{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r07-convex-set-0001","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6396aae96f97fa75365b77b7b888ece02358365be174724dfbb623da825b895a","family_id":"qf-mod24-convexset-interval-certificate","seed":202610247001,"semantic_template":"qf-mod24-convexset-interval-certificate/{\"lower\":-2,\"upper\":3}"},"fields":{"title":"A Universal Closed-Interval Convexity Certificate","stem":"Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. Nonempty closed interval [lower,upper] has lower=-2, upper=3, allowing equality. Give (lower,upper,all x,y in interval and all lambda in [0,1],convex?).","answer":{"kind":"expression","value":"(-2, 3, all x,y in interval and all lambda in [0,1], Yes)","explanation":"For arbitrary x,y in the interval and 0≤lambda≤1, multiplying both endpoint inequalities by nonnegative lambda and 1−lambda and adding keeps their weighted sum between the bounds. This covers every endpoint pair and lambda, not samples. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (-2, 3, all x,y in interval and all lambda in [0,1], Yes)."},"skills":["A Universal Closed-Interval Convexity 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-20261024-r07-convex-set-0002","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4041dfb7a97d08cc7d4add749dfd85e4d227524caf721ce9769638483c7f0786","family_id":"qf-mod24-convexset-interval-certificate","seed":202610247002,"semantic_template":"qf-mod24-convexset-interval-certificate/{\"lower\":-3,\"upper\":4}"},"fields":{"title":"A Universal Closed-Interval Convexity Certificate","stem":"Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. Nonempty closed interval [lower,upper] has lower=-3, upper=4, allowing equality. Give (lower,upper,all x,y in interval and all lambda in [0,1],convex?).","answer":{"kind":"expression","value":"(-3, 4, all x,y in interval and all lambda in [0,1], Yes)","explanation":"For arbitrary x,y in the interval and 0≤lambda≤1, multiplying both endpoint inequalities by nonnegative lambda and 1−lambda and adding keeps their weighted sum between the bounds. This covers every endpoint pair and lambda, not samples. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (-3, 4, all x,y in interval and all lambda in [0,1], Yes)."},"skills":["A Universal Closed-Interval Convexity 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-20261024-r07-convex-set-0003","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"29983a20c23e401009a787befe80acccffaddd528aeb27c11b8b7fff61e4290e","family_id":"qf-mod24-convexset-interval-certificate","seed":202610247003,"semantic_template":"qf-mod24-convexset-interval-certificate/{\"lower\":-4,\"upper\":5}"},"fields":{"title":"A Universal Closed-Interval Convexity Certificate","stem":"Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. Nonempty closed interval [lower,upper] has lower=-4, upper=5, allowing equality. Give (lower,upper,all x,y in interval and all lambda in [0,1],convex?).","answer":{"kind":"expression","value":"(-4, 5, all x,y in interval and all lambda in [0,1], Yes)","explanation":"For arbitrary x,y in the interval and 0≤lambda≤1, multiplying both endpoint inequalities by nonnegative lambda and 1−lambda and adding keeps their weighted sum between the bounds. This covers every endpoint pair and lambda, not samples. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (-4, 5, all x,y in interval and all lambda in [0,1], Yes)."},"skills":["A Universal Closed-Interval Convexity 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-20261024-r07-convex-set-0004","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e7a648fab0dcdcd75f1fec879db6530a43a0f9a8a660c74f5dc454fde4671f60","family_id":"qf-mod24-convexset-interval-certificate","seed":202610247004,"semantic_template":"qf-mod24-convexset-interval-certificate/{\"lower\":-5,\"upper\":6}"},"fields":{"title":"A Universal Closed-Interval Convexity Certificate","stem":"Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. Nonempty closed interval [lower,upper] has lower=-5, upper=6, allowing equality. Give (lower,upper,all x,y in interval and all lambda in [0,1],convex?).","answer":{"kind":"expression","value":"(-5, 6, all x,y in interval and all lambda in [0,1], Yes)","explanation":"For arbitrary x,y in the interval and 0≤lambda≤1, multiplying both endpoint inequalities by nonnegative lambda and 1−lambda and adding keeps their weighted sum between the bounds. This covers every endpoint pair and lambda, not samples. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (-5, 6, all x,y in interval and all lambda in [0,1], Yes)."},"skills":["A Universal Closed-Interval Convexity 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-20261024-r07-convex-set-0005","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2aa495c654a2e436c36def5b850511f6977bc700a97e52e166740000e8b7d36c","family_id":"qf-mod24-convexset-interval-certificate","seed":202610247005,"semantic_template":"qf-mod24-convexset-interval-certificate/{\"lower\":-6,\"upper\":7}"},"fields":{"title":"A Universal Closed-Interval Convexity Certificate","stem":"Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. Nonempty closed interval [lower,upper] has lower=-6, upper=7, allowing equality. Give (lower,upper,all x,y in interval and all lambda in [0,1],convex?).","answer":{"kind":"expression","value":"(-6, 7, all x,y in interval and all lambda in [0,1], Yes)","explanation":"For arbitrary x,y in the interval and 0≤lambda≤1, multiplying both endpoint inequalities by nonnegative lambda and 1−lambda and adding keeps their weighted sum between the bounds. This covers every endpoint pair and lambda, not samples. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (-6, 7, all x,y in interval and all lambda in [0,1], Yes)."},"skills":["A Universal Closed-Interval Convexity 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-20261024-r07-convex-set-0006","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"493775250c8fe94700e4587547245de0e11bfa8b22cd08a7077191db366d7de5","family_id":"qf-mod24-convexset-interval-certificate","seed":202610247006,"semantic_template":"qf-mod24-convexset-interval-certificate/{\"lower\":-7,\"upper\":8}"},"fields":{"title":"A Universal Closed-Interval Convexity Certificate","stem":"Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. Nonempty closed interval [lower,upper] has lower=-7, upper=8, allowing equality. Give (lower,upper,all x,y in interval and all lambda in [0,1],convex?).","answer":{"kind":"expression","value":"(-7, 8, all x,y in interval and all lambda in [0,1], Yes)","explanation":"For arbitrary x,y in the interval and 0≤lambda≤1, multiplying both endpoint inequalities by nonnegative lambda and 1−lambda and adding keeps their weighted sum between the bounds. This covers every endpoint pair and lambda, not samples. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (-7, 8, all x,y in interval and all lambda in [0,1], Yes)."},"skills":["A Universal Closed-Interval Convexity 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-20261024-r07-convex-set-0007","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5f80a86a2a2f934bf81f2443f49cf4aed50d0e2201e1f41009a0340b1bbd44c8","family_id":"qf-mod24-convexset-interval-certificate","seed":202610247007,"semantic_template":"qf-mod24-convexset-interval-certificate/{\"lower\":-8,\"upper\":9}"},"fields":{"title":"A Universal Closed-Interval Convexity Certificate","stem":"Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. Nonempty closed interval [lower,upper] has lower=-8, upper=9, allowing equality. Give (lower,upper,all x,y in interval and all lambda in [0,1],convex?).","answer":{"kind":"expression","value":"(-8, 9, all x,y in interval and all lambda in [0,1], Yes)","explanation":"For arbitrary x,y in the interval and 0≤lambda≤1, multiplying both endpoint inequalities by nonnegative lambda and 1−lambda and adding keeps their weighted sum between the bounds. This covers every endpoint pair and lambda, not samples. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (-8, 9, all x,y in interval and all lambda in [0,1], Yes)."},"skills":["A Universal Closed-Interval Convexity 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-20261024-r07-convex-set-0008","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ec28a7261e85b6b43246f39c48f5c4453912b0e7a057c27a17efc2f4a5e05524","family_id":"qf-mod24-convexset-interval-certificate","seed":202610247008,"semantic_template":"qf-mod24-convexset-interval-certificate/{\"lower\":-9,\"upper\":10}"},"fields":{"title":"A Universal Closed-Interval Convexity Certificate","stem":"Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. Nonempty closed interval [lower,upper] has lower=-9, upper=10, allowing equality. Give (lower,upper,all x,y in interval and all lambda in [0,1],convex?).","answer":{"kind":"expression","value":"(-9, 10, all x,y in interval and all lambda in [0,1], Yes)","explanation":"For arbitrary x,y in the interval and 0≤lambda≤1, multiplying both endpoint inequalities by nonnegative lambda and 1−lambda and adding keeps their weighted sum between the bounds. This covers every endpoint pair and lambda, not samples. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (-9, 10, all x,y in interval and all lambda in [0,1], Yes)."},"skills":["A Universal Closed-Interval Convexity 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-20261024-r07-convex-set-0009","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e7133856ed55337eb2553b5c30602d2c507aa41b93d59d31fef8ff99249c6667","family_id":"qf-mod24-convexset-interval-certificate","seed":202610247009,"semantic_template":"qf-mod24-convexset-interval-certificate/{\"lower\":-10,\"upper\":11}"},"fields":{"title":"A Universal Closed-Interval Convexity Certificate","stem":"Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. Nonempty closed interval [lower,upper] has lower=-10, upper=11, allowing equality. Give (lower,upper,all x,y in interval and all lambda in [0,1],convex?).","answer":{"kind":"expression","value":"(-10, 11, all x,y in interval and all lambda in [0,1], Yes)","explanation":"For arbitrary x,y in the interval and 0≤lambda≤1, multiplying both endpoint inequalities by nonnegative lambda and 1−lambda and adding keeps their weighted sum between the bounds. This covers every endpoint pair and lambda, not samples. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (-10, 11, all x,y in interval and all lambda in [0,1], Yes)."},"skills":["A Universal Closed-Interval Convexity 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-20261024-r07-convex-set-0010","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"50206e9a66e379e2bd9089e53cd0fa04acfdc1c3468c883dab4a517577925bb0","family_id":"qf-mod24-convexset-interval-certificate","seed":202610247010,"semantic_template":"qf-mod24-convexset-interval-certificate/{\"lower\":-11,\"upper\":12}"},"fields":{"title":"A Universal Closed-Interval Convexity Certificate","stem":"Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. Nonempty closed interval [lower,upper] has lower=-11, upper=12, allowing equality. Give (lower,upper,all x,y in interval and all lambda in [0,1],convex?).","answer":{"kind":"expression","value":"(-11, 12, all x,y in interval and all lambda in [0,1], Yes)","explanation":"For arbitrary x,y in the interval and 0≤lambda≤1, multiplying both endpoint inequalities by nonnegative lambda and 1−lambda and adding keeps their weighted sum between the bounds. This covers every endpoint pair and lambda, not samples. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (-11, 12, all x,y in interval and all lambda in [0,1], Yes)."},"skills":["A Universal Closed-Interval Convexity 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-20261024-r07-convex-set-0011","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d5ab392698a8d64dca213a312fd71a170799e2fe3fb2bece3b4459ec8f59d5ee","family_id":"qf-mod24-convexset-interval-certificate","seed":202610247011,"semantic_template":"qf-mod24-convexset-interval-certificate/{\"lower\":-12,\"upper\":13}"},"fields":{"title":"A Universal Closed-Interval Convexity Certificate","stem":"Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. Nonempty closed interval [lower,upper] has lower=-12, upper=13, allowing equality. Give (lower,upper,all x,y in interval and all lambda in [0,1],convex?).","answer":{"kind":"expression","value":"(-12, 13, all x,y in interval and all lambda in [0,1], Yes)","explanation":"For arbitrary x,y in the interval and 0≤lambda≤1, multiplying both endpoint inequalities by nonnegative lambda and 1−lambda and adding keeps their weighted sum between the bounds. This covers every endpoint pair and lambda, not samples. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (-12, 13, all x,y in interval and all lambda in [0,1], Yes)."},"skills":["A Universal Closed-Interval Convexity 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-20261024-r07-convex-set-0012","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5d4eef8e7fa2fa60309d4e295d474bf274e5a50b95823221cd3b43013574e951","family_id":"qf-mod24-convexset-interval-certificate","seed":202610247012,"semantic_template":"qf-mod24-convexset-interval-certificate/{\"lower\":-13,\"upper\":14}"},"fields":{"title":"A Universal Closed-Interval Convexity Certificate","stem":"Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. Nonempty closed interval [lower,upper] has lower=-13, upper=14, allowing equality. Give (lower,upper,all x,y in interval and all lambda in [0,1],convex?).","answer":{"kind":"expression","value":"(-13, 14, all x,y in interval and all lambda in [0,1], Yes)","explanation":"For arbitrary x,y in the interval and 0≤lambda≤1, multiplying both endpoint inequalities by nonnegative lambda and 1−lambda and adding keeps their weighted sum between the bounds. This covers every endpoint pair and lambda, not samples. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (-13, 14, all x,y in interval and all lambda in [0,1], Yes)."},"skills":["A Universal Closed-Interval Convexity 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-20261024-r07-convex-set-0013","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"178497a3dbab597892f6a3d1428097a2c0b7145c1593c044862071c183e209be","family_id":"qf-mod24-convexset-interval-certificate","seed":202610247013,"semantic_template":"qf-mod24-convexset-interval-certificate/{\"lower\":-14,\"upper\":15}"},"fields":{"title":"A Universal Closed-Interval Convexity Certificate","stem":"Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. Nonempty closed interval [lower,upper] has lower=-14, upper=15, allowing equality. Give (lower,upper,all x,y in interval and all lambda in [0,1],convex?).","answer":{"kind":"expression","value":"(-14, 15, all x,y in interval and all lambda in [0,1], Yes)","explanation":"For arbitrary x,y in the interval and 0≤lambda≤1, multiplying both endpoint inequalities by nonnegative lambda and 1−lambda and adding keeps their weighted sum between the bounds. This covers every endpoint pair and lambda, not samples. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (-14, 15, all x,y in interval and all lambda in [0,1], Yes)."},"skills":["A Universal Closed-Interval Convexity 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-20261024-r07-convex-set-0014","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9dfe2069ea0d577100c412874edcace30bc0f4ecea70e4f10c48c101b47e1247","family_id":"qf-mod24-convexset-interval-certificate","seed":202610247014,"semantic_template":"qf-mod24-convexset-interval-certificate/{\"lower\":-15,\"upper\":16}"},"fields":{"title":"A Universal Closed-Interval Convexity Certificate","stem":"Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. Nonempty closed interval [lower,upper] has lower=-15, upper=16, allowing equality. Give (lower,upper,all x,y in interval and all lambda in [0,1],convex?).","answer":{"kind":"expression","value":"(-15, 16, all x,y in interval and all lambda in [0,1], Yes)","explanation":"For arbitrary x,y in the interval and 0≤lambda≤1, multiplying both endpoint inequalities by nonnegative lambda and 1−lambda and adding keeps their weighted sum between the bounds. This covers every endpoint pair and lambda, not samples. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (-15, 16, all x,y in interval and all lambda in [0,1], Yes)."},"skills":["A Universal Closed-Interval Convexity 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-20261024-r07-convex-set-0015","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"039b18f5183576068fe19aee6a86b8100bdb006fa17c2922562decc4e8dff3f6","family_id":"qf-mod24-convexset-interval-certificate","seed":202610247015,"semantic_template":"qf-mod24-convexset-interval-certificate/{\"lower\":-16,\"upper\":17}"},"fields":{"title":"A Universal Closed-Interval Convexity Certificate","stem":"Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. Nonempty closed interval [lower,upper] has lower=-16, upper=17, allowing equality. Give (lower,upper,all x,y in interval and all lambda in [0,1],convex?).","answer":{"kind":"expression","value":"(-16, 17, all x,y in interval and all lambda in [0,1], Yes)","explanation":"For arbitrary x,y in the interval and 0≤lambda≤1, multiplying both endpoint inequalities by nonnegative lambda and 1−lambda and adding keeps their weighted sum between the bounds. This covers every endpoint pair and lambda, not samples. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (-16, 17, all x,y in interval and all lambda in [0,1], Yes)."},"skills":["A Universal Closed-Interval Convexity 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-20261024-r07-convex-set-0016","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"73fee514e6691eb0a0c2c6e7e0fb80911a6c6386293943543a96f5565f3f07bb","family_id":"qf-mod24-convexset-interval-certificate","seed":202610247016,"semantic_template":"qf-mod24-convexset-interval-certificate/{\"lower\":-17,\"upper\":18}"},"fields":{"title":"A Universal Closed-Interval Convexity Certificate","stem":"Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. Nonempty closed interval [lower,upper] has lower=-17, upper=18, allowing equality. Give (lower,upper,all x,y in interval and all lambda in [0,1],convex?).","answer":{"kind":"expression","value":"(-17, 18, all x,y in interval and all lambda in [0,1], Yes)","explanation":"For arbitrary x,y in the interval and 0≤lambda≤1, multiplying both endpoint inequalities by nonnegative lambda and 1−lambda and adding keeps their weighted sum between the bounds. This covers every endpoint pair and lambda, not samples. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (-17, 18, all x,y in interval and all lambda in [0,1], Yes)."},"skills":["A Universal Closed-Interval Convexity 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-20261024-r07-convex-set-0017","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5fb086d543fe66f3eeae0f1a27e3ae914eec38f9a6cec180c748838a1bfdbdbc","family_id":"qf-mod24-convexset-interval-certificate","seed":202610247017,"semantic_template":"qf-mod24-convexset-interval-certificate/{\"lower\":-18,\"upper\":19}"},"fields":{"title":"A Universal Closed-Interval Convexity Certificate","stem":"Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. Nonempty closed interval [lower,upper] has lower=-18, upper=19, allowing equality. Give (lower,upper,all x,y in interval and all lambda in [0,1],convex?).","answer":{"kind":"expression","value":"(-18, 19, all x,y in interval and all lambda in [0,1], Yes)","explanation":"For arbitrary x,y in the interval and 0≤lambda≤1, multiplying both endpoint inequalities by nonnegative lambda and 1−lambda and adding keeps their weighted sum between the bounds. This covers every endpoint pair and lambda, not samples. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (-18, 19, all x,y in interval and all lambda in [0,1], Yes)."},"skills":["A Universal Closed-Interval Convexity 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-20261024-r07-convex-set-0018","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f3dd3d9684df1966c5110f14f90d3b64955fac6577d48103dc50f6c539bfe01a","family_id":"qf-mod24-convexset-interval-certificate","seed":202610247018,"semantic_template":"qf-mod24-convexset-interval-certificate/{\"lower\":-19,\"upper\":20}"},"fields":{"title":"A Universal Closed-Interval Convexity Certificate","stem":"Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. Nonempty closed interval [lower,upper] has lower=-19, upper=20, allowing equality. Give (lower,upper,all x,y in interval and all lambda in [0,1],convex?).","answer":{"kind":"expression","value":"(-19, 20, all x,y in interval and all lambda in [0,1], Yes)","explanation":"For arbitrary x,y in the interval and 0≤lambda≤1, multiplying both endpoint inequalities by nonnegative lambda and 1−lambda and adding keeps their weighted sum between the bounds. This covers every endpoint pair and lambda, not samples. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (-19, 20, all x,y in interval and all lambda in [0,1], Yes)."},"skills":["A Universal Closed-Interval Convexity 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-20261024-r07-convex-set-0019","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"53f0551703df6e5994190a8a63c824e5fec797bdd47722ec238548fe04235338","family_id":"qf-mod24-convexset-interval-certificate","seed":202610247019,"semantic_template":"qf-mod24-convexset-interval-certificate/{\"lower\":-20,\"upper\":21}"},"fields":{"title":"A Universal Closed-Interval Convexity Certificate","stem":"Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. Nonempty closed interval [lower,upper] has lower=-20, upper=21, allowing equality. Give (lower,upper,all x,y in interval and all lambda in [0,1],convex?).","answer":{"kind":"expression","value":"(-20, 21, all x,y in interval and all lambda in [0,1], Yes)","explanation":"For arbitrary x,y in the interval and 0≤lambda≤1, multiplying both endpoint inequalities by nonnegative lambda and 1−lambda and adding keeps their weighted sum between the bounds. This covers every endpoint pair and lambda, not samples. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (-20, 21, all x,y in interval and all lambda in [0,1], Yes)."},"skills":["A Universal Closed-Interval Convexity 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-20261024-r07-convex-set-0020","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"26cc9cf637556bec388a6be8ddeed741e2d02c2c5dec6069b8030bc8d645fa28","family_id":"qf-mod24-convexset-interval-certificate","seed":202610247020,"semantic_template":"qf-mod24-convexset-interval-certificate/{\"lower\":-21,\"upper\":22}"},"fields":{"title":"A Universal Closed-Interval Convexity Certificate","stem":"Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. Nonempty closed interval [lower,upper] has lower=-21, upper=22, allowing equality. Give (lower,upper,all x,y in interval and all lambda in [0,1],convex?).","answer":{"kind":"expression","value":"(-21, 22, all x,y in interval and all lambda in [0,1], Yes)","explanation":"For arbitrary x,y in the interval and 0≤lambda≤1, multiplying both endpoint inequalities by nonnegative lambda and 1−lambda and adding keeps their weighted sum between the bounds. This covers every endpoint pair and lambda, not samples. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (-21, 22, all x,y in interval and all lambda in [0,1], Yes)."},"skills":["A Universal Closed-Interval Convexity 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-20261024-r07-convex-set-0021","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a9d802e7d7c8d0131e8cb6ff165af658db0da0cadbcaa346510a8c5739b6ea82","family_id":"qf-mod24-convexset-interval-certificate","seed":202610247021,"semantic_template":"qf-mod24-convexset-interval-certificate/{\"lower\":-22,\"upper\":23}"},"fields":{"title":"A Universal Closed-Interval Convexity Certificate","stem":"Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. Nonempty closed interval [lower,upper] has lower=-22, upper=23, allowing equality. Give (lower,upper,all x,y in interval and all lambda in [0,1],convex?).","answer":{"kind":"expression","value":"(-22, 23, all x,y in interval and all lambda in [0,1], Yes)","explanation":"For arbitrary x,y in the interval and 0≤lambda≤1, multiplying both endpoint inequalities by nonnegative lambda and 1−lambda and adding keeps their weighted sum between the bounds. This covers every endpoint pair and lambda, not samples. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (-22, 23, all x,y in interval and all lambda in [0,1], Yes)."},"skills":["A Universal Closed-Interval Convexity 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-20261024-r07-convex-set-0022","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"86780be23bf132e37aeb14806d70d13b3b853bcfc615d9ca829380ac6a2d32a0","family_id":"qf-mod24-convexset-interval-certificate","seed":202610247022,"semantic_template":"qf-mod24-convexset-interval-certificate/{\"lower\":-23,\"upper\":24}"},"fields":{"title":"A Universal Closed-Interval Convexity Certificate","stem":"Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. Nonempty closed interval [lower,upper] has lower=-23, upper=24, allowing equality. Give (lower,upper,all x,y in interval and all lambda in [0,1],convex?).","answer":{"kind":"expression","value":"(-23, 24, all x,y in interval and all lambda in [0,1], Yes)","explanation":"For arbitrary x,y in the interval and 0≤lambda≤1, multiplying both endpoint inequalities by nonnegative lambda and 1−lambda and adding keeps their weighted sum between the bounds. This covers every endpoint pair and lambda, not samples. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (-23, 24, all x,y in interval and all lambda in [0,1], Yes)."},"skills":["A Universal Closed-Interval Convexity 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-20261024-r07-convex-set-0023","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2312f444c972a04c882ac1fca3048edfcacb8f60beeb57d5ac2cefe619b17380","family_id":"qf-mod24-convexset-interval-certificate","seed":202610247023,"semantic_template":"qf-mod24-convexset-interval-certificate/{\"lower\":-24,\"upper\":25}"},"fields":{"title":"A Universal Closed-Interval Convexity Certificate","stem":"Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. Nonempty closed interval [lower,upper] has lower=-24, upper=25, allowing equality. Give (lower,upper,all x,y in interval and all lambda in [0,1],convex?).","answer":{"kind":"expression","value":"(-24, 25, all x,y in interval and all lambda in [0,1], Yes)","explanation":"For arbitrary x,y in the interval and 0≤lambda≤1, multiplying both endpoint inequalities by nonnegative lambda and 1−lambda and adding keeps their weighted sum between the bounds. This covers every endpoint pair and lambda, not samples. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (-24, 25, all x,y in interval and all lambda in [0,1], Yes)."},"skills":["A Universal Closed-Interval Convexity 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-20261024-r07-convex-set-0024","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6532c463b008f80e03ffd7422e2e61e075e367205aa9da510958036eb3bb3ab8","family_id":"qf-mod24-convexset-interval-certificate","seed":202610247024,"semantic_template":"qf-mod24-convexset-interval-certificate/{\"lower\":-25,\"upper\":26}"},"fields":{"title":"A Universal Closed-Interval Convexity Certificate","stem":"Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. Nonempty closed interval [lower,upper] has lower=-25, upper=26, allowing equality. Give (lower,upper,all x,y in interval and all lambda in [0,1],convex?).","answer":{"kind":"expression","value":"(-25, 26, all x,y in interval and all lambda in [0,1], Yes)","explanation":"For arbitrary x,y in the interval and 0≤lambda≤1, multiplying both endpoint inequalities by nonnegative lambda and 1−lambda and adding keeps their weighted sum between the bounds. This covers every endpoint pair and lambda, not samples. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (-25, 26, all x,y in interval and all lambda in [0,1], Yes)."},"skills":["A Universal Closed-Interval Convexity 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-20261024-r07-convex-set-0025","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d43d47e5e068621958c2e5b657131811ba91b0c640ed4b3399112ec275b6f8c2","family_id":"qf-mod24-convexset-interval-certificate","seed":202610247025,"semantic_template":"qf-mod24-convexset-interval-certificate/{\"lower\":-26,\"upper\":27}"},"fields":{"title":"A Universal Closed-Interval Convexity Certificate","stem":"Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. Nonempty closed interval [lower,upper] has lower=-26, upper=27, allowing equality. Give (lower,upper,all x,y in interval and all lambda in [0,1],convex?).","answer":{"kind":"expression","value":"(-26, 27, all x,y in interval and all lambda in [0,1], Yes)","explanation":"For arbitrary x,y in the interval and 0≤lambda≤1, multiplying both endpoint inequalities by nonnegative lambda and 1−lambda and adding keeps their weighted sum between the bounds. This covers every endpoint pair and lambda, not samples. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (-26, 27, all x,y in interval and all lambda in [0,1], Yes)."},"skills":["A Universal Closed-Interval Convexity 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-20261024-r07-convex-set-0026","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ae25a807df624409007db694c4e5f4f599087a1a27e9ed26dd5d346d82cf11a7","family_id":"qf-mod24-convexset-box-certificate","seed":202610247026,"semantic_template":"qf-mod24-convexset-box-certificate/{\"bounds\":[-2,3,-1,4]}"},"fields":{"title":"Coordinatewise Universal Convexity of a Closed Box","stem":"Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. Bounds [x_min,x_max,y_min,y_max]=[-2, 3, -1, 4] are ordered and may have zero widths. Give (bounds,coordinatewise interval certificate for every lambda in [0,1],convex?).","answer":{"kind":"expression","value":"([-2, 3, -1, 4], coordinatewise interval certificate for every lambda in [0,1], Yes)","explanation":"Apply the universal interval inequality independently in each coordinate. Their conjunction covers every point pair and every permitted lambda in the box. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-2, 3, -1, 4], coordinatewise interval certificate for every lambda in [0,1], Yes)."},"skills":["Coordinatewise Universal Convexity of a Closed Box"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261024-r07-convex-set-0027","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d753ded42d117b0dc54acca9e19645a814e5dae1758c857a6e17edb25104361a","family_id":"qf-mod24-convexset-box-certificate","seed":202610247027,"semantic_template":"qf-mod24-convexset-box-certificate/{\"bounds\":[-3,4,-2,5]}"},"fields":{"title":"Coordinatewise Universal Convexity of a Closed Box","stem":"Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. Bounds [x_min,x_max,y_min,y_max]=[-3, 4, -2, 5] are ordered and may have zero widths. Give (bounds,coordinatewise interval certificate for every lambda in [0,1],convex?).","answer":{"kind":"expression","value":"([-3, 4, -2, 5], coordinatewise interval certificate for every lambda in [0,1], Yes)","explanation":"Apply the universal interval inequality independently in each coordinate. Their conjunction covers every point pair and every permitted lambda in the box. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-3, 4, -2, 5], coordinatewise interval certificate for every lambda in [0,1], Yes)."},"skills":["Coordinatewise Universal Convexity of a Closed Box"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261024-r07-convex-set-0028","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d72b001c3d278fc8fe98518319627093a7bf16db1e0298b843a6f0cc9aa20ebb","family_id":"qf-mod24-convexset-box-certificate","seed":202610247028,"semantic_template":"qf-mod24-convexset-box-certificate/{\"bounds\":[-4,5,-3,6]}"},"fields":{"title":"Coordinatewise Universal Convexity of a Closed Box","stem":"Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. Bounds [x_min,x_max,y_min,y_max]=[-4, 5, -3, 6] are ordered and may have zero widths. Give (bounds,coordinatewise interval certificate for every lambda in [0,1],convex?).","answer":{"kind":"expression","value":"([-4, 5, -3, 6], coordinatewise interval certificate for every lambda in [0,1], Yes)","explanation":"Apply the universal interval inequality independently in each coordinate. Their conjunction covers every point pair and every permitted lambda in the box. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-4, 5, -3, 6], coordinatewise interval certificate for every lambda in [0,1], Yes)."},"skills":["Coordinatewise Universal Convexity of a Closed Box"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261024-r07-convex-set-0029","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"75341abe40fb4a1438c8efb4fdf1f6f43cff305da8d11204edfe0ff6f734244a","family_id":"qf-mod24-convexset-box-certificate","seed":202610247029,"semantic_template":"qf-mod24-convexset-box-certificate/{\"bounds\":[-5,6,-4,7]}"},"fields":{"title":"Coordinatewise Universal Convexity of a Closed Box","stem":"Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. Bounds [x_min,x_max,y_min,y_max]=[-5, 6, -4, 7] are ordered and may have zero widths. Give (bounds,coordinatewise interval certificate for every lambda in [0,1],convex?).","answer":{"kind":"expression","value":"([-5, 6, -4, 7], coordinatewise interval certificate for every lambda in [0,1], Yes)","explanation":"Apply the universal interval inequality independently in each coordinate. Their conjunction covers every point pair and every permitted lambda in the box. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-5, 6, -4, 7], coordinatewise interval certificate for every lambda in [0,1], Yes)."},"skills":["Coordinatewise Universal Convexity of a Closed Box"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261024-r07-convex-set-0030","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5402147feca3dc1e5a8202b0cdfb202d370e89bb140e9d4ce507ea8f739bb51d","family_id":"qf-mod24-convexset-box-certificate","seed":202610247030,"semantic_template":"qf-mod24-convexset-box-certificate/{\"bounds\":[-6,7,-5,8]}"},"fields":{"title":"Coordinatewise Universal Convexity of a Closed Box","stem":"Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. Bounds [x_min,x_max,y_min,y_max]=[-6, 7, -5, 8] are ordered and may have zero widths. Give (bounds,coordinatewise interval certificate for every lambda in [0,1],convex?).","answer":{"kind":"expression","value":"([-6, 7, -5, 8], coordinatewise interval certificate for every lambda in [0,1], Yes)","explanation":"Apply the universal interval inequality independently in each coordinate. Their conjunction covers every point pair and every permitted lambda in the box. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-6, 7, -5, 8], coordinatewise interval certificate for every lambda in [0,1], Yes)."},"skills":["Coordinatewise Universal Convexity of a Closed Box"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261024-r07-convex-set-0031","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2f2ca2486bb8195dce93d82adca288de7febbc7c75ab89fdaa060bb6b2c4a676","family_id":"qf-mod24-convexset-box-certificate","seed":202610247031,"semantic_template":"qf-mod24-convexset-box-certificate/{\"bounds\":[-7,8,-6,9]}"},"fields":{"title":"Coordinatewise Universal Convexity of a Closed Box","stem":"Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. Bounds [x_min,x_max,y_min,y_max]=[-7, 8, -6, 9] are ordered and may have zero widths. Give (bounds,coordinatewise interval certificate for every lambda in [0,1],convex?).","answer":{"kind":"expression","value":"([-7, 8, -6, 9], coordinatewise interval certificate for every lambda in [0,1], Yes)","explanation":"Apply the universal interval inequality independently in each coordinate. Their conjunction covers every point pair and every permitted lambda in the box. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-7, 8, -6, 9], coordinatewise interval certificate for every lambda in [0,1], Yes)."},"skills":["Coordinatewise Universal Convexity of a Closed Box"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261024-r07-convex-set-0032","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0ab91fbf0e681a768dfcb6d5b807ddbf2e074dad1a9252158e792cb50db79f44","family_id":"qf-mod24-convexset-box-certificate","seed":202610247032,"semantic_template":"qf-mod24-convexset-box-certificate/{\"bounds\":[-8,9,-7,10]}"},"fields":{"title":"Coordinatewise Universal Convexity of a Closed Box","stem":"Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. Bounds [x_min,x_max,y_min,y_max]=[-8, 9, -7, 10] are ordered and may have zero widths. Give (bounds,coordinatewise interval certificate for every lambda in [0,1],convex?).","answer":{"kind":"expression","value":"([-8, 9, -7, 10], coordinatewise interval certificate for every lambda in [0,1], Yes)","explanation":"Apply the universal interval inequality independently in each coordinate. Their conjunction covers every point pair and every permitted lambda in the box. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-8, 9, -7, 10], coordinatewise interval certificate for every lambda in [0,1], Yes)."},"skills":["Coordinatewise Universal Convexity of a Closed Box"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261024-r07-convex-set-0033","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3432f7251aa60bd8e45ba29168b741b6ac1bf1081c6357d837caa3020e8e4520","family_id":"qf-mod24-convexset-box-certificate","seed":202610247033,"semantic_template":"qf-mod24-convexset-box-certificate/{\"bounds\":[-9,10,-8,11]}"},"fields":{"title":"Coordinatewise Universal Convexity of a Closed Box","stem":"Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. Bounds [x_min,x_max,y_min,y_max]=[-9, 10, -8, 11] are ordered and may have zero widths. Give (bounds,coordinatewise interval certificate for every lambda in [0,1],convex?).","answer":{"kind":"expression","value":"([-9, 10, -8, 11], coordinatewise interval certificate for every lambda in [0,1], Yes)","explanation":"Apply the universal interval inequality independently in each coordinate. Their conjunction covers every point pair and every permitted lambda in the box. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-9, 10, -8, 11], coordinatewise interval certificate for every lambda in [0,1], Yes)."},"skills":["Coordinatewise Universal Convexity of a Closed Box"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261024-r07-convex-set-0034","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0875d9868fd179f092f2e898132946f07400952b204c6ba15b122fd1d5763078","family_id":"qf-mod24-convexset-box-certificate","seed":202610247034,"semantic_template":"qf-mod24-convexset-box-certificate/{\"bounds\":[-10,11,-9,12]}"},"fields":{"title":"Coordinatewise Universal Convexity of a Closed Box","stem":"Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. Bounds [x_min,x_max,y_min,y_max]=[-10, 11, -9, 12] are ordered and may have zero widths. Give (bounds,coordinatewise interval certificate for every lambda in [0,1],convex?).","answer":{"kind":"expression","value":"([-10, 11, -9, 12], coordinatewise interval certificate for every lambda in [0,1], Yes)","explanation":"Apply the universal interval inequality independently in each coordinate. Their conjunction covers every point pair and every permitted lambda in the box. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-10, 11, -9, 12], coordinatewise interval certificate for every lambda in [0,1], Yes)."},"skills":["Coordinatewise Universal Convexity of a Closed Box"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261024-r07-convex-set-0035","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1c6371cf5cf0bfc3c2bf0f7fdbcc0af511d92d0f128d92e24c33382c4f645baa","family_id":"qf-mod24-convexset-box-certificate","seed":202610247035,"semantic_template":"qf-mod24-convexset-box-certificate/{\"bounds\":[-11,12,-10,13]}"},"fields":{"title":"Coordinatewise Universal Convexity of a Closed Box","stem":"Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. Bounds [x_min,x_max,y_min,y_max]=[-11, 12, -10, 13] are ordered and may have zero widths. Give (bounds,coordinatewise interval certificate for every lambda in [0,1],convex?).","answer":{"kind":"expression","value":"([-11, 12, -10, 13], coordinatewise interval certificate for every lambda in [0,1], Yes)","explanation":"Apply the universal interval inequality independently in each coordinate. Their conjunction covers every point pair and every permitted lambda in the box. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-11, 12, -10, 13], coordinatewise interval certificate for every lambda in [0,1], Yes)."},"skills":["Coordinatewise Universal Convexity of a Closed Box"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261024-r07-convex-set-0036","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"133a7ea41a473a9ef25914c1c2147f68ac596f6d041a5cefc6573067b57c5c19","family_id":"qf-mod24-convexset-box-certificate","seed":202610247036,"semantic_template":"qf-mod24-convexset-box-certificate/{\"bounds\":[-12,13,-11,14]}"},"fields":{"title":"Coordinatewise Universal Convexity of a Closed Box","stem":"Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. Bounds [x_min,x_max,y_min,y_max]=[-12, 13, -11, 14] are ordered and may have zero widths. Give (bounds,coordinatewise interval certificate for every lambda in [0,1],convex?).","answer":{"kind":"expression","value":"([-12, 13, -11, 14], coordinatewise interval certificate for every lambda in [0,1], Yes)","explanation":"Apply the universal interval inequality independently in each coordinate. Their conjunction covers every point pair and every permitted lambda in the box. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-12, 13, -11, 14], coordinatewise interval certificate for every lambda in [0,1], Yes)."},"skills":["Coordinatewise Universal Convexity of a Closed Box"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261024-r07-convex-set-0037","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c9705eb12e87b94aa150ce0c00426c8ea74255c4104757466f4f437467eee4df","family_id":"qf-mod24-convexset-box-certificate","seed":202610247037,"semantic_template":"qf-mod24-convexset-box-certificate/{\"bounds\":[-13,14,-12,15]}"},"fields":{"title":"Coordinatewise Universal Convexity of a Closed Box","stem":"Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. Bounds [x_min,x_max,y_min,y_max]=[-13, 14, -12, 15] are ordered and may have zero widths. Give (bounds,coordinatewise interval certificate for every lambda in [0,1],convex?).","answer":{"kind":"expression","value":"([-13, 14, -12, 15], coordinatewise interval certificate for every lambda in [0,1], Yes)","explanation":"Apply the universal interval inequality independently in each coordinate. Their conjunction covers every point pair and every permitted lambda in the box. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-13, 14, -12, 15], coordinatewise interval certificate for every lambda in [0,1], Yes)."},"skills":["Coordinatewise Universal Convexity of a Closed Box"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261024-r07-convex-set-0038","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d0926f9dff3e38067370bd98480e72b4a182a76fa68ab59e9d0d6090c610ffcb","family_id":"qf-mod24-convexset-box-certificate","seed":202610247038,"semantic_template":"qf-mod24-convexset-box-certificate/{\"bounds\":[-14,15,-13,16]}"},"fields":{"title":"Coordinatewise Universal Convexity of a Closed Box","stem":"Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. Bounds [x_min,x_max,y_min,y_max]=[-14, 15, -13, 16] are ordered and may have zero widths. Give (bounds,coordinatewise interval certificate for every lambda in [0,1],convex?).","answer":{"kind":"expression","value":"([-14, 15, -13, 16], coordinatewise interval certificate for every lambda in [0,1], Yes)","explanation":"Apply the universal interval inequality independently in each coordinate. Their conjunction covers every point pair and every permitted lambda in the box. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-14, 15, -13, 16], coordinatewise interval certificate for every lambda in [0,1], Yes)."},"skills":["Coordinatewise Universal Convexity of a Closed Box"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261024-r07-convex-set-0039","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f97d4380ebafdf6deeed4f677c0b775be07cb17d6fa589f73f217ead39529359","family_id":"qf-mod24-convexset-box-certificate","seed":202610247039,"semantic_template":"qf-mod24-convexset-box-certificate/{\"bounds\":[-15,16,-14,17]}"},"fields":{"title":"Coordinatewise Universal Convexity of a Closed Box","stem":"Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. Bounds [x_min,x_max,y_min,y_max]=[-15, 16, -14, 17] are ordered and may have zero widths. Give (bounds,coordinatewise interval certificate for every lambda in [0,1],convex?).","answer":{"kind":"expression","value":"([-15, 16, -14, 17], coordinatewise interval certificate for every lambda in [0,1], Yes)","explanation":"Apply the universal interval inequality independently in each coordinate. Their conjunction covers every point pair and every permitted lambda in the box. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-15, 16, -14, 17], coordinatewise interval certificate for every lambda in [0,1], Yes)."},"skills":["Coordinatewise Universal Convexity of a Closed Box"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261024-r07-convex-set-0040","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ee69ae3845de097a401d5ec3a1037001d2c43416e92a3d7f96214e5570e3e934","family_id":"qf-mod24-convexset-box-certificate","seed":202610247040,"semantic_template":"qf-mod24-convexset-box-certificate/{\"bounds\":[-16,17,-15,18]}"},"fields":{"title":"Coordinatewise Universal Convexity of a Closed Box","stem":"Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. Bounds [x_min,x_max,y_min,y_max]=[-16, 17, -15, 18] are ordered and may have zero widths. Give (bounds,coordinatewise interval certificate for every lambda in [0,1],convex?).","answer":{"kind":"expression","value":"([-16, 17, -15, 18], coordinatewise interval certificate for every lambda in [0,1], Yes)","explanation":"Apply the universal interval inequality independently in each coordinate. Their conjunction covers every point pair and every permitted lambda in the box. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-16, 17, -15, 18], coordinatewise interval certificate for every lambda in [0,1], Yes)."},"skills":["Coordinatewise Universal Convexity of a Closed Box"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261024-r07-convex-set-0041","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b20db7d489da00109705c0eb3ceaded0648c38e08333182dc229a33bd1707561","family_id":"qf-mod24-convexset-box-certificate","seed":202610247041,"semantic_template":"qf-mod24-convexset-box-certificate/{\"bounds\":[-17,18,-16,19]}"},"fields":{"title":"Coordinatewise Universal Convexity of a Closed Box","stem":"Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. Bounds [x_min,x_max,y_min,y_max]=[-17, 18, -16, 19] are ordered and may have zero widths. Give (bounds,coordinatewise interval certificate for every lambda in [0,1],convex?).","answer":{"kind":"expression","value":"([-17, 18, -16, 19], coordinatewise interval certificate for every lambda in [0,1], Yes)","explanation":"Apply the universal interval inequality independently in each coordinate. Their conjunction covers every point pair and every permitted lambda in the box. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-17, 18, -16, 19], coordinatewise interval certificate for every lambda in [0,1], Yes)."},"skills":["Coordinatewise Universal Convexity of a Closed Box"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261024-r07-convex-set-0042","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b3c616d4f108ff6d79feaaa40846891dc9465410270bda37e1665cbf804ab52a","family_id":"qf-mod24-convexset-box-certificate","seed":202610247042,"semantic_template":"qf-mod24-convexset-box-certificate/{\"bounds\":[-18,19,-17,20]}"},"fields":{"title":"Coordinatewise Universal Convexity of a Closed Box","stem":"Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. Bounds [x_min,x_max,y_min,y_max]=[-18, 19, -17, 20] are ordered and may have zero widths. Give (bounds,coordinatewise interval certificate for every lambda in [0,1],convex?).","answer":{"kind":"expression","value":"([-18, 19, -17, 20], coordinatewise interval certificate for every lambda in [0,1], Yes)","explanation":"Apply the universal interval inequality independently in each coordinate. Their conjunction covers every point pair and every permitted lambda in the box. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-18, 19, -17, 20], coordinatewise interval certificate for every lambda in [0,1], Yes)."},"skills":["Coordinatewise Universal Convexity of a Closed Box"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261024-r07-convex-set-0043","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d33f1fe7cdb08980361ba0c571466485f4f456869d99006cbc8b41f4552ad0fd","family_id":"qf-mod24-convexset-box-certificate","seed":202610247043,"semantic_template":"qf-mod24-convexset-box-certificate/{\"bounds\":[-19,20,-18,21]}"},"fields":{"title":"Coordinatewise Universal Convexity of a Closed Box","stem":"Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. Bounds [x_min,x_max,y_min,y_max]=[-19, 20, -18, 21] are ordered and may have zero widths. Give (bounds,coordinatewise interval certificate for every lambda in [0,1],convex?).","answer":{"kind":"expression","value":"([-19, 20, -18, 21], coordinatewise interval certificate for every lambda in [0,1], Yes)","explanation":"Apply the universal interval inequality independently in each coordinate. Their conjunction covers every point pair and every permitted lambda in the box. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-19, 20, -18, 21], coordinatewise interval certificate for every lambda in [0,1], Yes)."},"skills":["Coordinatewise Universal Convexity of a Closed Box"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261024-r07-convex-set-0044","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"be32afbe12372072a6ec7c3b982f0b51947b844059fda1e866808dbfd045072e","family_id":"qf-mod24-convexset-box-certificate","seed":202610247044,"semantic_template":"qf-mod24-convexset-box-certificate/{\"bounds\":[-20,21,-19,22]}"},"fields":{"title":"Coordinatewise Universal Convexity of a Closed Box","stem":"Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. Bounds [x_min,x_max,y_min,y_max]=[-20, 21, -19, 22] are ordered and may have zero widths. Give (bounds,coordinatewise interval certificate for every lambda in [0,1],convex?).","answer":{"kind":"expression","value":"([-20, 21, -19, 22], coordinatewise interval certificate for every lambda in [0,1], Yes)","explanation":"Apply the universal interval inequality independently in each coordinate. Their conjunction covers every point pair and every permitted lambda in the box. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-20, 21, -19, 22], coordinatewise interval certificate for every lambda in [0,1], Yes)."},"skills":["Coordinatewise Universal Convexity of a Closed Box"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261024-r07-convex-set-0045","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dc992ddc9f264ac39d61d4c1c808572a7f90c3183366efe4c7c97595bdd705b3","family_id":"qf-mod24-convexset-box-certificate","seed":202610247045,"semantic_template":"qf-mod24-convexset-box-certificate/{\"bounds\":[-21,22,-20,23]}"},"fields":{"title":"Coordinatewise Universal Convexity of a Closed Box","stem":"Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. Bounds [x_min,x_max,y_min,y_max]=[-21, 22, -20, 23] are ordered and may have zero widths. Give (bounds,coordinatewise interval certificate for every lambda in [0,1],convex?).","answer":{"kind":"expression","value":"([-21, 22, -20, 23], coordinatewise interval certificate for every lambda in [0,1], Yes)","explanation":"Apply the universal interval inequality independently in each coordinate. Their conjunction covers every point pair and every permitted lambda in the box. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-21, 22, -20, 23], coordinatewise interval certificate for every lambda in [0,1], Yes)."},"skills":["Coordinatewise Universal Convexity of a Closed Box"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261024-r07-convex-set-0046","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"69d679be0fdafcc77c4d91e13f1de02673a72c7b48cd9df33a950c3e0ced9c24","family_id":"qf-mod24-convexset-box-certificate","seed":202610247046,"semantic_template":"qf-mod24-convexset-box-certificate/{\"bounds\":[-22,23,-21,24]}"},"fields":{"title":"Coordinatewise Universal Convexity of a Closed Box","stem":"Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. Bounds [x_min,x_max,y_min,y_max]=[-22, 23, -21, 24] are ordered and may have zero widths. Give (bounds,coordinatewise interval certificate for every lambda in [0,1],convex?).","answer":{"kind":"expression","value":"([-22, 23, -21, 24], coordinatewise interval certificate for every lambda in [0,1], Yes)","explanation":"Apply the universal interval inequality independently in each coordinate. Their conjunction covers every point pair and every permitted lambda in the box. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-22, 23, -21, 24], coordinatewise interval certificate for every lambda in [0,1], Yes)."},"skills":["Coordinatewise Universal Convexity of a Closed Box"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261024-r07-convex-set-0047","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0040bd09b81cd7106d2fbd111084d732f992339e97671ec5b4fddbc578ec5f2e","family_id":"qf-mod24-convexset-box-certificate","seed":202610247047,"semantic_template":"qf-mod24-convexset-box-certificate/{\"bounds\":[-23,24,-22,25]}"},"fields":{"title":"Coordinatewise Universal Convexity of a Closed Box","stem":"Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. Bounds [x_min,x_max,y_min,y_max]=[-23, 24, -22, 25] are ordered and may have zero widths. Give (bounds,coordinatewise interval certificate for every lambda in [0,1],convex?).","answer":{"kind":"expression","value":"([-23, 24, -22, 25], coordinatewise interval certificate for every lambda in [0,1], Yes)","explanation":"Apply the universal interval inequality independently in each coordinate. Their conjunction covers every point pair and every permitted lambda in the box. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-23, 24, -22, 25], coordinatewise interval certificate for every lambda in [0,1], Yes)."},"skills":["Coordinatewise Universal Convexity of a Closed Box"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261024-r07-convex-set-0048","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"79b67a394ec4442f794ebe2bfd9f2ac95b045b3b633798ab8530ab54c863eafc","family_id":"qf-mod24-convexset-box-certificate","seed":202610247048,"semantic_template":"qf-mod24-convexset-box-certificate/{\"bounds\":[-24,25,-23,26]}"},"fields":{"title":"Coordinatewise Universal Convexity of a Closed Box","stem":"Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. Bounds [x_min,x_max,y_min,y_max]=[-24, 25, -23, 26] are ordered and may have zero widths. Give (bounds,coordinatewise interval certificate for every lambda in [0,1],convex?).","answer":{"kind":"expression","value":"([-24, 25, -23, 26], coordinatewise interval certificate for every lambda in [0,1], Yes)","explanation":"Apply the universal interval inequality independently in each coordinate. Their conjunction covers every point pair and every permitted lambda in the box. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-24, 25, -23, 26], coordinatewise interval certificate for every lambda in [0,1], Yes)."},"skills":["Coordinatewise Universal Convexity of a Closed Box"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261024-r07-convex-set-0049","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a93376a61ddc5711edd49987f2b69775fafba15d3c797f7cda4f472f1acefa74","family_id":"qf-mod24-convexset-box-certificate","seed":202610247049,"semantic_template":"qf-mod24-convexset-box-certificate/{\"bounds\":[-25,26,-24,27]}"},"fields":{"title":"Coordinatewise Universal Convexity of a Closed Box","stem":"Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. Bounds [x_min,x_max,y_min,y_max]=[-25, 26, -24, 27] are ordered and may have zero widths. Give (bounds,coordinatewise interval certificate for every lambda in [0,1],convex?).","answer":{"kind":"expression","value":"([-25, 26, -24, 27], coordinatewise interval certificate for every lambda in [0,1], Yes)","explanation":"Apply the universal interval inequality independently in each coordinate. Their conjunction covers every point pair and every permitted lambda in the box. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-25, 26, -24, 27], coordinatewise interval certificate for every lambda in [0,1], Yes)."},"skills":["Coordinatewise Universal Convexity of a Closed Box"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261024-r07-convex-set-0050","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e2cd3a560442b88e092dd134a7a1a5744736dad80966f1ed665cb1878361507c","family_id":"qf-mod24-convexset-box-certificate","seed":202610247050,"semantic_template":"qf-mod24-convexset-box-certificate/{\"bounds\":[-26,27,-25,28]}"},"fields":{"title":"Coordinatewise Universal Convexity of a Closed Box","stem":"Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. Bounds [x_min,x_max,y_min,y_max]=[-26, 27, -25, 28] are ordered and may have zero widths. Give (bounds,coordinatewise interval certificate for every lambda in [0,1],convex?).","answer":{"kind":"expression","value":"([-26, 27, -25, 28], coordinatewise interval certificate for every lambda in [0,1], Yes)","explanation":"Apply the universal interval inequality independently in each coordinate. Their conjunction covers every point pair and every permitted lambda in the box. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-26, 27, -25, 28], coordinatewise interval certificate for every lambda in [0,1], Yes)."},"skills":["Coordinatewise Universal Convexity of a Closed Box"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261024-r07-convex-set-0051","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"000148dc39c4d2550b0836dc6e28ffc6c83625d0b78f5042cb45470e53223f37","family_id":"qf-mod24-convexset-halfspace-combination","seed":202610247051,"semantic_template":"qf-mod24-convexset-halfspace-combination/{\"bound\":2,\"lambda\":[1,3],\"left\":[0,0],\"normal\":[1,1],\"right\":[1,0]}"},"fields":{"title":"Linear Halfspace Closure at Qualified Endpoints","stem":"Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. normal=[1, 1], bound=2, left=[0, 0], right=[1, 0], lambda=1/3∈[0,1]. Both endpoints must satisfy normal·point≤bound. Give (normal,bound,left,right,left dot product,right dot product,lambda,weighted point,its dot product,inside?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.","answer":{"kind":"expression","value":"([1, 1], 2, [0, 0], [1, 0], 0, 1, 1/3, [2/3, 0], 2/3, Yes)","explanation":"Linearity makes the weighted-point dot product the same nonnegative weighted sum of endpoint dot products, hence at most the bound. Boundary equality is included. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([1, 1], 2, [0, 0], [1, 0], 0, 1, 1/3, [2/3, 0], 2/3, Yes)."},"skills":["Linear Halfspace Closure at Qualified Endpoints"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261024-r07-convex-set-0052","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5380dafd4ddca59636c5aaa40f486992431d6d4c4cf214d25cf0f419d13da80b","family_id":"qf-mod24-convexset-halfspace-combination","seed":202610247052,"semantic_template":"qf-mod24-convexset-halfspace-combination/{\"bound\":3,\"lambda\":[2,4],\"left\":[0,0],\"normal\":[1,1],\"right\":[2,0]}"},"fields":{"title":"Linear Halfspace Closure at Qualified Endpoints","stem":"Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. normal=[1, 1], bound=3, left=[0, 0], right=[2, 0], lambda=2/4∈[0,1]. Both endpoints must satisfy normal·point≤bound. Give (normal,bound,left,right,left dot product,right dot product,lambda,weighted point,its dot product,inside?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.","answer":{"kind":"expression","value":"([1, 1], 3, [0, 0], [2, 0], 0, 2, 1/2, [1, 0], 1, Yes)","explanation":"Linearity makes the weighted-point dot product the same nonnegative weighted sum of endpoint dot products, hence at most the bound. Boundary equality is included. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([1, 1], 3, [0, 0], [2, 0], 0, 2, 1/2, [1, 0], 1, Yes)."},"skills":["Linear Halfspace Closure at Qualified Endpoints"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261024-r07-convex-set-0053","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"122643b0e20ba1b9cc5436b34e91d60ec6a69f80153bf36399049629f12a83be","family_id":"qf-mod24-convexset-halfspace-combination","seed":202610247053,"semantic_template":"qf-mod24-convexset-halfspace-combination/{\"bound\":4,\"lambda\":[3,5],\"left\":[0,0],\"normal\":[1,1],\"right\":[3,0]}"},"fields":{"title":"Linear Halfspace Closure at Qualified Endpoints","stem":"Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. normal=[1, 1], bound=4, left=[0, 0], right=[3, 0], lambda=3/5∈[0,1]. Both endpoints must satisfy normal·point≤bound. Give (normal,bound,left,right,left dot product,right dot product,lambda,weighted point,its dot product,inside?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.","answer":{"kind":"expression","value":"([1, 1], 4, [0, 0], [3, 0], 0, 3, 3/5, [6/5, 0], 6/5, Yes)","explanation":"Linearity makes the weighted-point dot product the same nonnegative weighted sum of endpoint dot products, hence at most the bound. Boundary equality is included. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([1, 1], 4, [0, 0], [3, 0], 0, 3, 3/5, [6/5, 0], 6/5, Yes)."},"skills":["Linear Halfspace Closure at Qualified Endpoints"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261024-r07-convex-set-0054","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8e9d71c9d6c662e7a26c7f8116d3e909a7145ed56166d4592c4ec006ab5e0a50","family_id":"qf-mod24-convexset-halfspace-combination","seed":202610247054,"semantic_template":"qf-mod24-convexset-halfspace-combination/{\"bound\":5,\"lambda\":[4,6],\"left\":[0,0],\"normal\":[1,1],\"right\":[4,0]}"},"fields":{"title":"Linear Halfspace Closure at Qualified Endpoints","stem":"Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. normal=[1, 1], bound=5, left=[0, 0], right=[4, 0], lambda=4/6∈[0,1]. Both endpoints must satisfy normal·point≤bound. Give (normal,bound,left,right,left dot product,right dot product,lambda,weighted point,its dot product,inside?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.","answer":{"kind":"expression","value":"([1, 1], 5, [0, 0], [4, 0], 0, 4, 2/3, [4/3, 0], 4/3, Yes)","explanation":"Linearity makes the weighted-point dot product the same nonnegative weighted sum of endpoint dot products, hence at most the bound. Boundary equality is included. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([1, 1], 5, [0, 0], [4, 0], 0, 4, 2/3, [4/3, 0], 4/3, Yes)."},"skills":["Linear Halfspace Closure at Qualified Endpoints"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261024-r07-convex-set-0055","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"96ae11eb6b68d237d5fdcb322ed2a3f3298ef2e802bb99bbc6e03e525560c5aa","family_id":"qf-mod24-convexset-halfspace-combination","seed":202610247055,"semantic_template":"qf-mod24-convexset-halfspace-combination/{\"bound\":6,\"lambda\":[5,7],\"left\":[0,0],\"normal\":[1,1],\"right\":[5,0]}"},"fields":{"title":"Linear Halfspace Closure at Qualified Endpoints","stem":"Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. normal=[1, 1], bound=6, left=[0, 0], right=[5, 0], lambda=5/7∈[0,1]. Both endpoints must satisfy normal·point≤bound. Give (normal,bound,left,right,left dot product,right dot product,lambda,weighted point,its dot product,inside?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.","answer":{"kind":"expression","value":"([1, 1], 6, [0, 0], [5, 0], 0, 5, 5/7, [10/7, 0], 10/7, Yes)","explanation":"Linearity makes the weighted-point dot product the same nonnegative weighted sum of endpoint dot products, hence at most the bound. Boundary equality is included. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([1, 1], 6, [0, 0], [5, 0], 0, 5, 5/7, [10/7, 0], 10/7, Yes)."},"skills":["Linear Halfspace Closure at Qualified Endpoints"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261024-r07-convex-set-0056","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"82196855cd652ade911e9433f4e3a449212e23ea7bfeb4b3605feef388acd826","family_id":"qf-mod24-convexset-halfspace-combination","seed":202610247056,"semantic_template":"qf-mod24-convexset-halfspace-combination/{\"bound\":7,\"lambda\":[6,8],\"left\":[0,0],\"normal\":[1,1],\"right\":[6,0]}"},"fields":{"title":"Linear Halfspace Closure at Qualified Endpoints","stem":"Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. normal=[1, 1], bound=7, left=[0, 0], right=[6, 0], lambda=6/8∈[0,1]. Both endpoints must satisfy normal·point≤bound. Give (normal,bound,left,right,left dot product,right dot product,lambda,weighted point,its dot product,inside?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.","answer":{"kind":"expression","value":"([1, 1], 7, [0, 0], [6, 0], 0, 6, 3/4, [3/2, 0], 3/2, Yes)","explanation":"Linearity makes the weighted-point dot product the same nonnegative weighted sum of endpoint dot products, hence at most the bound. Boundary equality is included. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([1, 1], 7, [0, 0], [6, 0], 0, 6, 3/4, [3/2, 0], 3/2, Yes)."},"skills":["Linear Halfspace Closure at Qualified Endpoints"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261024-r07-convex-set-0057","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a1fe945407e67164b78222c4c61a770b1a24303284cce4691d23689535bde054","family_id":"qf-mod24-convexset-halfspace-combination","seed":202610247057,"semantic_template":"qf-mod24-convexset-halfspace-combination/{\"bound\":8,\"lambda\":[7,9],\"left\":[0,0],\"normal\":[1,1],\"right\":[7,0]}"},"fields":{"title":"Linear Halfspace Closure at Qualified Endpoints","stem":"Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. normal=[1, 1], bound=8, left=[0, 0], right=[7, 0], lambda=7/9∈[0,1]. Both endpoints must satisfy normal·point≤bound. Give (normal,bound,left,right,left dot product,right dot product,lambda,weighted point,its dot product,inside?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.","answer":{"kind":"expression","value":"([1, 1], 8, [0, 0], [7, 0], 0, 7, 7/9, [14/9, 0], 14/9, Yes)","explanation":"Linearity makes the weighted-point dot product the same nonnegative weighted sum of endpoint dot products, hence at most the bound. Boundary equality is included. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([1, 1], 8, [0, 0], [7, 0], 0, 7, 7/9, [14/9, 0], 14/9, Yes)."},"skills":["Linear Halfspace Closure at Qualified Endpoints"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261024-r07-convex-set-0058","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"88477290e9cbaafa0d87baae470bf278d730eeff620c10830312356f8641c482","family_id":"qf-mod24-convexset-halfspace-combination","seed":202610247058,"semantic_template":"qf-mod24-convexset-halfspace-combination/{\"bound\":9,\"lambda\":[8,10],\"left\":[0,0],\"normal\":[1,1],\"right\":[8,0]}"},"fields":{"title":"Linear Halfspace Closure at Qualified Endpoints","stem":"Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. normal=[1, 1], bound=9, left=[0, 0], right=[8, 0], lambda=8/10∈[0,1]. Both endpoints must satisfy normal·point≤bound. Give (normal,bound,left,right,left dot product,right dot product,lambda,weighted point,its dot product,inside?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.","answer":{"kind":"expression","value":"([1, 1], 9, [0, 0], [8, 0], 0, 8, 4/5, [8/5, 0], 8/5, Yes)","explanation":"Linearity makes the weighted-point dot product the same nonnegative weighted sum of endpoint dot products, hence at most the bound. Boundary equality is included. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([1, 1], 9, [0, 0], [8, 0], 0, 8, 4/5, [8/5, 0], 8/5, Yes)."},"skills":["Linear Halfspace Closure at Qualified Endpoints"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261024-r07-convex-set-0059","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"36fec5ac43b0e41d530ba42db140cab08f559731f77c8522bca3b9a9d64f8616","family_id":"qf-mod24-convexset-halfspace-combination","seed":202610247059,"semantic_template":"qf-mod24-convexset-halfspace-combination/{\"bound\":10,\"lambda\":[9,11],\"left\":[0,0],\"normal\":[1,1],\"right\":[9,0]}"},"fields":{"title":"Linear Halfspace Closure at Qualified Endpoints","stem":"Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. normal=[1, 1], bound=10, left=[0, 0], right=[9, 0], lambda=9/11∈[0,1]. Both endpoints must satisfy normal·point≤bound. Give (normal,bound,left,right,left dot product,right dot product,lambda,weighted point,its dot product,inside?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.","answer":{"kind":"expression","value":"([1, 1], 10, [0, 0], [9, 0], 0, 9, 9/11, [18/11, 0], 18/11, Yes)","explanation":"Linearity makes the weighted-point dot product the same nonnegative weighted sum of endpoint dot products, hence at most the bound. Boundary equality is included. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([1, 1], 10, [0, 0], [9, 0], 0, 9, 9/11, [18/11, 0], 18/11, Yes)."},"skills":["Linear Halfspace Closure at Qualified Endpoints"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261024-r07-convex-set-0060","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"26ce28af03f03f7e02d8947ee70ea7fce30c6cc7ec3d1f1795717ef8ee3e149d","family_id":"qf-mod24-convexset-halfspace-combination","seed":202610247060,"semantic_template":"qf-mod24-convexset-halfspace-combination/{\"bound\":11,\"lambda\":[10,12],\"left\":[0,0],\"normal\":[1,1],\"right\":[10,0]}"},"fields":{"title":"Linear Halfspace Closure at Qualified Endpoints","stem":"Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. normal=[1, 1], bound=11, left=[0, 0], right=[10, 0], lambda=10/12∈[0,1]. Both endpoints must satisfy normal·point≤bound. Give (normal,bound,left,right,left dot product,right dot product,lambda,weighted point,its dot product,inside?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.","answer":{"kind":"expression","value":"([1, 1], 11, [0, 0], [10, 0], 0, 10, 5/6, [5/3, 0], 5/3, Yes)","explanation":"Linearity makes the weighted-point dot product the same nonnegative weighted sum of endpoint dot products, hence at most the bound. Boundary equality is included. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([1, 1], 11, [0, 0], [10, 0], 0, 10, 5/6, [5/3, 0], 5/3, Yes)."},"skills":["Linear Halfspace Closure at Qualified Endpoints"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261024-r07-convex-set-0061","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"77cd1a4f45850a4f348f2c02b63b6eec1a27e3f5181cd25f0066fd84dbe8911f","family_id":"qf-mod24-convexset-halfspace-combination","seed":202610247061,"semantic_template":"qf-mod24-convexset-halfspace-combination/{\"bound\":12,\"lambda\":[11,13],\"left\":[0,0],\"normal\":[1,1],\"right\":[11,0]}"},"fields":{"title":"Linear Halfspace Closure at Qualified Endpoints","stem":"Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. normal=[1, 1], bound=12, left=[0, 0], right=[11, 0], lambda=11/13∈[0,1]. Both endpoints must satisfy normal·point≤bound. Give (normal,bound,left,right,left dot product,right dot product,lambda,weighted point,its dot product,inside?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.","answer":{"kind":"expression","value":"([1, 1], 12, [0, 0], [11, 0], 0, 11, 11/13, [22/13, 0], 22/13, Yes)","explanation":"Linearity makes the weighted-point dot product the same nonnegative weighted sum of endpoint dot products, hence at most the bound. Boundary equality is included. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([1, 1], 12, [0, 0], [11, 0], 0, 11, 11/13, [22/13, 0], 22/13, Yes)."},"skills":["Linear Halfspace Closure at Qualified Endpoints"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261024-r07-convex-set-0062","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"26d589d20cbc00877ec4efb71f27c9dc43ad33f06f46d60a77326c8f77ea2990","family_id":"qf-mod24-convexset-halfspace-combination","seed":202610247062,"semantic_template":"qf-mod24-convexset-halfspace-combination/{\"bound\":13,\"lambda\":[12,14],\"left\":[0,0],\"normal\":[1,1],\"right\":[12,0]}"},"fields":{"title":"Linear Halfspace Closure at Qualified Endpoints","stem":"Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. normal=[1, 1], bound=13, left=[0, 0], right=[12, 0], lambda=12/14∈[0,1]. Both endpoints must satisfy normal·point≤bound. Give (normal,bound,left,right,left dot product,right dot product,lambda,weighted point,its dot product,inside?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.","answer":{"kind":"expression","value":"([1, 1], 13, [0, 0], [12, 0], 0, 12, 6/7, [12/7, 0], 12/7, Yes)","explanation":"Linearity makes the weighted-point dot product the same nonnegative weighted sum of endpoint dot products, hence at most the bound. Boundary equality is included. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([1, 1], 13, [0, 0], [12, 0], 0, 12, 6/7, [12/7, 0], 12/7, Yes)."},"skills":["Linear Halfspace Closure at Qualified Endpoints"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261024-r07-convex-set-0063","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0cb318e121c9a9e46dc2fa32be04e8a221e14948ad22a0bad667ef10d2d616f2","family_id":"qf-mod24-convexset-halfspace-combination","seed":202610247063,"semantic_template":"qf-mod24-convexset-halfspace-combination/{\"bound\":14,\"lambda\":[13,15],\"left\":[0,0],\"normal\":[1,1],\"right\":[13,0]}"},"fields":{"title":"Linear Halfspace Closure at Qualified Endpoints","stem":"Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. normal=[1, 1], bound=14, left=[0, 0], right=[13, 0], lambda=13/15∈[0,1]. Both endpoints must satisfy normal·point≤bound. Give (normal,bound,left,right,left dot product,right dot product,lambda,weighted point,its dot product,inside?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.","answer":{"kind":"expression","value":"([1, 1], 14, [0, 0], [13, 0], 0, 13, 13/15, [26/15, 0], 26/15, Yes)","explanation":"Linearity makes the weighted-point dot product the same nonnegative weighted sum of endpoint dot products, hence at most the bound. Boundary equality is included. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([1, 1], 14, [0, 0], [13, 0], 0, 13, 13/15, [26/15, 0], 26/15, Yes)."},"skills":["Linear Halfspace Closure at Qualified Endpoints"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261024-r07-convex-set-0064","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d6f8fdb2c83b4e8d6e6b6c215a0446fb64e02765f8be27f8682ee4416f6c4459","family_id":"qf-mod24-convexset-halfspace-combination","seed":202610247064,"semantic_template":"qf-mod24-convexset-halfspace-combination/{\"bound\":15,\"lambda\":[14,16],\"left\":[0,0],\"normal\":[1,1],\"right\":[14,0]}"},"fields":{"title":"Linear Halfspace Closure at Qualified Endpoints","stem":"Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. normal=[1, 1], bound=15, left=[0, 0], right=[14, 0], lambda=14/16∈[0,1]. Both endpoints must satisfy normal·point≤bound. Give (normal,bound,left,right,left dot product,right dot product,lambda,weighted point,its dot product,inside?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.","answer":{"kind":"expression","value":"([1, 1], 15, [0, 0], [14, 0], 0, 14, 7/8, [7/4, 0], 7/4, Yes)","explanation":"Linearity makes the weighted-point dot product the same nonnegative weighted sum of endpoint dot products, hence at most the bound. Boundary equality is included. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([1, 1], 15, [0, 0], [14, 0], 0, 14, 7/8, [7/4, 0], 7/4, Yes)."},"skills":["Linear Halfspace Closure at Qualified Endpoints"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261024-r07-convex-set-0065","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e011bb590838b08362872f5322fdc439357df2c8750a69017f91d25f9998755d","family_id":"qf-mod24-convexset-halfspace-combination","seed":202610247065,"semantic_template":"qf-mod24-convexset-halfspace-combination/{\"bound\":16,\"lambda\":[15,17],\"left\":[0,0],\"normal\":[1,1],\"right\":[15,0]}"},"fields":{"title":"Linear Halfspace Closure at Qualified Endpoints","stem":"Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. normal=[1, 1], bound=16, left=[0, 0], right=[15, 0], lambda=15/17∈[0,1]. Both endpoints must satisfy normal·point≤bound. Give (normal,bound,left,right,left dot product,right dot product,lambda,weighted point,its dot product,inside?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.","answer":{"kind":"expression","value":"([1, 1], 16, [0, 0], [15, 0], 0, 15, 15/17, [30/17, 0], 30/17, Yes)","explanation":"Linearity makes the weighted-point dot product the same nonnegative weighted sum of endpoint dot products, hence at most the bound. Boundary equality is included. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([1, 1], 16, [0, 0], [15, 0], 0, 15, 15/17, [30/17, 0], 30/17, Yes)."},"skills":["Linear Halfspace Closure at Qualified Endpoints"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261024-r07-convex-set-0066","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"93314dc7d8b7041f8223bb3f02681914577725ec2cfc2edf5181fd2a202c6cf1","family_id":"qf-mod24-convexset-halfspace-combination","seed":202610247066,"semantic_template":"qf-mod24-convexset-halfspace-combination/{\"bound\":17,\"lambda\":[16,18],\"left\":[0,0],\"normal\":[1,1],\"right\":[16,0]}"},"fields":{"title":"Linear Halfspace Closure at Qualified Endpoints","stem":"Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. normal=[1, 1], bound=17, left=[0, 0], right=[16, 0], lambda=16/18∈[0,1]. Both endpoints must satisfy normal·point≤bound. Give (normal,bound,left,right,left dot product,right dot product,lambda,weighted point,its dot product,inside?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.","answer":{"kind":"expression","value":"([1, 1], 17, [0, 0], [16, 0], 0, 16, 8/9, [16/9, 0], 16/9, Yes)","explanation":"Linearity makes the weighted-point dot product the same nonnegative weighted sum of endpoint dot products, hence at most the bound. Boundary equality is included. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([1, 1], 17, [0, 0], [16, 0], 0, 16, 8/9, [16/9, 0], 16/9, Yes)."},"skills":["Linear Halfspace Closure at Qualified Endpoints"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261024-r07-convex-set-0067","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"95c6b017183b88c9a616d3b16b71c2d07c3daf8f7e998496334e257e6f250152","family_id":"qf-mod24-convexset-halfspace-combination","seed":202610247067,"semantic_template":"qf-mod24-convexset-halfspace-combination/{\"bound\":18,\"lambda\":[17,19],\"left\":[0,0],\"normal\":[1,1],\"right\":[17,0]}"},"fields":{"title":"Linear Halfspace Closure at Qualified Endpoints","stem":"Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. normal=[1, 1], bound=18, left=[0, 0], right=[17, 0], lambda=17/19∈[0,1]. Both endpoints must satisfy normal·point≤bound. Give (normal,bound,left,right,left dot product,right dot product,lambda,weighted point,its dot product,inside?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.","answer":{"kind":"expression","value":"([1, 1], 18, [0, 0], [17, 0], 0, 17, 17/19, [34/19, 0], 34/19, Yes)","explanation":"Linearity makes the weighted-point dot product the same nonnegative weighted sum of endpoint dot products, hence at most the bound. Boundary equality is included. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([1, 1], 18, [0, 0], [17, 0], 0, 17, 17/19, [34/19, 0], 34/19, Yes)."},"skills":["Linear Halfspace Closure at Qualified Endpoints"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261024-r07-convex-set-0068","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"af748537c138b23fffe59769d47605f705b7956e756c3825f7cbe58665c879f8","family_id":"qf-mod24-convexset-halfspace-combination","seed":202610247068,"semantic_template":"qf-mod24-convexset-halfspace-combination/{\"bound\":19,\"lambda\":[18,20],\"left\":[0,0],\"normal\":[1,1],\"right\":[18,0]}"},"fields":{"title":"Linear Halfspace Closure at Qualified Endpoints","stem":"Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. normal=[1, 1], bound=19, left=[0, 0], right=[18, 0], lambda=18/20∈[0,1]. Both endpoints must satisfy normal·point≤bound. Give (normal,bound,left,right,left dot product,right dot product,lambda,weighted point,its dot product,inside?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.","answer":{"kind":"expression","value":"([1, 1], 19, [0, 0], [18, 0], 0, 18, 9/10, [9/5, 0], 9/5, Yes)","explanation":"Linearity makes the weighted-point dot product the same nonnegative weighted sum of endpoint dot products, hence at most the bound. Boundary equality is included. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([1, 1], 19, [0, 0], [18, 0], 0, 18, 9/10, [9/5, 0], 9/5, Yes)."},"skills":["Linear Halfspace Closure at Qualified Endpoints"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261024-r07-convex-set-0069","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b97a1c4ba31394e3f512a0c1e8603de9c3dff99b52da2fff38adc60b46b945bd","family_id":"qf-mod24-convexset-halfspace-combination","seed":202610247069,"semantic_template":"qf-mod24-convexset-halfspace-combination/{\"bound\":20,\"lambda\":[19,21],\"left\":[0,0],\"normal\":[1,1],\"right\":[19,0]}"},"fields":{"title":"Linear Halfspace Closure at Qualified Endpoints","stem":"Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. normal=[1, 1], bound=20, left=[0, 0], right=[19, 0], lambda=19/21∈[0,1]. Both endpoints must satisfy normal·point≤bound. Give (normal,bound,left,right,left dot product,right dot product,lambda,weighted point,its dot product,inside?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.","answer":{"kind":"expression","value":"([1, 1], 20, [0, 0], [19, 0], 0, 19, 19/21, [38/21, 0], 38/21, Yes)","explanation":"Linearity makes the weighted-point dot product the same nonnegative weighted sum of endpoint dot products, hence at most the bound. Boundary equality is included. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([1, 1], 20, [0, 0], [19, 0], 0, 19, 19/21, [38/21, 0], 38/21, Yes)."},"skills":["Linear Halfspace Closure at Qualified Endpoints"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261024-r07-convex-set-0070","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e439dbeb768e78f364df94bccc9a9ed353673f8af1b6541065fe27bdcc6b61c9","family_id":"qf-mod24-convexset-halfspace-combination","seed":202610247070,"semantic_template":"qf-mod24-convexset-halfspace-combination/{\"bound\":21,\"lambda\":[20,22],\"left\":[0,0],\"normal\":[1,1],\"right\":[20,0]}"},"fields":{"title":"Linear Halfspace Closure at Qualified Endpoints","stem":"Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. normal=[1, 1], bound=21, left=[0, 0], right=[20, 0], lambda=20/22∈[0,1]. Both endpoints must satisfy normal·point≤bound. Give (normal,bound,left,right,left dot product,right dot product,lambda,weighted point,its dot product,inside?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.","answer":{"kind":"expression","value":"([1, 1], 21, [0, 0], [20, 0], 0, 20, 10/11, [20/11, 0], 20/11, Yes)","explanation":"Linearity makes the weighted-point dot product the same nonnegative weighted sum of endpoint dot products, hence at most the bound. Boundary equality is included. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([1, 1], 21, [0, 0], [20, 0], 0, 20, 10/11, [20/11, 0], 20/11, Yes)."},"skills":["Linear Halfspace Closure at Qualified Endpoints"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261024-r07-convex-set-0071","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b8d159b34514cffe7f209406bcd23c8dbb8183386a10d5afcc61a2bddb819675","family_id":"qf-mod24-convexset-halfspace-combination","seed":202610247071,"semantic_template":"qf-mod24-convexset-halfspace-combination/{\"bound\":22,\"lambda\":[21,23],\"left\":[0,0],\"normal\":[1,1],\"right\":[21,0]}"},"fields":{"title":"Linear Halfspace Closure at Qualified Endpoints","stem":"Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. normal=[1, 1], bound=22, left=[0, 0], right=[21, 0], lambda=21/23∈[0,1]. Both endpoints must satisfy normal·point≤bound. Give (normal,bound,left,right,left dot product,right dot product,lambda,weighted point,its dot product,inside?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.","answer":{"kind":"expression","value":"([1, 1], 22, [0, 0], [21, 0], 0, 21, 21/23, [42/23, 0], 42/23, Yes)","explanation":"Linearity makes the weighted-point dot product the same nonnegative weighted sum of endpoint dot products, hence at most the bound. Boundary equality is included. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([1, 1], 22, [0, 0], [21, 0], 0, 21, 21/23, [42/23, 0], 42/23, Yes)."},"skills":["Linear Halfspace Closure at Qualified Endpoints"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261024-r07-convex-set-0072","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"670492c0f112e169177d70f62da5146906212b84be73227d0ad967095cc49618","family_id":"qf-mod24-convexset-halfspace-combination","seed":202610247072,"semantic_template":"qf-mod24-convexset-halfspace-combination/{\"bound\":23,\"lambda\":[22,24],\"left\":[0,0],\"normal\":[1,1],\"right\":[22,0]}"},"fields":{"title":"Linear Halfspace Closure at Qualified Endpoints","stem":"Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. normal=[1, 1], bound=23, left=[0, 0], right=[22, 0], lambda=22/24∈[0,1]. Both endpoints must satisfy normal·point≤bound. Give (normal,bound,left,right,left dot product,right dot product,lambda,weighted point,its dot product,inside?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.","answer":{"kind":"expression","value":"([1, 1], 23, [0, 0], [22, 0], 0, 22, 11/12, [11/6, 0], 11/6, Yes)","explanation":"Linearity makes the weighted-point dot product the same nonnegative weighted sum of endpoint dot products, hence at most the bound. Boundary equality is included. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([1, 1], 23, [0, 0], [22, 0], 0, 22, 11/12, [11/6, 0], 11/6, Yes)."},"skills":["Linear Halfspace Closure at Qualified Endpoints"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261024-r07-convex-set-0073","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fe9ecc5ab89ae14b4b0b3b417b479391f8640488f1170da16c8192c38b5559a8","family_id":"qf-mod24-convexset-halfspace-combination","seed":202610247073,"semantic_template":"qf-mod24-convexset-halfspace-combination/{\"bound\":24,\"lambda\":[23,25],\"left\":[0,0],\"normal\":[1,1],\"right\":[23,0]}"},"fields":{"title":"Linear Halfspace Closure at Qualified Endpoints","stem":"Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. normal=[1, 1], bound=24, left=[0, 0], right=[23, 0], lambda=23/25∈[0,1]. Both endpoints must satisfy normal·point≤bound. Give (normal,bound,left,right,left dot product,right dot product,lambda,weighted point,its dot product,inside?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.","answer":{"kind":"expression","value":"([1, 1], 24, [0, 0], [23, 0], 0, 23, 23/25, [46/25, 0], 46/25, Yes)","explanation":"Linearity makes the weighted-point dot product the same nonnegative weighted sum of endpoint dot products, hence at most the bound. Boundary equality is included. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([1, 1], 24, [0, 0], [23, 0], 0, 23, 23/25, [46/25, 0], 46/25, Yes)."},"skills":["Linear Halfspace Closure at Qualified Endpoints"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261024-r07-convex-set-0074","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"03d8ace86f0391db5e3d72bb23c2d711a65db1c6c07845c9f2e0f8722dfa1ced","family_id":"qf-mod24-convexset-halfspace-combination","seed":202610247074,"semantic_template":"qf-mod24-convexset-halfspace-combination/{\"bound\":25,\"lambda\":[24,26],\"left\":[0,0],\"normal\":[1,1],\"right\":[24,0]}"},"fields":{"title":"Linear Halfspace Closure at Qualified Endpoints","stem":"Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. normal=[1, 1], bound=25, left=[0, 0], right=[24, 0], lambda=24/26∈[0,1]. Both endpoints must satisfy normal·point≤bound. Give (normal,bound,left,right,left dot product,right dot product,lambda,weighted point,its dot product,inside?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.","answer":{"kind":"expression","value":"([1, 1], 25, [0, 0], [24, 0], 0, 24, 12/13, [24/13, 0], 24/13, Yes)","explanation":"Linearity makes the weighted-point dot product the same nonnegative weighted sum of endpoint dot products, hence at most the bound. Boundary equality is included. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([1, 1], 25, [0, 0], [24, 0], 0, 24, 12/13, [24/13, 0], 24/13, Yes)."},"skills":["Linear Halfspace Closure at Qualified Endpoints"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261024-r07-convex-set-0075","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"af916b7bed29aa6d22427d56df99e83097725e6e0d03f520170d7effad05782e","family_id":"qf-mod24-convexset-halfspace-combination","seed":202610247075,"semantic_template":"qf-mod24-convexset-halfspace-combination/{\"bound\":26,\"lambda\":[25,27],\"left\":[0,0],\"normal\":[1,1],\"right\":[25,0]}"},"fields":{"title":"Linear Halfspace Closure at Qualified Endpoints","stem":"Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. normal=[1, 1], bound=26, left=[0, 0], right=[25, 0], lambda=25/27∈[0,1]. Both endpoints must satisfy normal·point≤bound. Give (normal,bound,left,right,left dot product,right dot product,lambda,weighted point,its dot product,inside?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.","answer":{"kind":"expression","value":"([1, 1], 26, [0, 0], [25, 0], 0, 25, 25/27, [50/27, 0], 50/27, Yes)","explanation":"Linearity makes the weighted-point dot product the same nonnegative weighted sum of endpoint dot products, hence at most the bound. Boundary equality is included. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([1, 1], 26, [0, 0], [25, 0], 0, 25, 25/27, [50/27, 0], 50/27, Yes)."},"skills":["Linear Halfspace Closure at Qualified Endpoints"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261024-r07-convex-set-0076","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"077df9aab245fef266ca44c2a55d670823b5dd94019e1414d747d1ce116128a8","family_id":"qf-mod24-convexset-finite-midpoint-counterexample","seed":202610247076,"semantic_template":"qf-mod24-convexset-finite-midpoint-counterexample/{\"left\":[2,0],\"points\":[[0,0],[0,2],[2,0]],\"right\":[0,2]}"},"fields":{"title":"A Missing Midpoint Refutes Convexity, but a Present One Does Not Prove It","stem":"Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. Distinct finite points=[[0, 0], [2, 0], [0, 2]] retain input order; endpoints left=[2, 0], right=[0, 2] must belong to the set. Give (points,left,right,exact midpoint,midpoint present?,is this a nonconvexity witness?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.","answer":{"kind":"expression","value":"([[0, 0], [2, 0], [0, 2]], [2, 0], [0, 2], [1, 1], No, Yes)","explanation":"An absent midpoint between two members is one exact counterexample. A present midpoint for this pair does not prove the whole set convex; another pair or weight may still fail. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[0, 0], [2, 0], [0, 2]], [2, 0], [0, 2], [1, 1], No, Yes)."},"skills":["A Missing Midpoint Refutes Convexity, but a Present One Does Not Prove It"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261024-r07-convex-set-0077","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"928fd0272a60818070e8b01c869f7e53ad43d46fb0d19b123c1fe98224231ec3","family_id":"qf-mod24-convexset-finite-midpoint-counterexample","seed":202610247077,"semantic_template":"qf-mod24-convexset-finite-midpoint-counterexample/{\"left\":[4,0],\"points\":[[0,0],[0,4],[4,0]],\"right\":[0,4]}"},"fields":{"title":"A Missing Midpoint Refutes Convexity, but a Present One Does Not Prove It","stem":"Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. Distinct finite points=[[0, 0], [4, 0], [0, 4]] retain input order; endpoints left=[4, 0], right=[0, 4] must belong to the set. Give (points,left,right,exact midpoint,midpoint present?,is this a nonconvexity witness?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.","answer":{"kind":"expression","value":"([[0, 0], [4, 0], [0, 4]], [4, 0], [0, 4], [2, 2], No, Yes)","explanation":"An absent midpoint between two members is one exact counterexample. A present midpoint for this pair does not prove the whole set convex; another pair or weight may still fail. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[0, 0], [4, 0], [0, 4]], [4, 0], [0, 4], [2, 2], No, Yes)."},"skills":["A Missing Midpoint Refutes Convexity, but a Present One Does Not Prove It"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261024-r07-convex-set-0078","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"20f34d30c906b9e6dce399081e2ae141751fff35486fd636108f64fa409e2b72","family_id":"qf-mod24-convexset-finite-midpoint-counterexample","seed":202610247078,"semantic_template":"qf-mod24-convexset-finite-midpoint-counterexample/{\"left\":[6,0],\"points\":[[0,0],[0,6],[6,0]],\"right\":[0,6]}"},"fields":{"title":"A Missing Midpoint Refutes Convexity, but a Present One Does Not Prove It","stem":"Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. Distinct finite points=[[0, 0], [6, 0], [0, 6]] retain input order; endpoints left=[6, 0], right=[0, 6] must belong to the set. Give (points,left,right,exact midpoint,midpoint present?,is this a nonconvexity witness?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.","answer":{"kind":"expression","value":"([[0, 0], [6, 0], [0, 6]], [6, 0], [0, 6], [3, 3], No, Yes)","explanation":"An absent midpoint between two members is one exact counterexample. A present midpoint for this pair does not prove the whole set convex; another pair or weight may still fail. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[0, 0], [6, 0], [0, 6]], [6, 0], [0, 6], [3, 3], No, Yes)."},"skills":["A Missing Midpoint Refutes Convexity, but a Present One Does Not Prove It"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261024-r07-convex-set-0079","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"916a86c94686a7f2c36bfdd5bd7b1c835f23ec2cf70b796d3ee92d9f341af605","family_id":"qf-mod24-convexset-finite-midpoint-counterexample","seed":202610247079,"semantic_template":"qf-mod24-convexset-finite-midpoint-counterexample/{\"left\":[8,0],\"points\":[[0,0],[0,8],[8,0]],\"right\":[0,8]}"},"fields":{"title":"A Missing Midpoint Refutes Convexity, but a Present One Does Not Prove It","stem":"Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. Distinct finite points=[[0, 0], [8, 0], [0, 8]] retain input order; endpoints left=[8, 0], right=[0, 8] must belong to the set. Give (points,left,right,exact midpoint,midpoint present?,is this a nonconvexity witness?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.","answer":{"kind":"expression","value":"([[0, 0], [8, 0], [0, 8]], [8, 0], [0, 8], [4, 4], No, Yes)","explanation":"An absent midpoint between two members is one exact counterexample. A present midpoint for this pair does not prove the whole set convex; another pair or weight may still fail. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[0, 0], [8, 0], [0, 8]], [8, 0], [0, 8], [4, 4], No, Yes)."},"skills":["A Missing Midpoint Refutes Convexity, but a Present One Does Not Prove It"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261024-r07-convex-set-0080","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7942630650647fad744212cb612e91056286b9b1fe146aeb6c05bacc0eb2d878","family_id":"qf-mod24-convexset-finite-midpoint-counterexample","seed":202610247080,"semantic_template":"qf-mod24-convexset-finite-midpoint-counterexample/{\"left\":[10,0],\"points\":[[0,0],[0,10],[10,0]],\"right\":[0,10]}"},"fields":{"title":"A Missing Midpoint Refutes Convexity, but a Present One Does Not Prove It","stem":"Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. Distinct finite points=[[0, 0], [10, 0], [0, 10]] retain input order; endpoints left=[10, 0], right=[0, 10] must belong to the set. Give (points,left,right,exact midpoint,midpoint present?,is this a nonconvexity witness?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.","answer":{"kind":"expression","value":"([[0, 0], [10, 0], [0, 10]], [10, 0], [0, 10], [5, 5], No, Yes)","explanation":"An absent midpoint between two members is one exact counterexample. A present midpoint for this pair does not prove the whole set convex; another pair or weight may still fail. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[0, 0], [10, 0], [0, 10]], [10, 0], [0, 10], [5, 5], No, Yes)."},"skills":["A Missing Midpoint Refutes Convexity, but a Present One Does Not Prove It"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261024-r07-convex-set-0081","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"22717a220174905fa5162b3f7dc58d74fb8f14e81937d416494875d487f06f17","family_id":"qf-mod24-convexset-finite-midpoint-counterexample","seed":202610247081,"semantic_template":"qf-mod24-convexset-finite-midpoint-counterexample/{\"left\":[12,0],\"points\":[[0,0],[0,12],[12,0]],\"right\":[0,12]}"},"fields":{"title":"A Missing Midpoint Refutes Convexity, but a Present One Does Not Prove It","stem":"Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. Distinct finite points=[[0, 0], [12, 0], [0, 12]] retain input order; endpoints left=[12, 0], right=[0, 12] must belong to the set. Give (points,left,right,exact midpoint,midpoint present?,is this a nonconvexity witness?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.","answer":{"kind":"expression","value":"([[0, 0], [12, 0], [0, 12]], [12, 0], [0, 12], [6, 6], No, Yes)","explanation":"An absent midpoint between two members is one exact counterexample. A present midpoint for this pair does not prove the whole set convex; another pair or weight may still fail. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[0, 0], [12, 0], [0, 12]], [12, 0], [0, 12], [6, 6], No, Yes)."},"skills":["A Missing Midpoint Refutes Convexity, but a Present One Does Not Prove It"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261024-r07-convex-set-0082","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"163a490ae93190a0210be4df6c4d4954eb8dd61dde4a366493065e2606264363","family_id":"qf-mod24-convexset-finite-midpoint-counterexample","seed":202610247082,"semantic_template":"qf-mod24-convexset-finite-midpoint-counterexample/{\"left\":[14,0],\"points\":[[0,0],[0,14],[14,0]],\"right\":[0,14]}"},"fields":{"title":"A Missing Midpoint Refutes Convexity, but a Present One Does Not Prove It","stem":"Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. Distinct finite points=[[0, 0], [14, 0], [0, 14]] retain input order; endpoints left=[14, 0], right=[0, 14] must belong to the set. Give (points,left,right,exact midpoint,midpoint present?,is this a nonconvexity witness?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.","answer":{"kind":"expression","value":"([[0, 0], [14, 0], [0, 14]], [14, 0], [0, 14], [7, 7], No, Yes)","explanation":"An absent midpoint between two members is one exact counterexample. A present midpoint for this pair does not prove the whole set convex; another pair or weight may still fail. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[0, 0], [14, 0], [0, 14]], [14, 0], [0, 14], [7, 7], No, Yes)."},"skills":["A Missing Midpoint Refutes Convexity, but a Present One Does Not Prove It"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261024-r07-convex-set-0083","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8a596766bc4875b20ff6807294e6d00442c4cfe67ab95820832015794f9a4d7e","family_id":"qf-mod24-convexset-finite-midpoint-counterexample","seed":202610247083,"semantic_template":"qf-mod24-convexset-finite-midpoint-counterexample/{\"left\":[16,0],\"points\":[[0,0],[0,16],[16,0]],\"right\":[0,16]}"},"fields":{"title":"A Missing Midpoint Refutes Convexity, but a Present One Does Not Prove It","stem":"Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. Distinct finite points=[[0, 0], [16, 0], [0, 16]] retain input order; endpoints left=[16, 0], right=[0, 16] must belong to the set. Give (points,left,right,exact midpoint,midpoint present?,is this a nonconvexity witness?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.","answer":{"kind":"expression","value":"([[0, 0], [16, 0], [0, 16]], [16, 0], [0, 16], [8, 8], No, Yes)","explanation":"An absent midpoint between two members is one exact counterexample. A present midpoint for this pair does not prove the whole set convex; another pair or weight may still fail. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[0, 0], [16, 0], [0, 16]], [16, 0], [0, 16], [8, 8], No, Yes)."},"skills":["A Missing Midpoint Refutes Convexity, but a Present One Does Not Prove It"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261024-r07-convex-set-0084","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"78f4abe58e9037e7398dd639d8b38f74baddc3118b32228bc2de3723e76c31ee","family_id":"qf-mod24-convexset-finite-midpoint-counterexample","seed":202610247084,"semantic_template":"qf-mod24-convexset-finite-midpoint-counterexample/{\"left\":[18,0],\"points\":[[0,0],[0,18],[18,0]],\"right\":[0,18]}"},"fields":{"title":"A Missing Midpoint Refutes Convexity, but a Present One Does Not Prove It","stem":"Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. Distinct finite points=[[0, 0], [18, 0], [0, 18]] retain input order; endpoints left=[18, 0], right=[0, 18] must belong to the set. Give (points,left,right,exact midpoint,midpoint present?,is this a nonconvexity witness?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.","answer":{"kind":"expression","value":"([[0, 0], [18, 0], [0, 18]], [18, 0], [0, 18], [9, 9], No, Yes)","explanation":"An absent midpoint between two members is one exact counterexample. A present midpoint for this pair does not prove the whole set convex; another pair or weight may still fail. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[0, 0], [18, 0], [0, 18]], [18, 0], [0, 18], [9, 9], No, Yes)."},"skills":["A Missing Midpoint Refutes Convexity, but a Present One Does Not Prove It"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261024-r07-convex-set-0085","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"00430768eb87d48a889658644c70f65cba2dc76b7b6db2d83b3c1b549285bc8a","family_id":"qf-mod24-convexset-finite-midpoint-counterexample","seed":202610247085,"semantic_template":"qf-mod24-convexset-finite-midpoint-counterexample/{\"left\":[20,0],\"points\":[[0,0],[0,20],[20,0]],\"right\":[0,20]}"},"fields":{"title":"A Missing Midpoint Refutes Convexity, but a Present One Does Not Prove It","stem":"Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. Distinct finite points=[[0, 0], [20, 0], [0, 20]] retain input order; endpoints left=[20, 0], right=[0, 20] must belong to the set. Give (points,left,right,exact midpoint,midpoint present?,is this a nonconvexity witness?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.","answer":{"kind":"expression","value":"([[0, 0], [20, 0], [0, 20]], [20, 0], [0, 20], [10, 10], No, Yes)","explanation":"An absent midpoint between two members is one exact counterexample. A present midpoint for this pair does not prove the whole set convex; another pair or weight may still fail. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[0, 0], [20, 0], [0, 20]], [20, 0], [0, 20], [10, 10], No, Yes)."},"skills":["A Missing Midpoint Refutes Convexity, but a Present One Does Not Prove It"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261024-r07-convex-set-0086","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"42fa216ece3f5a336444abe9147056e24b3654dd0a094566c6fc0a7de8067ae8","family_id":"qf-mod24-convexset-finite-midpoint-counterexample","seed":202610247086,"semantic_template":"qf-mod24-convexset-finite-midpoint-counterexample/{\"left\":[22,0],\"points\":[[0,0],[0,22],[22,0]],\"right\":[0,22]}"},"fields":{"title":"A Missing Midpoint Refutes Convexity, but a Present One Does Not Prove It","stem":"Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. Distinct finite points=[[0, 0], [22, 0], [0, 22]] retain input order; endpoints left=[22, 0], right=[0, 22] must belong to the set. Give (points,left,right,exact midpoint,midpoint present?,is this a nonconvexity witness?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.","answer":{"kind":"expression","value":"([[0, 0], [22, 0], [0, 22]], [22, 0], [0, 22], [11, 11], No, Yes)","explanation":"An absent midpoint between two members is one exact counterexample. A present midpoint for this pair does not prove the whole set convex; another pair or weight may still fail. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[0, 0], [22, 0], [0, 22]], [22, 0], [0, 22], [11, 11], No, Yes)."},"skills":["A Missing Midpoint Refutes Convexity, but a Present One Does Not Prove It"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261024-r07-convex-set-0087","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5f7663a866fd42710098be4a94b91d26ff74d56da8a2530a1c5cce65f0444d58","family_id":"qf-mod24-convexset-finite-midpoint-counterexample","seed":202610247087,"semantic_template":"qf-mod24-convexset-finite-midpoint-counterexample/{\"left\":[24,0],\"points\":[[0,0],[0,24],[24,0]],\"right\":[0,24]}"},"fields":{"title":"A Missing Midpoint Refutes Convexity, but a Present One Does Not Prove It","stem":"Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. Distinct finite points=[[0, 0], [24, 0], [0, 24]] retain input order; endpoints left=[24, 0], right=[0, 24] must belong to the set. Give (points,left,right,exact midpoint,midpoint present?,is this a nonconvexity witness?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.","answer":{"kind":"expression","value":"([[0, 0], [24, 0], [0, 24]], [24, 0], [0, 24], [12, 12], No, Yes)","explanation":"An absent midpoint between two members is one exact counterexample. A present midpoint for this pair does not prove the whole set convex; another pair or weight may still fail. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[0, 0], [24, 0], [0, 24]], [24, 0], [0, 24], [12, 12], No, Yes)."},"skills":["A Missing Midpoint Refutes Convexity, but a Present One Does Not Prove It"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261024-r07-convex-set-0088","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bee3e31194bc61c560148c7da8f07a9e776a2388c6e67ed7d6ada8a1927eda88","family_id":"qf-mod24-convexset-finite-midpoint-counterexample","seed":202610247088,"semantic_template":"qf-mod24-convexset-finite-midpoint-counterexample/{\"left\":[26,0],\"points\":[[0,0],[0,26],[26,0]],\"right\":[0,26]}"},"fields":{"title":"A Missing Midpoint Refutes Convexity, but a Present One Does Not Prove It","stem":"Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. Distinct finite points=[[0, 0], [26, 0], [0, 26]] retain input order; endpoints left=[26, 0], right=[0, 26] must belong to the set. Give (points,left,right,exact midpoint,midpoint present?,is this a nonconvexity witness?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.","answer":{"kind":"expression","value":"([[0, 0], [26, 0], [0, 26]], [26, 0], [0, 26], [13, 13], No, Yes)","explanation":"An absent midpoint between two members is one exact counterexample. A present midpoint for this pair does not prove the whole set convex; another pair or weight may still fail. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[0, 0], [26, 0], [0, 26]], [26, 0], [0, 26], [13, 13], No, Yes)."},"skills":["A Missing Midpoint Refutes Convexity, but a Present One Does Not Prove It"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261024-r07-convex-set-0089","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3617027b19f7d9fcc6af86b9a2de694ac7300683ecb9c0c42daecdf739ed5132","family_id":"qf-mod24-convexset-finite-midpoint-counterexample","seed":202610247089,"semantic_template":"qf-mod24-convexset-finite-midpoint-counterexample/{\"left\":[28,0],\"points\":[[0,0],[0,28],[28,0]],\"right\":[0,28]}"},"fields":{"title":"A Missing Midpoint Refutes Convexity, but a Present One Does Not Prove It","stem":"Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. Distinct finite points=[[0, 0], [28, 0], [0, 28]] retain input order; endpoints left=[28, 0], right=[0, 28] must belong to the set. Give (points,left,right,exact midpoint,midpoint present?,is this a nonconvexity witness?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.","answer":{"kind":"expression","value":"([[0, 0], [28, 0], [0, 28]], [28, 0], [0, 28], [14, 14], No, Yes)","explanation":"An absent midpoint between two members is one exact counterexample. A present midpoint for this pair does not prove the whole set convex; another pair or weight may still fail. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[0, 0], [28, 0], [0, 28]], [28, 0], [0, 28], [14, 14], No, Yes)."},"skills":["A Missing Midpoint Refutes Convexity, but a Present One Does Not Prove It"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261024-r07-convex-set-0090","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f37d7513b0c50f53a4df251161f5b550f4bf2751e53c609419464ea47606d95f","family_id":"qf-mod24-convexset-finite-midpoint-counterexample","seed":202610247090,"semantic_template":"qf-mod24-convexset-finite-midpoint-counterexample/{\"left\":[30,0],\"points\":[[0,0],[0,30],[30,0]],\"right\":[0,30]}"},"fields":{"title":"A Missing Midpoint Refutes Convexity, but a Present One Does Not Prove It","stem":"Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. Distinct finite points=[[0, 0], [30, 0], [0, 30]] retain input order; endpoints left=[30, 0], right=[0, 30] must belong to the set. Give (points,left,right,exact midpoint,midpoint present?,is this a nonconvexity witness?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.","answer":{"kind":"expression","value":"([[0, 0], [30, 0], [0, 30]], [30, 0], [0, 30], [15, 15], No, Yes)","explanation":"An absent midpoint between two members is one exact counterexample. A present midpoint for this pair does not prove the whole set convex; another pair or weight may still fail. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[0, 0], [30, 0], [0, 30]], [30, 0], [0, 30], [15, 15], No, Yes)."},"skills":["A Missing Midpoint Refutes Convexity, but a Present One Does Not Prove It"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261024-r07-convex-set-0091","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"af0ce0ebd50b4d0dd24f962df93d4f8c0e01676087a5965c54f6e79617de1d08","family_id":"qf-mod24-convexset-finite-midpoint-counterexample","seed":202610247091,"semantic_template":"qf-mod24-convexset-finite-midpoint-counterexample/{\"left\":[32,0],\"points\":[[0,0],[0,32],[32,0]],\"right\":[0,32]}"},"fields":{"title":"A Missing Midpoint Refutes Convexity, but a Present One Does Not Prove It","stem":"Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. Distinct finite points=[[0, 0], [32, 0], [0, 32]] retain input order; endpoints left=[32, 0], right=[0, 32] must belong to the set. Give (points,left,right,exact midpoint,midpoint present?,is this a nonconvexity witness?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.","answer":{"kind":"expression","value":"([[0, 0], [32, 0], [0, 32]], [32, 0], [0, 32], [16, 16], No, Yes)","explanation":"An absent midpoint between two members is one exact counterexample. A present midpoint for this pair does not prove the whole set convex; another pair or weight may still fail. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[0, 0], [32, 0], [0, 32]], [32, 0], [0, 32], [16, 16], No, Yes)."},"skills":["A Missing Midpoint Refutes Convexity, but a Present One Does Not Prove It"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261024-r07-convex-set-0092","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dd7b03d78b4ef02ecb4e9875c4e6bad42c75f9b740da632dd5d87a167ee369ce","family_id":"qf-mod24-convexset-finite-midpoint-counterexample","seed":202610247092,"semantic_template":"qf-mod24-convexset-finite-midpoint-counterexample/{\"left\":[34,0],\"points\":[[0,0],[0,34],[34,0]],\"right\":[0,34]}"},"fields":{"title":"A Missing Midpoint Refutes Convexity, but a Present One Does Not Prove It","stem":"Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. Distinct finite points=[[0, 0], [34, 0], [0, 34]] retain input order; endpoints left=[34, 0], right=[0, 34] must belong to the set. Give (points,left,right,exact midpoint,midpoint present?,is this a nonconvexity witness?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.","answer":{"kind":"expression","value":"([[0, 0], [34, 0], [0, 34]], [34, 0], [0, 34], [17, 17], No, Yes)","explanation":"An absent midpoint between two members is one exact counterexample. A present midpoint for this pair does not prove the whole set convex; another pair or weight may still fail. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[0, 0], [34, 0], [0, 34]], [34, 0], [0, 34], [17, 17], No, Yes)."},"skills":["A Missing Midpoint Refutes Convexity, but a Present One Does Not Prove It"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261024-r07-convex-set-0093","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"294cf660c851806289d718ab21b4055d33002996621fae3d8c357009534b0823","family_id":"qf-mod24-convexset-finite-midpoint-counterexample","seed":202610247093,"semantic_template":"qf-mod24-convexset-finite-midpoint-counterexample/{\"left\":[36,0],\"points\":[[0,0],[0,36],[36,0]],\"right\":[0,36]}"},"fields":{"title":"A Missing Midpoint Refutes Convexity, but a Present One Does Not Prove It","stem":"Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. Distinct finite points=[[0, 0], [36, 0], [0, 36]] retain input order; endpoints left=[36, 0], right=[0, 36] must belong to the set. Give (points,left,right,exact midpoint,midpoint present?,is this a nonconvexity witness?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.","answer":{"kind":"expression","value":"([[0, 0], [36, 0], [0, 36]], [36, 0], [0, 36], [18, 18], No, Yes)","explanation":"An absent midpoint between two members is one exact counterexample. A present midpoint for this pair does not prove the whole set convex; another pair or weight may still fail. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[0, 0], [36, 0], [0, 36]], [36, 0], [0, 36], [18, 18], No, Yes)."},"skills":["A Missing Midpoint Refutes Convexity, but a Present One Does Not Prove It"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261024-r07-convex-set-0094","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"41d3814e8b9378137d0db6c5053ae859b9fcc566ed1cc8555e8755d7b862f361","family_id":"qf-mod24-convexset-finite-midpoint-counterexample","seed":202610247094,"semantic_template":"qf-mod24-convexset-finite-midpoint-counterexample/{\"left\":[38,0],\"points\":[[0,0],[0,38],[38,0]],\"right\":[0,38]}"},"fields":{"title":"A Missing Midpoint Refutes Convexity, but a Present One Does Not Prove It","stem":"Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. Distinct finite points=[[0, 0], [38, 0], [0, 38]] retain input order; endpoints left=[38, 0], right=[0, 38] must belong to the set. Give (points,left,right,exact midpoint,midpoint present?,is this a nonconvexity witness?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.","answer":{"kind":"expression","value":"([[0, 0], [38, 0], [0, 38]], [38, 0], [0, 38], [19, 19], No, Yes)","explanation":"An absent midpoint between two members is one exact counterexample. A present midpoint for this pair does not prove the whole set convex; another pair or weight may still fail. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[0, 0], [38, 0], [0, 38]], [38, 0], [0, 38], [19, 19], No, Yes)."},"skills":["A Missing Midpoint Refutes Convexity, but a Present One Does Not Prove It"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261024-r07-convex-set-0095","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"51a3a00a75c57c1478bf425d01768287537291b38f6a653e83a5e2ebe397ef40","family_id":"qf-mod24-convexset-finite-midpoint-counterexample","seed":202610247095,"semantic_template":"qf-mod24-convexset-finite-midpoint-counterexample/{\"left\":[40,0],\"points\":[[0,0],[0,40],[40,0]],\"right\":[0,40]}"},"fields":{"title":"A Missing Midpoint Refutes Convexity, but a Present One Does Not Prove It","stem":"Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. Distinct finite points=[[0, 0], [40, 0], [0, 40]] retain input order; endpoints left=[40, 0], right=[0, 40] must belong to the set. Give (points,left,right,exact midpoint,midpoint present?,is this a nonconvexity witness?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.","answer":{"kind":"expression","value":"([[0, 0], [40, 0], [0, 40]], [40, 0], [0, 40], [20, 20], No, Yes)","explanation":"An absent midpoint between two members is one exact counterexample. A present midpoint for this pair does not prove the whole set convex; another pair or weight may still fail. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[0, 0], [40, 0], [0, 40]], [40, 0], [0, 40], [20, 20], No, Yes)."},"skills":["A Missing Midpoint Refutes Convexity, but a Present One Does Not Prove It"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261024-r07-convex-set-0096","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"be94b78fe0211f9970433ec3058882eae8cff31703d519b215cf08f66855e010","family_id":"qf-mod24-convexset-finite-midpoint-counterexample","seed":202610247096,"semantic_template":"qf-mod24-convexset-finite-midpoint-counterexample/{\"left\":[42,0],\"points\":[[0,0],[0,42],[42,0]],\"right\":[0,42]}"},"fields":{"title":"A Missing Midpoint Refutes Convexity, but a Present One Does Not Prove It","stem":"Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. Distinct finite points=[[0, 0], [42, 0], [0, 42]] retain input order; endpoints left=[42, 0], right=[0, 42] must belong to the set. Give (points,left,right,exact midpoint,midpoint present?,is this a nonconvexity witness?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.","answer":{"kind":"expression","value":"([[0, 0], [42, 0], [0, 42]], [42, 0], [0, 42], [21, 21], No, Yes)","explanation":"An absent midpoint between two members is one exact counterexample. A present midpoint for this pair does not prove the whole set convex; another pair or weight may still fail. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[0, 0], [42, 0], [0, 42]], [42, 0], [0, 42], [21, 21], No, Yes)."},"skills":["A Missing Midpoint Refutes Convexity, but a Present One Does Not Prove It"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261024-r07-convex-set-0097","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4b0c9161bab801473bc977ccc0893d73a793afc3f4f456f5db050e3857befca2","family_id":"qf-mod24-convexset-finite-midpoint-counterexample","seed":202610247097,"semantic_template":"qf-mod24-convexset-finite-midpoint-counterexample/{\"left\":[44,0],\"points\":[[0,0],[0,44],[44,0]],\"right\":[0,44]}"},"fields":{"title":"A Missing Midpoint Refutes Convexity, but a Present One Does Not Prove It","stem":"Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. Distinct finite points=[[0, 0], [44, 0], [0, 44]] retain input order; endpoints left=[44, 0], right=[0, 44] must belong to the set. Give (points,left,right,exact midpoint,midpoint present?,is this a nonconvexity witness?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.","answer":{"kind":"expression","value":"([[0, 0], [44, 0], [0, 44]], [44, 0], [0, 44], [22, 22], No, Yes)","explanation":"An absent midpoint between two members is one exact counterexample. A present midpoint for this pair does not prove the whole set convex; another pair or weight may still fail. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[0, 0], [44, 0], [0, 44]], [44, 0], [0, 44], [22, 22], No, Yes)."},"skills":["A Missing Midpoint Refutes Convexity, but a Present One Does Not Prove It"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261024-r07-convex-set-0098","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2def9b72dc9b1ea5125f3aa5a406023b76a04525d48b2b0ac0d9ae7a31addb61","family_id":"qf-mod24-convexset-finite-midpoint-counterexample","seed":202610247098,"semantic_template":"qf-mod24-convexset-finite-midpoint-counterexample/{\"left\":[46,0],\"points\":[[0,0],[0,46],[46,0]],\"right\":[0,46]}"},"fields":{"title":"A Missing Midpoint Refutes Convexity, but a Present One Does Not Prove It","stem":"Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. Distinct finite points=[[0, 0], [46, 0], [0, 46]] retain input order; endpoints left=[46, 0], right=[0, 46] must belong to the set. Give (points,left,right,exact midpoint,midpoint present?,is this a nonconvexity witness?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.","answer":{"kind":"expression","value":"([[0, 0], [46, 0], [0, 46]], [46, 0], [0, 46], [23, 23], No, Yes)","explanation":"An absent midpoint between two members is one exact counterexample. A present midpoint for this pair does not prove the whole set convex; another pair or weight may still fail. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[0, 0], [46, 0], [0, 46]], [46, 0], [0, 46], [23, 23], No, Yes)."},"skills":["A Missing Midpoint Refutes Convexity, but a Present One Does Not Prove It"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261024-r07-convex-set-0099","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5db97a482a4a15192869a404e503e77984b4f11b010e4badced2af92dfb425e6","family_id":"qf-mod24-convexset-finite-midpoint-counterexample","seed":202610247099,"semantic_template":"qf-mod24-convexset-finite-midpoint-counterexample/{\"left\":[48,0],\"points\":[[0,0],[0,48],[48,0]],\"right\":[0,48]}"},"fields":{"title":"A Missing Midpoint Refutes Convexity, but a Present One Does Not Prove It","stem":"Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. Distinct finite points=[[0, 0], [48, 0], [0, 48]] retain input order; endpoints left=[48, 0], right=[0, 48] must belong to the set. Give (points,left,right,exact midpoint,midpoint present?,is this a nonconvexity witness?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.","answer":{"kind":"expression","value":"([[0, 0], [48, 0], [0, 48]], [48, 0], [0, 48], [24, 24], No, Yes)","explanation":"An absent midpoint between two members is one exact counterexample. A present midpoint for this pair does not prove the whole set convex; another pair or weight may still fail. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[0, 0], [48, 0], [0, 48]], [48, 0], [0, 48], [24, 24], No, Yes)."},"skills":["A Missing Midpoint Refutes Convexity, but a Present One Does Not Prove It"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261024-r07-convex-set-0100","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"76b3a7c9a9877efe7c5b9ba88120605e2ea93fd9bd819cb853396fe81a97a462","family_id":"qf-mod24-convexset-finite-midpoint-counterexample","seed":202610247100,"semantic_template":"qf-mod24-convexset-finite-midpoint-counterexample/{\"left\":[50,0],\"points\":[[0,0],[0,50],[50,0]],\"right\":[0,50]}"},"fields":{"title":"A Missing Midpoint Refutes Convexity, but a Present One Does Not Prove It","stem":"Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. Distinct finite points=[[0, 0], [50, 0], [0, 50]] retain input order; endpoints left=[50, 0], right=[0, 50] must belong to the set. Give (points,left,right,exact midpoint,midpoint present?,is this a nonconvexity witness?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.","answer":{"kind":"expression","value":"([[0, 0], [50, 0], [0, 50]], [50, 0], [0, 50], [25, 25], No, Yes)","explanation":"An absent midpoint between two members is one exact counterexample. A present midpoint for this pair does not prove the whole set convex; another pair or weight may still fail. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[0, 0], [50, 0], [0, 50]], [50, 0], [0, 50], [25, 25], No, Yes)."},"skills":["A Missing Midpoint Refutes Convexity, but a Present One Does Not Prove It"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}