{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r02-real-number-0001","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f3a9f732ddd84c9060f171ab16b9b212bba8c66d8acf91f0c696167d59ea7451","family_id":"qf-density14-real-ordered-midpoint","seed":202610142001,"semantic_template":"qf-density14-real-ordered-midpoint/[\"0/1\",\"0/1\"]"},"fields":{"title":"Ordered Rational Midpoint and Endpoint Gaps","stem":"Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. a=0/1≤b=0/1. Give (midpoint, midpoint−a, b−midpoint). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(0/1, 0/1, 0/1)","explanation":"Add and divide by two exactly, then subtract each endpoint separately; no irrational classification or real-completeness proof is claimed. The exact result, retaining the requested order, is (0/1, 0/1, 0/1)."},"skills":["Ordered Rational Midpoint and Endpoint Gaps"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r02-real-number-0002","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c193f185f8fe47b2101ecc2635c79a219be0efc22058990e90fc54aed49abd03","family_id":"qf-density14-real-ordered-midpoint","seed":202610142002,"semantic_template":"qf-density14-real-ordered-midpoint/[\"-1/2\",\"1/2\"]"},"fields":{"title":"Ordered Rational Midpoint and Endpoint Gaps","stem":"Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. a=-1/2≤b=1/2. Give (midpoint, midpoint−a, b−midpoint). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(0/1, 1/2, 1/2)","explanation":"Add and divide by two exactly, then subtract each endpoint separately; no irrational classification or real-completeness proof is claimed. The exact result, retaining the requested order, is (0/1, 1/2, 1/2)."},"skills":["Ordered Rational Midpoint and Endpoint Gaps"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r02-real-number-0003","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"58820e35b56a45faaa2392d1a1cb80e105fad9151eefce98773a5aaf94f6728f","family_id":"qf-density14-real-ordered-midpoint","seed":202610142003,"semantic_template":"qf-density14-real-ordered-midpoint/[\"-1/1\",\"1/1\"]"},"fields":{"title":"Ordered Rational Midpoint and Endpoint Gaps","stem":"Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. a=-2/2≤b=3/3. Give (midpoint, midpoint−a, b−midpoint). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(0/1, 1/1, 1/1)","explanation":"Add and divide by two exactly, then subtract each endpoint separately; no irrational classification or real-completeness proof is claimed. The exact result, retaining the requested order, is (0/1, 1/1, 1/1)."},"skills":["Ordered Rational Midpoint and Endpoint Gaps"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r02-real-number-0004","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"264e636c3c47200cf6e22ff6a8c95689928bc5e1295ea9a47d095c7ffec14fe9","family_id":"qf-density14-real-ordered-midpoint","seed":202610142004,"semantic_template":"qf-density14-real-ordered-midpoint/[\"-3/2\",\"4/3\"]"},"fields":{"title":"Ordered Rational Midpoint and Endpoint Gaps","stem":"Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. a=-3/2≤b=4/3. Give (midpoint, midpoint−a, b−midpoint). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(-1/12, 17/12, 17/12)","explanation":"Add and divide by two exactly, then subtract each endpoint separately; no irrational classification or real-completeness proof is claimed. The exact result, retaining the requested order, is (-1/12, 17/12, 17/12)."},"skills":["Ordered Rational Midpoint and Endpoint Gaps"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r02-real-number-0005","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"74109256c61133308612ef4ea8f049842f110e495c715ef7d004d9d53695e9d5","family_id":"qf-density14-real-ordered-midpoint","seed":202610142005,"semantic_template":"qf-density14-real-ordered-midpoint/[\"-2/1\",\"5/3\"]"},"fields":{"title":"Ordered Rational Midpoint and Endpoint Gaps","stem":"Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. a=-4/2≤b=5/3. Give (midpoint, midpoint−a, b−midpoint). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(-1/6, 11/6, 11/6)","explanation":"Add and divide by two exactly, then subtract each endpoint separately; no irrational classification or real-completeness proof is claimed. The exact result, retaining the requested order, is (-1/6, 11/6, 11/6)."},"skills":["Ordered Rational Midpoint and Endpoint Gaps"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r02-real-number-0006","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"94a54167303f819ad05d9abad8528f3984290aeb6789c1a3d35b718fe7147351","family_id":"qf-density14-real-ordered-midpoint","seed":202610142006,"semantic_template":"qf-density14-real-ordered-midpoint/[\"-5/2\",\"2/1\"]"},"fields":{"title":"Ordered Rational Midpoint and Endpoint Gaps","stem":"Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. a=-5/2≤b=6/3. Give (midpoint, midpoint−a, b−midpoint). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(-1/4, 9/4, 9/4)","explanation":"Add and divide by two exactly, then subtract each endpoint separately; no irrational classification or real-completeness proof is claimed. The exact result, retaining the requested order, is (-1/4, 9/4, 9/4)."},"skills":["Ordered Rational Midpoint and Endpoint Gaps"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r02-real-number-0007","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1831f6f762f9525a9079362f954494fe787866f46e62cf80fe5af36c0da03b30","family_id":"qf-density14-real-ordered-midpoint","seed":202610142007,"semantic_template":"qf-density14-real-ordered-midpoint/[\"-3/1\",\"7/3\"]"},"fields":{"title":"Ordered Rational Midpoint and Endpoint Gaps","stem":"Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. a=-6/2≤b=7/3. Give (midpoint, midpoint−a, b−midpoint). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(-1/3, 8/3, 8/3)","explanation":"Add and divide by two exactly, then subtract each endpoint separately; no irrational classification or real-completeness proof is claimed. The exact result, retaining the requested order, is (-1/3, 8/3, 8/3)."},"skills":["Ordered Rational Midpoint and Endpoint Gaps"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r02-real-number-0008","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8910d0df1e079908f3ec94fb3373bd54f3e818b4831ebdca9d8180d35b3ee314","family_id":"qf-density14-real-ordered-midpoint","seed":202610142008,"semantic_template":"qf-density14-real-ordered-midpoint/[\"-7/2\",\"8/3\"]"},"fields":{"title":"Ordered Rational Midpoint and Endpoint Gaps","stem":"Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. a=-7/2≤b=8/3. Give (midpoint, midpoint−a, b−midpoint). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(-5/12, 37/12, 37/12)","explanation":"Add and divide by two exactly, then subtract each endpoint separately; no irrational classification or real-completeness proof is claimed. The exact result, retaining the requested order, is (-5/12, 37/12, 37/12)."},"skills":["Ordered Rational Midpoint and Endpoint Gaps"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r02-real-number-0009","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"da78280979d8c0feb01c53342019cc531d68034f4887cb2b9e10732d16f22bbf","family_id":"qf-density14-real-ordered-midpoint","seed":202610142009,"semantic_template":"qf-density14-real-ordered-midpoint/[\"-4/1\",\"3/1\"]"},"fields":{"title":"Ordered Rational Midpoint and Endpoint Gaps","stem":"Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. a=-8/2≤b=9/3. Give (midpoint, midpoint−a, b−midpoint). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(-1/2, 7/2, 7/2)","explanation":"Add and divide by two exactly, then subtract each endpoint separately; no irrational classification or real-completeness proof is claimed. The exact result, retaining the requested order, is (-1/2, 7/2, 7/2)."},"skills":["Ordered Rational Midpoint and Endpoint Gaps"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r02-real-number-0010","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"59173de65564a1124d3e9cfb6635fb71fcb4c03436b9684d78d9171b24d4d66e","family_id":"qf-density14-real-ordered-midpoint","seed":202610142010,"semantic_template":"qf-density14-real-ordered-midpoint/[\"-9/2\",\"10/3\"]"},"fields":{"title":"Ordered Rational Midpoint and Endpoint Gaps","stem":"Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. a=-9/2≤b=10/3. Give (midpoint, midpoint−a, b−midpoint). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(-7/12, 47/12, 47/12)","explanation":"Add and divide by two exactly, then subtract each endpoint separately; no irrational classification or real-completeness proof is claimed. The exact result, retaining the requested order, is (-7/12, 47/12, 47/12)."},"skills":["Ordered Rational Midpoint and Endpoint Gaps"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r02-real-number-0011","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2cce6c48df57aa601252834493f9a45602dca92411b91a1776a36bbd53c9bd99","family_id":"qf-density14-real-ordered-midpoint","seed":202610142011,"semantic_template":"qf-density14-real-ordered-midpoint/[\"-5/1\",\"11/3\"]"},"fields":{"title":"Ordered Rational Midpoint and Endpoint Gaps","stem":"Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. a=-10/2≤b=11/3. Give (midpoint, midpoint−a, b−midpoint). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(-2/3, 13/3, 13/3)","explanation":"Add and divide by two exactly, then subtract each endpoint separately; no irrational classification or real-completeness proof is claimed. The exact result, retaining the requested order, is (-2/3, 13/3, 13/3)."},"skills":["Ordered Rational Midpoint and Endpoint Gaps"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r02-real-number-0012","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dbf2c9d96a2934bda5db31d3f14d9a51847ebe4f0353c8b3ee66842ab1e9fc62","family_id":"qf-density14-real-ordered-midpoint","seed":202610142012,"semantic_template":"qf-density14-real-ordered-midpoint/[\"-11/2\",\"4/1\"]"},"fields":{"title":"Ordered Rational Midpoint and Endpoint Gaps","stem":"Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. a=-11/2≤b=12/3. Give (midpoint, midpoint−a, b−midpoint). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(-3/4, 19/4, 19/4)","explanation":"Add and divide by two exactly, then subtract each endpoint separately; no irrational classification or real-completeness proof is claimed. The exact result, retaining the requested order, is (-3/4, 19/4, 19/4)."},"skills":["Ordered Rational Midpoint and Endpoint Gaps"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r02-real-number-0013","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"adfc2da32d49afff3abecbc7ff868683da32f4a3c98bbdc068e56d3e51c85b49","family_id":"qf-density14-real-ordered-midpoint","seed":202610142013,"semantic_template":"qf-density14-real-ordered-midpoint/[\"-6/1\",\"13/3\"]"},"fields":{"title":"Ordered Rational Midpoint and Endpoint Gaps","stem":"Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. a=-12/2≤b=13/3. Give (midpoint, midpoint−a, b−midpoint). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(-5/6, 31/6, 31/6)","explanation":"Add and divide by two exactly, then subtract each endpoint separately; no irrational classification or real-completeness proof is claimed. The exact result, retaining the requested order, is (-5/6, 31/6, 31/6)."},"skills":["Ordered Rational Midpoint and Endpoint Gaps"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r02-real-number-0014","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"815471213a42eba97b0d4c05abecb237f932323228eb5b845176d6e6f93bf289","family_id":"qf-density14-real-ordered-midpoint","seed":202610142014,"semantic_template":"qf-density14-real-ordered-midpoint/[\"-13/2\",\"14/3\"]"},"fields":{"title":"Ordered Rational Midpoint and Endpoint Gaps","stem":"Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. a=-13/2≤b=14/3. Give (midpoint, midpoint−a, b−midpoint). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(-11/12, 67/12, 67/12)","explanation":"Add and divide by two exactly, then subtract each endpoint separately; no irrational classification or real-completeness proof is claimed. The exact result, retaining the requested order, is (-11/12, 67/12, 67/12)."},"skills":["Ordered Rational Midpoint and Endpoint Gaps"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r02-real-number-0015","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"53366e0000c7dc0eb58796e29174ebcf097f055c2845c780f37b4cded59cb4cf","family_id":"qf-density14-real-ordered-midpoint","seed":202610142015,"semantic_template":"qf-density14-real-ordered-midpoint/[\"-7/1\",\"5/1\"]"},"fields":{"title":"Ordered Rational Midpoint and Endpoint Gaps","stem":"Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. a=-14/2≤b=15/3. Give (midpoint, midpoint−a, b−midpoint). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(-1/1, 6/1, 6/1)","explanation":"Add and divide by two exactly, then subtract each endpoint separately; no irrational classification or real-completeness proof is claimed. The exact result, retaining the requested order, is (-1/1, 6/1, 6/1)."},"skills":["Ordered Rational Midpoint and Endpoint Gaps"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r02-real-number-0016","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"334987f8c04a14c5b456eb135c4bd9180bbf76204c4e8d404c49fe15166d5061","family_id":"qf-density14-real-ordered-midpoint","seed":202610142016,"semantic_template":"qf-density14-real-ordered-midpoint/[\"-15/2\",\"16/3\"]"},"fields":{"title":"Ordered Rational Midpoint and Endpoint Gaps","stem":"Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. a=-15/2≤b=16/3. Give (midpoint, midpoint−a, b−midpoint). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(-13/12, 77/12, 77/12)","explanation":"Add and divide by two exactly, then subtract each endpoint separately; no irrational classification or real-completeness proof is claimed. The exact result, retaining the requested order, is (-13/12, 77/12, 77/12)."},"skills":["Ordered Rational Midpoint and Endpoint Gaps"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r02-real-number-0017","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b9fe69e2e65480387e769b49c7f81475717c66c86039c511abda8a63fe9a7357","family_id":"qf-density14-real-ordered-midpoint","seed":202610142017,"semantic_template":"qf-density14-real-ordered-midpoint/[\"-8/1\",\"17/3\"]"},"fields":{"title":"Ordered Rational Midpoint and Endpoint Gaps","stem":"Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. a=-16/2≤b=17/3. Give (midpoint, midpoint−a, b−midpoint). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(-7/6, 41/6, 41/6)","explanation":"Add and divide by two exactly, then subtract each endpoint separately; no irrational classification or real-completeness proof is claimed. The exact result, retaining the requested order, is (-7/6, 41/6, 41/6)."},"skills":["Ordered Rational Midpoint and Endpoint Gaps"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r02-real-number-0018","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ac8ee59c3f4af3eef4d73418acb6e9cff9752f72016500f9f536577df6988742","family_id":"qf-density14-real-ordered-midpoint","seed":202610142018,"semantic_template":"qf-density14-real-ordered-midpoint/[\"-17/2\",\"6/1\"]"},"fields":{"title":"Ordered Rational Midpoint and Endpoint Gaps","stem":"Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. a=-17/2≤b=18/3. Give (midpoint, midpoint−a, b−midpoint). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(-5/4, 29/4, 29/4)","explanation":"Add and divide by two exactly, then subtract each endpoint separately; no irrational classification or real-completeness proof is claimed. The exact result, retaining the requested order, is (-5/4, 29/4, 29/4)."},"skills":["Ordered Rational Midpoint and Endpoint Gaps"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r02-real-number-0019","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1e588ca361e6b4482cbbe32b10eb7e05dab97ab980f15b128c149f3a16403f01","family_id":"qf-density14-real-ordered-midpoint","seed":202610142019,"semantic_template":"qf-density14-real-ordered-midpoint/[\"-9/1\",\"19/3\"]"},"fields":{"title":"Ordered Rational Midpoint and Endpoint Gaps","stem":"Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. a=-18/2≤b=19/3. Give (midpoint, midpoint−a, b−midpoint). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(-4/3, 23/3, 23/3)","explanation":"Add and divide by two exactly, then subtract each endpoint separately; no irrational classification or real-completeness proof is claimed. The exact result, retaining the requested order, is (-4/3, 23/3, 23/3)."},"skills":["Ordered Rational Midpoint and Endpoint Gaps"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r02-real-number-0020","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"187520347ef7789b6915768e5d568fda794e27162a68e56f38f334d96b9b93ba","family_id":"qf-density14-real-ordered-midpoint","seed":202610142020,"semantic_template":"qf-density14-real-ordered-midpoint/[\"-19/2\",\"20/3\"]"},"fields":{"title":"Ordered Rational Midpoint and Endpoint Gaps","stem":"Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. a=-19/2≤b=20/3. Give (midpoint, midpoint−a, b−midpoint). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(-17/12, 97/12, 97/12)","explanation":"Add and divide by two exactly, then subtract each endpoint separately; no irrational classification or real-completeness proof is claimed. The exact result, retaining the requested order, is (-17/12, 97/12, 97/12)."},"skills":["Ordered Rational Midpoint and Endpoint Gaps"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r02-real-number-0021","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bdffdb05adf382d05d1b1434a53c84362b1574a186647a8b74ac39245cd2a259","family_id":"qf-density14-real-ordered-midpoint","seed":202610142021,"semantic_template":"qf-density14-real-ordered-midpoint/[\"-10/1\",\"7/1\"]"},"fields":{"title":"Ordered Rational Midpoint and Endpoint Gaps","stem":"Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. a=-20/2≤b=21/3. Give (midpoint, midpoint−a, b−midpoint). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(-3/2, 17/2, 17/2)","explanation":"Add and divide by two exactly, then subtract each endpoint separately; no irrational classification or real-completeness proof is claimed. The exact result, retaining the requested order, is (-3/2, 17/2, 17/2)."},"skills":["Ordered Rational Midpoint and Endpoint Gaps"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r02-real-number-0022","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1b10afcad8b2957a3e7a2b72cc454c62dd032f57c150a9c9553d4f737d4b17d7","family_id":"qf-density14-real-ordered-midpoint","seed":202610142022,"semantic_template":"qf-density14-real-ordered-midpoint/[\"-21/2\",\"22/3\"]"},"fields":{"title":"Ordered Rational Midpoint and Endpoint Gaps","stem":"Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. a=-21/2≤b=22/3. Give (midpoint, midpoint−a, b−midpoint). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(-19/12, 107/12, 107/12)","explanation":"Add and divide by two exactly, then subtract each endpoint separately; no irrational classification or real-completeness proof is claimed. The exact result, retaining the requested order, is (-19/12, 107/12, 107/12)."},"skills":["Ordered Rational Midpoint and Endpoint Gaps"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r02-real-number-0023","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c77fb327af01507791ca77fc0d3e2c7737a77e365bc76696757ea7f6cbb1f99b","family_id":"qf-density14-real-ordered-midpoint","seed":202610142023,"semantic_template":"qf-density14-real-ordered-midpoint/[\"-11/1\",\"23/3\"]"},"fields":{"title":"Ordered Rational Midpoint and Endpoint Gaps","stem":"Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. a=-22/2≤b=23/3. Give (midpoint, midpoint−a, b−midpoint). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(-5/3, 28/3, 28/3)","explanation":"Add and divide by two exactly, then subtract each endpoint separately; no irrational classification or real-completeness proof is claimed. The exact result, retaining the requested order, is (-5/3, 28/3, 28/3)."},"skills":["Ordered Rational Midpoint and Endpoint Gaps"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r02-real-number-0024","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2ad74ef8ac5fe9721d5a2afa8b2f202a267b4ca896c329bb94cde0cf1ba0cccd","family_id":"qf-density14-real-ordered-midpoint","seed":202610142024,"semantic_template":"qf-density14-real-ordered-midpoint/[\"-23/2\",\"8/1\"]"},"fields":{"title":"Ordered Rational Midpoint and Endpoint Gaps","stem":"Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. a=-23/2≤b=24/3. Give (midpoint, midpoint−a, b−midpoint). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(-7/4, 39/4, 39/4)","explanation":"Add and divide by two exactly, then subtract each endpoint separately; no irrational classification or real-completeness proof is claimed. The exact result, retaining the requested order, is (-7/4, 39/4, 39/4)."},"skills":["Ordered Rational Midpoint and Endpoint Gaps"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r02-real-number-0025","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e8ef20f87a8d987470d0c78af677b556c7022e5f74f34074b35d58c397e1f3cf","family_id":"qf-density14-real-ordered-midpoint","seed":202610142025,"semantic_template":"qf-density14-real-ordered-midpoint/[\"-12/1\",\"25/3\"]"},"fields":{"title":"Ordered Rational Midpoint and Endpoint Gaps","stem":"Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. a=-24/2≤b=25/3. Give (midpoint, midpoint−a, b−midpoint). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(-11/6, 61/6, 61/6)","explanation":"Add and divide by two exactly, then subtract each endpoint separately; no irrational classification or real-completeness proof is claimed. The exact result, retaining the requested order, is (-11/6, 61/6, 61/6)."},"skills":["Ordered Rational Midpoint and Endpoint Gaps"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r02-real-number-0026","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"04bda6dc04bdc363469489cc075f4bd629e70e01dd62f73a094903773b122d83","family_id":"qf-density14-real-point-closed-hull","seed":202610142026,"semantic_template":"qf-density14-real-point-closed-hull/[\"0/1\"]"},"fields":{"title":"Finite Distinct Points and Least Containing Closed Interval","stem":"Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. Ordered points=[0/1, 0/-2, 0/3]. Give (increasing distinct points, minimum lo, maximum hi, hi−lo); the containing interval is [lo,hi], not the point set’s topological closure.","answer":{"kind":"expression","value":"([0/1], 0/1, 0/1, 0/1)","explanation":"Sort and deduplicate by exact rational value, then take the extreme endpoints; the least containing closed interval differs from topological closure of a finite point set. The exact result, retaining the requested order, is ([0/1], 0/1, 0/1, 0/1)."},"skills":["Finite Distinct Points and Least Containing Closed Interval"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r02-real-number-0027","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5d39095f628d3b4011a07f0305e67f954b0cd11a7956e6d8cec7de9298de0c5a","family_id":"qf-density14-real-point-closed-hull","seed":202610142027,"semantic_template":"qf-density14-real-point-closed-hull/[\"-1/2\",\"1/2\"]"},"fields":{"title":"Finite Distinct Points and Least Containing Closed Interval","stem":"Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. Ordered points=[1/2, -1/2, 1/2]. Give (increasing distinct points, minimum lo, maximum hi, hi−lo); the containing interval is [lo,hi], not the point set’s topological closure.","answer":{"kind":"expression","value":"([-1/2, 1/2], -1/2, 1/2, 1/1)","explanation":"Sort and deduplicate by exact rational value, then take the extreme endpoints; the least containing closed interval differs from topological closure of a finite point set. The exact result, retaining the requested order, is ([-1/2, 1/2], -1/2, 1/2, 1/1)."},"skills":["Finite Distinct Points and Least Containing Closed Interval"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r02-real-number-0028","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"963401913ab74d4448550901c849e02d12cf6cd21f72610aa531bf33e98ceaa7","family_id":"qf-density14-real-point-closed-hull","seed":202610142028,"semantic_template":"qf-density14-real-point-closed-hull/[\"-1/1\",\"0/1\",\"1/1\"]"},"fields":{"title":"Finite Distinct Points and Least Containing Closed Interval","stem":"Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. Ordered points=[-2/2, 0/1, 3/3]. Give (increasing distinct points, minimum lo, maximum hi, hi−lo); the containing interval is [lo,hi], not the point set’s topological closure.","answer":{"kind":"expression","value":"([-1/1, 0/1, 1/1], -1/1, 1/1, 2/1)","explanation":"Sort and deduplicate by exact rational value, then take the extreme endpoints; the least containing closed interval differs from topological closure of a finite point set. The exact result, retaining the requested order, is ([-1/1, 0/1, 1/1], -1/1, 1/1, 2/1)."},"skills":["Finite Distinct Points and Least Containing Closed Interval"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r02-real-number-0029","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3be5c40be76738a4628e07a260366eefa5e066f4b526ded6e3f2f2b2203fda74","family_id":"qf-density14-real-point-closed-hull","seed":202610142029,"semantic_template":"qf-density14-real-point-closed-hull/[\"-3/2\",\"0/1\",\"4/3\"]"},"fields":{"title":"Finite Distinct Points and Least Containing Closed Interval","stem":"Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. Ordered points=[-3/2, 0/1, 4/3]. Give (increasing distinct points, minimum lo, maximum hi, hi−lo); the containing interval is [lo,hi], not the point set’s topological closure.","answer":{"kind":"expression","value":"([-3/2, 0/1, 4/3], -3/2, 4/3, 17/6)","explanation":"Sort and deduplicate by exact rational value, then take the extreme endpoints; the least containing closed interval differs from topological closure of a finite point set. The exact result, retaining the requested order, is ([-3/2, 0/1, 4/3], -3/2, 4/3, 17/6)."},"skills":["Finite Distinct Points and Least Containing Closed Interval"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r02-real-number-0030","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"aad9618ddd3d6ce8568767d6d10685c18d5b626f19f517404e07f5c18445a99b","family_id":"qf-density14-real-point-closed-hull","seed":202610142030,"semantic_template":"qf-density14-real-point-closed-hull/[\"-2/1\",\"0/1\",\"5/3\"]"},"fields":{"title":"Finite Distinct Points and Least Containing Closed Interval","stem":"Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. Ordered points=[-4/2, 0/1, 5/3]. Give (increasing distinct points, minimum lo, maximum hi, hi−lo); the containing interval is [lo,hi], not the point set’s topological closure.","answer":{"kind":"expression","value":"([-2/1, 0/1, 5/3], -2/1, 5/3, 11/3)","explanation":"Sort and deduplicate by exact rational value, then take the extreme endpoints; the least containing closed interval differs from topological closure of a finite point set. The exact result, retaining the requested order, is ([-2/1, 0/1, 5/3], -2/1, 5/3, 11/3)."},"skills":["Finite Distinct Points and Least Containing Closed Interval"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r02-real-number-0031","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"54d279be963da38bd6d86ced6ebb7e52ff2488d62c8ade9573d393bbb48f23fe","family_id":"qf-density14-real-point-closed-hull","seed":202610142031,"semantic_template":"qf-density14-real-point-closed-hull/[\"-5/2\",\"0/1\",\"2/1\"]"},"fields":{"title":"Finite Distinct Points and Least Containing Closed Interval","stem":"Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. Ordered points=[-5/2, 0/1, 6/3]. Give (increasing distinct points, minimum lo, maximum hi, hi−lo); the containing interval is [lo,hi], not the point set’s topological closure.","answer":{"kind":"expression","value":"([-5/2, 0/1, 2/1], -5/2, 2/1, 9/2)","explanation":"Sort and deduplicate by exact rational value, then take the extreme endpoints; the least containing closed interval differs from topological closure of a finite point set. The exact result, retaining the requested order, is ([-5/2, 0/1, 2/1], -5/2, 2/1, 9/2)."},"skills":["Finite Distinct Points and Least Containing Closed Interval"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r02-real-number-0032","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b3afb82a75abad16f64728acbf143cb6d721c1680cd3c983c8c38d1cd20ad16b","family_id":"qf-density14-real-point-closed-hull","seed":202610142032,"semantic_template":"qf-density14-real-point-closed-hull/[\"-3/1\",\"0/1\",\"7/3\"]"},"fields":{"title":"Finite Distinct Points and Least Containing Closed Interval","stem":"Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. Ordered points=[-6/2, 0/1, 7/3]. Give (increasing distinct points, minimum lo, maximum hi, hi−lo); the containing interval is [lo,hi], not the point set’s topological closure.","answer":{"kind":"expression","value":"([-3/1, 0/1, 7/3], -3/1, 7/3, 16/3)","explanation":"Sort and deduplicate by exact rational value, then take the extreme endpoints; the least containing closed interval differs from topological closure of a finite point set. The exact result, retaining the requested order, is ([-3/1, 0/1, 7/3], -3/1, 7/3, 16/3)."},"skills":["Finite Distinct Points and Least Containing Closed Interval"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r02-real-number-0033","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6107f99009601e98705b5d7216e476142d27c57e33ae522799bb4091d158f670","family_id":"qf-density14-real-point-closed-hull","seed":202610142033,"semantic_template":"qf-density14-real-point-closed-hull/[\"-7/2\",\"0/1\",\"8/3\"]"},"fields":{"title":"Finite Distinct Points and Least Containing Closed Interval","stem":"Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. Ordered points=[-7/2, 0/1, 8/3]. Give (increasing distinct points, minimum lo, maximum hi, hi−lo); the containing interval is [lo,hi], not the point set’s topological closure.","answer":{"kind":"expression","value":"([-7/2, 0/1, 8/3], -7/2, 8/3, 37/6)","explanation":"Sort and deduplicate by exact rational value, then take the extreme endpoints; the least containing closed interval differs from topological closure of a finite point set. The exact result, retaining the requested order, is ([-7/2, 0/1, 8/3], -7/2, 8/3, 37/6)."},"skills":["Finite Distinct Points and Least Containing Closed Interval"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r02-real-number-0034","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e2717237cbe840cc9538791febcf2f1830b5517609828916f0350a845a83cd38","family_id":"qf-density14-real-point-closed-hull","seed":202610142034,"semantic_template":"qf-density14-real-point-closed-hull/[\"-4/1\",\"0/1\",\"3/1\"]"},"fields":{"title":"Finite Distinct Points and Least Containing Closed Interval","stem":"Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. Ordered points=[-8/2, 0/1, 9/3]. Give (increasing distinct points, minimum lo, maximum hi, hi−lo); the containing interval is [lo,hi], not the point set’s topological closure.","answer":{"kind":"expression","value":"([-4/1, 0/1, 3/1], -4/1, 3/1, 7/1)","explanation":"Sort and deduplicate by exact rational value, then take the extreme endpoints; the least containing closed interval differs from topological closure of a finite point set. The exact result, retaining the requested order, is ([-4/1, 0/1, 3/1], -4/1, 3/1, 7/1)."},"skills":["Finite Distinct Points and Least Containing Closed Interval"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r02-real-number-0035","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ad5375aec34467cc9c95ba104d7d52c552f4ddc058f2346dc92aa439fd9e7ce9","family_id":"qf-density14-real-point-closed-hull","seed":202610142035,"semantic_template":"qf-density14-real-point-closed-hull/[\"-9/2\",\"0/1\",\"10/3\"]"},"fields":{"title":"Finite Distinct Points and Least Containing Closed Interval","stem":"Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. Ordered points=[-9/2, 0/1, 10/3]. Give (increasing distinct points, minimum lo, maximum hi, hi−lo); the containing interval is [lo,hi], not the point set’s topological closure.","answer":{"kind":"expression","value":"([-9/2, 0/1, 10/3], -9/2, 10/3, 47/6)","explanation":"Sort and deduplicate by exact rational value, then take the extreme endpoints; the least containing closed interval differs from topological closure of a finite point set. The exact result, retaining the requested order, is ([-9/2, 0/1, 10/3], -9/2, 10/3, 47/6)."},"skills":["Finite Distinct Points and Least Containing Closed Interval"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r02-real-number-0036","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7b41080db1851488d7e5ea145bf5bcb69b4a4017697d06695f7c8222e9df094b","family_id":"qf-density14-real-point-closed-hull","seed":202610142036,"semantic_template":"qf-density14-real-point-closed-hull/[\"-5/1\",\"0/1\",\"11/3\"]"},"fields":{"title":"Finite Distinct Points and Least Containing Closed Interval","stem":"Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. Ordered points=[-10/2, 0/1, 11/3]. Give (increasing distinct points, minimum lo, maximum hi, hi−lo); the containing interval is [lo,hi], not the point set’s topological closure.","answer":{"kind":"expression","value":"([-5/1, 0/1, 11/3], -5/1, 11/3, 26/3)","explanation":"Sort and deduplicate by exact rational value, then take the extreme endpoints; the least containing closed interval differs from topological closure of a finite point set. The exact result, retaining the requested order, is ([-5/1, 0/1, 11/3], -5/1, 11/3, 26/3)."},"skills":["Finite Distinct Points and Least Containing Closed Interval"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r02-real-number-0037","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0d2233c14d35700e3c0e00be9aa61c3b0304e38a510328e0193a4eaf4cc47d47","family_id":"qf-density14-real-point-closed-hull","seed":202610142037,"semantic_template":"qf-density14-real-point-closed-hull/[\"-11/2\",\"0/1\",\"4/1\"]"},"fields":{"title":"Finite Distinct Points and Least Containing Closed Interval","stem":"Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. Ordered points=[-11/2, 0/1, 12/3]. Give (increasing distinct points, minimum lo, maximum hi, hi−lo); the containing interval is [lo,hi], not the point set’s topological closure.","answer":{"kind":"expression","value":"([-11/2, 0/1, 4/1], -11/2, 4/1, 19/2)","explanation":"Sort and deduplicate by exact rational value, then take the extreme endpoints; the least containing closed interval differs from topological closure of a finite point set. The exact result, retaining the requested order, is ([-11/2, 0/1, 4/1], -11/2, 4/1, 19/2)."},"skills":["Finite Distinct Points and Least Containing Closed Interval"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r02-real-number-0038","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"227d89fa7e4c9c98a5ccd87bfa174ff5dd0e828a6a671d5afe7f4181990045b8","family_id":"qf-density14-real-point-closed-hull","seed":202610142038,"semantic_template":"qf-density14-real-point-closed-hull/[\"-6/1\",\"0/1\",\"13/3\"]"},"fields":{"title":"Finite Distinct Points and Least Containing Closed Interval","stem":"Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. Ordered points=[-12/2, 0/1, 13/3]. Give (increasing distinct points, minimum lo, maximum hi, hi−lo); the containing interval is [lo,hi], not the point set’s topological closure.","answer":{"kind":"expression","value":"([-6/1, 0/1, 13/3], -6/1, 13/3, 31/3)","explanation":"Sort and deduplicate by exact rational value, then take the extreme endpoints; the least containing closed interval differs from topological closure of a finite point set. The exact result, retaining the requested order, is ([-6/1, 0/1, 13/3], -6/1, 13/3, 31/3)."},"skills":["Finite Distinct Points and Least Containing Closed Interval"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r02-real-number-0039","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c4aa8f2433395f2c3f635ef42ed23081139f62c7c510a9787a118e58c51d2b13","family_id":"qf-density14-real-point-closed-hull","seed":202610142039,"semantic_template":"qf-density14-real-point-closed-hull/[\"-13/2\",\"0/1\",\"14/3\"]"},"fields":{"title":"Finite Distinct Points and Least Containing Closed Interval","stem":"Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. Ordered points=[-13/2, 0/1, 14/3]. Give (increasing distinct points, minimum lo, maximum hi, hi−lo); the containing interval is [lo,hi], not the point set’s topological closure.","answer":{"kind":"expression","value":"([-13/2, 0/1, 14/3], -13/2, 14/3, 67/6)","explanation":"Sort and deduplicate by exact rational value, then take the extreme endpoints; the least containing closed interval differs from topological closure of a finite point set. The exact result, retaining the requested order, is ([-13/2, 0/1, 14/3], -13/2, 14/3, 67/6)."},"skills":["Finite Distinct Points and Least Containing Closed Interval"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r02-real-number-0040","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3a75b07c902db0c588cdbdaa4611e7260749d2b8dff0e77f31a044f58eaffe25","family_id":"qf-density14-real-point-closed-hull","seed":202610142040,"semantic_template":"qf-density14-real-point-closed-hull/[\"-7/1\",\"0/1\",\"5/1\"]"},"fields":{"title":"Finite Distinct Points and Least Containing Closed Interval","stem":"Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. Ordered points=[-14/2, 0/1, 15/3]. Give (increasing distinct points, minimum lo, maximum hi, hi−lo); the containing interval is [lo,hi], not the point set’s topological closure.","answer":{"kind":"expression","value":"([-7/1, 0/1, 5/1], -7/1, 5/1, 12/1)","explanation":"Sort and deduplicate by exact rational value, then take the extreme endpoints; the least containing closed interval differs from topological closure of a finite point set. The exact result, retaining the requested order, is ([-7/1, 0/1, 5/1], -7/1, 5/1, 12/1)."},"skills":["Finite Distinct Points and Least Containing Closed Interval"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r02-real-number-0041","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bb1fcc843c875191048bc9ca5137bfb47cdc37f6f0c08e3a755de22e47271c3b","family_id":"qf-density14-real-point-closed-hull","seed":202610142041,"semantic_template":"qf-density14-real-point-closed-hull/[\"-15/2\",\"0/1\",\"16/3\"]"},"fields":{"title":"Finite Distinct Points and Least Containing Closed Interval","stem":"Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. Ordered points=[-15/2, 0/1, 16/3]. Give (increasing distinct points, minimum lo, maximum hi, hi−lo); the containing interval is [lo,hi], not the point set’s topological closure.","answer":{"kind":"expression","value":"([-15/2, 0/1, 16/3], -15/2, 16/3, 77/6)","explanation":"Sort and deduplicate by exact rational value, then take the extreme endpoints; the least containing closed interval differs from topological closure of a finite point set. The exact result, retaining the requested order, is ([-15/2, 0/1, 16/3], -15/2, 16/3, 77/6)."},"skills":["Finite Distinct Points and Least Containing Closed Interval"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r02-real-number-0042","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7459c0162c4037fae5fc8f8499488578daa183a2292fc2d85ccf5b5f801ff59d","family_id":"qf-density14-real-point-closed-hull","seed":202610142042,"semantic_template":"qf-density14-real-point-closed-hull/[\"-8/1\",\"0/1\",\"17/3\"]"},"fields":{"title":"Finite Distinct Points and Least Containing Closed Interval","stem":"Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. Ordered points=[-16/2, 0/1, 17/3]. Give (increasing distinct points, minimum lo, maximum hi, hi−lo); the containing interval is [lo,hi], not the point set’s topological closure.","answer":{"kind":"expression","value":"([-8/1, 0/1, 17/3], -8/1, 17/3, 41/3)","explanation":"Sort and deduplicate by exact rational value, then take the extreme endpoints; the least containing closed interval differs from topological closure of a finite point set. The exact result, retaining the requested order, is ([-8/1, 0/1, 17/3], -8/1, 17/3, 41/3)."},"skills":["Finite Distinct Points and Least Containing Closed Interval"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r02-real-number-0043","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"19ce2af04a21f60571811dccda0e17b605bad688ad65a7abd35a1552ea0b8316","family_id":"qf-density14-real-point-closed-hull","seed":202610142043,"semantic_template":"qf-density14-real-point-closed-hull/[\"-17/2\",\"0/1\",\"6/1\"]"},"fields":{"title":"Finite Distinct Points and Least Containing Closed Interval","stem":"Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. Ordered points=[-17/2, 0/1, 18/3]. Give (increasing distinct points, minimum lo, maximum hi, hi−lo); the containing interval is [lo,hi], not the point set’s topological closure.","answer":{"kind":"expression","value":"([-17/2, 0/1, 6/1], -17/2, 6/1, 29/2)","explanation":"Sort and deduplicate by exact rational value, then take the extreme endpoints; the least containing closed interval differs from topological closure of a finite point set. The exact result, retaining the requested order, is ([-17/2, 0/1, 6/1], -17/2, 6/1, 29/2)."},"skills":["Finite Distinct Points and Least Containing Closed Interval"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r02-real-number-0044","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"af94325d137031464d07f64a9c91d090705ccd8bcf3d0983346f69b70ff381f5","family_id":"qf-density14-real-point-closed-hull","seed":202610142044,"semantic_template":"qf-density14-real-point-closed-hull/[\"-9/1\",\"0/1\",\"19/3\"]"},"fields":{"title":"Finite Distinct Points and Least Containing Closed Interval","stem":"Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. Ordered points=[-18/2, 0/1, 19/3]. Give (increasing distinct points, minimum lo, maximum hi, hi−lo); the containing interval is [lo,hi], not the point set’s topological closure.","answer":{"kind":"expression","value":"([-9/1, 0/1, 19/3], -9/1, 19/3, 46/3)","explanation":"Sort and deduplicate by exact rational value, then take the extreme endpoints; the least containing closed interval differs from topological closure of a finite point set. The exact result, retaining the requested order, is ([-9/1, 0/1, 19/3], -9/1, 19/3, 46/3)."},"skills":["Finite Distinct Points and Least Containing Closed Interval"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r02-real-number-0045","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"adf401d482901d4f32eec8996e6905a27cb5db47ff942454615d1be88c02dead","family_id":"qf-density14-real-point-closed-hull","seed":202610142045,"semantic_template":"qf-density14-real-point-closed-hull/[\"-19/2\",\"0/1\",\"20/3\"]"},"fields":{"title":"Finite Distinct Points and Least Containing Closed Interval","stem":"Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. Ordered points=[-19/2, 0/1, 20/3]. Give (increasing distinct points, minimum lo, maximum hi, hi−lo); the containing interval is [lo,hi], not the point set’s topological closure.","answer":{"kind":"expression","value":"([-19/2, 0/1, 20/3], -19/2, 20/3, 97/6)","explanation":"Sort and deduplicate by exact rational value, then take the extreme endpoints; the least containing closed interval differs from topological closure of a finite point set. The exact result, retaining the requested order, is ([-19/2, 0/1, 20/3], -19/2, 20/3, 97/6)."},"skills":["Finite Distinct Points and Least Containing Closed Interval"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r02-real-number-0046","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dc6221c8c47663abc4bf26417dbd660377c5084c344b3329ab7a0c75dc114779","family_id":"qf-density14-real-point-closed-hull","seed":202610142046,"semantic_template":"qf-density14-real-point-closed-hull/[\"-10/1\",\"0/1\",\"7/1\"]"},"fields":{"title":"Finite Distinct Points and Least Containing Closed Interval","stem":"Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. Ordered points=[-20/2, 0/1, 21/3]. Give (increasing distinct points, minimum lo, maximum hi, hi−lo); the containing interval is [lo,hi], not the point set’s topological closure.","answer":{"kind":"expression","value":"([-10/1, 0/1, 7/1], -10/1, 7/1, 17/1)","explanation":"Sort and deduplicate by exact rational value, then take the extreme endpoints; the least containing closed interval differs from topological closure of a finite point set. The exact result, retaining the requested order, is ([-10/1, 0/1, 7/1], -10/1, 7/1, 17/1)."},"skills":["Finite Distinct Points and Least Containing Closed Interval"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r02-real-number-0047","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7a0611a57476734bb16e0d661a89c7f68d029404ab10bc7c03fa991b965858c9","family_id":"qf-density14-real-point-closed-hull","seed":202610142047,"semantic_template":"qf-density14-real-point-closed-hull/[\"-21/2\",\"0/1\",\"22/3\"]"},"fields":{"title":"Finite Distinct Points and Least Containing Closed Interval","stem":"Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. Ordered points=[-21/2, 0/1, 22/3]. Give (increasing distinct points, minimum lo, maximum hi, hi−lo); the containing interval is [lo,hi], not the point set’s topological closure.","answer":{"kind":"expression","value":"([-21/2, 0/1, 22/3], -21/2, 22/3, 107/6)","explanation":"Sort and deduplicate by exact rational value, then take the extreme endpoints; the least containing closed interval differs from topological closure of a finite point set. The exact result, retaining the requested order, is ([-21/2, 0/1, 22/3], -21/2, 22/3, 107/6)."},"skills":["Finite Distinct Points and Least Containing Closed Interval"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r02-real-number-0048","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f6736baeaeb113238436a71d58b8f973c7cde1fb2180abd450d006eb8296b077","family_id":"qf-density14-real-point-closed-hull","seed":202610142048,"semantic_template":"qf-density14-real-point-closed-hull/[\"-11/1\",\"0/1\",\"23/3\"]"},"fields":{"title":"Finite Distinct Points and Least Containing Closed Interval","stem":"Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. Ordered points=[-22/2, 0/1, 23/3]. Give (increasing distinct points, minimum lo, maximum hi, hi−lo); the containing interval is [lo,hi], not the point set’s topological closure.","answer":{"kind":"expression","value":"([-11/1, 0/1, 23/3], -11/1, 23/3, 56/3)","explanation":"Sort and deduplicate by exact rational value, then take the extreme endpoints; the least containing closed interval differs from topological closure of a finite point set. The exact result, retaining the requested order, is ([-11/1, 0/1, 23/3], -11/1, 23/3, 56/3)."},"skills":["Finite Distinct Points and Least Containing Closed Interval"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r02-real-number-0049","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"df696f0a4c7492f43b97b2929205921b0a097f5f4aec20580b7492e8d714482d","family_id":"qf-density14-real-point-closed-hull","seed":202610142049,"semantic_template":"qf-density14-real-point-closed-hull/[\"-23/2\",\"0/1\",\"8/1\"]"},"fields":{"title":"Finite Distinct Points and Least Containing Closed Interval","stem":"Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. Ordered points=[-23/2, 0/1, 24/3]. Give (increasing distinct points, minimum lo, maximum hi, hi−lo); the containing interval is [lo,hi], not the point set’s topological closure.","answer":{"kind":"expression","value":"([-23/2, 0/1, 8/1], -23/2, 8/1, 39/2)","explanation":"Sort and deduplicate by exact rational value, then take the extreme endpoints; the least containing closed interval differs from topological closure of a finite point set. The exact result, retaining the requested order, is ([-23/2, 0/1, 8/1], -23/2, 8/1, 39/2)."},"skills":["Finite Distinct Points and Least Containing Closed Interval"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r02-real-number-0050","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"14a5d2f35571c32985260d6ae6f027299216878d0aeb83c3e8984ff152f8c086","family_id":"qf-density14-real-point-closed-hull","seed":202610142050,"semantic_template":"qf-density14-real-point-closed-hull/[\"-12/1\",\"0/1\",\"25/3\"]"},"fields":{"title":"Finite Distinct Points and Least Containing Closed Interval","stem":"Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. Ordered points=[-24/2, 0/1, 25/3]. Give (increasing distinct points, minimum lo, maximum hi, hi−lo); the containing interval is [lo,hi], not the point set’s topological closure.","answer":{"kind":"expression","value":"([-12/1, 0/1, 25/3], -12/1, 25/3, 61/3)","explanation":"Sort and deduplicate by exact rational value, then take the extreme endpoints; the least containing closed interval differs from topological closure of a finite point set. The exact result, retaining the requested order, is ([-12/1, 0/1, 25/3], -12/1, 25/3, 61/3)."},"skills":["Finite Distinct Points and Least Containing Closed Interval"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r02-real-number-0051","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"752191794f7d75c23ad52a3e71bdb0e33abb0ea4403078d27136fffc66c9dc80","family_id":"qf-density14-real-overlap-closed-intervals","seed":202610142051,"semantic_template":"qf-density14-real-overlap-closed-intervals/[\"intersection\",\"1/1\",\"1/1\"]"},"fields":{"title":"Closed Interval Intersection and Separation Gap","stem":"Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. I=[0/1, 1/1], J=[1/1, 2/1] have ordered inclusive endpoints. Give (is the intersection nonempty?, intersection [lo,hi] or [], separation gap). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(Yes, [1/1, 1/1], 0/1)","explanation":"Take the larger lower endpoint and smaller upper endpoint; intersecting intervals have gap 0/1, and touching at one included point is not an empty intersection. The exact result, retaining the requested order, is (Yes, [1/1, 1/1], 0/1)."},"skills":["Closed Interval Intersection and Separation Gap"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r02-real-number-0052","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"91626b10352a9f17e4060d76432e7153f4a49262df9309963e31bb2b51c78286","family_id":"qf-density14-real-overlap-closed-intervals","seed":202610142052,"semantic_template":"qf-density14-real-overlap-closed-intervals/[\"disjoint\",\"1/1\"]"},"fields":{"title":"Closed Interval Intersection and Separation Gap","stem":"Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. I=[0/1, 1/1], J=[2/1, 3/1] have ordered inclusive endpoints. Give (is the intersection nonempty?, intersection [lo,hi] or [], separation gap). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(No, [], 1/1)","explanation":"Take the larger lower endpoint and smaller upper endpoint; intersecting intervals have gap 0/1, and touching at one included point is not an empty intersection. The exact result, retaining the requested order, is (No, [], 1/1)."},"skills":["Closed Interval Intersection and Separation Gap"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r02-real-number-0053","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e140cd8530a90c3d4060cf1ea4dc9099b422acc3e4ef2f990fc38dc51550dd41","family_id":"qf-density14-real-overlap-closed-intervals","seed":202610142053,"semantic_template":"qf-density14-real-overlap-closed-intervals/[\"intersection\",\"0/1\",\"0/1\"]"},"fields":{"title":"Closed Interval Intersection and Separation Gap","stem":"Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. I=[0/1, 0/1], J=[0/1, 1/1] have ordered inclusive endpoints. Give (is the intersection nonempty?, intersection [lo,hi] or [], separation gap). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(Yes, [0/1, 0/1], 0/1)","explanation":"Take the larger lower endpoint and smaller upper endpoint; intersecting intervals have gap 0/1, and touching at one included point is not an empty intersection. The exact result, retaining the requested order, is (Yes, [0/1, 0/1], 0/1)."},"skills":["Closed Interval Intersection and Separation Gap"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r02-real-number-0054","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"32fbf1e97cce46d40eaf39c68f181cb932cad04196f0baa0ccd7d21aac9952a5","family_id":"qf-density14-real-overlap-closed-intervals","seed":202610142054,"semantic_template":"qf-density14-real-overlap-closed-intervals/[\"intersection\",\"0/1\",\"3/2\"]"},"fields":{"title":"Closed Interval Intersection and Separation Gap","stem":"Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. I=[-3/2, 3/2], J=[0/1, 5/2] have ordered inclusive endpoints. Give (is the intersection nonempty?, intersection [lo,hi] or [], separation gap). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(Yes, [0/1, 3/2], 0/1)","explanation":"Take the larger lower endpoint and smaller upper endpoint; intersecting intervals have gap 0/1, and touching at one included point is not an empty intersection. The exact result, retaining the requested order, is (Yes, [0/1, 3/2], 0/1)."},"skills":["Closed Interval Intersection and Separation Gap"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r02-real-number-0055","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"be40abdfdc810b6494d5407a8fa1c76680226fd9be336eab0029e594dbd8a6cc","family_id":"qf-density14-real-overlap-closed-intervals","seed":202610142055,"semantic_template":"qf-density14-real-overlap-closed-intervals/[\"intersection\",\"0/1\",\"2/1\"]"},"fields":{"title":"Closed Interval Intersection and Separation Gap","stem":"Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. I=[-4/2, 4/2], J=[0/1, 6/2] have ordered inclusive endpoints. Give (is the intersection nonempty?, intersection [lo,hi] or [], separation gap). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(Yes, [0/1, 2/1], 0/1)","explanation":"Take the larger lower endpoint and smaller upper endpoint; intersecting intervals have gap 0/1, and touching at one included point is not an empty intersection. The exact result, retaining the requested order, is (Yes, [0/1, 2/1], 0/1)."},"skills":["Closed Interval Intersection and Separation Gap"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r02-real-number-0056","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8401c79a7e60d05a67bf2f722f1893284514f7c78720d7a4fcb866bc55eb2653","family_id":"qf-density14-real-overlap-closed-intervals","seed":202610142056,"semantic_template":"qf-density14-real-overlap-closed-intervals/[\"intersection\",\"0/1\",\"5/2\"]"},"fields":{"title":"Closed Interval Intersection and Separation Gap","stem":"Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. I=[-5/2, 5/2], J=[0/1, 7/2] have ordered inclusive endpoints. Give (is the intersection nonempty?, intersection [lo,hi] or [], separation gap). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(Yes, [0/1, 5/2], 0/1)","explanation":"Take the larger lower endpoint and smaller upper endpoint; intersecting intervals have gap 0/1, and touching at one included point is not an empty intersection. The exact result, retaining the requested order, is (Yes, [0/1, 5/2], 0/1)."},"skills":["Closed Interval Intersection and Separation Gap"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r02-real-number-0057","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"17ae93a6af26431a2458cc59a00154fd44323f6bc2993ebc52bf48ad3705a4c2","family_id":"qf-density14-real-overlap-closed-intervals","seed":202610142057,"semantic_template":"qf-density14-real-overlap-closed-intervals/[\"intersection\",\"0/1\",\"3/1\"]"},"fields":{"title":"Closed Interval Intersection and Separation Gap","stem":"Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. I=[-6/2, 6/2], J=[0/1, 8/2] have ordered inclusive endpoints. Give (is the intersection nonempty?, intersection [lo,hi] or [], separation gap). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(Yes, [0/1, 3/1], 0/1)","explanation":"Take the larger lower endpoint and smaller upper endpoint; intersecting intervals have gap 0/1, and touching at one included point is not an empty intersection. The exact result, retaining the requested order, is (Yes, [0/1, 3/1], 0/1)."},"skills":["Closed Interval Intersection and Separation Gap"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r02-real-number-0058","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1a6af99d8784c895329d6f29a4233825a413b4883fc2397899b7f757dc025ef2","family_id":"qf-density14-real-overlap-closed-intervals","seed":202610142058,"semantic_template":"qf-density14-real-overlap-closed-intervals/[\"intersection\",\"0/1\",\"7/2\"]"},"fields":{"title":"Closed Interval Intersection and Separation Gap","stem":"Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. I=[-7/2, 7/2], J=[0/1, 9/2] have ordered inclusive endpoints. Give (is the intersection nonempty?, intersection [lo,hi] or [], separation gap). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(Yes, [0/1, 7/2], 0/1)","explanation":"Take the larger lower endpoint and smaller upper endpoint; intersecting intervals have gap 0/1, and touching at one included point is not an empty intersection. The exact result, retaining the requested order, is (Yes, [0/1, 7/2], 0/1)."},"skills":["Closed Interval Intersection and Separation Gap"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r02-real-number-0059","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5f71d39f31f3c12ab45ca5f3a11a90ce5ef793b5189bb6abbcc49f8c314302af","family_id":"qf-density14-real-overlap-closed-intervals","seed":202610142059,"semantic_template":"qf-density14-real-overlap-closed-intervals/[\"intersection\",\"0/1\",\"4/1\"]"},"fields":{"title":"Closed Interval Intersection and Separation Gap","stem":"Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. I=[-8/2, 8/2], J=[0/1, 10/2] have ordered inclusive endpoints. Give (is the intersection nonempty?, intersection [lo,hi] or [], separation gap). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(Yes, [0/1, 4/1], 0/1)","explanation":"Take the larger lower endpoint and smaller upper endpoint; intersecting intervals have gap 0/1, and touching at one included point is not an empty intersection. The exact result, retaining the requested order, is (Yes, [0/1, 4/1], 0/1)."},"skills":["Closed Interval Intersection and Separation Gap"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r02-real-number-0060","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a74febc4e42e5c6c0b4b4043a60c9cbded368a5188c73bf3f852441a78ca6e74","family_id":"qf-density14-real-overlap-closed-intervals","seed":202610142060,"semantic_template":"qf-density14-real-overlap-closed-intervals/[\"intersection\",\"0/1\",\"9/2\"]"},"fields":{"title":"Closed Interval Intersection and Separation Gap","stem":"Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. I=[-9/2, 9/2], J=[0/1, 11/2] have ordered inclusive endpoints. Give (is the intersection nonempty?, intersection [lo,hi] or [], separation gap). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(Yes, [0/1, 9/2], 0/1)","explanation":"Take the larger lower endpoint and smaller upper endpoint; intersecting intervals have gap 0/1, and touching at one included point is not an empty intersection. The exact result, retaining the requested order, is (Yes, [0/1, 9/2], 0/1)."},"skills":["Closed Interval Intersection and Separation Gap"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r02-real-number-0061","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"de1974c892e9a06300aa181ef05428c97fb4f3e34c7c964f53bcc2ada5ca4891","family_id":"qf-density14-real-overlap-closed-intervals","seed":202610142061,"semantic_template":"qf-density14-real-overlap-closed-intervals/[\"intersection\",\"0/1\",\"5/1\"]"},"fields":{"title":"Closed Interval Intersection and Separation Gap","stem":"Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. I=[-10/2, 10/2], J=[0/1, 12/2] have ordered inclusive endpoints. Give (is the intersection nonempty?, intersection [lo,hi] or [], separation gap). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(Yes, [0/1, 5/1], 0/1)","explanation":"Take the larger lower endpoint and smaller upper endpoint; intersecting intervals have gap 0/1, and touching at one included point is not an empty intersection. The exact result, retaining the requested order, is (Yes, [0/1, 5/1], 0/1)."},"skills":["Closed Interval Intersection and Separation Gap"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r02-real-number-0062","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"536aa6a62704f9d577827ffb9188fcbbbd3cb317af6ce757a650fff2cb380992","family_id":"qf-density14-real-overlap-closed-intervals","seed":202610142062,"semantic_template":"qf-density14-real-overlap-closed-intervals/[\"intersection\",\"0/1\",\"11/2\"]"},"fields":{"title":"Closed Interval Intersection and Separation Gap","stem":"Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. I=[-11/2, 11/2], J=[0/1, 13/2] have ordered inclusive endpoints. Give (is the intersection nonempty?, intersection [lo,hi] or [], separation gap). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(Yes, [0/1, 11/2], 0/1)","explanation":"Take the larger lower endpoint and smaller upper endpoint; intersecting intervals have gap 0/1, and touching at one included point is not an empty intersection. The exact result, retaining the requested order, is (Yes, [0/1, 11/2], 0/1)."},"skills":["Closed Interval Intersection and Separation Gap"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r02-real-number-0063","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e64deb9bae6fda725ed87d87ce6283d7d151d6bd4eae2b0f4995b9e77372d8af","family_id":"qf-density14-real-overlap-closed-intervals","seed":202610142063,"semantic_template":"qf-density14-real-overlap-closed-intervals/[\"intersection\",\"0/1\",\"6/1\"]"},"fields":{"title":"Closed Interval Intersection and Separation Gap","stem":"Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. I=[-12/2, 12/2], J=[0/1, 14/2] have ordered inclusive endpoints. Give (is the intersection nonempty?, intersection [lo,hi] or [], separation gap). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(Yes, [0/1, 6/1], 0/1)","explanation":"Take the larger lower endpoint and smaller upper endpoint; intersecting intervals have gap 0/1, and touching at one included point is not an empty intersection. The exact result, retaining the requested order, is (Yes, [0/1, 6/1], 0/1)."},"skills":["Closed Interval Intersection and Separation Gap"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r02-real-number-0064","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"562392af2a658fe6afb4be2c5dc69d42a70e8af7231dd9aa9c15c3c38d147b94","family_id":"qf-density14-real-overlap-closed-intervals","seed":202610142064,"semantic_template":"qf-density14-real-overlap-closed-intervals/[\"intersection\",\"0/1\",\"13/2\"]"},"fields":{"title":"Closed Interval Intersection and Separation Gap","stem":"Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. I=[-13/2, 13/2], J=[0/1, 15/2] have ordered inclusive endpoints. Give (is the intersection nonempty?, intersection [lo,hi] or [], separation gap). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(Yes, [0/1, 13/2], 0/1)","explanation":"Take the larger lower endpoint and smaller upper endpoint; intersecting intervals have gap 0/1, and touching at one included point is not an empty intersection. The exact result, retaining the requested order, is (Yes, [0/1, 13/2], 0/1)."},"skills":["Closed Interval Intersection and Separation Gap"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r02-real-number-0065","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"24065f9f37de8741b9e1f89a0322418cc9803e1bddbe9a0d7c68f308289f5843","family_id":"qf-density14-real-overlap-closed-intervals","seed":202610142065,"semantic_template":"qf-density14-real-overlap-closed-intervals/[\"intersection\",\"0/1\",\"7/1\"]"},"fields":{"title":"Closed Interval Intersection and Separation Gap","stem":"Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. I=[-14/2, 14/2], J=[0/1, 16/2] have ordered inclusive endpoints. Give (is the intersection nonempty?, intersection [lo,hi] or [], separation gap). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(Yes, [0/1, 7/1], 0/1)","explanation":"Take the larger lower endpoint and smaller upper endpoint; intersecting intervals have gap 0/1, and touching at one included point is not an empty intersection. The exact result, retaining the requested order, is (Yes, [0/1, 7/1], 0/1)."},"skills":["Closed Interval Intersection and Separation Gap"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r02-real-number-0066","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5dde85e8f0d3f7a53ef416af73812fd1f0be27b77974734b78e8e21fb6721481","family_id":"qf-density14-real-overlap-closed-intervals","seed":202610142066,"semantic_template":"qf-density14-real-overlap-closed-intervals/[\"intersection\",\"0/1\",\"15/2\"]"},"fields":{"title":"Closed Interval Intersection and Separation Gap","stem":"Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. I=[-15/2, 15/2], J=[0/1, 17/2] have ordered inclusive endpoints. Give (is the intersection nonempty?, intersection [lo,hi] or [], separation gap). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(Yes, [0/1, 15/2], 0/1)","explanation":"Take the larger lower endpoint and smaller upper endpoint; intersecting intervals have gap 0/1, and touching at one included point is not an empty intersection. The exact result, retaining the requested order, is (Yes, [0/1, 15/2], 0/1)."},"skills":["Closed Interval Intersection and Separation Gap"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r02-real-number-0067","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fc7383e3f6183fcd05951f726d3310f7a6097010f20045dd24490a108916be4b","family_id":"qf-density14-real-overlap-closed-intervals","seed":202610142067,"semantic_template":"qf-density14-real-overlap-closed-intervals/[\"intersection\",\"0/1\",\"8/1\"]"},"fields":{"title":"Closed Interval Intersection and Separation Gap","stem":"Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. I=[-16/2, 16/2], J=[0/1, 18/2] have ordered inclusive endpoints. Give (is the intersection nonempty?, intersection [lo,hi] or [], separation gap). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(Yes, [0/1, 8/1], 0/1)","explanation":"Take the larger lower endpoint and smaller upper endpoint; intersecting intervals have gap 0/1, and touching at one included point is not an empty intersection. The exact result, retaining the requested order, is (Yes, [0/1, 8/1], 0/1)."},"skills":["Closed Interval Intersection and Separation Gap"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r02-real-number-0068","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"aba6af06f77470e935a1f9230fe1c169970001015eb4c9785c49052e6569597f","family_id":"qf-density14-real-overlap-closed-intervals","seed":202610142068,"semantic_template":"qf-density14-real-overlap-closed-intervals/[\"intersection\",\"0/1\",\"17/2\"]"},"fields":{"title":"Closed Interval Intersection and Separation Gap","stem":"Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. I=[-17/2, 17/2], J=[0/1, 19/2] have ordered inclusive endpoints. Give (is the intersection nonempty?, intersection [lo,hi] or [], separation gap). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(Yes, [0/1, 17/2], 0/1)","explanation":"Take the larger lower endpoint and smaller upper endpoint; intersecting intervals have gap 0/1, and touching at one included point is not an empty intersection. The exact result, retaining the requested order, is (Yes, [0/1, 17/2], 0/1)."},"skills":["Closed Interval Intersection and Separation Gap"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r02-real-number-0069","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d37f69486f873129ad5965ed45e6481ea231131f6a50cbed01b52cf0ae14ccf7","family_id":"qf-density14-real-overlap-closed-intervals","seed":202610142069,"semantic_template":"qf-density14-real-overlap-closed-intervals/[\"intersection\",\"0/1\",\"9/1\"]"},"fields":{"title":"Closed Interval Intersection and Separation Gap","stem":"Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. I=[-18/2, 18/2], J=[0/1, 20/2] have ordered inclusive endpoints. Give (is the intersection nonempty?, intersection [lo,hi] or [], separation gap). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(Yes, [0/1, 9/1], 0/1)","explanation":"Take the larger lower endpoint and smaller upper endpoint; intersecting intervals have gap 0/1, and touching at one included point is not an empty intersection. The exact result, retaining the requested order, is (Yes, [0/1, 9/1], 0/1)."},"skills":["Closed Interval Intersection and Separation Gap"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r02-real-number-0070","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cfb68fa6a29b2d65a5d91060cd349e07d974a4f715b5f179d56e1aae992f9c8f","family_id":"qf-density14-real-overlap-closed-intervals","seed":202610142070,"semantic_template":"qf-density14-real-overlap-closed-intervals/[\"intersection\",\"0/1\",\"19/2\"]"},"fields":{"title":"Closed Interval Intersection and Separation Gap","stem":"Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. I=[-19/2, 19/2], J=[0/1, 21/2] have ordered inclusive endpoints. Give (is the intersection nonempty?, intersection [lo,hi] or [], separation gap). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(Yes, [0/1, 19/2], 0/1)","explanation":"Take the larger lower endpoint and smaller upper endpoint; intersecting intervals have gap 0/1, and touching at one included point is not an empty intersection. The exact result, retaining the requested order, is (Yes, [0/1, 19/2], 0/1)."},"skills":["Closed Interval Intersection and Separation Gap"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r02-real-number-0071","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9e2d09c8d86be27a6a6d3f8edb80ef9535024b451bf4e2971d6d1a2f80d08deb","family_id":"qf-density14-real-overlap-closed-intervals","seed":202610142071,"semantic_template":"qf-density14-real-overlap-closed-intervals/[\"intersection\",\"0/1\",\"10/1\"]"},"fields":{"title":"Closed Interval Intersection and Separation Gap","stem":"Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. I=[-20/2, 20/2], J=[0/1, 22/2] have ordered inclusive endpoints. Give (is the intersection nonempty?, intersection [lo,hi] or [], separation gap). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(Yes, [0/1, 10/1], 0/1)","explanation":"Take the larger lower endpoint and smaller upper endpoint; intersecting intervals have gap 0/1, and touching at one included point is not an empty intersection. The exact result, retaining the requested order, is (Yes, [0/1, 10/1], 0/1)."},"skills":["Closed Interval Intersection and Separation Gap"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r02-real-number-0072","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"928b87157643d6a2e2078cd8716388259cdbdda739df5d304cde41ee3247a1ac","family_id":"qf-density14-real-overlap-closed-intervals","seed":202610142072,"semantic_template":"qf-density14-real-overlap-closed-intervals/[\"intersection\",\"0/1\",\"21/2\"]"},"fields":{"title":"Closed Interval Intersection and Separation Gap","stem":"Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. I=[-21/2, 21/2], J=[0/1, 23/2] have ordered inclusive endpoints. Give (is the intersection nonempty?, intersection [lo,hi] or [], separation gap). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(Yes, [0/1, 21/2], 0/1)","explanation":"Take the larger lower endpoint and smaller upper endpoint; intersecting intervals have gap 0/1, and touching at one included point is not an empty intersection. The exact result, retaining the requested order, is (Yes, [0/1, 21/2], 0/1)."},"skills":["Closed Interval Intersection and Separation Gap"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r02-real-number-0073","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1c781f1cef678a30537c2b1dcd796ff2d046d9a6e74da099cf72a52ac587a219","family_id":"qf-density14-real-overlap-closed-intervals","seed":202610142073,"semantic_template":"qf-density14-real-overlap-closed-intervals/[\"intersection\",\"0/1\",\"11/1\"]"},"fields":{"title":"Closed Interval Intersection and Separation Gap","stem":"Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. I=[-22/2, 22/2], J=[0/1, 24/2] have ordered inclusive endpoints. Give (is the intersection nonempty?, intersection [lo,hi] or [], separation gap). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(Yes, [0/1, 11/1], 0/1)","explanation":"Take the larger lower endpoint and smaller upper endpoint; intersecting intervals have gap 0/1, and touching at one included point is not an empty intersection. The exact result, retaining the requested order, is (Yes, [0/1, 11/1], 0/1)."},"skills":["Closed Interval Intersection and Separation Gap"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r02-real-number-0074","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"961bb311b6116ae90e1b4094f4a06bf4dab01fabcfd22c81c340c68b89101c91","family_id":"qf-density14-real-overlap-closed-intervals","seed":202610142074,"semantic_template":"qf-density14-real-overlap-closed-intervals/[\"intersection\",\"0/1\",\"23/2\"]"},"fields":{"title":"Closed Interval Intersection and Separation Gap","stem":"Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. I=[-23/2, 23/2], J=[0/1, 25/2] have ordered inclusive endpoints. Give (is the intersection nonempty?, intersection [lo,hi] or [], separation gap). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(Yes, [0/1, 23/2], 0/1)","explanation":"Take the larger lower endpoint and smaller upper endpoint; intersecting intervals have gap 0/1, and touching at one included point is not an empty intersection. The exact result, retaining the requested order, is (Yes, [0/1, 23/2], 0/1)."},"skills":["Closed Interval Intersection and Separation Gap"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r02-real-number-0075","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0f1bb5ee090fbd7fc0a317e9b14091cc6ef65a8d8863cf39955aa7c50bd39976","family_id":"qf-density14-real-overlap-closed-intervals","seed":202610142075,"semantic_template":"qf-density14-real-overlap-closed-intervals/[\"intersection\",\"0/1\",\"12/1\"]"},"fields":{"title":"Closed Interval Intersection and Separation Gap","stem":"Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. I=[-24/2, 24/2], J=[0/1, 26/2] have ordered inclusive endpoints. Give (is the intersection nonempty?, intersection [lo,hi] or [], separation gap). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(Yes, [0/1, 12/1], 0/1)","explanation":"Take the larger lower endpoint and smaller upper endpoint; intersecting intervals have gap 0/1, and touching at one included point is not an empty intersection. The exact result, retaining the requested order, is (Yes, [0/1, 12/1], 0/1)."},"skills":["Closed Interval Intersection and Separation Gap"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r02-real-number-0076","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"19334134ae585f55fddcee5629fb136ac782b86590365462839c8ea714d612ac","family_id":"qf-density14-real-point-offset-distances","seed":202610142076,"semantic_template":"qf-density14-real-point-offset-distances/[\"-4/1\",\"-1/2\",\"0/1\",\"2/3\"]"},"fields":{"title":"Ordered Rational Offsets and Absolute Distances","stem":"Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. origin=-12/2, ordered points=[-10/1, -13/2, -6/1, -16/3]. Give (each point−origin, each offset’s absolute value), retaining the four-field order.","answer":{"kind":"expression","value":"([-4/1, -1/2, 0/1, 2/3], [4/1, 1/2, 0/1, 2/3])","explanation":"Subtract exactly to retain direction, then take the nonnegative absolute value; directed offsets and distances are separate. The exact result, retaining the requested order, is ([-4/1, -1/2, 0/1, 2/3], [4/1, 1/2, 0/1, 2/3])."},"skills":["Ordered Rational Offsets and Absolute Distances"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r02-real-number-0077","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"aeb69e89f896fd097807f725011e13c86f9a6689bf03ada2cd117ddd3b8761d5","family_id":"qf-density14-real-point-offset-distances","seed":202610142077,"semantic_template":"qf-density14-real-point-offset-distances/[\"-11/3\",\"-1/2\",\"0/1\",\"2/3\"]"},"fields":{"title":"Ordered Rational Offsets and Absolute Distances","stem":"Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. origin=-11/2, ordered points=[-55/6, -6/1, -11/2, -29/6]. Give (each point−origin, each offset’s absolute value), retaining the four-field order.","answer":{"kind":"expression","value":"([-11/3, -1/2, 0/1, 2/3], [11/3, 1/2, 0/1, 2/3])","explanation":"Subtract exactly to retain direction, then take the nonnegative absolute value; directed offsets and distances are separate. The exact result, retaining the requested order, is ([-11/3, -1/2, 0/1, 2/3], [11/3, 1/2, 0/1, 2/3])."},"skills":["Ordered Rational Offsets and Absolute Distances"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r02-real-number-0078","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e2fb08d4422616f8cb1b002cd8f63a869066f2d1a488ac41ae637005b551b2e6","family_id":"qf-density14-real-point-offset-distances","seed":202610142078,"semantic_template":"qf-density14-real-point-offset-distances/[\"-10/3\",\"-1/2\",\"0/1\",\"2/3\"]"},"fields":{"title":"Ordered Rational Offsets and Absolute Distances","stem":"Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. origin=-10/2, ordered points=[-25/3, -11/2, -5/1, -13/3]. Give (each point−origin, each offset’s absolute value), retaining the four-field order.","answer":{"kind":"expression","value":"([-10/3, -1/2, 0/1, 2/3], [10/3, 1/2, 0/1, 2/3])","explanation":"Subtract exactly to retain direction, then take the nonnegative absolute value; directed offsets and distances are separate. The exact result, retaining the requested order, is ([-10/3, -1/2, 0/1, 2/3], [10/3, 1/2, 0/1, 2/3])."},"skills":["Ordered Rational Offsets and Absolute Distances"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r02-real-number-0079","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9a8f07b7acb225ce4d071c39d627ceb302111b15abdf20508a4b5d69ee67200a","family_id":"qf-density14-real-point-offset-distances","seed":202610142079,"semantic_template":"qf-density14-real-point-offset-distances/[\"-3/1\",\"-1/2\",\"0/1\",\"2/3\"]"},"fields":{"title":"Ordered Rational Offsets and Absolute Distances","stem":"Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. origin=-9/2, ordered points=[-15/2, -5/1, -9/2, -23/6]. Give (each point−origin, each offset’s absolute value), retaining the four-field order.","answer":{"kind":"expression","value":"([-3/1, -1/2, 0/1, 2/3], [3/1, 1/2, 0/1, 2/3])","explanation":"Subtract exactly to retain direction, then take the nonnegative absolute value; directed offsets and distances are separate. The exact result, retaining the requested order, is ([-3/1, -1/2, 0/1, 2/3], [3/1, 1/2, 0/1, 2/3])."},"skills":["Ordered Rational Offsets and Absolute Distances"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r02-real-number-0080","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c565264acec23d97f9530c24dfc5dac98ba3bb8c689cea0efc1d5b6dab9d5b87","family_id":"qf-density14-real-point-offset-distances","seed":202610142080,"semantic_template":"qf-density14-real-point-offset-distances/[\"-8/3\",\"-1/2\",\"0/1\",\"2/3\"]"},"fields":{"title":"Ordered Rational Offsets and Absolute Distances","stem":"Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. origin=-8/2, ordered points=[-20/3, -9/2, -4/1, -10/3]. Give (each point−origin, each offset’s absolute value), retaining the four-field order.","answer":{"kind":"expression","value":"([-8/3, -1/2, 0/1, 2/3], [8/3, 1/2, 0/1, 2/3])","explanation":"Subtract exactly to retain direction, then take the nonnegative absolute value; directed offsets and distances are separate. The exact result, retaining the requested order, is ([-8/3, -1/2, 0/1, 2/3], [8/3, 1/2, 0/1, 2/3])."},"skills":["Ordered Rational Offsets and Absolute Distances"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r02-real-number-0081","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"70f1a2e848e73f6c49ab449e9dc993aaf8c55e1be9d19a668f4d42d5a1162291","family_id":"qf-density14-real-point-offset-distances","seed":202610142081,"semantic_template":"qf-density14-real-point-offset-distances/[\"-7/3\",\"-1/2\",\"0/1\",\"2/3\"]"},"fields":{"title":"Ordered Rational Offsets and Absolute Distances","stem":"Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. origin=-7/2, ordered points=[-35/6, -4/1, -7/2, -17/6]. Give (each point−origin, each offset’s absolute value), retaining the four-field order.","answer":{"kind":"expression","value":"([-7/3, -1/2, 0/1, 2/3], [7/3, 1/2, 0/1, 2/3])","explanation":"Subtract exactly to retain direction, then take the nonnegative absolute value; directed offsets and distances are separate. The exact result, retaining the requested order, is ([-7/3, -1/2, 0/1, 2/3], [7/3, 1/2, 0/1, 2/3])."},"skills":["Ordered Rational Offsets and Absolute Distances"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r02-real-number-0082","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"145aaecf73681a3a575fc74a05f20d3aeecf243fbe3897780c57631c412a3c2e","family_id":"qf-density14-real-point-offset-distances","seed":202610142082,"semantic_template":"qf-density14-real-point-offset-distances/[\"-2/1\",\"-1/2\",\"0/1\",\"2/3\"]"},"fields":{"title":"Ordered Rational Offsets and Absolute Distances","stem":"Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. origin=-6/2, ordered points=[-5/1, -7/2, -3/1, -7/3]. Give (each point−origin, each offset’s absolute value), retaining the four-field order.","answer":{"kind":"expression","value":"([-2/1, -1/2, 0/1, 2/3], [2/1, 1/2, 0/1, 2/3])","explanation":"Subtract exactly to retain direction, then take the nonnegative absolute value; directed offsets and distances are separate. The exact result, retaining the requested order, is ([-2/1, -1/2, 0/1, 2/3], [2/1, 1/2, 0/1, 2/3])."},"skills":["Ordered Rational Offsets and Absolute Distances"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r02-real-number-0083","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"177b3bb2f3c8773f6b8ee79eec17432f4fdab646f2eae6282c56edab0110a95f","family_id":"qf-density14-real-point-offset-distances","seed":202610142083,"semantic_template":"qf-density14-real-point-offset-distances/[\"-5/3\",\"-1/2\",\"0/1\",\"2/3\"]"},"fields":{"title":"Ordered Rational Offsets and Absolute Distances","stem":"Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. origin=-5/2, ordered points=[-25/6, -3/1, -5/2, -11/6]. Give (each point−origin, each offset’s absolute value), retaining the four-field order.","answer":{"kind":"expression","value":"([-5/3, -1/2, 0/1, 2/3], [5/3, 1/2, 0/1, 2/3])","explanation":"Subtract exactly to retain direction, then take the nonnegative absolute value; directed offsets and distances are separate. The exact result, retaining the requested order, is ([-5/3, -1/2, 0/1, 2/3], [5/3, 1/2, 0/1, 2/3])."},"skills":["Ordered Rational Offsets and Absolute Distances"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r02-real-number-0084","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7b0384aca739dfe2a5bf461038563c5760b795e03aebcec474fd062ca37b099c","family_id":"qf-density14-real-point-offset-distances","seed":202610142084,"semantic_template":"qf-density14-real-point-offset-distances/[\"-4/3\",\"-1/2\",\"0/1\",\"2/3\"]"},"fields":{"title":"Ordered Rational Offsets and Absolute Distances","stem":"Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. origin=-4/2, ordered points=[-10/3, -5/2, -2/1, -4/3]. Give (each point−origin, each offset’s absolute value), retaining the four-field order.","answer":{"kind":"expression","value":"([-4/3, -1/2, 0/1, 2/3], [4/3, 1/2, 0/1, 2/3])","explanation":"Subtract exactly to retain direction, then take the nonnegative absolute value; directed offsets and distances are separate. The exact result, retaining the requested order, is ([-4/3, -1/2, 0/1, 2/3], [4/3, 1/2, 0/1, 2/3])."},"skills":["Ordered Rational Offsets and Absolute Distances"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r02-real-number-0085","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4d0863bf981b79ddc4657c9166ff5ed58f54efadb64d7551d284d4df6043208b","family_id":"qf-density14-real-point-offset-distances","seed":202610142085,"semantic_template":"qf-density14-real-point-offset-distances/[\"-1/1\",\"-1/2\",\"0/1\",\"2/3\"]"},"fields":{"title":"Ordered Rational Offsets and Absolute Distances","stem":"Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. origin=-3/2, ordered points=[-5/2, -2/1, -3/2, -5/6]. Give (each point−origin, each offset’s absolute value), retaining the four-field order.","answer":{"kind":"expression","value":"([-1/1, -1/2, 0/1, 2/3], [1/1, 1/2, 0/1, 2/3])","explanation":"Subtract exactly to retain direction, then take the nonnegative absolute value; directed offsets and distances are separate. The exact result, retaining the requested order, is ([-1/1, -1/2, 0/1, 2/3], [1/1, 1/2, 0/1, 2/3])."},"skills":["Ordered Rational Offsets and Absolute Distances"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r02-real-number-0086","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cd87fe0d876522e867e3cb1530b5ba7f8bb2d1d32c86fa174b19ca604d6a106b","family_id":"qf-density14-real-point-offset-distances","seed":202610142086,"semantic_template":"qf-density14-real-point-offset-distances/[\"-2/3\",\"-1/2\",\"0/1\",\"2/3\"]"},"fields":{"title":"Ordered Rational Offsets and Absolute Distances","stem":"Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. origin=-2/2, ordered points=[-5/3, -3/2, -1/1, -1/3]. Give (each point−origin, each offset’s absolute value), retaining the four-field order.","answer":{"kind":"expression","value":"([-2/3, -1/2, 0/1, 2/3], [2/3, 1/2, 0/1, 2/3])","explanation":"Subtract exactly to retain direction, then take the nonnegative absolute value; directed offsets and distances are separate. The exact result, retaining the requested order, is ([-2/3, -1/2, 0/1, 2/3], [2/3, 1/2, 0/1, 2/3])."},"skills":["Ordered Rational Offsets and Absolute Distances"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r02-real-number-0087","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6c4084452a3d6eae8ae4a606cbb3a649b03c5588098e0ecf8f762100698ec1f7","family_id":"qf-density14-real-point-offset-distances","seed":202610142087,"semantic_template":"qf-density14-real-point-offset-distances/[\"-1/3\",\"-1/2\",\"0/1\",\"2/3\"]"},"fields":{"title":"Ordered Rational Offsets and Absolute Distances","stem":"Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. origin=-1/2, ordered points=[-5/6, -1/1, -1/2, 1/6]. Give (each point−origin, each offset’s absolute value), retaining the four-field order.","answer":{"kind":"expression","value":"([-1/3, -1/2, 0/1, 2/3], [1/3, 1/2, 0/1, 2/3])","explanation":"Subtract exactly to retain direction, then take the nonnegative absolute value; directed offsets and distances are separate. The exact result, retaining the requested order, is ([-1/3, -1/2, 0/1, 2/3], [1/3, 1/2, 0/1, 2/3])."},"skills":["Ordered Rational Offsets and Absolute Distances"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r02-real-number-0088","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0aa7e096a8b25efa8aecead509a0c1525ccf80d2a125148d81b0245b63e98257","family_id":"qf-density14-real-point-offset-distances","seed":202610142088,"semantic_template":"qf-density14-real-point-offset-distances/[\"0/1\",\"-1/2\",\"0/1\",\"2/3\"]"},"fields":{"title":"Ordered Rational Offsets and Absolute Distances","stem":"Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. origin=0/2, ordered points=[0/1, -1/2, 0/1, 2/3]. Give (each point−origin, each offset’s absolute value), retaining the four-field order.","answer":{"kind":"expression","value":"([0/1, -1/2, 0/1, 2/3], [0/1, 1/2, 0/1, 2/3])","explanation":"Subtract exactly to retain direction, then take the nonnegative absolute value; directed offsets and distances are separate. The exact result, retaining the requested order, is ([0/1, -1/2, 0/1, 2/3], [0/1, 1/2, 0/1, 2/3])."},"skills":["Ordered Rational Offsets and Absolute Distances"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r02-real-number-0089","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6a3017473711dd495d99262000c37bf50d5f9f948df9aefe3f62045b8d592388","family_id":"qf-density14-real-point-offset-distances","seed":202610142089,"semantic_template":"qf-density14-real-point-offset-distances/[\"1/3\",\"-1/2\",\"0/1\",\"2/3\"]"},"fields":{"title":"Ordered Rational Offsets and Absolute Distances","stem":"Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. origin=1/2, ordered points=[5/6, 0/1, 1/2, 7/6]. Give (each point−origin, each offset’s absolute value), retaining the four-field order.","answer":{"kind":"expression","value":"([1/3, -1/2, 0/1, 2/3], [1/3, 1/2, 0/1, 2/3])","explanation":"Subtract exactly to retain direction, then take the nonnegative absolute value; directed offsets and distances are separate. The exact result, retaining the requested order, is ([1/3, -1/2, 0/1, 2/3], [1/3, 1/2, 0/1, 2/3])."},"skills":["Ordered Rational Offsets and Absolute Distances"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r02-real-number-0090","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b92e9786ac816293208ab83209237002c185edc1f34e1fe4289245027368a3df","family_id":"qf-density14-real-point-offset-distances","seed":202610142090,"semantic_template":"qf-density14-real-point-offset-distances/[\"2/3\",\"-1/2\",\"0/1\",\"2/3\"]"},"fields":{"title":"Ordered Rational Offsets and Absolute Distances","stem":"Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. origin=2/2, ordered points=[5/3, 1/2, 1/1, 5/3]. Give (each point−origin, each offset’s absolute value), retaining the four-field order.","answer":{"kind":"expression","value":"([2/3, -1/2, 0/1, 2/3], [2/3, 1/2, 0/1, 2/3])","explanation":"Subtract exactly to retain direction, then take the nonnegative absolute value; directed offsets and distances are separate. The exact result, retaining the requested order, is ([2/3, -1/2, 0/1, 2/3], [2/3, 1/2, 0/1, 2/3])."},"skills":["Ordered Rational Offsets and Absolute Distances"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r02-real-number-0091","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5a991ca6b154bf0d01feb3bf363c9f3b7ac22f5381e6ab08a4f99d8f8fbea6fd","family_id":"qf-density14-real-point-offset-distances","seed":202610142091,"semantic_template":"qf-density14-real-point-offset-distances/[\"1/1\",\"-1/2\",\"0/1\",\"2/3\"]"},"fields":{"title":"Ordered Rational Offsets and Absolute Distances","stem":"Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. origin=3/2, ordered points=[5/2, 1/1, 3/2, 13/6]. Give (each point−origin, each offset’s absolute value), retaining the four-field order.","answer":{"kind":"expression","value":"([1/1, -1/2, 0/1, 2/3], [1/1, 1/2, 0/1, 2/3])","explanation":"Subtract exactly to retain direction, then take the nonnegative absolute value; directed offsets and distances are separate. The exact result, retaining the requested order, is ([1/1, -1/2, 0/1, 2/3], [1/1, 1/2, 0/1, 2/3])."},"skills":["Ordered Rational Offsets and Absolute Distances"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r02-real-number-0092","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f80561edfb43a24a4de24d774ff470f06aee453fdce64abd8534751e6669969b","family_id":"qf-density14-real-point-offset-distances","seed":202610142092,"semantic_template":"qf-density14-real-point-offset-distances/[\"4/3\",\"-1/2\",\"0/1\",\"2/3\"]"},"fields":{"title":"Ordered Rational Offsets and Absolute Distances","stem":"Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. origin=4/2, ordered points=[10/3, 3/2, 2/1, 8/3]. Give (each point−origin, each offset’s absolute value), retaining the four-field order.","answer":{"kind":"expression","value":"([4/3, -1/2, 0/1, 2/3], [4/3, 1/2, 0/1, 2/3])","explanation":"Subtract exactly to retain direction, then take the nonnegative absolute value; directed offsets and distances are separate. The exact result, retaining the requested order, is ([4/3, -1/2, 0/1, 2/3], [4/3, 1/2, 0/1, 2/3])."},"skills":["Ordered Rational Offsets and Absolute Distances"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r02-real-number-0093","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4f689a7c01eee1df91579b02cd0edc280562d8a0e122b081970450ef11a21897","family_id":"qf-density14-real-point-offset-distances","seed":202610142093,"semantic_template":"qf-density14-real-point-offset-distances/[\"5/3\",\"-1/2\",\"0/1\",\"2/3\"]"},"fields":{"title":"Ordered Rational Offsets and Absolute Distances","stem":"Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. origin=5/2, ordered points=[25/6, 2/1, 5/2, 19/6]. Give (each point−origin, each offset’s absolute value), retaining the four-field order.","answer":{"kind":"expression","value":"([5/3, -1/2, 0/1, 2/3], [5/3, 1/2, 0/1, 2/3])","explanation":"Subtract exactly to retain direction, then take the nonnegative absolute value; directed offsets and distances are separate. The exact result, retaining the requested order, is ([5/3, -1/2, 0/1, 2/3], [5/3, 1/2, 0/1, 2/3])."},"skills":["Ordered Rational Offsets and Absolute Distances"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r02-real-number-0094","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"457a19dbba86896f0959d974ee4bffd0cda8912615be89fad3d07e4a479d5e57","family_id":"qf-density14-real-point-offset-distances","seed":202610142094,"semantic_template":"qf-density14-real-point-offset-distances/[\"2/1\",\"-1/2\",\"0/1\",\"2/3\"]"},"fields":{"title":"Ordered Rational Offsets and Absolute Distances","stem":"Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. origin=6/2, ordered points=[5/1, 5/2, 3/1, 11/3]. Give (each point−origin, each offset’s absolute value), retaining the four-field order.","answer":{"kind":"expression","value":"([2/1, -1/2, 0/1, 2/3], [2/1, 1/2, 0/1, 2/3])","explanation":"Subtract exactly to retain direction, then take the nonnegative absolute value; directed offsets and distances are separate. The exact result, retaining the requested order, is ([2/1, -1/2, 0/1, 2/3], [2/1, 1/2, 0/1, 2/3])."},"skills":["Ordered Rational Offsets and Absolute Distances"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r02-real-number-0095","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9fe455fe8712d7a8694ebf850e6a7aa622af8f31b551faf401735fa89d1940ec","family_id":"qf-density14-real-point-offset-distances","seed":202610142095,"semantic_template":"qf-density14-real-point-offset-distances/[\"7/3\",\"-1/2\",\"0/1\",\"2/3\"]"},"fields":{"title":"Ordered Rational Offsets and Absolute Distances","stem":"Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. origin=7/2, ordered points=[35/6, 3/1, 7/2, 25/6]. Give (each point−origin, each offset’s absolute value), retaining the four-field order.","answer":{"kind":"expression","value":"([7/3, -1/2, 0/1, 2/3], [7/3, 1/2, 0/1, 2/3])","explanation":"Subtract exactly to retain direction, then take the nonnegative absolute value; directed offsets and distances are separate. The exact result, retaining the requested order, is ([7/3, -1/2, 0/1, 2/3], [7/3, 1/2, 0/1, 2/3])."},"skills":["Ordered Rational Offsets and Absolute Distances"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r02-real-number-0096","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ee49bf0315a690cc578d22225203626c67c887b2bd490756235099a21680d8c8","family_id":"qf-density14-real-point-offset-distances","seed":202610142096,"semantic_template":"qf-density14-real-point-offset-distances/[\"8/3\",\"-1/2\",\"0/1\",\"2/3\"]"},"fields":{"title":"Ordered Rational Offsets and Absolute Distances","stem":"Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. origin=8/2, ordered points=[20/3, 7/2, 4/1, 14/3]. Give (each point−origin, each offset’s absolute value), retaining the four-field order.","answer":{"kind":"expression","value":"([8/3, -1/2, 0/1, 2/3], [8/3, 1/2, 0/1, 2/3])","explanation":"Subtract exactly to retain direction, then take the nonnegative absolute value; directed offsets and distances are separate. The exact result, retaining the requested order, is ([8/3, -1/2, 0/1, 2/3], [8/3, 1/2, 0/1, 2/3])."},"skills":["Ordered Rational Offsets and Absolute Distances"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r02-real-number-0097","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6ebd34b743ce506c7b192c8ad4bc5f22a650b7b5307d450e9c3b643bf8d7aef3","family_id":"qf-density14-real-point-offset-distances","seed":202610142097,"semantic_template":"qf-density14-real-point-offset-distances/[\"3/1\",\"-1/2\",\"0/1\",\"2/3\"]"},"fields":{"title":"Ordered Rational Offsets and Absolute Distances","stem":"Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. origin=9/2, ordered points=[15/2, 4/1, 9/2, 31/6]. Give (each point−origin, each offset’s absolute value), retaining the four-field order.","answer":{"kind":"expression","value":"([3/1, -1/2, 0/1, 2/3], [3/1, 1/2, 0/1, 2/3])","explanation":"Subtract exactly to retain direction, then take the nonnegative absolute value; directed offsets and distances are separate. The exact result, retaining the requested order, is ([3/1, -1/2, 0/1, 2/3], [3/1, 1/2, 0/1, 2/3])."},"skills":["Ordered Rational Offsets and Absolute Distances"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r02-real-number-0098","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1872ecb14704eaea7f965bd5c92d1e99c296037fbd282db84ceff69d3fbb858a","family_id":"qf-density14-real-point-offset-distances","seed":202610142098,"semantic_template":"qf-density14-real-point-offset-distances/[\"10/3\",\"-1/2\",\"0/1\",\"2/3\"]"},"fields":{"title":"Ordered Rational Offsets and Absolute Distances","stem":"Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. origin=10/2, ordered points=[25/3, 9/2, 5/1, 17/3]. Give (each point−origin, each offset’s absolute value), retaining the four-field order.","answer":{"kind":"expression","value":"([10/3, -1/2, 0/1, 2/3], [10/3, 1/2, 0/1, 2/3])","explanation":"Subtract exactly to retain direction, then take the nonnegative absolute value; directed offsets and distances are separate. The exact result, retaining the requested order, is ([10/3, -1/2, 0/1, 2/3], [10/3, 1/2, 0/1, 2/3])."},"skills":["Ordered Rational Offsets and Absolute Distances"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r02-real-number-0099","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"39222c6f6fe4c358b009519d925d0dc383aa242e58a4d8b7ade8ad6479a5b471","family_id":"qf-density14-real-point-offset-distances","seed":202610142099,"semantic_template":"qf-density14-real-point-offset-distances/[\"11/3\",\"-1/2\",\"0/1\",\"2/3\"]"},"fields":{"title":"Ordered Rational Offsets and Absolute Distances","stem":"Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. origin=11/2, ordered points=[55/6, 5/1, 11/2, 37/6]. Give (each point−origin, each offset’s absolute value), retaining the four-field order.","answer":{"kind":"expression","value":"([11/3, -1/2, 0/1, 2/3], [11/3, 1/2, 0/1, 2/3])","explanation":"Subtract exactly to retain direction, then take the nonnegative absolute value; directed offsets and distances are separate. The exact result, retaining the requested order, is ([11/3, -1/2, 0/1, 2/3], [11/3, 1/2, 0/1, 2/3])."},"skills":["Ordered Rational Offsets and Absolute Distances"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r02-real-number-0100","batch_id":"batch-20261010-advance-20261014-r02-real-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"aeafc28bd008144deba8081dc4738566e69290124498745f88bd48cc17cdf343","family_id":"qf-density14-real-point-offset-distances","seed":202610142100,"semantic_template":"qf-density14-real-point-offset-distances/[\"4/1\",\"-1/2\",\"0/1\",\"2/3\"]"},"fields":{"title":"Ordered Rational Offsets and Absolute Distances","stem":"Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. origin=12/2, ordered points=[10/1, 11/2, 6/1, 20/3]. Give (each point−origin, each offset’s absolute value), retaining the four-field order.","answer":{"kind":"expression","value":"([4/1, -1/2, 0/1, 2/3], [4/1, 1/2, 0/1, 2/3])","explanation":"Subtract exactly to retain direction, then take the nonnegative absolute value; directed offsets and distances are separate. The exact result, retaining the requested order, is ([4/1, -1/2, 0/1, 2/3], [4/1, 1/2, 0/1, 2/3])."},"skills":["Ordered Rational Offsets and Absolute Distances"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}