18297c6389f619431079dd23a96263104a17fcbb57b587c26b4def0f757","family_id":"qf-mod25-convexfn-affine-universal-certificate","seed":202610252050,"semantic_template":"qf-mod25-convexfn-affine-universal-certificate/{\"coefficients\":[12,2],\"interval\":[-26,27]}"},"fields":{"title":"An Affine Jensen Identity on a Qualified Interval","stem":"Use f(t)=a*t+b with coefficients [a,b]=[12, 2] and interval=[-26, 27]. Give (coefficients,interval,ordered nonempty domain?,f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y),qualified convex certificate?). A singleton is allowed; a reversed interval record is unqualified. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.","answer":{"kind":"expression","value":"([12, 2], [-26, 27], Yes, f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y), Yes)","explanation":"Distributivity proves the equality for every pair of domain points and every permitted weight. No finite sampling is used. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([12, 2], [-26, 27], Yes, f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y), Yes)."},"skills":["An Affine Jensen Identity on a Qualified Interval"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r02-convex-function-0051","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ab43b3b09c5ee3a81c8178569619bd44b26a75311014619989558ab72443d487","family_id":"qf-mod25-convexfn-quadratic-universal-certificate","seed":202610252051,"semantic_template":"qf-mod25-convexfn-quadratic-universal-certificate/{\"coefficients\":[-1,-12,-3],\"interval\":[-2,3]}"},"fields":{"title":"A Nondegenerate-Interval Quadratic Jensen-Gap Certificate","stem":"Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[-1, -12, -3], not ascending-power order. The declared interval [lower,upper]=[-2, 3] belongs to the function data. Give (coefficients,interval,lower<upper?,leading coefficient,rhs-lhs=a*lambda*(1-lambda)*(x-y)^2,qualified convex certificate?). This family requires a nondegenerate interval and a≥0. A No for a singleton or reversed interval means this certificate is unqualified, not that the function on a singleton is nonconvex. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.","answer":{"kind":"expression","value":"([-1, -12, -3], [-2, 3], Yes, -1, rhs-lhs=a*lambda*(1-lambda)*(x-y)^2, No)","explanation":"Expand the Jensen gap symbolically. On a nondegenerate interval its sign is controlled by a, since lambda*(1−lambda)*(x−y)² is nonnegative. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-1, -12, -3], [-2, 3], Yes, -1, rhs-lhs=a*lambda*(1-lambda)*(x-y)^2, No)."},"skills":["A Nondegenerate-Interval Quadratic Jensen-Gap Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r02-convex-function-0052","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"50d4a403e634a61d6678717e0cdda7b81b732756b4b7126762e2f4c1ad562971","family_id":"qf-mod25-convexfn-quadratic-universal-certificate","seed":202610252052,"semantic_template":"qf-mod25-convexfn-quadratic-universal-certificate/{\"coefficients\":[1,-11,-2],\"interval\":[-3,4]}"},"fields":{"title":"A Nondegenerate-Interval Quadratic Jensen-Gap Certificate","stem":"Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[1, -11, -2], not ascending-power order. The declared interval [lower,upper]=[-3, 4] belongs to the function data. Give (coefficients,interval,lower<upper?,leading coefficient,rhs-lhs=a*lambda*(1-lambda)*(x-y)^2,qualified convex certificate?). This family requires a nondegenerate interval and a≥0. A No for a singleton or reversed interval means this certificate is unqualified, not that the function on a singleton is nonconvex. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.","answer":{"kind":"expression","value":"([1, -11, -2], [-3, 4], Yes, 1, rhs-lhs=a*lambda*(1-lambda)*(x-y)^2, Yes)","explanation":"Expand the Jensen gap symbolically. On a nondegenerate interval its sign is controlled by a, since lambda*(1−lambda)*(x−y)² is nonnegative. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([1, -11, -2], [-3, 4], Yes, 1, rhs-lhs=a*lambda*(1-lambda)*(x-y)^2, Yes)."},"skills":["A Nondegenerate-Interval Quadratic Jensen-Gap Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r02-convex-function-0053","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"63cb5934d379ca294f874f696be12719d2a363bf528222d6ed02acab94b8b07d","family_id":"qf-mod25-convexfn-quadratic-universal-certificate","seed":202610252053,"semantic_template":"qf-mod25-convexfn-quadratic-universal-certificate/{\"coefficients\":[2,-10,-1],\"interval\":[-4,5]}"},"fields":{"title":"A Nondegenerate-Interval Quadratic Jensen-Gap Certificate","stem":"Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[2, -10, -1], not ascending-power order. The declared interval [lower,upper]=[-4, 5] belongs to the function data. Give (coefficients,interval,lower<upper?,leading coefficient,rhs-lhs=a*lambda*(1-lambda)*(x-y)^2,qualified convex certificate?). This family requires a nondegenerate interval and a≥0. A No for a singleton or reversed interval means this certificate is unqualified, not that the function on a singleton is nonconvex. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.","answer":{"kind":"expression","value":"([2, -10, -1], [-4, 5], Yes, 2, rhs-lhs=a*lambda*(1-lambda)*(x-y)^2, Yes)","explanation":"Expand the Jensen gap symbolically. On a nondegenerate interval its sign is controlled by a, since lambda*(1−lambda)*(x−y)² is nonnegative. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([2, -10, -1], [-4, 5], Yes, 2, rhs-lhs=a*lambda*(1-lambda)*(x-y)^2, Yes)."},"skills":["A Nondegenerate-Interval Quadratic Jensen-Gap Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r02-convex-function-0054","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c99e6a0b8f227483a2f8dfb66159e4648df7830b5087c211c5e059d7058afc2f","family_id":"qf-mod25-convexfn-quadratic-universal-certificate","seed":202610252054,"semantic_template":"qf-mod25-convexfn-quadratic-universal-certificate/{\"coefficients\":[3,-9,0],\"interval\":[-5,6]}"},"fields":{"title":"A Nondegenerate-Interval Quadratic Jensen-Gap Certificate","stem":"Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[3, -9, 0], not ascending-power order. The declared interval [lower,upper]=[-5, 6] belongs to the function data. Give (coefficients,interval,lower<upper?,leading coefficient,rhs-lhs=a*lambda*(1-lambda)*(x-y)^2,qualified convex certificate?). This family requires a nondegenerate interval and a≥0. A No for a singleton or reversed interval means this certificate is unqualified, not that the function on a singleton is nonconvex. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.","answer":{"kind":"expression","value":"([3, -9, 0], [-5, 6], Yes, 3, rhs-lhs=a*lambda*(1-lambda)*(x-y)^2, Yes)","explanation":"Expand the Jensen gap symbolically. On a nondegenerate interval its sign is controlled by a, since lambda*(1−lambda)*(x−y)² is nonnegative. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([3, -9, 0], [-5, 6], Yes, 3, rhs-lhs=a*lambda*(1-lambda)*(x-y)^2, Yes)."},"skills":["A Nondegenerate-Interval Quadratic Jensen-Gap Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r02-convex-function-0055","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d87be818d768a887a3191990b0cb979caed1f816e181619869babebb7d9477e5","family_id":"qf-mod25-convexfn-quadratic-universal-certificate","seed":202610252055,"semantic_template":"qf-mod25-convexfn-quadratic-universal-certificate/{\"coefficients\":[0,-8,1],\"interval\":[-6,7]}"},"fields":{"title":"A Nondegenerate-Interval Quadratic Jensen-Gap Certificate","stem":"Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[0, -8, 1], not ascending-power order. The declared interval [lower,upper]=[-6, 7] belongs to the function data. Give (coefficients,interval,lower<upper?,leading coefficient,rhs-lhs=a*lambda*(1-lambda)*(x-y)^2,qualified convex certificate?). This family requires a nondegenerate interval and a≥0. A No for a singleton or reversed interval means this certificate is unqualified, not that the function on a singleton is nonconvex. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.","answer":{"kind":"expression","value":"([0, -8, 1], [-6, 7], Yes, 0, rhs-lhs=a*lambda*(1-lambda)*(x-y)^2, Yes)","explanation":"Expand the Jensen gap symbolically. On a nondegenerate interval its sign is controlled by a, since lambda*(1−lambda)*(x−y)² is nonnegative. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([0, -8, 1], [-6, 7], Yes, 0, rhs-lhs=a*lambda*(1-lambda)*(x-y)^2, Yes)."},"skills":["A Nondegenerate-Interval Quadratic Jensen-Gap Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r02-convex-function-0056","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1b11274cee42ec56d74c08282f005295e6713bd1da3e325d0461b7604359135c","family_id":"qf-mod25-convexfn-quadratic-universal-certificate","seed":202610252056,"semantic_template":"qf-mod25-convexfn-quadratic-universal-certificate/{\"coefficients\":[-1,-7,2],\"interval\":[-7,8]}"},"fields":{"title":"A Nondegenerate-Interval Quadratic Jensen-Gap Certificate","stem":"Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[-1, -7, 2], not ascending-power order. The declared interval [lower,upper]=[-7, 8] belongs to the function data. Give (coefficients,interval,lower<upper?,leading coefficient,rhs-lhs=a*lambda*(1-lambda)*(x-y)^2,qualified convex certificate?). This family requires a nondegenerate interval and a≥0. A No for a singleton or reversed interval means this certificate is unqualified, not that the function on a singleton is nonconvex. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.","answer":{"kind":"expression","value":"([-1, -7, 2], [-7, 8], Yes, -1, rhs-lhs=a*lambda*(1-lambda)*(x-y)^2, No)","explanation":"Expand the Jensen gap symbolically. On a nondegenerate interval its sign is controlled by a, since lambda*(1−lambda)*(x−y)² is nonnegative. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-1, -7, 2], [-7, 8], Yes, -1, rhs-lhs=a*lambda*(1-lambda)*(x-y)^2, No)."},"skills":["A Nondegenerate-Interval Quadratic Jensen-Gap Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r02-convex-function-0057","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7638e2f24316bf8d72351066899f1d338b30517f9d45313c7a2d8dfef7293380","family_id":"qf-mod25-convexfn-quadratic-universal-certificate","seed":202610252057,"semantic_template":"qf-mod25-convexfn-quadratic-universal-certificate/{\"coefficients\":[2,-6,3],\"interval\":[-8,9]}"},"fields":{"title":"A Nondegenerate-Interval Quadratic Jensen-Gap Certificate","stem":"Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[2, -6, 3], not ascending-power order. The declared interval [lower,upper]=[-8, 9] belongs to the function data. Give (coefficients,interval,lower<upper?,leading coefficient,rhs-lhs=a*lambda*(1-lambda)*(x-y)^2,qualified convex certificate?). This family requires a nondegenerate interval and a≥0. A No for a singleton or reversed interval means this certificate is unqualified, not that the function on a singleton is nonconvex. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.","answer":{"kind":"expression","value":"([2, -6, 3], [-8, 9], Yes, 2, rhs-lhs=a*lambda*(1-lambda)*(x-y)^2, Yes)","explanation":"Expand the Jensen gap symbolically. On a nondegenerate interval its sign is controlled by a, since lambda*(1−lambda)*(x−y)² is nonnegative. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([2, -6, 3], [-8, 9], Yes, 2, rhs-lhs=a*lambda*(1-lambda)*(x-y)^2, Yes)."},"skills":["A Nondegenerate-Interval Quadratic Jensen-Gap Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r02-convex-function-0058","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d8213752cccce2ebe7b547c63857b7e5ae8480465025276922c264dd5c13f4e5","family_id":"qf-mod25-convexfn-quadratic-universal-certificate","seed":202610252058,"semantic_template":"qf-mod25-convexfn-quadratic-universal-certificate/{\"coefficients\":[3,-5,-3],\"interval\":[-9,10]}"},"fields":{"title":"A Nondegenerate-Interval Quadratic Jensen-Gap Certificate","stem":"Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[3, -5, -3], not ascending-power order. The declared interval [lower,upper]=[-9, 10] belongs to the function data. Give (coefficients,interval,lower<upper?,leading coefficient,rhs-lhs=a*lambda*(1-lambda)*(x-y)^2,qualified convex certificate?). This family requires a nondegenerate interval and a≥0. A No for a singleton or reversed interval means this certificate is unqualified, not that the function on a singleton is nonconvex. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.","answer":{"kind":"expression","value":"([3, -5, -3], [-9, 10], Yes, 3, rhs-lhs=a*lambda*(1-lambda)*(x-y)^2, Yes)","explanation":"Expand the Jensen gap symbolically. On a nondegenerate interval its sign is controlled by a, since lambda*(1−lambda)*(x−y)² is nonnegative. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([3, -5, -3], [-9, 10], Yes, 3, rhs-lhs=a*lambda*(1-lambda)*(x-y)^2, Yes)."},"skills":["A Nondegenerate-Interval Quadratic Jensen-Gap Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r02-convex-function-0059","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"471973bd24b29ddf843066883921fe3c7034386c9880f257b3ce04aa087c6423","family_id":"qf-mod25-convexfn-quadratic-universal-certificate","seed":202610252059,"semantic_template":"qf-mod25-convexfn-quadratic-universal-certificate/{\"coefficients\":[0,-4,-2],\"interval\":[-10,11]}"},"fields":{"title":"A Nondegenerate-Interval Quadratic Jensen-Gap Certificate","stem":"Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[0, -4, -2], not ascending-power order. The declared interval [lower,upper]=[-10, 11] belongs to the function data. Give (coefficients,interval,lower<upper?,leading coefficient,rhs-lhs=a*lambda*(1-lambda)*(x-y)^2,qualified convex certificate?). This family requires a nondegenerate interval and a≥0. A No for a singleton or reversed interval means this certificate is unqualified, not that the function on a singleton is nonconvex. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.","answer":{"kind":"expression","value":"([0, -4, -2], [-10, 11], Yes, 0, rhs-lhs=a*lambda*(1-lambda)*(x-y)^2, Yes)","explanation":"Expand the Jensen gap symbolically. On a nondegenerate interval its sign is controlled by a, since lambda*(1−lambda)*(x−y)² is nonnegative. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([0, -4, -2], [-10, 11], Yes, 0, rhs-lhs=a*lambda*(1-lambda)*(x-y)^2, Yes)."},"skills":["A Nondegenerate-Interval Quadratic Jensen-Gap Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r02-convex-function-0060","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fcac3110a516717f052ec9bbc8484efdf79fc1ae21d5e8e467577de6541e8b4d","family_id":"qf-mod25-convexfn-quadratic-universal-certificate","seed":202610252060,"semantic_template":"qf-mod25-convexfn-quadratic-universal-certificate/{\"coefficients\":[1,-3,-1],\"interval\":[-11,12]}"},"fields":{"title":"A Nondegenerate-Interval Quadratic Jensen-Gap Certificate","stem":"Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[1, -3, -1], not ascending-power order. The declared interval [lower,upper]=[-11, 12] belongs to the function data. Give (coefficients,interval,lower<upper?,leading coefficient,rhs-lhs=a*lambda*(1-lambda)*(x-y)^2,qualified convex certificate?). This family requires a nondegenerate interval and a≥0. A No for a singleton or reversed interval means this certificate is unqualified, not that the function on a singleton is nonconvex. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.","answer":{"kind":"expression","value":"([1, -3, -1], [-11, 12], Yes, 1, rhs-lhs=a*lambda*(1-lambda)*(x-y)^2, Yes)","explanation":"Expand the Jensen gap symbolically. On a nondegenerate interval its sign is controlled by a, since lambda*(1−lambda)*(x−y)² is nonnegative. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([1, -3, -1], [-11, 12], Yes, 1, rhs-lhs=a*lambda*(1-lambda)*(x-y)^2, Yes)."},"skills":["A Nondegenerate-Interval Quadratic Jensen-Gap Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r02-convex-function-0061","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8187cc28daedabc21e6b4b00c4b8dbf09531ac0755d6d302e9b59a817aa8fd33","family_id":"qf-mod25-convexfn-quadratic-universal-certificate","seed":202610252061,"semantic_template":"qf-mod25-convexfn-quadratic-universal-certificate/{\"coefficients\":[-1,-2,0],\"interval\":[-12,13]}"},"fields":{"title":"A Nondegenerate-Interval Quadratic Jensen-Gap Certificate","stem":"Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[-1, -2, 0], not ascending-power order. The declared interval [lower,upper]=[-12, 13] belongs to the function data. Give (coefficients,interval,lower<upper?,leading coefficient,rhs-lhs=a*lambda*(1-lambda)*(x-y)^2,qualified convex certificate?). This family requires a nondegenerate interval and a≥0. A No for a singleton or reversed interval means this certificate is unqualified, not that the function on a singleton is nonconvex. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.","answer":{"kind":"expression","value":"([-1, -2, 0], [-12, 13], Yes, -1, rhs-lhs=a*lambda*(1-lambda)*(x-y)^2, No)","explanation":"Expand the Jensen gap symbolically. On a nondegenerate interval its sign is controlled by a, since lambda*(1−lambda)*(x−y)² is nonnegative. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-1, -2, 0], [-12, 13], Yes, -1, rhs-lhs=a*lambda*(1-lambda)*(x-y)^2, No)."},"skills":["A Nondegenerate-Interval Quadratic Jensen-Gap Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r02-convex-function-0062","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3ef59fb95cd4fac02edd7aaac54c87149c4b25a151cb43fa954c64b21a96ad0b","family_id":"qf-mod25-convexfn-quadratic-universal-certificate","seed":202610252062,"semantic_template":"qf-mod25-convexfn-quadratic-universal-certificate/{\"coefficients\":[3,-1,1],\"interval\":[-13,14]}"},"fields":{"title":"A Nondegenerate-Interval Quadratic Jensen-Gap Certificate","stem":"Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[3, -1, 1], not ascending-power order. The declared interval [lower,upper]=[-13, 14] belongs to the function data. Give (coefficients,interval,lower<upper?,leading coefficient,rhs-lhs=a*lambda*(1-lambda)*(x-y)^2,qualified convex certificate?). This family requires a nondegenerate interval and a≥0. A No for a singleton or reversed interval means this certificate is unqualified, not that the function on a singleton is nonconvex. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.","answer":{"kind":"expression","value":"([3, -1, 1], [-13, 14], Yes, 3, rhs-lhs=a*lambda*(1-lambda)*(x-y)^2, Yes)","explanation":"Expand the Jensen gap symbolically. On a nondegenerate interval its sign is controlled by a, since lambda*(1−lambda)*(x−y)² is nonnegative. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([3, -1, 1], [-13, 14], Yes, 3, rhs-lhs=a*lambda*(1-lambda)*(x-y)^2, Yes)."},"skills":["A Nondegenerate-Interval Quadratic Jensen-Gap Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r02-convex-function-0063","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4d3dc71a0175e28752e9ce7bc6756fb877c052d1d796ae6d40aeacb65e16e45a","family_id":"qf-mod25-convexfn-quadratic-universal-certificate","seed":202610252063,"semantic_template":"qf-mod25-convexfn-quadratic-universal-certificate/{\"coefficients\":[0,0,2],\"interval\":[-14,15]}"},"fields":{"title":"A Nondegenerate-Interval Quadratic Jensen-Gap Certificate","stem":"Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[0, 0, 2], not ascending-power order. The declared interval [lower,upper]=[-14, 15] belongs to the function data. Give (coefficients,interval,lower<upper?,leading coefficient,rhs-lhs=a*lambda*(1-lambda)*(x-y)^2,qualified convex certificate?). This family requires a nondegenerate interval and a≥0. A No for a singleton or reversed interval means this certificate is unqualified, not that the function on a singleton is nonconvex. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.","answer":{"kind":"expression","value":"([0, 0, 2], [-14, 15], Yes, 0, rhs-lhs=a*lambda*(1-lambda)*(x-y)^2, Yes)","explanation":"Expand the Jensen gap symbolically. On a nondegenerate interval its sign is controlled by a, since lambda*(1−lambda)*(x−y)² is nonnegative. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([0, 0, 2], [-14, 15], Yes, 0, rhs-lhs=a*lambda*(1-lambda)*(x-y)^2, Yes)."},"skills":["A Nondegenerate-Interval Quadratic Jensen-Gap Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r02-convex-function-0064","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"34b8b400a92a8306efd40f8cf5429e5e03975103e7b155fc5953931c10bbc157","family_id":"qf-mod25-convexfn-quadratic-universal-certificate","seed":202610252064,"semantic_template":"qf-mod25-convexfn-quadratic-universal-certificate/{\"coefficients\":[1,1,3],\"interval\":[-15,16]}"},"fields":{"title":"A Nondegenerate-Interval Quadratic Jensen-Gap Certificate","stem":"Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[1, 1, 3], not ascending-power order. The declared interval [lower,upper]=[-15, 16] belongs to the function data. Give (coefficients,interval,lower<upper?,leading coefficient,rhs-lhs=a*lambda*(1-lambda)*(x-y)^2,qualified convex certificate?). This family requires a nondegenerate interval and a≥0. A No for a singleton or reversed interval means this certificate is unqualified, not that the function on a singleton is nonconvex. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.","answer":{"kind":"expression","value":"([1, 1, 3], [-15, 16], Yes, 1, rhs-lhs=a*lambda*(1-lambda)*(x-y)^2, Yes)","explanation":"Expand the Jensen gap symbolically. On a nondegenerate interval its sign is controlled by a, since lambda*(1−lambda)*(x−y)² is nonnegative. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([1, 1, 3], [-15, 16], Yes, 1, rhs-lhs=a*lambda*(1-lambda)*(x-y)^2, Yes)."},"skills":["A Nondegenerate-Interval Quadratic Jensen-Gap Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r02-convex-function-0065","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4ced87881ecaa0129ba71575b69bbd5462512d622625f5b4d4c39d7f67b83bc7","family_id":"qf-mod25-convexfn-quadratic-universal-certificate","seed":202610252065,"semantic_template":"qf-mod25-convexfn-quadratic-universal-certificate/{\"coefficients\":[2,2,-3],\"interval\":[-16,17]}"},"fields":{"title":"A Nondegenerate-Interval Quadratic Jensen-Gap Certificate","stem":"Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[2, 2, -3], not ascending-power order. The declared interval [lower,upper]=[-16, 17] belongs to the function data. Give (coefficients,interval,lower<upper?,leading coefficient,rhs-lhs=a*lambda*(1-lambda)*(x-y)^2,qualified convex certificate?). This family requires a nondegenerate interval and a≥0. A No for a singleton or reversed interval means this certificate is unqualified, not that the function on a singleton is nonconvex. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.","answer":{"kind":"expression","value":"([2, 2, -3], [-16, 17], Yes, 2, rhs-lhs=a*lambda*(1-lambda)*(x-y)^2, Yes)","explanation":"Expand the Jensen gap symbolically. On a nondegenerate interval its sign is controlled by a, since lambda*(1−lambda)*(x−y)² is nonnegative. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([2, 2, -3], [-16, 17], Yes, 2, rhs-lhs=a*lambda*(1-lambda)*(x-y)^2, Yes)."},"skills":["A Nondegenerate-Interval Quadratic Jensen-Gap Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r02-convex-function-0066","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1056b428d068f7efb78cf753a18c83dd1b4d681047d6545ef6c1ae7fd48981e6","family_id":"qf-mod25-convexfn-quadratic-universal-certificate","seed":202610252066,"semantic_template":"qf-mod25-convexfn-quadratic-universal-certificate/{\"coefficients\":[-1,3,-2],\"interval\":[-17,18]}"},"fields":{"title":"A Nondegenerate-Interval Quadratic Jensen-Gap Certificate","stem":"Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[-1, 3, -2], not ascending-power order. The declared interval [lower,upper]=[-17, 18] belongs to the function data. Give (coefficients,interval,lower<upper?,leading coefficient,rhs-lhs=a*lambda*(1-lambda)*(x-y)^2,qualified convex certificate?). This family requires a nondegenerate interval and a≥0. A No for a singleton or reversed interval means this certificate is unqualified, not that the function on a singleton is nonconvex. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.","answer":{"kind":"expression","value":"([-1, 3, -2], [-17, 18], Yes, -1, rhs-lhs=a*lambda*(1-lambda)*(x-y)^2, No)","explanation":"Expand the Jensen gap symbolically. On a nondegenerate interval its sign is controlled by a, since lambda*(1−lambda)*(x−y)² is nonnegative. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-1, 3, -2], [-17, 18], Yes, -1, rhs-lhs=a*lambda*(1-lambda)*(x-y)^2, No)."},"skills":["A Nondegenerate-Interval Quadratic Jensen-Gap Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r02-convex-function-0067","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c783b3987008061e0b9b57ba09e8d29b0d96838487bd1fbe39ed7876ab91b822","family_id":"qf-mod25-convexfn-quadratic-universal-certificate","seed":202610252067,"semantic_template":"qf-mod25-convexfn-quadratic-universal-certificate/{\"coefficients\":[0,4,-1],\"interval\":[-18,19]}"},"fields":{"title":"A Nondegenerate-Interval Quadratic Jensen-Gap Certificate","stem":"Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[0, 4, -1], not ascending-power order. The declared interval [lower,upper]=[-18, 19] belongs to the function data. Give (coefficients,interval,lower<upper?,leading coefficient,rhs-lhs=a*lambda*(1-lambda)*(x-y)^2,qualified convex certificate?). This family requires a nondegenerate interval and a≥0. A No for a singleton or reversed interval means this certificate is unqualified, not that the function on a singleton is nonconvex. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.","answer":{"kind":"expression","value":"([0, 4, -1], [-18, 19], Yes, 0, rhs-lhs=a*lambda*(1-lambda)*(x-y)^2, Yes)","explanation":"Expand the Jensen gap symbolically. On a nondegenerate interval its sign is controlled by a, since lambda*(1−lambda)*(x−y)² is nonnegative. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([0, 4, -1], [-18, 19], Yes, 0, rhs-lhs=a*lambda*(1-lambda)*(x-y)^2, Yes)."},"skills":["A Nondegenerate-Interval Quadratic Jensen-Gap Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r02-convex-function-0068","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"46f96405afaddbb21d3bcd8b0b2d2e27a5b595d702898a105071c1b42efc6c55","family_id":"qf-mod25-convexfn-quadratic-universal-certificate","seed":202610252068,"semantic_template":"qf-mod25-convexfn-quadratic-universal-certificate/{\"coefficients\":[1,5,0],\"interval\":[-19,20]}"},"fields":{"title":"A Nondegenerate-Interval Quadratic Jensen-Gap Certificate","stem":"Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[1, 5, 0], not ascending-power order. The declared interval [lower,upper]=[-19, 20] belongs to the function data. Give (coefficients,interval,lower<upper?,leading coefficient,rhs-lhs=a*lambda*(1-lambda)*(x-y)^2,qualified convex certificate?). This family requires a nondegenerate interval and a≥0. A No for a singleton or reversed interval means this certificate is unqualified, not that the function on a singleton is nonconvex. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.","answer":{"kind":"expression","value":"([1, 5, 0], [-19, 20], Yes, 1, rhs-lhs=a*lambda*(1-lambda)*(x-y)^2, Yes)","explanation":"Expand the Jensen gap symbolically. On a nondegenerate interval its sign is controlled by a, since lambda*(1−lambda)*(x−y)² is nonnegative. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([1, 5, 0], [-19, 20], Yes, 1, rhs-lhs=a*lambda*(1-lambda)*(x-y)^2, Yes)."},"skills":["A Nondegenerate-Interval Quadratic Jensen-Gap Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r02-convex-function-0069","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3720e71a00bb0eb2eef6a81c40f535363800d1215731885ba6d2e3914380b5fa","family_id":"qf-mod25-convexfn-quadratic-universal-certificate","seed":202610252069,"semantic_template":"qf-mod25-convexfn-quadratic-universal-certificate/{\"coefficients\":[2,6,1],\"interval\":[-20,21]}"},"fields":{"title":"A Nondegenerate-Interval Quadratic Jensen-Gap Certificate","stem":"Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[2, 6, 1], not ascending-power order. The declared interval [lower,upper]=[-20, 21] belongs to the function data. Give (coefficients,interval,lower<upper?,leading coefficient,rhs-lhs=a*lambda*(1-lambda)*(x-y)^2,qualified convex certificate?). This family requires a nondegenerate interval and a≥0. A No for a singleton or reversed interval means this certificate is unqualified, not that the function on a singleton is nonconvex. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.","answer":{"kind":"expression","value":"([2, 6, 1], [-20, 21], Yes, 2, rhs-lhs=a*lambda*(1-lambda)*(x-y)^2, Yes)","explanation":"Expand the Jensen gap symbolically. On a nondegenerate interval its sign is controlled by a, since lambda*(1−lambda)*(x−y)² is nonnegative. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([2, 6, 1], [-20, 21], Yes, 2, rhs-lhs=a*lambda*(1-lambda)*(x-y)^2, Yes)."},"skills":["A Nondegenerate-Interval Quadratic Jensen-Gap Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r02-convex-function-0070","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"406ed2c2d86fbfe1373e40d23df85c766f09e5edacb1945eb9a7a84a9bdaa7fe","family_id":"qf-mod25-convexfn-quadratic-universal-certificate","seed":202610252070,"semantic_template":"qf-mod25-convexfn-quadratic-universal-certificate/{\"coefficients\":[3,7,2],\"interval\":[-21,22]}"},"fields":{"title":"A Nondegenerate-Interval Quadratic Jensen-Gap Certificate","stem":"Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[3, 7, 2], not ascending-power order. The declared interval [lower,upper]=[-21, 22] belongs to the function data. Give (coefficients,interval,lower<upper?,leading coefficient,rhs-lhs=a*lambda*(1-lambda)*(x-y)^2,qualified convex certificate?). This family requires a nondegenerate interval and a≥0. A No for a singleton or reversed interval means this certificate is unqualified, not that the function on a singleton is nonconvex. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.","answer":{"kind":"expression","value":"([3, 7, 2], [-21, 22], Yes, 3, rhs-lhs=a*lambda*(1-lambda)*(x-y)^2, Yes)","explanation":"Expand the Jensen gap symbolically. On a nondegenerate interval its sign is controlled by a, since lambda*(1−lambda)*(x−y)² is nonnegative. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([3, 7, 2], [-21, 22], Yes, 3, rhs-lhs=a*lambda*(1-lambda)*(x-y)^2, Yes)."},"skills":["A Nondegenerate-Interval Quadratic Jensen-Gap Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r02-convex-function-0071","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2e0234432693a47a363717042450827b341a642c27bd76f511e999b73ee843f2","family_id":"qf-mod25-convexfn-quadratic-universal-certificate","seed":202610252071,"semantic_template":"qf-mod25-convexfn-quadratic-universal-certificate/{\"coefficients\":[-1,8,3],\"interval\":[-22,23]}"},"fields":{"title":"A Nondegenerate-Interval Quadratic Jensen-Gap Certificate","stem":"Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[-1, 8, 3], not ascending-power order. The declared interval [lower,upper]=[-22, 23] belongs to the function data. Give (coefficients,interval,lower<upper?,leading coefficient,rhs-lhs=a*lambda*(1-lambda)*(x-y)^2,qualified convex certificate?). This family requires a nondegenerate interval and a≥0. A No for a singleton or reversed interval means this certificate is unqualified, not that the function on a singleton is nonconvex. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.","answer":{"kind":"expression","value":"([-1, 8, 3], [-22, 23], Yes, -1, rhs-lhs=a*lambda*(1-lambda)*(x-y)^2, No)","explanation":"Expand the Jensen gap symbolically. On a nondegenerate interval its sign is controlled by a, since lambda*(1−lambda)*(x−y)² is nonnegative. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-1, 8, 3], [-22, 23], Yes, -1, rhs-lhs=a*lambda*(1-lambda)*(x-y)^2, No)."},"skills":["A Nondegenerate-Interval Quadratic Jensen-Gap Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r02-convex-function-0072","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"24248015a0875c6b33932d60e89f383ff3a34492f9fe792c7219aa593b89cbc0","family_id":"qf-mod25-convexfn-quadratic-universal-certificate","seed":202610252072,"semantic_template":"qf-mod25-convexfn-quadratic-universal-certificate/{\"coefficients\":[1,9,-3],\"interval\":[-23,24]}"},"fields":{"title":"A Nondegenerate-Interval Quadratic Jensen-Gap Certificate","stem":"Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[1, 9, -3], not ascending-power order. The declared interval [lower,upper]=[-23, 24] belongs to the function data. Give (coefficients,interval,lower<upper?,leading coefficient,rhs-lhs=a*lambda*(1-lambda)*(x-y)^2,qualified convex certificate?). This family requires a nondegenerate interval and a≥0. A No for a singleton or reversed interval means this certificate is unqualified, not that the function on a singleton is nonconvex. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.","answer":{"kind":"expression","value":"([1, 9, -3], [-23, 24], Yes, 1, rhs-lhs=a*lambda*(1-lambda)*(x-y)^2, Yes)","explanation":"Expand the Jensen gap symbolically. On a nondegenerate interval its sign is controlled by a, since lambda*(1−lambda)*(x−y)² is nonnegative. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([1, 9, -3], [-23, 24], Yes, 1, rhs-lhs=a*lambda*(1-lambda)*(x-y)^2, Yes)."},"skills":["A Nondegenerate-Interval Quadratic Jensen-Gap Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r02-convex-function-0073","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3a94518de73a150a8027a24938ee2a0514e6a521f887ce49e5a312232a28f632","family_id":"qf-mod25-convexfn-quadratic-universal-certificate","seed":202610252073,"semantic_template":"qf-mod25-convexfn-quadratic-universal-certificate/{\"coefficients\":[2,10,-2],\"interval\":[-24,25]}"},"fields":{"title":"A Nondegenerate-Interval Quadratic Jensen-Gap Certificate","stem":"Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[2, 10, -2], not ascending-power order. The declared interval [lower,upper]=[-24, 25] belongs to the function data. Give (coefficients,interval,lower<upper?,leading coefficient,rhs-lhs=a*lambda*(1-lambda)*(x-y)^2,qualified convex certificate?). This family requires a nondegenerate interval and a≥0. A No for a singleton or reversed interval means this certificate is unqualified, not that the function on a singleton is nonconvex. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.","answer":{"kind":"expression","value":"([2, 10, -2], [-24, 25], Yes, 2, rhs-lhs=a*lambda*(1-lambda)*(x-y)^2, Yes)","explanation":"Expand the Jensen gap symbolically. On a nondegenerate interval its sign is controlled by a, since lambda*(1−lambda)*(x−y)² is nonnegative. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([2, 10, -2], [-24, 25], Yes, 2, rhs-lhs=a*lambda*(1-lambda)*(x-y)^2, Yes)."},"skills":["A Nondegenerate-Interval Quadratic Jensen-Gap Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r02-convex-function-0074","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ca95eee2d7a8a3ef55e8d5149a836cedbf1c3b5334153c9f5f5c5bf4f08bb6f7","family_id":"qf-mod25-convexfn-quadratic-universal-certificate","seed":202610252074,"semantic_template":"qf-mod25-convexfn-quadratic-universal-certificate/{\"coefficients\":[3,11,-1],\"interval\":[-25,26]}"},"fields":{"title":"A Nondegenerate-Interval Quadratic Jensen-Gap Certificate","stem":"Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[3, 11, -1], not ascending-power order. The declared interval [lower,upper]=[-25, 26] belongs to the function data. Give (coefficients,interval,lower<upper?,leading coefficient,rhs-lhs=a*lambda*(1-lambda)*(x-y)^2,qualified convex certificate?). This family requires a nondegenerate interval and a≥0. A No for a singleton or reversed interval means this certificate is unqualified, not that the function on a singleton is nonconvex. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.","answer":{"kind":"expression","value":"([3, 11, -1], [-25, 26], Yes, 3, rhs-lhs=a*lambda*(1-lambda)*(x-y)^2, Yes)","explanation":"Expand the Jensen gap symbolically. On a nondegenerate interval its sign is controlled by a, since lambda*(1−lambda)*(x−y)² is nonnegative. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([3, 11, -1], [-25, 26], Yes, 3, rhs-lhs=a*lambda*(1-lambda)*(x-y)^2, Yes)."},"skills":["A Nondegenerate-Interval Quadratic Jensen-Gap Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r02-convex-function-0075","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e4804198ce58023c70df9eaa83a92512ef828a91e3205a921f276c23c0007b58","family_id":"qf-mod25-convexfn-quadratic-universal-certificate","seed":202610252075,"semantic_template":"qf-mod25-convexfn-quadratic-universal-certificate/{\"coefficients\":[0,12,0],\"interval\":[-26,27]}"},"fields":{"title":"A Nondegenerate-Interval Quadratic Jensen-Gap Certificate","stem":"Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[0, 12, 0], not ascending-power order. The declared interval [lower,upper]=[-26, 27] belongs to the function data. Give (coefficients,interval,lower<upper?,leading coefficient,rhs-lhs=a*lambda*(1-lambda)*(x-y)^2,qualified convex certificate?). This family requires a nondegenerate interval and a≥0. A No for a singleton or reversed interval means this certificate is unqualified, not that the function on a singleton is nonconvex. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.","answer":{"kind":"expression","value":"([0, 12, 0], [-26, 27], Yes, 0, rhs-lhs=a*lambda*(1-lambda)*(x-y)^2, Yes)","explanation":"Expand the Jensen gap symbolically. On a nondegenerate interval its sign is controlled by a, since lambda*(1−lambda)*(x−y)² is nonnegative. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([0, 12, 0], [-26, 27], Yes, 0, rhs-lhs=a*lambda*(1-lambda)*(x-y)^2, Yes)."},"skills":["A Nondegenerate-Interval Quadratic Jensen-Gap Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r02-convex-function-0076","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"25c3cad6e941a8ef0a5fc5cd095d7fd661be8d6ce63fe3e0e08c9251236ebb4d","family_id":"qf-mod25-convexfn-jensen-instance-audit","seed":202610252076,"semantic_template":"qf-mod25-convexfn-jensen-instance-audit/{\"coefficients\":[-1,-12,-3],\"interval\":[-5,5],\"lambda\":[1,3],\"x\":[-2,1],\"y\":[-3,1]}"},"fields":{"title":"One Exact Jensen Instance: a Counterexample or Only an Example","stem":"Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[-1, -12, -3], not ascending-power order. The declared interval [lower,upper]=[-5, 5] belongs to the function data. x=-2/1, y=-3/1, lambda=1/3 must lie in the declared domain and closed unit weight interval. Give (coefficients,interval,x,y,lambda,weighted input,lhs=f(weighted input),rhs=weighted endpoint values,rhs−lhs,inequality passes?,nonconvexity_witness if the gap is negative, otherwise single_instance_only). Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.","answer":{"kind":"expression","value":"([-1, -12, -3], [-5, 5], -2, -3, 1/3, -8/3, 197/9, 65/3, -2/9, No, nonconvexity_witness)","explanation":"Compute both sides exactly. A negative gap refutes convexity on the declared domain; a nonnegative gap proves only this one instance. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-1, -12, -3], [-5, 5], -2, -3, 1/3, -8/3, 197/9, 65/3, -2/9, No, nonconvexity_witness)."},"skills":["One Exact Jensen Instance: a Counterexample or Only an Example"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r02-convex-function-0077","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1ef48de14f115fbbd40906876463d6b689978cb964f47a9dbbf015bcd930fc70","family_id":"qf-mod25-convexfn-jensen-instance-audit","seed":202610252077,"semantic_template":"qf-mod25-convexfn-jensen-instance-audit/{\"coefficients\":[1,-11,-2],\"interval\":[-5,5],\"lambda\":[2,4],\"x\":[-1,1],\"y\":[-2,1]}"},"fields":{"title":"One Exact Jensen Instance: a Counterexample or Only an Example","stem":"Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[1, -11, -2], not ascending-power order. The declared interval [lower,upper]=[-5, 5] belongs to the function data. x=-1/1, y=-2/1, lambda=2/4 must lie in the declared domain and closed unit weight interval. Give (coefficients,interval,x,y,lambda,weighted input,lhs=f(weighted input),rhs=weighted endpoint values,rhs−lhs,inequality passes?,nonconvexity_witness if the gap is negative, otherwise single_instance_only). Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.","answer":{"kind":"expression","value":"([1, -11, -2], [-5, 5], -1, -2, 1/2, -3/2, 67/4, 17, 1/4, Yes, single_instance_only)","explanation":"Compute both sides exactly. A negative gap refutes convexity on the declared domain; a nonnegative gap proves only this one instance. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([1, -11, -2], [-5, 5], -1, -2, 1/2, -3/2, 67/4, 17, 1/4, Yes, single_instance_only)."},"skills":["One Exact Jensen Instance: a Counterexample or Only an Example"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r02-convex-function-0078","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"074287d8886ad35a4fe2923b1c1bb220a6f3e9d20341800d21bb0c3806f29cbe","family_id":"qf-mod25-convexfn-jensen-instance-audit","seed":202610252078,"semantic_template":"qf-mod25-convexfn-jensen-instance-audit/{\"coefficients\":[2,-10,-1],\"interval\":[-5,5],\"lambda\":[3,5],\"x\":[0,1],\"y\":[-1,1]}"},"fields":{"title":"One Exact Jensen Instance: a Counterexample or Only an Example","stem":"Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[2, -10, -1], not ascending-power order. The declared interval [lower,upper]=[-5, 5] belongs to the function data. x=0/1, y=-1/1, lambda=3/5 must lie in the declared domain and closed unit weight interval. Give (coefficients,interval,x,y,lambda,weighted input,lhs=f(weighted input),rhs=weighted endpoint values,rhs−lhs,inequality passes?,nonconvexity_witness if the gap is negative, otherwise single_instance_only). Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.","answer":{"kind":"expression","value":"([2, -10, -1], [-5, 5], 0, -1, 3/5, -2/5, 83/25, 19/5, 12/25, Yes, single_instance_only)","explanation":"Compute both sides exactly. A negative gap refutes convexity on the declared domain; a nonnegative gap proves only this one instance. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([2, -10, -1], [-5, 5], 0, -1, 3/5, -2/5, 83/25, 19/5, 12/25, Yes, single_instance_only)."},"skills":["One Exact Jensen Instance: a Counterexample or Only an Example"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r02-convex-function-0079","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cd0393819a2c5c6e7f41bd02163d59ff7ea4262de85b9658146930923a00210b","family_id":"qf-mod25-convexfn-jensen-instance-audit","seed":202610252079,"semantic_template":"qf-mod25-convexfn-jensen-instance-audit/{\"coefficients\":[3,-9,0],\"interval\":[-5,5],\"lambda\":[4,6],\"x\":[1,1],\"y\":[0,1]}"},"fields":{"title":"One Exact Jensen Instance: a Counterexample or Only an Example","stem":"Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[3, -9, 0], not ascending-power order. The declared interval [lower,upper]=[-5, 5] belongs to the function data. x=1/1, y=0/1, lambda=4/6 must lie in the declared domain and closed unit weight interval. Give (coefficients,interval,x,y,lambda,weighted input,lhs=f(weighted input),rhs=weighted endpoint values,rhs−lhs,inequality passes?,nonconvexity_witness if the gap is negative, otherwise single_instance_only). Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.","answer":{"kind":"expression","value":"([3, -9, 0], [-5, 5], 1, 0, 2/3, 2/3, -14/3, -4, 2/3, Yes, single_instance_only)","explanation":"Compute both sides exactly. A negative gap refutes convexity on the declared domain; a nonnegative gap proves only this one instance. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([3, -9, 0], [-5, 5], 1, 0, 2/3, 2/3, -14/3, -4, 2/3, Yes, single_instance_only)."},"skills":["One Exact Jensen Instance: a Counterexample or Only an Example"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r02-convex-function-0080","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"54234f9cb4131defb4e90bd9ffcfb588d245bdb2ae95dda51b57d079caeac2ef","family_id":"qf-mod25-convexfn-jensen-instance-audit","seed":202610252080,"semantic_template":"qf-mod25-convexfn-jensen-instance-audit/{\"coefficients\":[0,-8,1],\"interval\":[-5,5],\"lambda\":[5,7],\"x\":[2,1],\"y\":[1,1]}"},"fields":{"title":"One Exact Jensen Instance: a Counterexample or Only an Example","stem":"Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[0, -8, 1], not ascending-power order. The declared interval [lower,upper]=[-5, 5] belongs to the function data. x=2/1, y=1/1, lambda=5/7 must lie in the declared domain and closed unit weight interval. Give (coefficients,interval,x,y,lambda,weighted input,lhs=f(weighted input),rhs=weighted endpoint values,rhs−lhs,inequality passes?,nonconvexity_witness if the gap is negative, otherwise single_instance_only). Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.","answer":{"kind":"expression","value":"([0, -8, 1], [-5, 5], 2, 1, 5/7, 12/7, -89/7, -89/7, 0, Yes, single_instance_only)","explanation":"Compute both sides exactly. A negative gap refutes convexity on the declared domain; a nonnegative gap proves only this one instance. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([0, -8, 1], [-5, 5], 2, 1, 5/7, 12/7, -89/7, -89/7, 0, Yes, single_instance_only)."},"skills":["One Exact Jensen Instance: a Counterexample or Only an Example"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r02-convex-function-0081","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3a6b2ceaffeffb8f577a66d1b9e19680ba419bbe990e5fb671f332353b33ae70","family_id":"qf-mod25-convexfn-jensen-instance-audit","seed":202610252081,"semantic_template":"qf-mod25-convexfn-jensen-instance-audit/{\"coefficients\":[1,-7,2],\"interval\":[-5,5],\"lambda\":[6,8],\"x\":[-2,1],\"y\":[2,1]}"},"fields":{"title":"One Exact Jensen Instance: a Counterexample or Only an Example","stem":"Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[1, -7, 2], not ascending-power order. The declared interval [lower,upper]=[-5, 5] belongs to the function data. x=-2/1, y=2/1, lambda=6/8 must lie in the declared domain and closed unit weight interval. Give (coefficients,interval,x,y,lambda,weighted input,lhs=f(weighted input),rhs=weighted endpoint values,rhs−lhs,inequality passes?,nonconvexity_witness if the gap is negative, otherwise single_instance_only). Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.","answer":{"kind":"expression","value":"([1, -7, 2], [-5, 5], -2, 2, 3/4, -1, 10, 13, 3, Yes, single_instance_only)","explanation":"Compute both sides exactly. A negative gap refutes convexity on the declared domain; a nonnegative gap proves only this one instance. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([1, -7, 2], [-5, 5], -2, 2, 3/4, -1, 10, 13, 3, Yes, single_instance_only)."},"skills":["One Exact Jensen Instance: a Counterexample or Only an Example"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r02-convex-function-0082","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0668c3366e4497420ef6c9682921affbac7726ce82ee667ce5848d067dcc61be","family_id":"qf-mod25-convexfn-jensen-instance-audit","seed":202610252082,"semantic_template":"qf-mod25-convexfn-jensen-instance-audit/{\"coefficients\":[-1,-6,3],\"interval\":[-5,5],\"lambda\":[7,9],\"x\":[-1,1],\"y\":[3,1]}"},"fields":{"title":"One Exact Jensen Instance: a Counterexample or Only an Example","stem":"Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[-1, -6, 3], not ascending-power order. The declared interval [lower,upper]=[-5, 5] belongs to the function data. x=-1/1, y=3/1, lambda=7/9 must lie in the declared domain and closed unit weight interval. Give (coefficients,interval,x,y,lambda,weighted input,lhs=f(weighted input),rhs=weighted endpoint values,rhs−lhs,inequality passes?,nonconvexity_witness if the gap is negative, otherwise single_instance_only). Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.","answer":{"kind":"expression","value":"([-1, -6, 3], [-5, 5], -1, 3, 7/9, -1/9, 296/81, 8/9, -224/81, No, nonconvexity_witness)","explanation":"Compute both sides exactly. A negative gap refutes convexity on the declared domain; a nonnegative gap proves only this one instance. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-1, -6, 3], [-5, 5], -1, 3, 7/9, -1/9, 296/81, 8/9, -224/81, No, nonconvexity_witness)."},"skills":["One Exact Jensen Instance: a Counterexample or Only an Example"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r02-convex-function-0083","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8ac012fcfb1ddbcc98803bad902370a0f6ec00a0b9ca77d201e319cc331260d0","family_id":"qf-mod25-convexfn-jensen-instance-audit","seed":202610252083,"semantic_template":"qf-mod25-convexfn-jensen-instance-audit/{\"coefficients\":[3,-5,-3],\"interval\":[-5,5],\"lambda\":[8,10],\"x\":[0,1],\"y\":[-3,1]}"},"fields":{"title":"One Exact Jensen Instance: a Counterexample or Only an Example","stem":"Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[3, -5, -3], not ascending-power order. The declared interval [lower,upper]=[-5, 5] belongs to the function data. x=0/1, y=-3/1, lambda=8/10 must lie in the declared domain and closed unit weight interval. Give (coefficients,interval,x,y,lambda,weighted input,lhs=f(weighted input),rhs=weighted endpoint values,rhs−lhs,inequality passes?,nonconvexity_witness if the gap is negative, otherwise single_instance_only). Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.","answer":{"kind":"expression","value":"([3, -5, -3], [-5, 5], 0, -3, 4/5, -3/5, 27/25, 27/5, 108/25, Yes, single_instance_only)","explanation":"Compute both sides exactly. A negative gap refutes convexity on the declared domain; a nonnegative gap proves only this one instance. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([3, -5, -3], [-5, 5], 0, -3, 4/5, -3/5, 27/25, 27/5, 108/25, Yes, single_instance_only)."},"skills":["One Exact Jensen Instance: a Counterexample or Only an Example"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r02-convex-function-0084","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"075384c92b1463842ce225a0e548ea54f475e8002e8a02cef17c01a448f8adb0","family_id":"qf-mod25-convexfn-jensen-instance-audit","seed":202610252084,"semantic_template":"qf-mod25-convexfn-jensen-instance-audit/{\"coefficients\":[0,-4,-2],\"interval\":[-5,5],\"lambda\":[9,11],\"x\":[1,1],\"y\":[-2,1]}"},"fields":{"title":"One Exact Jensen Instance: a Counterexample or Only an Example","stem":"Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[0, -4, -2], not ascending-power order. The declared interval [lower,upper]=[-5, 5] belongs to the function data. x=1/1, y=-2/1, lambda=9/11 must lie in the declared domain and closed unit weight interval. Give (coefficients,interval,x,y,lambda,weighted input,lhs=f(weighted input),rhs=weighted endpoint values,rhs−lhs,inequality passes?,nonconvexity_witness if the gap is negative, otherwise single_instance_only). Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.","answer":{"kind":"expression","value":"([0, -4, -2], [-5, 5], 1, -2, 9/11, 5/11, -42/11, -42/11, 0, Yes, single_instance_only)","explanation":"Compute both sides exactly. A negative gap refutes convexity on the declared domain; a nonnegative gap proves only this one instance. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([0, -4, -2], [-5, 5], 1, -2, 9/11, 5/11, -42/11, -42/11, 0, Yes, single_instance_only)."},"skills":["One Exact Jensen Instance: a Counterexample or Only an Example"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r02-convex-function-0085","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fa356853bf3770627429a7c7d9292593eafbe085b61fe0250afb8b4cbf412a2c","family_id":"qf-mod25-convexfn-jensen-instance-audit","seed":202610252085,"semantic_template":"qf-mod25-convexfn-jensen-instance-audit/{\"coefficients\":[1,-3,-1],\"interval\":[-5,5],\"lambda\":[10,12],\"x\":[2,1],\"y\":[-1,1]}"},"fields":{"title":"One Exact Jensen Instance: a Counterexample or Only an Example","stem":"Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[1, -3, -1], not ascending-power order. The declared interval [lower,upper]=[-5, 5] belongs to the function data. x=2/1, y=-1/1, lambda=10/12 must lie in the declared domain and closed unit weight interval. Give (coefficients,interval,x,y,lambda,weighted input,lhs=f(weighted input),rhs=weighted endpoint values,rhs−lhs,inequality passes?,nonconvexity_witness if the gap is negative, otherwise single_instance_only). Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.","answer":{"kind":"expression","value":"([1, -3, -1], [-5, 5], 2, -1, 5/6, 3/2, -13/4, -2, 5/4, Yes, single_instance_only)","explanation":"Compute both sides exactly. A negative gap refutes convexity on the declared domain; a nonnegative gap proves only this one instance. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([1, -3, -1], [-5, 5], 2, -1, 5/6, 3/2, -13/4, -2, 5/4, Yes, single_instance_only)."},"skills":["One Exact Jensen Instance: a Counterexample or Only an Example"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r02-convex-function-0086","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"915e8e86812c260c6ef5aef50b933f4d2ff6f2170be32dc7bd1195e30643b61b","family_id":"qf-mod25-convexfn-jensen-instance-audit","seed":202610252086,"semantic_template":"qf-mod25-convexfn-jensen-instance-audit/{\"coefficients\":[2,-2,0],\"interval\":[-5,5],\"lambda\":[11,13],\"x\":[-2,1],\"y\":[0,1]}"},"fields":{"title":"One Exact Jensen Instance: a Counterexample or Only an Example","stem":"Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[2, -2, 0], not ascending-power order. The declared interval [lower,upper]=[-5, 5] belongs to the function data. x=-2/1, y=0/1, lambda=11/13 must lie in the declared domain and closed unit weight interval. Give (coefficients,interval,x,y,lambda,weighted input,lhs=f(weighted input),rhs=weighted endpoint values,rhs−lhs,inequality passes?,nonconvexity_witness if the gap is negative, otherwise single_instance_only). Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.","answer":{"kind":"expression","value":"([2, -2, 0], [-5, 5], -2, 0, 11/13, -22/13, 1540/169, 132/13, 176/169, Yes, single_instance_only)","explanation":"Compute both sides exactly. A negative gap refutes convexity on the declared domain; a nonnegative gap proves only this one instance. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([2, -2, 0], [-5, 5], -2, 0, 11/13, -22/13, 1540/169, 132/13, 176/169, Yes, single_instance_only)."},"skills":["One Exact Jensen Instance: a Counterexample or Only an Example"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r02-convex-function-0087","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"404706add32d7af8d3646fbadbebb1e639b9cf121e74572043be5ddf82b896c5","family_id":"qf-mod25-convexfn-jensen-instance-audit","seed":202610252087,"semantic_template":"qf-mod25-convexfn-jensen-instance-audit/{\"coefficients\":[3,-1,1],\"interval\":[-5,5],\"lambda\":[12,14],\"x\":[-1,1],\"y\":[1,1]}"},"fields":{"title":"One Exact Jensen Instance: a Counterexample or Only an Example","stem":"Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[3, -1, 1], not ascending-power order. The declared interval [lower,upper]=[-5, 5] belongs to the function data. x=-1/1, y=1/1, lambda=12/14 must lie in the declared domain and closed unit weight interval. Give (coefficients,interval,x,y,lambda,weighted input,lhs=f(weighted input),rhs=weighted endpoint values,rhs−lhs,inequality passes?,nonconvexity_witness if the gap is negative, otherwise single_instance_only). Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.","answer":{"kind":"expression","value":"([3, -1, 1], [-5, 5], -1, 1, 6/7, -5/7, 159/49, 33/7, 72/49, Yes, single_instance_only)","explanation":"Compute both sides exactly. A negative gap refutes convexity on the declared domain; a nonnegative gap proves only this one instance. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([3, -1, 1], [-5, 5], -1, 1, 6/7, -5/7, 159/49, 33/7, 72/49, Yes, single_instance_only)."},"skills":["One Exact Jensen Instance: a Counterexample or Only an Example"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r02-convex-function-0088","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6082834ecf4c9b0350fe55c03dd4d221226b85a9bf2aa146cd6a319039b90ac2","family_id":"qf-mod25-convexfn-jensen-instance-audit","seed":202610252088,"semantic_template":"qf-mod25-convexfn-jensen-instance-audit/{\"coefficients\":[-1,0,2],\"interval\":[-5,5],\"lambda\":[13,15],\"x\":[0,1],\"y\":[2,1]}"},"fields":{"title":"One Exact Jensen Instance: a Counterexample or Only an Example","stem":"Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[-1, 0, 2], not ascending-power order. The declared interval [lower,upper]=[-5, 5] belongs to the function data. x=0/1, y=2/1, lambda=13/15 must lie in the declared domain and closed unit weight interval. Give (coefficients,interval,x,y,lambda,weighted input,lhs=f(weighted input),rhs=weighted endpoint values,rhs−lhs,inequality passes?,nonconvexity_witness if the gap is negative, otherwise single_instance_only). Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.","answer":{"kind":"expression","value":"([-1, 0, 2], [-5, 5], 0, 2, 13/15, 4/15, 434/225, 22/15, -104/225, No, nonconvexity_witness)","explanation":"Compute both sides exactly. A negative gap refutes convexity on the declared domain; a nonnegative gap proves only this one instance. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-1, 0, 2], [-5, 5], 0, 2, 13/15, 4/15, 434/225, 22/15, -104/225, No, nonconvexity_witness)."},"skills":["One Exact Jensen Instance: a Counterexample or Only an Example"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r02-convex-function-0089","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5f374057573fcb2b4a36d956b88c4ca6f917e9927afd2fe11493b91a62c0ccdd","family_id":"qf-mod25-convexfn-jensen-instance-audit","seed":202610252089,"semantic_template":"qf-mod25-convexfn-jensen-instance-audit/{\"coefficients\":[1,1,3],\"interval\":[-5,5],\"lambda\":[14,16],\"x\":[1,1],\"y\":[3,1]}"},"fields":{"title":"One Exact Jensen Instance: a Counterexample or Only an Example","stem":"Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[1, 1, 3], not ascending-power order. The declared interval [lower,upper]=[-5, 5] belongs to the function data. x=1/1, y=3/1, lambda=14/16 must lie in the declared domain and closed unit weight interval. Give (coefficients,interval,x,y,lambda,weighted input,lhs=f(weighted input),rhs=weighted endpoint values,rhs−lhs,inequality passes?,nonconvexity_witness if the gap is negative, otherwise single_instance_only). Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.","answer":{"kind":"expression","value":"([1, 1, 3], [-5, 5], 1, 3, 7/8, 5/4, 93/16, 25/4, 7/16, Yes, single_instance_only)","explanation":"Compute both sides exactly. A negative gap refutes convexity on the declared domain; a nonnegative gap proves only this one instance. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([1, 1, 3], [-5, 5], 1, 3, 7/8, 5/4, 93/16, 25/4, 7/16, Yes, single_instance_only)."},"skills":["One Exact Jensen Instance: a Counterexample or Only an Example"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r02-convex-function-0090","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d354a94ae5e6733f14852953d47739a479bfa470c34417ea01ee667ba173514d","family_id":"qf-mod25-convexfn-jensen-instance-audit","seed":202610252090,"semantic_template":"qf-mod25-convexfn-jensen-instance-audit/{\"coefficients\":[2,2,-3],\"interval\":[-5,5],\"lambda\":[15,17],\"x\":[2,1],\"y\":[-3,1]}"},"fields":{"title":"One Exact Jensen Instance: a Counterexample or Only an Example","stem":"Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[2, 2, -3], not ascending-power order. The declared interval [lower,upper]=[-5, 5] belongs to the function data. x=2/1, y=-3/1, lambda=15/17 must lie in the declared domain and closed unit weight interval. Give (coefficients,interval,x,y,lambda,weighted input,lhs=f(weighted input),rhs=weighted endpoint values,rhs−lhs,inequality passes?,nonconvexity_witness if the gap is negative, otherwise single_instance_only). Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.","answer":{"kind":"expression","value":"([2, 2, -3], [-5, 5], 2, -3, 15/17, 24/17, 1101/289, 9, 1500/289, Yes, single_instance_only)","explanation":"Compute both sides exactly. A negative gap refutes convexity on the declared domain; a nonnegative gap proves only this one instance. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([2, 2, -3], [-5, 5], 2, -3, 15/17, 24/17, 1101/289, 9, 1500/289, Yes, single_instance_only)."},"skills":["One Exact Jensen Instance: a Counterexample or Only an Example"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r02-convex-function-0091","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"175afbc1109116d717a34227a368f870d02dbe70edb7bbc82fc67a7e6d39b939","family_id":"qf-mod25-convexfn-jensen-instance-audit","seed":202610252091,"semantic_template":"qf-mod25-convexfn-jensen-instance-audit/{\"coefficients\":[3,3,-2],\"interval\":[-5,5],\"lambda\":[16,18],\"x\":[-2,1],\"y\":[-2,1]}"},"fields":{"title":"One Exact Jensen Instance: a Counterexample or Only an Example","stem":"Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[3, 3, -2], not ascending-power order. The declared interval [lower,upper]=[-5, 5] belongs to the function data. x=-2/1, y=-2/1, lambda=16/18 must lie in the declared domain and closed unit weight interval. Give (coefficients,interval,x,y,lambda,weighted input,lhs=f(weighted input),rhs=weighted endpoint values,rhs−lhs,inequality passes?,nonconvexity_witness if the gap is negative, otherwise single_instance_only). Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.","answer":{"kind":"expression","value":"([3, 3, -2], [-5, 5], -2, -2, 8/9, -2, 4, 4, 0, Yes, single_instance_only)","explanation":"Compute both sides exactly. A negative gap refutes convexity on the declared domain; a nonnegative gap proves only this one instance. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([3, 3, -2], [-5, 5], -2, -2, 8/9, -2, 4, 4, 0, Yes, single_instance_only)."},"skills":["One Exact Jensen Instance: a Counterexample or Only an Example"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r02-convex-function-0092","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"af2e2331da4b05e8b2d23a5f70a346608438f519da49f04bb011cfd100031eb6","family_id":"qf-mod25-convexfn-jensen-instance-audit","seed":202610252092,"semantic_template":"qf-mod25-convexfn-jensen-instance-audit/{\"coefficients\":[0,4,-1],\"interval\":[-5,5],\"lambda\":[17,19],\"x\":[-1,1],\"y\":[-1,1]}"},"fields":{"title":"One Exact Jensen Instance: a Counterexample or Only an Example","stem":"Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[0, 4, -1], not ascending-power order. The declared interval [lower,upper]=[-5, 5] belongs to the function data. x=-1/1, y=-1/1, lambda=17/19 must lie in the declared domain and closed unit weight interval. Give (coefficients,interval,x,y,lambda,weighted input,lhs=f(weighted input),rhs=weighted endpoint values,rhs−lhs,inequality passes?,nonconvexity_witness if the gap is negative, otherwise single_instance_only). Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.","answer":{"kind":"expression","value":"([0, 4, -1], [-5, 5], -1, -1, 17/19, -1, -5, -5, 0, Yes, single_instance_only)","explanation":"Compute both sides exactly. A negative gap refutes convexity on the declared domain; a nonnegative gap proves only this one instance. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([0, 4, -1], [-5, 5], -1, -1, 17/19, -1, -5, -5, 0, Yes, single_instance_only)."},"skills":["One Exact Jensen Instance: a Counterexample or Only an Example"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r02-convex-function-0093","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4c755a89c51f7633dd60b4cae0f80ad4b00147574ed29a89d2ace2e7790a8962","family_id":"qf-mod25-convexfn-jensen-instance-audit","seed":202610252093,"semantic_template":"qf-mod25-convexfn-jensen-instance-audit/{\"coefficients\":[1,5,0],\"interval\":[-5,5],\"lambda\":[18,20],\"x\":[0,1],\"y\":[0,1]}"},"fields":{"title":"One Exact Jensen Instance: a Counterexample or Only an Example","stem":"Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[1, 5, 0], not ascending-power order. The declared interval [lower,upper]=[-5, 5] belongs to the function data. x=0/1, y=0/1, lambda=18/20 must lie in the declared domain and closed unit weight interval. Give (coefficients,interval,x,y,lambda,weighted input,lhs=f(weighted input),rhs=weighted endpoint values,rhs−lhs,inequality passes?,nonconvexity_witness if the gap is negative, otherwise single_instance_only). Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.","answer":{"kind":"expression","value":"([1, 5, 0], [-5, 5], 0, 0, 9/10, 0, 0, 0, 0, Yes, single_instance_only)","explanation":"Compute both sides exactly. A negative gap refutes convexity on the declared domain; a nonnegative gap proves only this one instance. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([1, 5, 0], [-5, 5], 0, 0, 9/10, 0, 0, 0, 0, Yes, single_instance_only)."},"skills":["One Exact Jensen Instance: a Counterexample or Only an Example"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r02-convex-function-0094","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3623ec687bfd162a8212fe8316ca5963e1aa4497ae4a14cc60a4d54fda0dc5e5","family_id":"qf-mod25-convexfn-jensen-instance-audit","seed":202610252094,"semantic_template":"qf-mod25-convexfn-jensen-instance-audit/{\"coefficients\":[-1,6,1],\"interval\":[-5,5],\"lambda\":[19,21],\"x\":[1,1],\"y\":[1,1]}"},"fields":{"title":"One Exact Jensen Instance: a Counterexample or Only an Example","stem":"Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[-1, 6, 1], not ascending-power order. The declared interval [lower,upper]=[-5, 5] belongs to the function data. x=1/1, y=1/1, lambda=19/21 must lie in the declared domain and closed unit weight interval. Give (coefficients,interval,x,y,lambda,weighted input,lhs=f(weighted input),rhs=weighted endpoint values,rhs−lhs,inequality passes?,nonconvexity_witness if the gap is negative, otherwise single_instance_only). Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.","answer":{"kind":"expression","value":"([-1, 6, 1], [-5, 5], 1, 1, 19/21, 1, 6, 6, 0, Yes, single_instance_only)","explanation":"Compute both sides exactly. A negative gap refutes convexity on the declared domain; a nonnegative gap proves only this one instance. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-1, 6, 1], [-5, 5], 1, 1, 19/21, 1, 6, 6, 0, Yes, single_instance_only)."},"skills":["One Exact Jensen Instance: a Counterexample or Only an Example"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r02-convex-function-0095","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"472738be81f2fb147ee840d3482909efbe309ce09ddfda4c6af3a8f4dff4c22c","family_id":"qf-mod25-convexfn-jensen-instance-audit","seed":202610252095,"semantic_template":"qf-mod25-convexfn-jensen-instance-audit/{\"coefficients\":[3,7,2],\"interval\":[-5,5],\"lambda\":[20,22],\"x\":[2,1],\"y\":[2,1]}"},"fields":{"title":"One Exact Jensen Instance: a Counterexample or Only an Example","stem":"Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[3, 7, 2], not ascending-power order. The declared interval [lower,upper]=[-5, 5] belongs to the function data. x=2/1, y=2/1, lambda=20/22 must lie in the declared domain and closed unit weight interval. Give (coefficients,interval,x,y,lambda,weighted input,lhs=f(weighted input),rhs=weighted endpoint values,rhs−lhs,inequality passes?,nonconvexity_witness if the gap is negative, otherwise single_instance_only). Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.","answer":{"kind":"expression","value":"([3, 7, 2], [-5, 5], 2, 2, 10/11, 2, 28, 28, 0, Yes, single_instance_only)","explanation":"Compute both sides exactly. A negative gap refutes convexity on the declared domain; a nonnegative gap proves only this one instance. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([3, 7, 2], [-5, 5], 2, 2, 10/11, 2, 28, 28, 0, Yes, single_instance_only)."},"skills":["One Exact Jensen Instance: a Counterexample or Only an Example"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r02-convex-function-0096","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3f710a089683bcd17974651650b07b288dbde8593e6530149826b4bfa6e3b9d2","family_id":"qf-mod25-convexfn-jensen-instance-audit","seed":202610252096,"semantic_template":"qf-mod25-convexfn-jensen-instance-audit/{\"coefficients\":[0,8,3],\"interval\":[-5,5],\"lambda\":[21,23],\"x\":[-2,1],\"y\":[3,1]}"},"fields":{"title":"One Exact Jensen Instance: a Counterexample or Only an Example","stem":"Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[0, 8, 3], not ascending-power order. The declared interval [lower,upper]=[-5, 5] belongs to the function data. x=-2/1, y=3/1, lambda=21/23 must lie in the declared domain and closed unit weight interval. Give (coefficients,interval,x,y,lambda,weighted input,lhs=f(weighted input),rhs=weighted endpoint values,rhs−lhs,inequality passes?,nonconvexity_witness if the gap is negative, otherwise single_instance_only). Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.","answer":{"kind":"expression","value":"([0, 8, 3], [-5, 5], -2, 3, 21/23, -36/23, -219/23, -219/23, 0, Yes, single_instance_only)","explanation":"Compute both sides exactly. A negative gap refutes convexity on the declared domain; a nonnegative gap proves only this one instance. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([0, 8, 3], [-5, 5], -2, 3, 21/23, -36/23, -219/23, -219/23, 0, Yes, single_instance_only)."},"skills":["One Exact Jensen Instance: a Counterexample or Only an Example"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r02-convex-function-0097","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c94138575b12104befce5ab79cda05506019853c05437fbab2d9e06c8c5b2054","family_id":"qf-mod25-convexfn-jensen-instance-audit","seed":202610252097,"semantic_template":"qf-mod25-convexfn-jensen-instance-audit/{\"coefficients\":[1,9,-3],\"interval\":[-5,5],\"lambda\":[22,24],\"x\":[-1,1],\"y\":[-3,1]}"},"fields":{"title":"One Exact Jensen Instance: a Counterexample or Only an Example","stem":"Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[1, 9, -3], not ascending-power order. The declared interval [lower,upper]=[-5, 5] belongs to the function data. x=-1/1, y=-3/1, lambda=22/24 must lie in the declared domain and closed unit weight interval. Give (coefficients,interval,x,y,lambda,weighted input,lhs=f(weighted input),rhs=weighted endpoint values,rhs−lhs,inequality passes?,nonconvexity_witness if the gap is negative, otherwise single_instance_only). Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.","answer":{"kind":"expression","value":"([1, 9, -3], [-5, 5], -1, -3, 11/12, -7/6, -437/36, -71/6, 11/36, Yes, single_instance_only)","explanation":"Compute both sides exactly. A negative gap refutes convexity on the declared domain; a nonnegative gap proves only this one instance. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([1, 9, -3], [-5, 5], -1, -3, 11/12, -7/6, -437/36, -71/6, 11/36, Yes, single_instance_only)."},"skills":["One Exact Jensen Instance: a Counterexample or Only an Example"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r02-convex-function-0098","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5266e0f05980947a2aa1062cdd094dc4767a4a3d6736e66e607cf1be524ea5aa","family_id":"qf-mod25-convexfn-jensen-instance-audit","seed":202610252098,"semantic_template":"qf-mod25-convexfn-jensen-instance-audit/{\"coefficients\":[2,10,-2],\"interval\":[-5,5],\"lambda\":[23,25],\"x\":[0,1],\"y\":[-2,1]}"},"fields":{"title":"One Exact Jensen Instance: a Counterexample or Only an Example","stem":"Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[2, 10, -2], not ascending-power order. The declared interval [lower,upper]=[-5, 5] belongs to the function data. x=0/1, y=-2/1, lambda=23/25 must lie in the declared domain and closed unit weight interval. Give (coefficients,interval,x,y,lambda,weighted input,lhs=f(weighted input),rhs=weighted endpoint values,rhs−lhs,inequality passes?,nonconvexity_witness if the gap is negative, otherwise single_instance_only). Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.","answer":{"kind":"expression","value":"([2, 10, -2], [-5, 5], 0, -2, 23/25, -4/25, -2218/625, -74/25, 368/625, Yes, single_instance_only)","explanation":"Compute both sides exactly. A negative gap refutes convexity on the declared domain; a nonnegative gap proves only this one instance. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([2, 10, -2], [-5, 5], 0, -2, 23/25, -4/25, -2218/625, -74/25, 368/625, Yes, single_instance_only)."},"skills":["One Exact Jensen Instance: a Counterexample or Only an Example"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r02-convex-function-0099","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b1034ee8b598f2284fee181b9c0d22b67f5e4cd4a4cc3e069bc27264f0c15626","family_id":"qf-mod25-convexfn-jensen-instance-audit","seed":202610252099,"semantic_template":"qf-mod25-convexfn-jensen-instance-audit/{\"coefficients\":[3,11,-1],\"interval\":[-5,5],\"lambda\":[24,26],\"x\":[1,1],\"y\":[-1,1]}"},"fields":{"title":"One Exact Jensen Instance: a Counterexample or Only an Example","stem":"Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[3, 11, -1], not ascending-power order. The declared interval [lower,upper]=[-5, 5] belongs to the function data. x=1/1, y=-1/1, lambda=24/26 must lie in the declared domain and closed unit weight interval. Give (coefficients,interval,x,y,lambda,weighted input,lhs=f(weighted input),rhs=weighted endpoint values,rhs−lhs,inequality passes?,nonconvexity_witness if the gap is negative, otherwise single_instance_only). Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.","answer":{"kind":"expression","value":"([3, 11, -1], [-5, 5], 1, -1, 12/13, 11/13, 1767/169, 147/13, 144/169, Yes, single_instance_only)","explanation":"Compute both sides exactly. A negative gap refutes convexity on the declared domain; a nonnegative gap proves only this one instance. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([3, 11, -1], [-5, 5], 1, -1, 12/13, 11/13, 1767/169, 147/13, 144/169, Yes, single_instance_only)."},"skills":["One Exact Jensen Instance: a Counterexample or Only an Example"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r02-convex-function-0100","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bd6d88af514ef06b355df3a2c5cfa93300b11e771772697e4c06f193a8d30398","family_id":"qf-mod25-convexfn-jensen-instance-audit","seed":202610252100,"semantic_template":"qf-mod25-convexfn-jensen-instance-audit/{\"coefficients\":[-1,12,0],\"interval\":[-5,5],\"lambda\":[25,27],\"x\":[2,1],\"y\":[0,1]}"},"fields":{"title":"One Exact Jensen Instance: a Counterexample or Only an Example","stem":"Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[-1, 12, 0], not ascending-power order. The declared interval [lower,upper]=[-5, 5] belongs to the function data. x=2/1, y=0/1, lambda=25/27 must lie in the declared domain and closed unit weight interval. Give (coefficients,interval,x,y,lambda,weighted input,lhs=f(weighted input),rhs=weighted endpoint values,rhs−lhs,inequality passes?,nonconvexity_witness if the gap is negative, otherwise single_instance_only). Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.","answer":{"kind":"expression","value":"([-1, 12, 0], [-5, 5], 2, 0, 25/27, 50/27, 13700/729, 500/27, -200/729, No, nonconvexity_witness)","explanation":"Compute both sides exactly. A negative gap refutes convexity on the declared domain; a nonnegative gap proves only this one instance. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-1, 12, 0], [-5, 5], 2, 0, 25/27, 50/27, 13700/729, 500/27, -200/729, No, nonconvexity_witness)."},"skills":["One Exact Jensen Instance: a Counterexample or Only an Example"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}